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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

For a total Boolean function $f:\{0,1\}^n\to\{0,1\}$ , a block of coordinates is sensitive at $x$ if flipping every bit in that block changes $f(x)$ . Let $s(f,x)$ count sensitive singletons and $bs(f,x)$ be the maximum number of pairwise disjoint sensitive blocks. Define $s(f)=\max_x s(f,x)$ and $bs(f)=\max_x bs(f,x)$ .

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

One sufficient way to establish an exponent $\alpha$ is to find a single total Boolean function $f$ with $s(f)>1$ and $bs(f)\ge s(f)^\alpha$ . Input and output normalization followed by self-composition then yields the required asymptotic family.

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

Harvey and van der Hoeven established an $O(n\log n)$ algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

A positive $\kappa$ beats $n\log n$ asymptotically; larger $\kappa$ is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Classical algorithms solve 3SUM in $O(n^2)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic $O(n^{1.9992})$ algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Classical algorithms solve all-pairs shortest paths in $O(n^3)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic $O(n^{2.99942})$ algorithm. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

The sharp Hlawka inequality for Schatten $p$ -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real $p\ge256$ . We conjecture that the same formula holds for all $p\ge2$ .

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

Schnirelmann (1930) showed some finite $k$ works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved $k = 5$ unconditionally. Helfgott (2013) proved that every odd number greater than $5$ is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: $27$ is neither prime nor $2$ + prime.

≤ 25Formalized record≤ 5Open frontier

38 provers on it14 of 16 missions formalized

Matrix multiplication exponent

Schoolbook matrix multiplication takes $n^3$ operations. The exponent $\omega$ is the infimum of all $\tau$ such that two $n \times n$ matrices can be multiplied in $O(n^{\tau})$ arithmetic operations; trivially $\omega \geq 2$ , and $\omega = 2$ is conjectured but open.

Strassen gave the first nontrivial bound, $\omega < 2.81$ , in 1969, and introduced the laser method in 1986 to reach $\omega < 2.48$ . Coppersmith and Winograd's 1990 bound of $2.376$ stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached $\omega < 2.371339$ in 2025, and the current record is $\omega < 2.371177$ , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

See all campaigns →

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

≤ 25Formalized record≤ 5Open frontier

38 provers on it14 of 16 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Economics Operations Research·Captain: mikedeng1

Offering Memories to Sell Goods? Pricing and Welfare Implications of Experiential Retail: A Sufficiently Low-Quality Free Experience Raises Both the Monopoly Retailer's Profit and Consumer WelfareResearch Paper

Motivation

Retailers increasingly pair products with free in-store experiences: yoga classes at Lululemon, playgrounds at Toys "R" Us, cooking classes at Whole Foods. The experience is free to anyone who travels to the store, so it attracts buyers and free riders (visitors who enjoy the experience without buying) alike. The retailer earns revenue only from product sales, but pays for every visit to the experience. Whether a free experience pays for itself, and whether consumers gain from it, is the question of Mutlu, El-Amine and Sahin (SSRN 4617261; published in Manufacturing & Service Operations Management, 2023, doi:10.1287/msom.2021.0339).

The paper models the joint choice of experience and product with the multivariate logit (MVL) model of Russell and Petersen (2000), which builds a global choice model over baskets from conditional logit choices, via Besag's (1974) characterization of joint distributions from conditionals and the autologistic model of Cox (1972). The monopoly analysis (§3.1) yields two results: Theorem 1 on the optimal price and profit as functions of the experience quality, and Theorem 2, which says that low-quality experiences benefit both the retailer and its customers.

Setting

A monopolist sells a main product at price $p$ and offers a free experience of quality $u^x\ge 0$ . A consumer chooses $z^x\in\{0,1\}$ (obtain the experience or not) and $z^m\in\{0,1\}$ (buy the product or not). The parameters are the product's nominal utility $u^m$ , the travel cost $t>0$ , the product's unit cost $c^m$ , a memorable-event utility $s>0$ , a synergy sensitivity $\lambda>0$ , and a cost sensitivity $\eta>0$ .

The baseline utilities are $v^x=u^x-t$ and $v^m=u^m-t-p$ . Obtaining both offerings together saves one travel cost and creates a brand synergy $\beta(u^x)=s(1-e^{-\lambda u^x})$ , so the interaction term is $\psi=t+\beta(u^x)$ . Each basket has utility $\mu_{z^xz^m}=v^xz^x+v^mz^m+\psi z^xz^m$ and probability

q_{z^xz^m}=\frac{e^{\mu_{z^xz^m}}}{\sum_{j,i\in\{0,1\}}e^{\mu_{ji}}}.

The experience costs $c(u^x)=\eta(u^x)^2$ per user, and the retailer's profit is

\pi(p)=(q_{01}+q_{11})(p-c^m)-(q_{10}+q_{11})\,c(u^x).

Without an experience the model is the binary logit: $\pi^N(p)=q_1(p-c^m)$ with $q_1=e^{v^m}/(e^{v^m}+1)$ .

An optimal price $p^*$ maximizes $\pi$ over all real $p$ , and $p^{*N}$ maximizes $\pi^N$ ; $\pi^*(u^x)$ and $\pi^{*N}$ are the optimal profits. Consumer welfare is the logit log-sum $CS=\ln\sum_{j,i}e^{\mu_{ji}}$ with the experience and $CS^N=\ln(1+e^{v^m})$ without it.

Formalization targets

Goal: Theorem 2 (p. 13)

There is a threshold $\hat u^x>0$ such that for every $0\le u^x\le\hat u^x$ ,

\pi(p^*)>\pi^N(p^{*N})\quad\text{and}\quad CS(p^*)>CS^N(p^{*N})

for all optimal prices $p^*$ , $p^{*N}$ , which exist. In addition, some quality $u^x\ge0$ leaves the retailer strictly worse off than offering no experience. The threshold is existential, as in the paper; no particular value is claimed.

Milestones: three clauses of Theorem 1 (p. 12)

$p^*>p^{*N}$ for every $u^x\ge 0$ (with existence of the optimal prices and attainment of $\pi^*$ , $\pi^{*N}$ ).
$\left.\partial\pi^*/\partial u^x\right|_{u^x=0}>0$ .
$\lim_{u^x\to\infty}\pi^*=-\infty$ and $\pi^{*N}>0$ .

Significance

The result. Theorem 2 identifies a regime in which a free experience is a win-win: the retailer raises its price (Theorem 1) yet sells enough more, through the complementarity between experience and product, to raise profit, and consumers gain despite the higher price. Theorem 1 marks the limits of that regime: experiences of high quality attract so many free riders that profits fall without bound. Together they explain why experiential retail is typically offered at modest quality, and they are the benchmark for the paper's duopoly (§3.2) and conditionally-free (§4) analyses.

Formalizing it. The paper's lemmas and proofs are in an appendix that is not part of the held working paper; the published proof has never been machine-checked. A complete development would give the first verified analysis of optimal pricing under the MVL basket model, including existence of optimal prices, which the paper presupposes when it writes $p^*$ . The milestones are the clauses of Theorem 1 that the goal rests on.

Difficulty

The optimal profit $\pi^*$ is a supremum over prices of a ratio of exponential sums, and the experience quality $u^x$ enters through three channels at once: the experience-only basket, the synergy $\beta(u^x)$ , and the cost $\eta(u^x)^2$ , which is paid also for free riders. The obvious route, solving the first-order condition in closed form, fails: unlike the binary logit, where the optimal markup solves a Lambert-type equation, the MVL first-order condition mixes the free-rider basket into the denominator, and the optimal price need not be monotone in $u^x$ (Figure 1(a)). Comparing welfare requires the optimal prices themselves, not only the optimal profits, and the comparison is between two different optimization problems.

Formalization scope

All quantities are real numbers. Parameters form a structure with fields $u^m,t,c^m,s,\lambda,\eta$ ; the standing assumptions $t>0$ , $\eta>0$ , $s>0$ , $\lambda>0$ are one predicate carried by every statement. $u^m$ and $c^m$ are unconstrained, and prices range over all of $\mathbb R$ . The model uses only $\beta(u^x)=s(1-e^{-\lambda u^x})$ and $c(u^x)=\eta(u^x)^2$ , not the constant, piecewise-linear or linear variants the paper mentions.

Conventions committed to:

$s>0$ and $\lambda>0$ are read from "β(u^x) is positively valued and concave increasing" (p. 10).
Consumer welfare is the log-sum, never defined in the paper; it reproduces Figure 1(d) ( $CS^{*N}=\ln2$ at the Figure 1 parameters). The identification with expected maximum utility under Gumbel shocks, up to an additive constant, is cited, not formalized.
Optimal prices are global maximizers; their existence is a conclusion of each statement, never a hypothesis, and uniqueness is never assumed, so every comparison holds for all optimal prices.
"Improves" is strict. The derivative at $u^x=0$ is the right derivative on $[0,\infty)$ .
The second sentence of Theorem 2 is formalized as its retailer half.

A statement at a fixed price, at $u^x=0$ alone, or under an assumed (possibly non-existent) optimum would be trivial or vacuous and is ruled out: the goal quantifies over all optimal prices, asserts their existence, and covers a whole interval $[0,\hat u^x]$ .

Not formalized: the remaining clauses of Theorem 1 ("if $p^*$ is decreasing in $u^x$ , then $\pi^*$ is also decreasing in $u^x$ ", which needs $p^*$ as a function, and the comparative statics in $\eta$ and $\lambda$ ), and Theorems 3–8 (duopoly and conditionally-free offerings).

Needed infrastructure: continuity and coercivity arguments for existence of a maximizer over $\mathbb R$ , a parametric envelope argument for $\pi^*$ , and logit inequalities. Proofs of the milestones, of the existence of optimal prices, and reusable lemmas on logit pricing are welcome.

Selected references

N. Mutlu, H. El-Amine, O. Sahin, Offering Memories to Sell Goods? Pricing and Welfare Implications of Experiential Retail, SSRN working paper 4617261, 2023. https://ssrn.com/abstract=4617261 ; published in Manufacturing & Service Operations Management, 2023, https://doi.org/10.1287/msom.2021.0339
G. J. Russell, A. Petersen, Analysis of cross category dependence in market basket selection, Journal of Retailing 76(3), 2000. https://doi.org/10.1016/S0022-4359(00)00030-0
J. Besag, Spatial interaction and the statistical analysis of lattice systems, Journal of the Royal Statistical Society B 36(2), 1974. https://doi.org/10.1111/j.2517-6161.1974.tb00999.x
D. R. Cox, The analysis of multivariate binary data, Journal of the Royal Statistical Society C 21(2), 1972. https://doi.org/10.2307/2346482

5 thms1 active userReviewed

Economics Operations Research Partial Differential Equations·Captain: mikedeng1

High Discounts and Low Fundamental Surplus: An Equivalence Result for Unemployment Fluctuations: Tightness with Risk Premia Equals Tightness of an Artificial Economy Without Risk PremiaResearch Paper

Motivation

In the search-and-matching model of the labor market (Diamond–Mortensen–Pissarides), firms post vacancies until the expected value of a vacancy is zero, so how many jobs exist depends on how firms value future match cashflows. Shimer (2005) showed that the standard calibration produces unemployment fluctuations far smaller than in the data. Two explanations have been proposed for the gap. Ljungqvist and Sargent (2017) argue that what matters is a small fundamental surplus, the part of output that the labor market can allocate to vacancy creation. Hall (2017) argues instead for time-varying discount rates: when risk premia rise, firms discount future match surplus more heavily and post fewer vacancies.

Mitra, Seo and Xu (FRB Atlanta WP 2021-22; Management Science 2024) show that the two explanations are formally related. An economy with risk premia has the same labor-market tightness as an artificial economy without risk premia, in which the productivity process and the structural parameters are changed. In the artificial economy, the value of leisure and the vacancy cost are scaled up relative to average productivity, and discount-rate shocks appear as shocks to long-run productivity. This mission formalizes that equivalence, Proposition 1 of the paper.

Setting

The state of the economy is a pair of reals. In the economy without risk premia (§2.1), log productivity $y$ mean-reverts toward a long-run level $\mu$ :

dy_t=\kappa_y(\mu_t-y_t)\,dt+\sigma_y\,dB^{y,\mathcal P}_t,\qquad d\mu_t=-\kappa_\mu\mu_t\,dt+\sigma_\mu\,dB^{\mu,\mathcal P}_t,

with $\kappa_y>0$ , $\kappa_\mu\ge0$ , $\sigma_\mu\ge0$ . Further parameters are the risk-free rate $r_f$ , the separation rate $s$ , the workers' bargaining power $\beta\in(0,1)$ , the flow vacancy cost $c$ and the value of leisure $z$ . Matching is summarized by a vacancy-filling rate $q(\theta)$ as a function of tightness $\theta$ (vacancies per unemployed worker), and the job-finding rate is $f(\theta)=\theta q(\theta)$ . The generator of the productivity process is

\mathscr L^{\mathcal P}[\kappa_y,\sigma_y,\kappa_\mu,\sigma_\mu]F=\kappa_y(\mu-y)\partial_1F+\tfrac12\sigma_y^2\partial_{11}F-\kappa_\mu\mu\,\partial_2F+\tfrac12\sigma_\mu^2\partial_{22}F.

An equilibrium is a match surplus $S(y,\mu)$ and a tightness $\theta(y,\mu)$ with

(r_f+s)S=e^y-z-\tfrac{\beta c}{1-\beta}\theta+\mathscr L^{\mathcal P}S,\qquad c=(1-\beta)q(\theta)S.\qquad(8)

The paper derives (8) from four value-function equations by free entry and Nash bargaining.

In the economy with risk premia (§2.2), productivity is $dy_t=-\kappa_yy_t\,dt+\sigma_y\,dB^{y,\mathcal P}_t$ . Cashflows are discounted by a stochastic discount factor whose market price of risk $\eta$ follows $d\eta_t=\kappa_\eta(\bar\eta-\eta_t)\,dt+\sigma_\eta\,dB^{\eta,\mathcal P}_t$ with $\kappa_\eta,\sigma_\eta>0$ . Under the risk-neutral measure, productivity has drift $-(\sigma_y\eta+\kappa_yy)$ and is driven by $B^{y,\mathcal Q}_t=\int_0^t\eta_u\,du+B^{y,\mathcal P}_t$ . The equilibrium (19) is (8) with the risk-neutral generator

\mathscr L^{\mathcal Q}F=-(\sigma_y\eta+\kappa_yy)\partial_1F+\tfrac12\sigma_y^2\partial_{11}F+\kappa_\eta(\bar\eta-\eta)\partial_2F+\tfrac12\sigma_\eta^2\partial_{22}F.

The dictionary uses the shift $a=\sigma_y\bar\eta/\kappa_y=-\mathbb E^{\mathcal Q}[y_t]$ . The artificial states are $y^{\mathcal Q}=y+a$ and $\mu^{\mathcal Q}=\sigma_y(\bar\eta-\eta)/\kappa_y$ . The artificial parameters are $z^{\mathcal Q}=e^az$ , $c^{\mathcal Q}=e^ac$ , $\kappa^{\mathcal Q}_y=\kappa_y$ , $\sigma^{\mathcal Q}_y=\sigma_y$ , $\kappa^{\mathcal Q}_\mu=\kappa_\eta$ , $\sigma^{\mathcal Q}_\mu=\sigma_\eta\sigma_y/\kappa_y$ , and $r_f,s,\beta,q$ are unchanged.

Formalization targets

Goal: Proposition 1

For surpluses and tightnesses related by $S^{\mathcal Q}(y^{\mathcal Q},\mu^{\mathcal Q})=e^aS(y,\eta)$ and $\theta^{\mathcal Q}(y^{\mathcal Q},\mu^{\mathcal Q})=\theta(y,\eta)$ ,

(S,\theta)\ \text{solves (19)}\iff(S^{\mathcal Q},\theta^{\mathcal Q})\ \text{solves (8) for the artificial parameters}.

The goal also states that the artificial parameters satisfy the standing assumptions of §2.1, and that the transformed productivity paths solve the artificial dynamics (26), driven by $B^{y,\mathcal Q}$ and $-B^{\eta,\mathcal P}$ .

Milestones

The surplus characterization (8) from the value-function equations (4), with the wage formula (§2.1).
The risk-neutral productivity equation (15)–(16), along paths.
The chain-rule facts expressing $S_y,S_{yy},S_\eta,S_{\eta\eta}$ through derivatives of $S^{\mathcal Q}$ .
The forward direction (19) ⇒ (25).
The artificial productivity process (26)–(27).

Significance

The result. Proposition 1 lets every tool built for search models with risk-neutral firms be applied to models with time-varying risk premia. In particular, the Ljungqvist–Sargent decomposition of unemployment volatility through the fundamental surplus carries over. The paper uses it to show that high discounts in the original economy are equivalent to low fundamental surpluses in the artificial economy, and that discount-rate shocks act like long-run productivity shocks. Equilibrium moments of tightness-based statistics can then be computed in the artificial economy.

Formalizing it. The result is proved in the paper at the level of PDEs and change of variables. It has not been machine-checked. The mission produces a checked statement of the dictionary with every sign and scaling made explicit, together with the converse direction, which the paper leaves implicit. It also produces a small reusable layer: two-factor Ornstein–Uhlenbeck generators written as partial-derivative operators, and pathwise integral forms of the dynamics.

Difficulty

The algebra is elementary, but the bookkeeping is where errors occur. Three signs or scalings are easy to get wrong:

The long-run factor $\mu^{\mathcal Q}$ decreases in $\eta$ , so its shock is $-B^{\eta,\mathcal P}$ .
The drift $-\sigma_y\eta$ must be absorbed into $\kappa_y(\mu^{\mathcal Q}-y^{\mathcal Q})$ with exactly the shift $a$ .
$e^{a}$ rescales $c$ and $z$ but not $\beta$ , $s$ or $r_f$ .

The converse direction needs the change of variables to be onto. The PDE identities need second derivatives of composed functions to be transported in both directions while regularity is preserved. A tempting shortcut is to define the artificial equilibrium as the pullback of (19). That makes the statement empty, and it is excluded here.

Formalization scope

All quantities are real numbers. States are curried functions $\mathbb R\to\mathbb R\to\mathbb R$ with productivity first, and partial derivatives are deriv of one-variable sections. Both equilibrium predicates require the surplus to be $C^2$ , because deriv returns $0$ at non-differentiable points. The vacancy-filling rate $q$ is an arbitrary function; no Cobb–Douglas form is imposed. No sign is imposed on $r_f$ , $s$ , $c$ or $z$ .

The hypothesis $\sigma_y>0$ is added to the standing assumptions of the economy with risk premia. The map (21b) is invertible only when $\sigma_y\neq0$ , and the paper's estimate is $\sigma_y=0.0179$ .

Probability is not formalized. The stochastic differential equations are written as pathwise integral equations for $t\ge0$ , driven by given paths through their increments $B_t-B_0$ , and the paths $y,\eta$ are continuous. The following are outside the mission's scope:

Girsanov's theorem, which makes $B^{y,\mathcal Q}$ a Brownian motion under the risk-neutral measure;
the Feynman–Kac representation linking value functions to the PDEs;
the stationary mean (22) under $\mathcal Q$ , whose value is used as the shift.

An equilibrium is its PDE characterization, as in the paper's derivation. The artificial equilibrium is the §2.1 predicate with the explicit generator $\mathscr L^{\mathcal P}$ , evaluated at the artificial parameters. It is defined independently of (19) and of the dictionary, so the goal is not true by definition.

Contributions are welcome on the chain-rule lemmas for deriv under affine changes of variables in two dimensions and on regularity transfer for $C^2$ functions under affine maps. Both are reusable beyond this mission.

Selected references

I. Mitra, T. Seo, Y. Xu, High Discounts and Low Fundamental Surplus: An Equivalence Result for Unemployment Fluctuations, FRB Atlanta Working Paper 2021-22, 2021. https://doi.org/10.29338/wp2021-22 (published in Management Science 70(6), 4051–4068, 2024, https://doi.org/10.1287/mnsc.2022.03712)
R. E. Hall, High Discounts and High Unemployment, American Economic Review 107(2), 305–330, 2017. https://doi.org/10.1257/aer.20141297
L. Ljungqvist, T. J. Sargent, The Fundamental Surplus, American Economic Review 107(9), 2630–2665, 2017. https://doi.org/10.1257/aer.20150233
R. Shimer, The Cyclical Behavior of Equilibrium Unemployment and Vacancies, American Economic Review 95(1), 25–49, 2005. https://doi.org/10.1257/0002828053828572

7 thms1 active userReviewed

Algorithmic Game Theory Economics Operations Research·Captain: mikedeng1

Selling Assets: Are Sellers Better Off with Strong Buyers?: Under Free Entry, a More Dominant Strong Buyer Lowers the Seller's RevenueResearch Paper

Why buyer strength can reduce a sale price

A seller who can attract a buyer with a higher valuation might expect a higher auction price. That intuition holds when the number of competing buyers is fixed. If other buyers must pay to learn their valuations and decide whether to enter, a stronger buyer can instead cause some of them to stay out. Marquez and Singh study this response in a second-price auction and conclude that the seller's expected revenue falls under free entry when the strong buyer becomes more dominant (working paper, §§4–5). The result separates two effects that are often combined in informal auction advice: the price effect of a better bidder and the competition effect of discouraging entry.

This mission targets Proposition 1 of the October 2021 SSRN working paper. The paper was subsequently published in Management Science in 2023, but the theorem numbers and pages here refer only to the pinned working paper. The fixed-number comparison in Lemma 1 and the free-entry response in Lemma 2 supply the immediate context. No separate historical timeline is needed: the mission addresses this paper's particular auction model, rather than a long-running conjecture.

Auction and entry setting

There is one strong buyer and a pool of symmetric regular buyers. A buyer pays an entry cost $c>0$ to learn the value of the asset, enters, and bids that value in a simultaneous second-price auction. The highest bidder wins and pays the second-highest bid. Every buyer is risk neutral and the seller values the asset at zero. The working paper ignores integer constraints in its free-entry calculations, so the number $N$ of regular buyers is treated as a real parameter after the auction formulas have been derived (§2, p. 9; §5, p. 13).

Each regular value has distribution function $F$ ; the strong value has distribution function $G$ . They are cumulative distribution functions of probability laws concentrated on $[0,1]$ . In the paper's first-order stochastic dominance convention, $G$ dominates $F$ when $G(z)\le F(z)$ for $0<z<1$ , with strict inequality on a positive-measure set. A further improvement replaces $G$ by $G'$ with $G'(z)\le G(z)$ and the same strictness condition. The CDF direction may look reversed: a more valuable distribution puts less probability below any fixed threshold.

For a real buyer count $N\ge1$ , the gross expected profit of one regular buyer and the seller's revenue are represented by the paper's equations (1) and (3):

\pi(N\mid G)=\int_0^1 (1-F(z))G(z)F(z)^{N-1}\,dz, \qquad R(N\mid G)=1-\int_0^1F(z)^N\,dz-N\pi(N\mid G).

The second formula comes from the auction's surplus identity. It is used as the definition of $R$ in this mission, with the same $\pi$ as the first formula. A free-entry root $N_G$ satisfies $\pi(N_G\mid G)=c$ and $N_G\ge1$ . The root is a witness to the zero-profit equation, not an infimum or an independently chosen optimum. The statements quantify over every root satisfying these conditions.

Formalization targets

The first target is the profit and entry chain in Lemmas 1 and 2. Increasing $N$ lowers each regular buyer's profit; a more dominant strong buyer also lowers it. At a common positive entry cost, the improved strong buyer therefore has a smaller free-entry root:

\pi(N\mid G')<\pi(N\mid G),\qquad N_{G'}<N_G.

The main target is the corrected strict form of Proposition 1 (p. 14). If the improvement from $G$ to $G'$ has a strict effect at values where $F(z)<1$ , then

R(N_{G'}\mid G')<R(N_G\mid G).

This positive-measure condition is stated separately in Lean. It is needed because strict CDF dominance can occur entirely where every regular value is already at or below $z$ , so $F(z)=1$ . In that case the change in $G$ does not affect regular-buyer profit, the roots can agree, and the printed strict revenue conclusion fails. For example, a regular-value law supported on $[0,1/2]$ and a shift of the strong-buyer law only above $1/2$ have this problem. The mission keeps the paper's FOSD premise and discloses the additional guard.

What the result supplies

The theorem identifies a precise limit on the seller's benefit from attracting a strong buyer. At a fixed number of bidders, the paper's Lemma 1 says a more dominant strong buyer raises revenue. When regular bidders pay an entry cost and the count adjusts, Proposition 1 gives the opposite strict comparison under the stated effective-shift condition. The result therefore distinguishes advice for a thin market, where the bidder pool is fixed, from advice for a deep market with endogenous participation (§§4–5).

The economic result is already proved in the paper, subject to the corrections disclosed here. The open work for this mission is a machine-checked proof of the stated inequalities from the CDF formulas and free-entry equations. A verified development would also make the analytic comparison reusable for other entry models in which one entrant changes the incentives of the rest. The mission does not claim that a published Lean proof of this result already exists; the local prior-art search found only auction and stochastic-order formalizations with different models or conclusions.

Analytical difficulty

The immediate fixed- $N$ comparison cannot settle the free-entry claim, because $G'$ changes the equilibrium buyer count at the same time that it changes a bidder's value. The revenue identity reduces the equilibrium comparison to how the expression $1-\int_0^1F(z)^N\,dz-Nc$ varies between the two roots. The paper calculates a positive derivative at $N_G$ , but its prose then says the expression is increasing in $N$ without a range. Global monotonicity is false: the $-Nc$ term eventually dominates for a uniform $F$ and positive $c$ . The milestone records the needed statement on $[1,N_G]$ and labels the correction.

There is also a strictness issue in Lemma 1. A positive-measure CDF shift alone need not change the integral defining regular profit: its factor $1-F(z)$ can vanish throughout the shift. The definitions and hypotheses expose exactly where the strict integrand must be positive. Differentiating the real power of a CDF also requires treating zero CDF values and the endpoint of the integration interval consistently.

Formalization scope

Lean uses ProbabilityTheory.cdf of probability measures on $\mathbb R$ concentrated on $[0,1]`. The dominance relation compares their real-valued CDFs on$ (0,1) $and measures strictness with Lebesgue volume. Both$ N_G $and$ N_{G'} $are real and at least one, as assumed on p. 13. The interval integral runs from zero to one. At$ F(z)=0 $, the formulas use$ 0^0=1 $for$ N=1 $and the limiting zero convention for$ F(z)^N\log F(z) $; the statements never use$ N<1$, where the profit integral may fail to be integrable.

The definitions keep revenue tied to the paper's profit formula. Defining revenue directly as $1-\int F^N-Nc$ at a free-entry root would erase the comparison the theorem is meant to establish. The theorem also requires actual CDFs and positive-measure strictness, rather than arbitrary functions satisfying an uncheckable existence assumption. A complete proof needs measure-theoretic CDF bounds, integral monotonicity and strictness, real-power differentiation, and monotonicity on a closed interval. These analytic results can be reused beyond this auction. Contributions on the five listed milestones and the goal are within scope; the separate fixed-entry revenue clauses, numerical illustrations, and later subsidy, reserve, and bundling results are not included.

Selected references

R. Marquez and R. Singh, Selling assets: Are sellers better off with strong buyers?, SSRN working paper 3949699, October 2021, §§2–5 and Appendix, pp. 9–14, 37–39. Working paper.
R. Marquez and R. Singh, Selling assets: Are sellers better off with strong buyers?, Management Science, 2023. DOI: 10.1287/mnsc.2023.4938. The mission's statements and page citations use the working paper above.

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Differential Geometry Numerical Analysis Optimization·Captain: mikedeng1

Active-Set Newton Methods and Partial Smoothness: Transversal C² Manifolds with a Constant-Rank Map Between Them Yield a Quadratically Convergent Active-Set Newton IterationResearch Paper

Motivation

Active-set methods for optimization and variational problems run in two phases. A globally convergent first phase identifies which constraints are active at the solution; once they are identified the problem simplifies locally, and a second phase converges fast, typically by linearizing the simplified problem. Sequential quadratic programming for nonlinear programming is the classical instance: near a solution satisfying the standard second-order conditions, the graph of the normal-cone operator and the space $\mathbf R^n\times\{\bar c\}$ are manifolds meeting at a single point, and one Newton step linearizes the first, intersects with the second, and restores the result.

The property behind the first phase, finite identification of an active manifold, is formalized by partial smoothness, introduced by Lewis (SIAM J. Optim., 2002) after Wright's identifiable surfaces (SIAM J. Control Optim., 1993); its constant-rank characterization is in Lewis and Liang, arXiv:1807.03134. Lewis and Wylie (arXiv:1902.00724) isolate what drives the second phase in the same geometric language: two manifolds intersecting transversally at an isolated point, a semi-linearization step, and a constant-rank map between the manifolds that splits the restoration step into a primal and a dual update. Their Theorem 5.1 is the general result; applied to the graph of a set-valued map it gives an active-set Newton method for generalized equations $0\in\Phi(u)$ with $\Phi$ partly smooth.

Setting

Let $\mathbf E$ be a Euclidean space with norm $|\cdot|$ . A set $\mathcal X\subset\mathbf E$ is a $C^{(r)}$ -manifold around a point $z\in\mathcal X$ if there are a $C^{(r)}$ -smooth map $F:\mathbf E\to\mathbf R^m$ with surjective derivative $DF(z)$ and an open set $V\ni z$ with $\mathcal X\cap V=F^{-1}(0)\cap V$ .

The tangent space $T_{\mathcal X}(x)$ is the linear span of the tangent cone of $\mathcal X$ at $x$ , and the normal space is $N_{\mathcal X}(x)=T_{\mathcal X}(x)^\perp$ . For a manifold, $T_{\mathcal X}(z)=\mathrm{Null}\,DF(z)$ . Two manifolds $\mathcal X$ , $\mathcal Y$ around $z$ intersect transversally at $z$ if $N_{\mathcal X}(z)\cap N_{\mathcal Y}(z)=\{0\}$ , and $z$ is an isolated point of intersection if $\mathcal X\cap\mathcal Y\cap V=\{z\}$ for some open $V\ni z$ .

The semi-linearization point $y(x)$ of $x\in\mathcal X$ is the unique point where the linear approximation $x+T_{\mathcal X}(x)$ meets $\mathcal Y$ near $z$ : only $\mathcal X$ is linearized.

A map $P$ from $\mathcal X$ into $\mathcal Y$ is $C^{(2)}$ -smooth and constant rank nearby $z$ if it is $C^{(2)}$ on a neighborhood of $z$ and the rank of $DP(x)$ on $T_{\mathcal X}(x)$ is the same for all $x\in\mathcal X$ near $z$ . For a neighborhood $W$ of $z$ the image $\mathcal M=P(\mathcal X\cap W)$ is the active manifold. $\mathrm{Proj}_A$ is the nearest-point projection onto a set $A$ . An iteration $x\leftarrow T(x)$ converges quadratically to $z$ if, from every start $x_0\in\mathcal X$ close enough to $z$ , the iterates stay in $\mathcal X$ near $z$ , converge to $z$ , and satisfy $|x_{n+1}-z|\le C|x_n-z|^2$ with one constant $C$ .

Formalization targets

Goal: Theorem 5.1 (Active set method for manifold intersection)

Let $z$ be an isolated, transversal point of intersection of two $C^{(2)}$ -manifolds $\mathcal X$ and $\mathcal Y$ around $z$ , and let $P:\mathcal X\to\mathcal Y$ fix $z$ and be $C^{(2)}$ -smooth and constant rank nearby. Then for every sufficiently small ball $W$ around $z$ :

\begin{aligned} &\text{(a)}\ \ \mathcal M=P(\mathcal X\cap W)\ \text{is a } C^{(2)}\text{-manifold around } z;\\ &\text{(b)}\ \ \exists\ S:\mathcal Y\cap W\to\mathcal M \text{ Lipschitz},\ S(z)=z;\ \text{near } z,\ S=\mathrm{Proj}_{\mathcal M}\ \text{is one};\\ &\text{(c)}\ \ \exists\ Q:\mathcal M\cap W\to\mathcal X \text{ Lipschitz},\ Q(z)=z,\ P(Q(w))=w;\ \text{near } z,\ Q(w)=\mathrm{Proj}_{P^{-1}(w)}(w)\ \text{is one};\\ &\text{(d)}\ \ (x+T_{\mathcal X}(x))\cap\mathcal Y\cap W=\{y(x)\}\ \text{for all } x\in\mathcal X\cap W; \end{aligned}

and for every such $S$ and $Q$ the iteration $x\leftarrow Q\big(S(y(x))\big)$ converges quadratically to $z$ .

Milestones

§2, p. 4: $T_{\mathcal X}(z)=\mathrm{Null}\,DF(z)=\mathrm{Range}\,DG(0)$ and $N_{\mathcal X}(z)=\mathrm{Range}\,DF(z)^*=\mathrm{Null}\,DG(0)^*$ , for every defining map $F$ and coordinate system $G$ .
Theorem 3.1: $y(x)$ exists uniquely and $|y(x)-z|=O(|x-z|^2)$ .
Corollary 3.5: for every Lipschitz $R$ into $\mathcal X$ fixing $z$ , $x\leftarrow R(y(x))$ converges quadratically.
Theorem 5.1(a), (b), (c) as separate statements.

Significance

The result. Theorem 5.1 turns an abstract Newton scheme into an active-set method. Corollary 3.5 needs a Lipschitz restoration map $R$ from $\mathcal Y$ back into $\mathcal X$ , which in applications is coupled and opaque; Theorem 5.1 builds $R=Q\circ S$ from a constant-rank map, so the primal variables are updated by projecting onto the active manifold and the dual variables by a fiber projection with the primal part held fixed. With $\mathcal X=\operatorname{gph}\Phi$ , $\mathcal Y=\mathbf U\times\{0\}$ and $P(u,v)=(u,0)$ this yields the paper's Theorem 5.5 for partly smooth generalized equations, and through it Newton-type methods for variational inequalities over partly smooth sets (§6).

Formalizing it. The theorems are proved in the paper; none is machine-checked. The mission produces a statement of Theorem 5.1 at the level of sets in a Euclidean space, with every hypothesis explicit, and the intermediate results it depends on. A proof needs, in Lean, the identification of the tangent cone of a zero-set manifold with the kernel of the derivative, the semi-linearization estimate via the implicit function theorem, the constant-rank theorem for maps between embedded submanifolds, and local Lipschitz continuity of nearest-point projections onto $C^{(2)}$ submanifolds. Several of these are absent from Mathlib in this form.

Difficulty

The convergence claim follows from Corollary 3.5 once (a)–(c) are available, and Corollary 3.5 follows from the quadratic estimate (3.2). The hard part is (a)–(c). Mathlib has no constant-rank theorem, which is what makes $\mathcal M$ a manifold. And the image $\mathcal M=P(\mathcal X\cap W)$ is not closed, so neither the nearest point of $\mathcal M$ to a point of $\mathcal Y\cap W$ nor the nearest point of a fiber $P^{-1}(w)$ need exist: both projections are single-valued and Lipschitz only near $z$ , and the obvious argument (project onto a closed set) does not apply. A second obstacle is that $T_{\mathcal X}(x)$ enters at points $x\ne z$ , so the identification of the tangent cone with the classical tangent space is needed at every point of $\mathcal X$ near $z$ , not only at $z$ .

Formalization scope

$\mathbf E$ is a finite-dimensional real inner-product space; $\mathbf R^m$ is EuclideanSpace ℝ (Fin m). Smoothness $C^{(r)}$ is ContDiff ℝ r with $r$ a natural number. All maps ( $P$ , $S$ , $Q$ , $R$ , $y$ ) are total functions on $\mathbf E$ ; mapping and Lipschitz conditions are imposed only on the paper's domains ( $\mathcal Y\cap W$ , $\mathcal M\cap W$ ), never globally.
" $F^{-1}(0)$ is a neighborhood of $z$ in $\mathcal X$ " is local equality on an open set. $T_{\mathcal X}(x)$ and $N_{\mathcal X}(x)$ come from the published definition ProjLikeRetr.Retractor.tangentBundle (span of tangentConeAt and its orthogonal complement), so they do not depend on a choice of $F$ .
" $C^{(2)}$ -smooth and constant rank nearby" is read as $C^{(2)}$ on an open set of $\mathbf E$ around $z$ with constant rank of $DP(x)|_{T_{\mathcal X}(x)}$ for $x\in\mathcal X$ near $z$ .
"Any sufficiently small neighborhood $W$ " ranges over open balls $B(z,\varepsilon)$ , $0<\varepsilon\le r$ : for open sets that are not star-shaped about $z$ , part (d) can fail.
The two "in particular" projection clauses are stated on a ball $B(z,\delta)$ for some $\delta>0$ ; the fiber of $P$ over $w$ is taken in $\mathcal X\cap W$ .
Corollary 3.5 assumes $R$ Lipschitz only on $\mathcal Y\cap W$ , which is the form Theorem 5.1 uses.
The existence half of (b) is witnessed by the constant map $S\equiv z$ , and an existential convergence claim would be witnessed by $S\equiv Q\equiv z$ . The goal therefore keeps the projection clauses and quantifies the convergence claim over every admissible $S$ and $Q$ .
Not formalized: Theorem 5.5 and §6 (coderivatives, partly smooth set-valued maps, variational inequalities).

Contributions welcome: the tangent-cone/kernel identity (reusable for any zero-set manifold), a constant-rank theorem for maps between embedded submanifolds of a Euclidean space, and local Lipschitz continuity of the projection onto a $C^{(2)}$ submanifold. All three are useful well beyond this mission.

Selected references

A.S. Lewis and C. Wylie, Active-set Newton methods and partial smoothness, Mathematics of Operations Research, 2021; preprint arXiv:1902.00724v1, 2019. https://arxiv.org/abs/1902.00724
A.S. Lewis and J. Liang, Partial smoothness and constant rank, arXiv:1807.03134, 2018. https://arxiv.org/abs/1807.03134
A.S. Lewis, Active sets, nonsmoothness, and sensitivity, SIAM J. Optim. 13:702–725, 2002.
S.J. Wright, Identifiable surfaces in constrained optimization, SIAM J. Control Optim. 31:1063–1079, 1993.
J.M. Lee, Introduction to Smooth Manifolds, Springer, 2003.
R.T. Rockafellar and R.J-B. Wets, Variational Analysis, Springer, 1998.

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Economics Operations Research·Captain: mikedeng1

Managerial Regret and Inventory Pricing: The Regretful Seller's Marginal Value of Inventory and Price Increase with Time-to-Go and Decrease with Remaining InventoryResearch Paper

Motivation

Revenue management studies how a seller with a fixed stock and a finite selling season should price over time. The classical models assume an expected-revenue maximizer: Gallego and van Ryzin (1994) in continuous time and Bitran and Mondschein (1997) in discrete time showed that the optimal price and the marginal value of inventory rise with the time left in the season and fall with the stock on hand. These monotonicity results are what make simple markdown policies defensible in practice.

Managers are not always expected-revenue maximizers. Regret theory, going back to Bell (1982), models a decision maker who compares the outcome with what the best choice would have yielded once the uncertainty is resolved. A pricing manager who sees a customer walk away regrets having priced too high; one who sees a sale regrets having left money on the table. Li and Liu (Management Science, forthcoming; accepted manuscript SSRN 3726966) build this anticipated regret into the single-period pricing problem and into the finite-horizon inventory pricing dynamic program, and ask whether the classical structure survives. This mission formalizes their answer for the structure of the dynamic policy.

Setting

A customer's willingness-to-pay $v$ has density $f$ and distribution function $F$ , supported on $[a,b]$ with $0\le a<b<\infty$ . Write $\bar F=1-F$ and the failure rate $h(x)=f(x)/\bar F(x)$ ; the law has an increasing failure rate (IFR) if $h$ is non-decreasing. Throughout, "increasing" means non-decreasing.

Single period. A seller with one unit and salvage value (opportunity cost) $z\in[0,b]$ posts price $p$ . Her revenue beyond the salvage value is

\pi(p,z)=(p-z)\bar F(p),

maximized by the unbiased price $p^*$ , with optimal value $\pi^*(z)=\max_p\pi(p,z)$ . If the customer does not buy, she regrets the best lower price she could have charged; if he buys, she regrets the best higher price. The overpricing regret and underpricing regret are

r_o(p,z)=\max_{y<p}\big[\pi(y,z)-(y-z)\bar F(p)\big],\qquad r_u(p,z)=\max_{y\ge p}\pi(y,z)-\pi(p,z),

and with overpricing parameter $\alpha\in[0,1]$ and underpricing parameter $\beta\in[0,1]$ the regretful seller maximizes

\hat\pi(p,z)=\pi(p,z)-\alpha\,r_o(p,z)-\beta\,r_u(p,z).

Its maximizer is the regretful price $\hat p$ .

Dynamic problem. The season has $T$ periods, indexed by the time-to-go $t$ ; in each period one customer arrives with probability $\lambda\le 1/2$ . With $x$ units left, the regretful seller's value-to-go satisfies

U_t(x)=U_{t-1}(x)+\lambda\max_p\hat\pi\big(p,\Delta U_{t-1}(x)\big),\qquad U_t(0)=U_0(x)=0,

where $\Delta U_t(x)=U_t(x)-U_t(x-1)$ is the marginal value of the $x$ -th unit. The price $\hat p_t(x)$ maximizes $\hat\pi(\cdot,\Delta U_{t-1}(x))$ : the dynamic seller acts as a single-period seller whose opportunity cost is the marginal value of inventory.

The paper imposes one regularity condition on the single-period problem: the regretful seller's marginal value from the opportunity cost, $\hat m(z,\alpha,\beta)=\partial\hat\pi(\hat p,z)/\partial z$ , is non-positive.

Formalization targets

Goal: Proposition 4

Under IFR, $\alpha,\beta\in[0,1]$ , $\lambda\le1/2$ and $\hat m\le 0$ , for every $t$ and $x\ge1$ ,

\Delta U_t(x)\le\Delta U_{t+1}(x),\qquad \Delta U_t(x+1)\le\Delta U_t(x),

the price problem of (10) has a unique solution, and

\hat p_t(x)\le\hat p_{t+1}(x),\qquad \hat p_t(x+1)\le\hat p_t(x).

Milestones

The single-period layer: under IFR the unbiased price is unique and revenue is unimodal (§2); Lemma 1 (monotonicity of $r_o$ and $r_u$ in $p$ , and $r_u=0$ above $p^*$ ); Proposition 1(a) (existence, uniqueness, corner case of $\hat p$ ); Proposition 2 ( $\hat p\le p^*$ ). The ingredients of the proof of Proposition 4: the envelope identity (14) $d\pi^*/dz=-\bar F(p^*)$ , the rewriting (15) of $\hat\pi$ , the monotonicity of $r_o$ in $z$ (step (g)), Proposition 4(a), and the monotonicity of the single-period regretful price $\hat p(z)$ in $z$ .

Significance

Proposition 4 shows that anticipated regret distorts the level of prices but not the shape of the optimal policy: a regretful seller still marks down as the season ends and as stock accumulates, exactly as the unbiased seller of Bitran and Mondschein does. This is what the paper's later results build on, namely the comparison $\hat p_t(x)\le p_t(x)$ of the regretful and unbiased dynamic prices (Proposition 5) and the observation that a regretful seller can earn less with more inventory (Proposition 6). It also tells a practitioner that markdown heuristics remain structurally appropriate under regret.

The result is proved in the paper; to our knowledge none of it is machine-checked. The mission produces a checked proof of the whole chain from the single-period model to the dynamic structure. The paper's identity (15) holds only for prices below $p^*$ , and its induction leaves the boundary $x=1$ implicit, so the formal proof also settles these points.

Difficulty

For the unbiased seller the argument is short: $\max_p\pi(p,z)$ is convex and decreasing in $z$ , and induction on $t$ closes. With regret, $\hat\pi(p,z)$ contains two inner maximizations whose arguments move with $p$ and $z$ , so neither convexity of $z\mapsto\max_p\hat\pi(p,z)$ nor a clean envelope formula is available, and the unbiased induction does not go through as written. The regret parameters also interact with the arrival probability: the paper's argument needs $\lambda(1+\beta)\le1$ , which is where $\lambda\le1/2$ enters. For part (b), $\hat\pi(p,z)$ has increasing differences in $(p,z)$ only on the range $p\le p^*(z)$ , not for all prices, so Topkis's monotone-argmax theorem does not apply directly. At the boundary $x=1$ the induction step refers to $\Delta U_t(0)$ , which (10) does not define.

Formalization scope

All quantities are real numbers. The willingness-to-pay law is a density $f$ on $\mathbb R$ with $f\ge0$ , $f=0$ off $[a,b]$ , $\int f=1$ ; continuity of $f$ on $[a,b]$ is added (the paper evaluates $f(a)$ and $h(a)$ and uses $F'=f$ ). IFR is required on $x<b$ , since $h$ is undefined where $\bar F=0$ . Every max of the paper is a real supremum over the indicated set; all families involved are bounded above. Optimal prices enter as argmax predicates, never as chosen functions, and the price statements quantify over every maximizer. The dynamic program is a recursion on $\mathbb N$ in $(t,x)$ ; the cap $x\le c$ on inventory is dropped, which only extends the domain; $\Delta U_t(x)$ is used for $x\ge1$ only.

The regularity condition $\hat m\le0$ is rendered as: $z\mapsto\max_p\hat\pi(p,z)$ is non-increasing on $[0,b]$ . It is a hypothesis about the single-period problem only; a version stated about $U$ or $\Delta U$ would assume part (a). The second regularity condition of the paper, $\partial\hat m/\partial\alpha\ge0$ , is not used by Proposition 4 and is not assumed. Uniqueness statements are posed for $z<b$ , since at $z=b$ every price above $b$ is optimal. The goal asserts existence and uniqueness of the price of (10) explicitly, so that its price-monotonicity clauses cannot hold vacuously. The identity (15) is stated on $p\le p^*$ together with the equality of maxima, its form in the proof.

Not formalized: the interior first-order characterization (5) of Proposition 1(a), Proposition 1(b), Proposition 3, Propositions 5–7, Lemmas 2–5, and the numerical study. The single-period layer (the regret objective, its monotonicity in price and opportunity cost) is reusable for the paper's other results; contributions proving any milestone independently are welcome.

Selected references

M. Li and Y. Liu, Managerial regret and inventory pricing, Management Science, forthcoming. https://doi.org/10.1287/mnsc.2021.4073; accepted manuscript: https://ssrn.com/abstract=3726966
G. R. Bitran and S. V. Mondschein, Periodic pricing of seasonal products in retailing, Management Science 43(1), 64–79, 1997. https://doi.org/10.1287/mnsc.43.1.64
G. Gallego and G. van Ryzin, Optimal dynamic pricing of inventories with stochastic demand over finite horizons, Management Science 40(8), 999–1020, 1994. https://doi.org/10.1287/mnsc.40.8.999
D. E. Bell, Regret in decision making under uncertainty, Operations Research 30(5), 961–981, 1982. https://doi.org/10.1287/opre.30.5.961

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Algorithmic Game Theory Economics Mechanism Design·Captain: mikedeng1

Interim Rationalizable Implementation of Functions: Weak Interim Rationalizable Monotonicity and the Weak No-Worst-Rule Condition Suffice for Implementation in Interim Rationalizable StrategiesResearch Paper

Motivation

A mechanism designer chooses a rule for turning agents’ messages into an outcome. When agents know their own types but have only beliefs about the others, the designer cannot assume that they use a single equilibrium strategy. Interim correlated rationalizability instead keeps every message that is a best response to some belief consistent with the agent’s information and with the other agents’ surviving messages. A rule that produces the desired outcome for all such messages is robust to this uncertainty about strategic play. Kunimoto, Saran, and Serrano study which social choice functions can be implemented under this solution concept. Their sufficiency result identifies two conditions on a suitable version of the target rule: weak interim rationalizable monotonicity and a weak no-worst-rule condition. The published article is Kunimoto, Saran, and Serrano (2024); this mission follows the authors’ October 2020 Bravo working-paper version, whose pagination and theorem labels are used below.

The paper separates necessity from sufficiency. Theorem 4.5 shows that implementability implies weak interim rationalizable monotonicity of some equivalent social choice function. Theorem 6.3 gives sufficiency when the equivalent function also satisfies weak no-worst-rule. Its Example 7.1 uses the sufficiency theorem to exhibit implementation in rationalizable strategies beyond the stronger conditions discussed in the earlier literature. This mission targets Theorem 6.3 and its stated supporting results.

Setting

There is a finite set $I$ of agents. Agent $i$ has a finite nonempty type set $T_i$ and, upon learning $t_i\in T_i$ , holds an interim belief $\pi_i(t_i)\in\Delta(T_{-i})$ about the other agents’ types. The full type space is $T=\prod_{i\in I}T_i$ . A finite set $A$ contains the pure outcomes, and $\Delta(A)$ is its set of lotteries. Agent $i$ evaluates a lottery by the expected value of a state-dependent utility $u_i(a,t)$ over pure outcomes $a\in A$ . A social choice function is $f:T\to\Delta(A)$ .

The designer cares about a set $T^*\subseteq T$ containing every state to which at least one agent assigns positive interim probability. Two social choice functions are equivalent, written $f\approx f'$ , when they agree on $T^*$ . A mechanism has a nonempty countable message space $M_i$ for every agent and an outcome rule $g:\prod_iM_i\to\Delta(A)$ .

For a proposed correspondence $S_i(t_i)\subseteq M_i$ , the operator $b$ retains a message $m_i$ if it is a best response to some probability distribution over $T_{-i}\times M_{-i}$ . This distribution must have type marginal $\pi_i(t_i)$ and assign positive weight only to the other agents’ messages in $S$ . The correspondence $S^\Gamma(T)$ is the largest fixed point of $b$ . The mechanism implements $f$ when there is an equivalent $\hat f$ such that $S_i^\Gamma(T)(t_i)$ is nonempty for every type and every rationalizable message profile at every state $t\in T$ yields $\hat f(t)$ . The uniqueness condition ranges over all of $T$ , including states outside $T^*$ .

Weak interim rationalizable monotonicity (weak IRM) is a condition on deceptions. A deception lets each true type $t_i$ be reported as one of several types, including itself. If some allowed report profile changes the chosen lottery, the deception is unacceptable. Weak IRM requires every unacceptable deception to be refutable: for some responsive misreport, and against every compatible belief over true other-agent types and announced full type profiles, there is an alternative social choice function that gives the true type a strict incentive to expose the deception while meeting the paper’s restrictions on each announced-type section.

The weak no-worst-rule condition (weak NWR) says that for every agent, type, and distribution over two copies of $T_{-i}$ , the lotteries in the weakly worse contour of $f$ do not all have the same expected utility under that distribution. The two distributions in this definition serve different roles: one describes the true other-agent types and the other indexes the outcome assigned by a candidate rule.

Formalization targets

Sufficiency for interim rationalizable implementation

Theorem 6.3 is the mission goal:

\bigl(\exists\hat f\approx f:\ \hat f\text{ satisfies weak IRM and weak NWR}\bigr) \quad\Longrightarrow\quad f\text{ is implementable in interim rationalizable strategies}.

The goal asks for an implementing mechanism with countable nonempty message spaces. It fixes no canonical-mechanism parameter or numerical bound. The milestone list follows the paper’s fixed-point observation, Lemmas 5.2 and 6.2, and Steps 1–3 of the proof of Theorem 6.3. Lemma 5.2 links weak IRM to strict-if-responsive Bayesian incentive compatibility. Lemma 6.2 supplies strict payoff comparisons from weak NWR. The steps establish the behavior and nonemptiness of rationalizable messages in the canonical mechanism.

Significance

The result gives a sufficient test for implementation when agents may choose any interim correlated rationalizable messages. It specifies conditions on a social choice function rather than asking the designer to inspect every equilibrium of an arbitrary mechanism. The paper uses it to certify its later example of a rule that is implementable under this solution concept while failing stronger implementation conditions.

Formalizing the result would add a reusable finite-type model of interim beliefs, correlated best-response beliefs over countable message spaces, and a greatest-fixed-point account of rationalizable messages. It would also add precise definitions of deceptions and weak refutability. The paper proves the theorem; the Lean statements in this proposal are open proof obligations. The definitions and their encoding of countable expected payoffs can be reused in later implementation-theory developments.

Difficulty

The main obstacle is the universal reach of the solution concept. To exclude a message from the greatest fixed point, the argument must address every compatible correlated belief, not one selected equilibrium belief. In the canonical mechanism, different messages activate different outcome rules, so a deviation that is profitable against one class of beliefs need not work against another. The proof also uses strict payoff comparisons after replacing arbitrary lotteries by a countable dense family of challenges. Simply restricting attention to truthful direct reports would miss the message profiles whose elimination is the point of rationalizable implementation.

Formalization scope

The Lean model uses finite agent, type, and outcome sets; each type set is nonempty because the paper’s interim probability beliefs presuppose that the other-type profile space can carry a distribution. A lottery is a real vector in the standard simplex, and utility on lotteries is linear in that vector. Other-agent profiles are dependent functions on indices distinct from $i$ . Beliefs over other-agent types and messages are nonnegative real functions with the required marginal expressed by a convergent sum. Expected payoffs use countable sums, and the best-response predicate explicitly requires their summability. Thus a non-summable family cannot obtain a default zero payoff and become a spurious best response.

The greatest rationalizable correspondence is the supremum of all post-fixed message correspondences for the mechanism’s actual best-response operator. It is never a free correspondence supplied as a hypothesis. Implementation keeps the paper’s countability, nonemptiness, and all-state uniqueness requirements. These choices rule out vacuous implementations obtained by using an empty message set, a least fixed point, or uniqueness only on $T^*$ .

The canonical proof uses a countable dense set $\Delta^*(A)$ , positive integers for the second message component, and full-support lotteries and challenge mixtures. Its Rule 2 weights are $m_i^2/(m_i^2+1)$ and $1/(m_i^2+1)$ ; Rule 3 assigns each agent’s pure outcome weight $m_j^2/((1+m_j^2)|I|)$ and the remainder to a full-support baseline lottery. The paper’s finite enumeration written as an infinite geometric series may not normalize, so the formalization takes normalized full-support mixtures with exactly the property used later. Steps 1–3 use at least two agents because Rule 2-1 does not choose a unique type with one agent. The main theorem has no such restriction; its one-agent case is a separate direct-mechanism proof obligation. Lemma 6.2(a) is stated in the cross-type form invoked in Step 1, which specializes to the printed same-type lemma. Step 3 assumes BIC, the part of Lemma 5.2 that its argument uses. Contributions toward density, summability, finite convex mixtures, and the one-agent case are welcome.

Selected references

T. Kunimoto, R. Saran, and R. Serrano, Interim Rationalizable Implementation of Functions, Bravo Working Paper # 2020-023, Brown University, October 2020; published in Mathematics of Operations Research (2024). DOI: 10.1287/moor.2022.0202.

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Operations Research Optimal Transport Probability+1·Captain: mikedeng1

Joint Mixability and Notions of Negative Dependence: The Minimizers of the Worst-Case Quadratic Cost over Subsets of Components Are Exactly the NCD Joint Mixes with Correlation Matrix P*_nResearch Paper

Motivation

A joint mix is a random vector $\mathbf X=(X_1,\dots,X_n)$ whose component sum $\sum_{i=1}^n X_i$ is almost surely constant. For fixed marginal distributions it minimizes the variance, the stop-loss premium and the Expected Shortfall of the aggregate, which comonotonicity maximizes, and for this reason it is regarded as the canonical form of extremal negative dependence (Puccetti & Wang, 2015). Joint mixability and its special case complete mixability (Wang & Wang, 2011; Wang & Wang, 2016) are the main tools for worst-case Value-at-Risk under dependence uncertainty and appear in bottleneck assignment and scheduling.

A joint mix need not be negatively dependent in any of the classical statistical senses: negative correlation, negative orthant dependence, or negative association (Joag-Dev & Proschan, 1983). Koike, Lin & Wang (arXiv:2204.11438v4) relate the two notions and ask whether the negatively dependent joint mixes are distinguished by an optimization property. They answer with a multi-marginal optimal transport problem (Pass, 2015) in which the decision maker does not know which subset of the $n$ components will be aggregated, and hedges against the worst case in the sense of Gilboa & Schmeidler (1989). Uncertainty about the aggregated subset models absent risks in an insurance pool, missing particles, or agents who may not participate in a risk-sharing arrangement. This mission formalizes the quadratic-cost part of that answer (§4.2 of the paper).

Setting

Let $n\ge 1$ , $[n]=\{1,\dots,n\}$ , and let $F$ be a distribution on $\mathbb R$ with finite second moment. A random vector $\mathbf X$ has marginals $F$ if $X_i\sim F$ for every $i\in[n]$ .

$\mathbf X$ is negative correlation dependent (NCD) if $\operatorname{Cov}(X_i,X_j)\le 0$ for all $i\ne j$ .
$F$ is $n$ -completely mixable if some joint mix has all marginals $F$ .
$\mathbf X$ is exchangeable if $\mathbf X\overset d=\mathbf X_\pi=(X_{\pi(1)},\dots,X_{\pi(n)})$ for every permutation $\pi\in\mathfrak S_n$ .
$P^*_n$ is the $n\times n$ correlation matrix with diagonal $1$ and off-diagonal entries $-1/(n-1)$ .

A probability on $2^{[n]}$ assigns weights $\mu(K)\ge 0$ to the subsets $K\subseteq[n]$ , summing to $1$ . For a cost $f:\mathbb R\to\mathbb R$ ,

C^f_\mu(\mathbf X)=\sum_{K\subseteq[n]}\mathbb E\Big[f\Big(\sum_{i\in K}X_i\Big)\Big]\mu(K),

with the empty sum equal to $0$ ; $C^2_\mu$ is the case $f(x)=x^2$ . An uncertainty set $\mathcal M$ is a set of probabilities on $2^{[n]}$ ; it is symmetric if $\mu\in\mathcal M$ implies $\mu_\pi\in\mathcal M$ for all $\pi$ , where $\mu_\pi(K)=\mu(\{\pi(i):i\in K\})$ . The three problems, each subject to $X_i\sim F$ for $i\in[n]$ , are

\text{(11)}\ \min\ \sup_{\mu\in\mathcal M}C^2_\mu(\mathbf X),\qquad \text{(12)}\ \min\ \max_{K\subseteq[n]}\mathbb E\Big[\Big(\sum_{i\in K}X_i\Big)^2\Big],\qquad \text{(13)}\ \min\ \max_{|K|=k}\mathbb E\Big[\Big(\sum_{i\in K}X_i\Big)^2\Big].

Problems (12) and (13) are (11) for $\mathcal M$ the set of all probabilities on $2^{[n]}$ and the set of those supported on $k$ -subsets.

Formalization targets

Goal: Theorem 5 (p. 12)

Let $n\ge 3$ and $F$ be $n$ -completely mixable with mean $0$ and finite positive variance. Then

\mathbf X\ \text{minimizes (12)}\iff \mathbf X\ \text{is an NCD joint mix with marginals } F \text{ and correlation matrix } P^*_n,

and the same equivalence holds for (13) with every $k\in[n]\setminus\{1,n-1,n\}$ . The goal fixes no constants beyond those of the paper.

Milestones

Proposition 5 (p. 11): for symmetric $\mathcal M$ and a joint mix $\mathbf X$ with marginals $F$ , one exchangeable NCD joint mix $\mathbf X^{\mathrm E}$ with correlation matrix $P^*_n$ satisfies $\sup_{\mathcal M}C^f_\mu(\mathbf X^{\mathrm E})\le\sup_{\mathcal M}C^f_\mu(\mathbf X)$ for all admissible $f$ .
Proof of Theorem 4 (p. 27): randomly permuting the components does not increase $\sup_{\mathcal M}C^2_\mu$ .
Proof of Theorem 4 (p. 27): an equicorrelation $\rho$ with identical marginals lies in $[-1/(n-1),1]$ , and $\rho=-1/(n-1)$ is attained when $F$ is $n$ -completely mixable.
Theorem 4 (p. 12): for symmetric $\mathcal M$ , every NCD joint mix with marginals $F$ and correlation matrix $P^*_n$ minimizes (11).
(18) (p. 28): under $P^*_n$ and mean $0$ , the value of (12) is $\operatorname{Var}(F)\,k^*(n-k^*)/(n-1)$ , $k^*=\lfloor n/2\rfloor$ .
(20) (p. 28): the average of $\operatorname{Var}(\sum_{i\in K}X_i)$ over $|K|=k$ is at least $\operatorname{Var}(F)\,k(n-k)/(n-1)$ .
After (21) (p. 28): a minimizer of (13) has $\operatorname{Var}(\sum_{i\in K}X_i)=\operatorname{Var}(F)\,k(n-k)/(n-1)$ for every $|K|=k$ .

Significance

Theorem 4 shows that NCD joint mixes with correlation matrix $P^*_n$ are optimal for the quadratic cost under every symmetric uncertainty set about the aggregated subset. Theorem 5 shows that for the two natural uncertainty sets, "no information on $K$ " and "only $|K|$ known", they are the only optimizers: every minimizer is a joint mix and every minimizer has the same correlation matrix. Negative dependence is therefore necessary, and not only sufficient, for robustness of the aggregate against uncertainty about which components are present. Remark 1 of the paper shows the excluded values $k\in\{1,n-1,n\}$ are exactly where uniqueness fails, so the statement is sharp in $k$ .

The results are proved in the paper; none of them, and none of the underlying notions (joint mix, complete mixability, NCD, worst-case costs over subsets), is formalized elsewhere. The mission produces machine-checked statements and proofs. One step of the published proof is incomplete: the derivation of $\Sigma=P^*_n$ for a minimizer of (13) is written out only for $k=2$ (p. 29), so for $3\le k\le n-2$ , and for (12) when $n\ge 6$ , a formal proof must supply the missing argument.

Difficulty

The competitor class is every law with marginals $F$ , an infinite-dimensional and non-parametric set, so the "if" direction cannot be settled by comparing finitely many candidates. In the "only if" direction the obvious reduction to exchangeable laws is unavailable: a minimizer need not be exchangeable, and its pairwise covariances must be recovered from the values of the subset objectives alone. With only $k$ -subset information, the variance of each $k$ -subset sum is pinned down, but turning these $\binom nk$ equalities into equality of all $\binom n2$ covariances is a combinatorial step that is immediate only for $k=2$ . For (12) and $n\ge 6$ it is the case $k=\lfloor n/2\rfloor\ge 3$ that is needed.

Formalization scope

Laws instead of random vectors. Competitors are probability measures $P$ on $\mathbb R^n$ (Fin n → ℝ, components 0-based) with every coordinate marginal equal to $F$ ; every objective depends on $\mathbf X$ only through its law. The goal is stated for a measurable random vector on an arbitrary probability space through its law. The competitor class of (11)–(13) is all such laws, not Gaussian, exchangeable, or joint-mix laws.
Finite second moment. §4's standing assumption ( $F\in L^2$ ) is a hypothesis of every theorem. Expectations are Bochner integrals, which return $0$ on non-integrable integrands; the $L^2$ hypothesis rules this out for the quadratic costs.
Proposition 5. "For all measurable $f$ " is read as: for all measurable $f$ such that $f(\sum_{i\in K}X_i)$ is integrable for every $K$ ; the conclusion includes that the same expectations exist for $\mathbf X^{\mathrm E}$ . The witness $\mathbf X^{\mathrm E}$ is existential and does not depend on $f$ .
Correlation matrix $P^*_n$ is expressed through covariances: $\operatorname{Cov}(X_i,X_j)=-\operatorname{Var}(F)/(n-1)$ for $i\ne j$ . Writing $-1/(n-1)$ without the variance factor would impose $\operatorname{Var}(F)=1$ , which the paper assumes only inside proofs.
Uncertainty sets are sets of functions Finset (Fin n) → ℝ contained in the probabilities on $2^{[n]}$ ; the supremum over $\mathcal M$ is the real sSup, which is finite for such sets. Problem (13) is only used with $k\le n$ .
Normalizations. Milestones 3, 5, 6 and 7 keep a general $\operatorname{Var}(F)$ where the paper normalizes it to $1$ .
Trivializations ruled out. Restricting the competitor class to Gaussian or exchangeable laws, dropping $L^2$ integrability, or dropping the variance factor in $P^*_n$ would each turn the goal into a different (weaker or false) statement; none is used.

Infrastructure that a complete development needs and that is reusable: the permutation average of a law on $\mathbb R^n$ and its marginals, variances of subset sums in terms of a covariance matrix, and the positive semidefiniteness bound for equicorrelation matrices. Contributions are welcome on each milestone and on the missing combinatorial step for $k\ge 3$ .

Selected references

T. Koike, L. Lin, R. Wang, Joint mixability and notions of negative dependence, Mathematics of Operations Research, 2024; arXiv:2204.11438v4. https://arxiv.org/abs/2204.11438
B. Wang, R. Wang, The complete mixability and convex minimization problems with monotone marginal densities, Journal of Multivariate Analysis 102(10), 2011. https://doi.org/10.1016/j.jmva.2011.05.002
B. Wang, R. Wang, Joint mixability, Mathematics of Operations Research 41(3), 2016. https://doi.org/10.1287/moor.2015.0755
G. Puccetti, R. Wang, Extremal dependence concepts, Statistical Science 30(4), 2015. https://doi.org/10.1214/15-STS525
K. Joag-Dev, F. Proschan, Negative association of random variables with applications, Annals of Statistics 11, 1983. https://doi.org/10.1214/aos/1176346079
I. Gilboa, D. Schmeidler, Maxmin expected utility with non-unique prior, Journal of Mathematical Economics 18, 1989. https://doi.org/10.1016/0304-4068(89)90018-9
B. Pass, Multi-marginal optimal transport: theory and applications, ESAIM: M2AN 49(6), 2015. https://doi.org/10.1051/m2an/2015020

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Combinatorics Graph Theory Linear Optimization+1·Captain: mikedeng1

Feasible Bases for a Polytope Related to the Hamilton Cycle Problem: Every Ultimate Feasible Basis Is a Good Augmented Spanning Tree Whose Thick Arcs Form the Unique Perfect MatchingResearch Paper

Why feasible bases matter

A Hamilton cycle visits every vertex of a directed graph once and returns to its start. Finding one is a standard combinatorial optimization problem, and linear programming provides a way to study it through extreme points. Kalinowski and Mohammadian examine a polytope introduced for this purpose and ask what its feasible bases look like when a discount parameter approaches one. The structure of those bases matters for interpreting computations on the polytope: it tells which variables can be basic, how their flows behave, and how the corresponding graph is connected. The paper proves structural results; this mission asks for their Lean formalization.

The polytope and its bases

Let G be a loopless directed graph on $V=[n]=\{1,\ldots,n\}$ , with distinguished node $1$ , and let $0<\beta<1$ . A flow variable $x_{ij}$ is assigned to each arc $(i,j)$ of $G$ . Write $\Phi(i)=\sum_{j:(j,i)\in E}x_{ji}$ for inflow and $\Psi(i)=\sum_{j:(i,j)\in E}x_{ij}$ for outflow. There is a slack variable $y_i$ for each $i\ne1$ . The polytope $\mathcal{WH}_{\beta}(G)$ consists of the pairs $(x,y)$ satisfying the following equations and bounds, taken from (2.10)–(2.15), p. 4:

\begin{aligned} \Psi(1)-\beta\Phi(1)&=1-\beta^n,&\Psi(1)&=1,\\ \Psi(i)-\beta\Phi(i)&=0,&\Psi(i)-y_i&=\beta^{n-1}\quad(i\ne1),\\ x_{ij}&\ge0,&0\le y_i&\le\beta-\beta^{n-1}. \end{aligned}

A basis is a triple $(B,L,U)$ . Its basic variables are $B=A\cup Y$ , where $A\subseteq E$ consists of basic flow variables and $Y\subseteq V\setminus\{1\}$ of basic slacks. The remaining slack indices are partitioned into $L$ and $U$ , fixed at zero and $\beta-\beta^{n-1}$ respectively. There are exactly $2n$ basic columns, and their matrix $M_B(\beta)$ is invertible. A basis is feasible if its basic solution also satisfies all the displayed bounds. The set of such bases is $\Gamma(G,\beta)$ .

The paper shows that $\Gamma(G,\beta)$ stabilizes for $\beta$ sufficiently close to one. An ultimate basis is a member of this stable set, denoted $\Gamma(G)$ . In Lean, membership means feasibility for every $\beta$ in some left neighborhood of one; Proposition 1 connects this definition to the paper's threshold $\alpha^*$ .

To inspect a basis as a graph, split every node $i$ into $v_i$ and $w_i$ . Replace each $(i,j)\in E$ by $(w_i,v_j)$ and add $(v_i,w_i)$ for $i\ne1$ . This gives $G'$ , and the basic variables give an arc set $B'$ . The polytope is a generalized network flow (GNF) polytope on $G'$ : an arc $(w_i,v_j)$ has multiplier $\beta$ and no upper capacity, while $(v_i,w_i)$ has multiplier one and capacity $\beta-\beta^{n-1}$ . Supplies are $1$ at $w_1$ , $\beta^{n-1}$ at the other $w_i$ , $-\beta^n$ at $v_1$ , and $-\beta^{n-1}$ at the other $v_i$ . The printed supply vector on pp. 6–7 gives $-\beta^{n-1}$ at $v_1$ too, but (2.10) and (2.12) imply $-\beta^n$ ; the Lean model follows those defining equations.

An augmented tree is a connected graph with one cycle. After orienting a traversal of that cycle, its cycle multiplier is the product of the multipliers on forward arcs divided by the product on backward arcs. The tree is good when this ratio differs from one. A thick arc of an ultimate basis has $x_{ij}=1-O(1-\beta)$ as $\beta\to1$ from below; a thin arc has $x_{ij}=O(1-\beta)$ (Definitions 4–7, pp. 7–12).

Formalization targets

The central target is Theorem 3.4(i)–(ii), p. 14. For every ultimate basis, $B'$ is one good augmented tree spanning all $2n$ split vertices. If $A'_1$ is the set of split thick arcs, then $A'_1$ is a perfect matching and is the unique perfect matching contained in $B'\setminus Y'$ , where $Y'$ is the set of basic split slack arcs:

B'\text{ is a good augmented spanning tree},\qquad A'_1\text{ is the unique perfect matching in }B'\setminus Y'.

The qualification on uniqueness follows the theorem's proof: the printed phrase “in $G'$ ” is false for graphs with multiple cycle covers. The paper's clause (iii), claiming a particular disjoint-path decomposition, is false as printed even for a nondegenerate basis and is not posed here. The milestones cover stabilization of feasible bases, the GNF interpretation, the GNF basis characterization, the equivalence between balanced and breakeven cycles, connectivity of ultimate bases, and the four parts of Theorem 3.3.

What the result establishes

The graph description narrows every ultimate feasible basis to a connected, one-cycle object, and the matching identifies one large-flow arc incident to every split vertex. Together these statements give a precise structure to a polytope motivated by the Hamilton cycle problem. They do not claim that an arbitrary ultimate basis is itself a Hamilton cycle or that the polytope solves the Hamilton cycle problem. The distinction matters: the thick arcs may form several directed cycles on the original graph (Theorem 3.3, p. 12).

A Lean development would put reusable GNF notions—capacity bounds, basis columns, oriented cycles, cycle multipliers and good augmented forests—alongside a precise model of this paper's polytope. The results are proved in the paper; the mission's open work is to supply machine-checked proofs of the stated Lean theorems, including the asymptotic statements for ultimate bases.

The difficult point

For an arbitrary GNF polytope, a nonsingular basis may have several augmented-tree components. The general GNF basis theorem therefore gives a good augmented forest, but it does not give the connected tree demanded here. The special supplies and slack bounds of $\mathcal{WH}_{\beta}(G)$ must rule out every disconnected feasible basis as $\beta$ approaches one. A second difficulty is distinguishing flow that tends to one from flow that tends to zero without replacing an asymptotic property by a condition at one fixed discount factor.

Formalization scope

Vertices of $G$ are Fin n, with paper node $1$ represented by 0; [NeZero n] ensures it exists. Arcs are a finite set of ordered pairs, so loops are excluded and parallel copies do not occur. The split vertices are Fin n ⊕ Fin n: inl i is $v_i$ and inr i is $w_i$ . Real numbers carry flows, multipliers and supplies. A GNF arc has an optional real upper capacity; none means unbounded. Its arc type distinguishes antiparallel and parallel arcs, so undirected two-edge cycles remain representable.

A basis includes its partition, exactly $2n$ columns, linear independence, all equality rows and fixed nonbasic values. Ultimate feasibility uses the left-neighborhood filter. Thick and thin mean eventual constant bounds against $1-\beta$ , quantified over basic solutions; uniqueness of those solutions follows from column independence. Connected components include every split vertex. The finite component edge count equals its vertex count, which is equivalent to one cycle in each connected component, and goodness requires that cycle's multiplier differ from one.

The goal retains degeneracy: a basic flow may be zero or a basic slack may be at a bound. It asserts connectivity and nonbreakeven of the cycle, as well as both existence and uniqueness of the specified perfect matching. The definitions of $\mathcal{WH}_{\beta}(G)$ , feasible and ultimate bases, GNF flows and cycles, and asymptotic arc classes are intended for reuse. Contributions toward the stabilization result, the GNF basis theorem, the connectivity theorem and the thick-arc theorem all support the goal.

Selected references

T. Kalinowski and S. Mohammadian, Feasible Bases for a Polytope Related to the Hamilton Cycle Problem, arXiv:1907.12691v1, 2019; published in Mathematics of Operations Research, 2021. arXiv

9 thms1 active userReviewed

Control Theory Dynamical Systems Operations Research·Captain: mikedeng1

Optimal Control of an Epidemic through Social Distancing: Under Limited Capacity Effects Every Optimal Policy Is Single-Peaked and Infections Are Log-Concave While Measures EscalateResearch Paper

Motivation

During the Covid-19 epidemic, governments had to decide not only whether to impose social distancing but how to schedule it over time: when to escalate, when to relax, and whether a relaxation might later have to be reversed. Epidemiology supplies the dynamics (the SIR model of Kermack and McKendrick, 1927), and optimal control supplies the language for trading off deaths against the economic cost of distancing. Most of the prior theoretical literature on optimal epidemic control assumed a cost of distancing that is linear in the reduction of the transmission rate, and focused on numerical solutions.

Kruse and Strack (Cowles Foundation Discussion Paper 2229R, 2020; later published as Optimal Dynamic Control of an Epidemic, Operations Research, 2024, doi:10.1287/opre.2022.0161) characterize the shape of every optimal policy for a general convex cost of distancing and a convex death-rate function. Their main structural result is that, when hospital-capacity effects are moderate, optimal policies are single-peaked: measures are escalated and then relaxed, never relaxed and re-escalated.

Setting

The population is split into a susceptible share $s(t)$ , an infected share $i(t)$ and the rest. A planner chooses a measurable transmission rate $\beta(t)\in B=[\underline b,\bar b]\subset(0,\infty)$ , with $\underline b<\bar b$ . With recovery rate $\gamma>0$ the shares follow

s'(t)=-\beta(t)i(t)s(t),\qquad i'(t)=\beta(t)i(t)s(t)-\gamma i(t),

from $s(0)=s_0$ , $i(0)=i_0$ , with $s_0,i_0\in(0,1)$ and $s_0+i_0\le1$ . A vaccine arrives at $T>0$ . The planner minimizes

J(\beta)=\int_0^T\big(v(i(t))+c(\beta(t))\big)\,dt+\bar v(i(T)),\qquad \bar v(x)=\int_0^x\frac{v(z)}{z\gamma}\,dz,

where the death-rate cost $v$ is convex, strictly increasing and $C^1$ with $v(0)=0$ . The cost of distancing $c$ is convex, non-increasing and continuous on $B$ , with $c(\bar b)=0<c(b)$ for $b<\bar b$ . A control is optimal if it minimizes $J$ over all admissible controls.

Write $M(y)=\min_{b\in B}[yb+c(b)]$ . Proposition 1 of the paper provides a positive costate $\eta^*$ along every optimal path, and the gain from reducing the transmission rate is $g^*(t)=\eta^*(t)i^*(t)s^*(t)$ ; the optimal control minimizes $g^*(t)b+c(b)$ over $B$ . Assumption 1 (limited capacity effects) asks that $v$ be $C^2$ with

\frac{v''(i)}{\gamma}<\frac{27}{4}\,\frac{\underline b}{\bar b-\underline b}\,\big|c'(\bar b-)\big|\qquad (i\in[0,1]),

where $c'(\bar b-)$ is the left derivative of $c$ at $\bar b$ .

Formalization targets

Goal: Proposition 7

Under Assumption 1, for every optimal control $\beta^*$ there is $t^*\in[0,T]$ such that

\beta^*\ \text{is non-increasing on }[0,t^*]\text{ and non-decreasing on }[t^*,T],\qquad \log i^*\ \text{is strictly concave on }[0,t^*].

Thus $i^*$ has at most one local maximum on $[0,t^*]$ . The monotonicity claim refers to the version of $\beta^*$ singled out by the paper's convention (see Formalization scope).

Milestones

In the paper's order of proof:

existence of an optimal policy and the Pontryagin multipliers (Lemma 9);
the behaviour of the control where the costate is non-positive (Lemma 10);
the reduced costate equation (Lemma 11) and positivity of the costate (Lemma 12, Proposition 1);
the choice of an everywhere-optimal version of the control (Lemma 13);
monotonicity and concavity of $M$ ;
the gain equation, bounds and monotone comparative statics (Proposition 2 (i)–(iii));
the inequality (33) $s^2i\le 4/27$ , the second-order inequality (37) for the gain;
strict quasi-concavity of the gain.

Significance

Proposition 7 rules out a whole class of policies (relax, then tighten again) for every convex cost of distancing, not only for the linear costs earlier work treated. The same argument also gives Proposition 8 of the paper, a three-phase bang-bang structure under linear cost. It also explains the factor $27/4$ : it is the reciprocal of $\max_{s\in[0,1]}s^2(1-s)$ .

The mission produces a machine-checked development of a deterministic optimal-control problem with measurable controls: existence of an optimal control, a Pontryagin principle with a state-dependent terminal cost, and a qualitative analysis of the adjoint system. The paper's result is proved on paper; none of it is formalized. The SIR model, its integral-form solutions and the costate machinery are reusable for other epidemic-control papers.

Difficulty

Existence and the necessary conditions rest on the existence theory and the Pontryagin principle for measurable controls, which Mathlib does not have. The general Pontryagin principles already on the platform do not apply, because they require a globally Lipschitz vector field or a cost that is smooth in the control.

The step specific to the paper is the reduction of the two multipliers to one costate, followed by the analysis of the gain $g^*$ . The gain is $C^1$ , but its derivative involves $M(g^*(t))$ and $\beta^*(t)$ , so $(g^*)''$ exists only almost everywhere. It must be bounded through a case split on the size of $g^*(t)$ relative to $-c'(\bar b-)$ . Showing that $g^*$ is single-peaked requires turning this almost-everywhere bound into a strict sign change of $(g^*)'$ at each critical point. A naive appeal to " $g^{*\prime\prime}<0$ at critical points" fails because $(g^*)''$ need not exist there.

Formalization scope

All objects live in EpiControl.SinglePeak; parameters are a structure Params with a predicate Standing for the assumptions of §2. Conventions:

Dynamics in integral form. Solutions are continuous on $[0,T]$ and satisfy the integral equations; adjoint equations are integral equations from the terminal value. This is how absolute continuity is carried.
Admissibility on $[0,T]$ . Controls are measurable with values in $B$ on $[0,T]$ , the only part entering $J$ .
Added hypotheses, all disclosed. $\underline b<\bar b$ ; $c>0$ on $[\underline b,\bar b)$ (the page prints the sign slip $0>c(b)$ ); global $C^1$ , resp. $C^2$ , smoothness of $v$ as an extension of the page's smoothness on $[0,1]$ .
$M$ is an infimum over the image of the compact interval $B$ , hence a minimum. $c'(\bar b-)$ is a real parameter with its defining one-sided limit.
The costate is a predicate (IsCostate) taken as a hypothesis in every statement about $\eta^*$ or $g^*$ , never an arbitrary function and never a Classical.choose.
Null sets. An optimal control is defined only up to a null set, so monotonicity of an arbitrary version is false. The paper fixes the version by requiring the argmin condition at every time (justified by Lemma 13). The goal therefore asserts the existence of an admissible $\hat\beta=\beta^*$ almost everywhere on $[0,T]$ with the stated monotonicity; milestones that need the convention take it as an explicit hypothesis. Stating monotonicity of $\beta^*$ itself, or dropping optimality, would make the goal false or vacuous; neither is done.

Infrastructure that a complete development needs: Carathéodory solutions of the SIR system, existence of optimal controls (a Filippov-type argument), a Pontryagin principle for measurable controls with terminal cost, and Lipschitz/almost-everywhere differentiability arguments. Contributions of these general results as separate theorems are welcome.

Selected references

T. Kruse and P. Strack, Optimal Control of an Epidemic through Social Distancing, Cowles Foundation Discussion Paper No. 2229R, 2020. https://ssrn.com/abstract=3581295
T. Kruse and P. Strack, Optimal Dynamic Control of an Epidemic, Operations Research, 2024. https://doi.org/10.1287/opre.2022.0161
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A 115, 1927. https://doi.org/10.1098/rspa.1927.0118
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer GTM 264, 2013. https://doi.org/10.1007/978-1-4471-4820-3

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Algebraic Geometry Optimization·Captain: mikedeng1

Exploiting Sign Symmetries in Minimizing Sums of Rational Functions: The Sign-Symmetry-Adapted Block-Diagonal SOS Hierarchy Converges Monotonically to the Global MinimumResearch Paper

Motivation

A sum of rational functions is a common objective when several performance measures each have a polynomial numerator and a positive polynomial denominator. Such problems appear, for example, in the applications to computer vision and signal processing cited by Guo, Wang and Zheng in their 2024 preprint. Even when all constraints and numerators are polynomial, the objective need not be convex, and its global minimum can be difficult to certify. Bugarin, Henrion and Lasserre studied a hierarchy based on a measure reformulation of this problem in Minimizing the sum of many rational functions. Guo, Wang and Zheng develop its polynomial dual and exploit sign symmetries to obtain a hierarchy with block-diagonal semidefinite matrices.

The mission targets the convergence assertion in Theorem 4.7(ii) of the pinned preprint, arXiv:2405.09419v1. Its mathematical question is precise: do the lower bounds from the sign-symmetry-adapted semidefinite relaxations increase to the true global infimum? The statement is already proved in the paper. Its formal proof and the supporting algebra are the work posed here.

Setting

Fix $n$ real variables, $N$ ratios, and $m$ polynomial inequality constraints. For polynomials $p_i,q_i,g_j\in\mathbb R[x_1,\ldots,x_n]$ , let

K=\{x\in\mathbb R^n:g_j(x)\geq0\text{ for every }j\in[m]\},\qquad \rho=\inf_{x\in K}\sum_{i=1}^{N}\frac{p_i(x)}{q_i(x)}.

Assumption 1.1 says that $K$ is compact and each $q_i$ is strictly positive on $K$ . The mission also requires $N\geq2$ : the paper's definition of the distinguished support set $A_1$ can omit the constraint supports when $N=1$ , and the advertised sparse convergence then fails. This repair is explicit in the formal goal.

A polynomial is a sum of squares (SOS) if it is a finite sum of squares of real polynomials. The quadratic module $Q(g)$ comprises $\sigma_0+\sum_j\sigma_jg_j$ with every $\sigma_j$ SOS. Its order- $k$ truncation $Q_k(g)$ restricts the degrees of $\sigma_0$ and $\sigma_jg_j$ to $2k$ . The Archimedean condition asks for some $M>0$ with $M-\sum_{a=1}^n x_a^2\in Q(g)$ ; it is a separate hypothesis in the convergence theorem.

For a finite monomial-exponent set $A\subseteq\mathbb N^n$ , the sign-symmetry group $R(A)$ consists of the vectors $r\in\{0,1\}^n$ for which $r^T\alpha$ is even for every $\alpha\in A$ . Its closure $\bar A$ contains every exponent fixed by all these parity tests. The sign action $\theta_r$ substitutes $(-1)^{r_a}x_a$ for $x_a$ . The paper forms $A_i$ from the supports of $p_i,q_i$ , and all $g_j$ for $i\geq2$ ; $A_1$ contains the first numerator and denominator supports and the other $A_i$ .

The sparse quadratic module $Q_k(g,A)$ represents each SOS multiplier with a positive semidefinite Gram matrix indexed by monomials of bounded degree. A matrix entry at exponents $\alpha,\beta$ must vanish when $\alpha+\beta\notin\bar A$ . The program (SDk) uses $Q_k(g,A_1)$ for the distinguished ratio and $Q_k(g,A_i)$ for each other ratio. Its auxiliary polynomials $h_i$ have support in $\bar A_i$ and degree at most $2k-\max\{\deg q_1,\deg q_i\}$ . The resulting optimal value is $\rho_k^s$ .

Formalization targets

Sparse convergence

The main theorem is the paper's Theorem 4.7(ii): under Assumption 1.1 and the Archimedean condition, the sparse bounds rise to the original optimum,

\rho_k^s\nearrow\rho\qquad(k\longrightarrow\infty).

The formal statement spells this out as monotonicity for $k\geq d_{\min}$ , the lower-bound inequality $\rho_k^s\leq\rho$ , and convergence to $\rho$ . Here $d_{\min}$ is the maximum of the rounded-up half-degrees of all numerators, denominators, and constraints, as defined on page 4 of the paper. No convergence rate is asserted.

Supporting results

The milestone list follows the source's claims: invariance of the data under sign flips (Lemma 4.2), cancellation of nonsymmetric monomials in the averaging argument, two-sided polynomial approximation in the proof of Theorem 3.1, equality of the polynomial dual value with $\rho$ (Theorem 3.1), equality of the sign-symmetric dual value with $\rho$ (Theorem 4.3), the sign-symmetric Positivstellensatz (Theorem 4.5), monotonicity in the proof of Theorem 4.7(ii), and the sparse-versus-dense inequality in Theorem 4.7(i). These are mathematical targets, not assumed facts in the goal.

Significance

A finite level of (SDk) supplies a certified lower bound on the global sum-of-ratios minimum. Theorem 4.7(ii) says that increasing the degree loses no value asymptotically while preserving the sign-induced zero pattern in the Gram matrices. The pattern can split a semidefinite constraint into smaller blocks; the paper demonstrates this computationally, but the theorem itself concerns correctness of the limiting bounds, not a runtime guarantee. The related term-sparsity framework is developed by Wang, Magron and Lasserre in TSSOS.

A Lean development of the known result would supply reusable definitions for support-based parity symmetries, sparse SOS Gram certificates, and extended-real optimal values of polynomial dual programs. None of this mission's proposed results has an existing machine-checked proof in the local prior-art search. The original paper establishes the mathematics; this mission asks solvers to formalize that proof and its stated intermediate results.

Difficulty

Ordinary SOS certificates do not automatically respect a prescribed zero pattern in their Gram matrices. Likewise, restricting the auxiliary polynomials to the symmetry-closed supports changes the dual feasible set. Thus the dense hierarchy's convergence theorem alone does not yield the sparse claim. The decisive issue is whether strictly positive polynomials with the relevant support admit sparse certificates, while the approximation steps retain the sign symmetries and the exact degree restrictions of (SDk).

Formalization scope

Lean uses MvPolynomial (Fin n) ℝ and points Fin n → ℝ. The paper's ratio $i=1$ is Lean index 0; $i\in[N]\setminus\{1\}$ means i ≠ 0. Sign vectors are Fin n → ZMod 2, monomial exponents are finitely supported natural-number vectors, and the Gram matrices are finite matrices over the monomials of degree at most the relevant order. The zero pattern is tested against $\bar A$ , not against $A$ . Every optimum is an EReal supremum or infimum, so an infeasible (SDk) has value $-\infty$ and an empty $K$ gives $\rho=+\infty$ . On $K$ , the positivity assumption makes every ratio well-defined as intended.

The $N\geq2$ hypothesis is necessary for the paper's $A_1$ formula to contain the constraint supports. It rules out a one-ratio counterexample to the sparse claim; it is disclosed in the goal and the affected milestones. Degrees use natural-number arithmetic only at orders $k\geq d_{\min}$ , where the subtractions match the paper's bounds. A valid formalization must retain positive semidefiniteness, the $\alpha+\beta\in\bar A$ matrix pattern, the correct support sets, and the exact $h_i$ degree bound. Dropping any of these would state a different hierarchy.

Reusable contributions include the polynomial and SOS definitions, the parity action, the Gram construction, and the sparse Positivstellensatz. Solvers may develop auxiliary lemmas for the monotone inclusions or the approximation argument, while keeping each audited theorem statement faithful to the source. The moment-side formulation, convergence rates, correlative sparsity, and numerical examples lie outside this mission.

Selected references

F. Guo, J. Wang and J. Zheng, Exploiting Sign Symmetries in Minimizing Sums of Rational Functions, arXiv:2405.09419v1, 2024. https://arxiv.org/abs/2405.09419v1
F. Bugarin, D. Henrion and J.-B. Lasserre, Minimizing the sum of many rational functions, Mathematical Programming Computation 8, 83–111, 2016. https://homepages.laas.fr/henrion/papers/rational.pdf
J. Wang, V. Magron and J.-B. Lasserre, TSSOS: A Moment-SOS Hierarchy That Exploits Term Sparsity, SIAM Journal on Optimization 31, 30–58, 2021. https://arxiv.org/abs/1912.08899

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Operations Research Optimization·Captain: mikedeng1

Sensitivity Analysis of the Maximal Value Function with Applications in Nonconvex Minimax Programs: Limiting and Horizon Subdifferentials of the Maximal Value Function via Wolfe DualityResearch Paper

Motivation

Value functions of parametric optimization problems appear wherever one optimization problem sits inside another. In a minimax program $\min_x \max_y f(x,y)$ the outer problem minimizes the maximal value function $V(x)=\max_y f(x,y)$ ; in bilevel programming and in the value-function approach to equilibrium problems the same object enters as a constraint. Optimality conditions for the outer problem therefore need generalized derivatives of $V$ , and $V$ is in general neither smooth, nor convex, nor locally Lipschitz.

Classical estimates of the subdifferential of a value function, going back to Clarke's Optimization and Nonsmooth Analysis (1983) and to Rockafellar and Wets' Variational Analysis (1998), take a convex hull over all optimal solutions, or assume inner semicontinuity of the solution map, or joint convexity. Guo, Ye and Zhang (arXiv:2303.01474, Mathematics of Operations Research, 2024) give estimates for the limiting and horizon subdifferentials of a maximal value function without any of these assumptions, when the inner problem is concave in $y$ . Their device is Wolfe's 1961 dual: weak duality bounds the maximal value from above by the value of a minimization problem over primal–dual pairs, to which minimal-value-function techniques apply. The estimates then feed into stationarity conditions for nonconvex–concave minimax programs (§4 of the paper).

Setting

Let $f:\mathbb R^n\times\mathbb R^m\to\mathbb R$ and $g=(g_1,\dots,g_q):\mathbb R^n\times\mathbb R^m\to\mathbb R^q$ . For a parameter $x\in\mathbb R^n$ the parametric maximization problem is

(P_x)\qquad \max_y f(x,y)\quad\text{s.t.}\quad y\in\mathcal F(x):=\{y: g(x,y)\le0\}.

The maximal value function is $V(x)=\sup\{f(x,y):y\in\mathcal F(x)\}\in[-\infty,\infty]$ (with $V(x)=-\infty$ on an empty feasible set), and the solution map is $\mathcal S(x)=\{y\in\mathcal F(x):f(x,y)=V(x)\}$ . The Lagrangian is $\mathcal L(x,y,\lambda)=f(x,y)-g(x,y)^\top\lambda$ , and $\mathcal L^0=-g^\top\lambda$ . The multiplier set of $(P_x)$ at $y$ is $\Sigma(x,y)=\{\lambda:\nabla_y\mathcal L(x,y,\lambda)=0,\ 0\le-g(x,y)\perp\lambda\ge0\}$ .

The Wolfe dual of $(P_x)$ is $\min_{y,\lambda}\mathcal L(x,y,\lambda)$ subject to $\nabla_y\mathcal L(x,y,\lambda)=0$ , $\lambda\ge0$ ; its value is $V_D(x)$ . For $r\in\{0,1\}$ its multiplier set at $(y,\lambda)$ is

\Xi^r(x,y,\lambda)=\{u\in\mathbb R^m:\ \nabla^2_{yy}\mathcal L(x,y,\lambda)u=0,\ 0\le-\nabla_yg(x,y)^\top u-r\,g(x,y)\perp\lambda\ge0\}.

For $\varphi:\mathbb R^n\to[-\infty,\infty]$ at a point where $\varphi(x)$ is finite, a regular subgradient $v\in\widehat\partial\varphi(x)$ satisfies $\varphi(x')\ge\varphi(x)+v^\top(x'-x)+o(\|x'-x\|)$ ; the limiting subdifferential $\partial\varphi(x)$ collects limits of $v^k\in\widehat\partial\varphi(x^k)$ with $x^k\to x$ , $\varphi(x^k)\to\varphi(x)$ ; the horizon subdifferential $\partial^\infty\varphi(x)$ collects limits of $t_kv^k$ with additionally $t_k\downarrow0$ .

Throughout, $f$ and $g$ are twice continuously differentiable, and $f(x,\cdot)$ is concave and every $g_i(x,\cdot)$ convex for $x$ near the reference point $\bar x$ . Restricted sup-compactness at $\bar x$ means $V(\bar x)$ is finite and some compact set meets $\mathcal S(x)$ for every $x$ near $\bar x$ with $V(x)>V(\bar x)-\epsilon$ . Two constraint qualifications enter: a stable parametric CQ for $(P_{\bar x})$ at a solution (Definition 2.3), and a stable CQ for the system $\nabla_y\mathcal L(x,y,\lambda)=0$ , $\lambda\ge0$ in $(x,y,\lambda)$ (Definition 2.4), each asking that a CQ hold at nearby points and that multipliers of nearby points have subsequences converging to multipliers at the limit.

Formalization targets

Goal: Theorem 3.3, (3.13)–(3.14)

Under the standing assumption, restricted sup-compactness at $\bar x$ , a stable parametric CQ at every $y\in\mathcal S(\bar x)$ , and a stable CQ for (3.11) at every $(\bar x,\tilde y,\tilde\lambda)$ with $\tilde y\in\mathcal S(\bar x)$ , $\tilde\lambda\in\Sigma(\bar x,\tilde y)$ :

\partial V(\bar x)\subseteq\bigcup_{y\in\mathcal S(\bar x)}\{\nabla_x\mathcal L(\bar x,y,\lambda)+\nabla^2_{xy}\mathcal L(\bar x,y,\lambda)u:\ u\in\Xi^1(\bar x,y,\lambda),\ \lambda\in\Sigma(\bar x,y)\},

\partial^\infty V(\bar x)\subseteq\bigcup_{y\in\mathcal S(\bar x)}\{\nabla^2_{xy}\mathcal L(\bar x,y,\lambda)u:\ u\in\Xi^0(\bar x,y,\lambda),\ \lambda\in\Sigma(\bar x,y)\}.

Milestones

Lemma 3.1: Wolfe weak duality $V(x)\le V_D(x)$ .
Proof of Theorem 3.3 (i): under a CQ, a local minimizer of $\mathcal L(x,y,\lambda)-\xi^\top(x-\bar x)+\epsilon\|x-\bar x\|$ subject to (3.11) is a KKT point, with multiplier in $\Xi^1$ up to an $\epsilon$ -ball.
Theorem 3.3 (3.12): $\widehat\partial V(\bar x)\subseteq\{\nabla_x\mathcal L(\bar x,\bar y,\bar\lambda)+\nabla^2_{xy}\mathcal L(\bar x,\bar y,\bar\lambda)u: u\in\Xi^1(\bar x,\bar y,\bar\lambda)\}$ for one solution and one multiplier.
Proof of Theorem 3.3 (ii): restricted sup-compactness gives solutions $y^k\in\mathcal S(x^k)$ converging, along a subsequence, to some $\tilde y\in\mathcal S(\bar x)$ .
Proof of Theorem 3.3 (ii): a stable parametric CQ gives multipliers $\lambda^k\in\Sigma(x^k,y^k)$ converging, along a subsequence, to some $\tilde\lambda\in\Sigma(\bar x,\tilde y)$ .

A supporting item (proof of Theorem 3.3 (i), p. 17) states that a regular subgradient $\xi$ of $V$ at $\bar x$ makes $(\bar x,\bar y,\bar\lambda)$ a local minimizer of that problem (3.16), for every $\epsilon>0$ .

Significance

The estimates (3.13)–(3.14) bound the limiting subdifferential of a maximal value function by a union over solutions rather than a convex hull, and they do not presuppose that $V$ is locally Lipschitz: (3.14) is exactly what is needed to derive Lipschitz continuity (Corollary 3.3 of the paper: if the right side of (3.14) is $\{0\}$ and $V$ is continuous, $V$ is Lipschitz near $\bar x$ ). The paper uses them to obtain stationarity conditions for nonconvex–concave minimax programs, including cases (Example 3.1) where the solution map is not inner semicontinuous and the problem is not jointly convex.

The result is proved in the paper; no machine-checked version exists. The mission formalizes the statement and its proof, which requires subdifferential calculus for extended-real functions, the exact-penalty consequence of a local error bound, and multiplier rules for problems with a nonsmooth term, none of which are in Mathlib today.

Difficulty

The obvious route to $\partial V$ differentiates the maximum directly (a Danskin-type argument), which needs compactness and continuity uniform in $y$ and yields a convex hull; it fails exactly when $V$ is not Lipschitz. The Wolfe-dual route replaces the maximization by a minimization over $(y,\lambda)$ , but the resulting problem (3.16) has a nonsmooth objective ( $\epsilon\|x-\bar x\|$ ) and a constraint system in which $x$ is a variable, so the KKT conditions follow only from a calmness/error-bound argument, not from a smooth Lagrange multiplier rule. The limiting estimate then requires passing to the limit simultaneously in solutions, primal multipliers $\lambda^k$ and dual multipliers $u^k$ , and the latter can be unbounded unless the stable CQ controls them; this is the role of Definition 2.3(ii).

Formalization scope

Parameters are in EuclideanSpace ℝ (Fin n), decisions in EuclideanSpace ℝ (Fin m), multipliers in Fin q → ℝ; $g$ is given by components $g_i$ . $V$ and $V_D$ are EReal-valued suprema/infima over the feasible sets. Regular subgradients and the limiting subdifferential are the published NonconvexSplitting.Shared.IsRegularSubgrad and LimitingSubdiff; the horizon subdifferential is defined from regular subgradients with $t_k>0$ , $t_k\to0$ . Second derivatives are gradient/fderiv of explicit functions of $C^2$ data.

The paper leaves "a CQ" abstract. Here it is the local error bound of (3.11), which every CQ listed in the paper implies (p. 8), so the formal statements are at least as strong as the paper's for each listed CQ. A stable CQ is the error bound at all feasible points near the reference point plus Definition 2.3(ii); a stable parametric CQ is Definition 2.3 with (i) read for local maximizers (the problem is a maximization) on a neighbourhood. Error-bound distances use the product sup norm, equivalent to the Euclidean one up to the constant.

A real-valued $V$ (junk value $0$ on empty or unbounded feasible sets) or a CQ hypothesis that already asserts the KKT conclusion of (3.12) would trivialize the goal; neither is used. A sanity file proves the standing assumption, a solution, a multiplier and restricted sup-compactness for Example 3.1 of the paper.

Needed infrastructure, reusable beyond this mission: Fermat's rule and sum rules for limiting subdifferentials of locally Lipschitz functions, exact penalization under a local error bound (Clarke calmness), and closedness of linear images of complementarity sets. Contributions on any of these are welcome.

Selected references

L. Guo, J. J. Ye, J. Zhang, Sensitivity analysis of the maximal value function with applications in nonconvex minimax programs, Mathematics of Operations Research, 2024; arXiv:2303.01474v1. https://arxiv.org/abs/2303.01474
P. Wolfe, A duality theorem for non-linear programming, Quarterly of Applied Mathematics 19 (1961), 239–244. https://doi.org/10.1090/qam/135625
F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM Classics in Applied Mathematics, 1990 (reprint of the Wiley 1983 edition). https://doi.org/10.1137/1.9781611971309
R. T. Rockafellar, R. J.-B. Wets, Variational Analysis, Springer, 1998. https://doi.org/10.1007/978-3-642-02431-3
J. J. Ye, J. Zhou, Verifiable sufficient conditions for the error bound property of second-order cone complementarity problems, Mathematical Programming 171 (2018), 361–395 (reference [52] of the paper; cited there for the fact that the listed CQs imply the error bound property).

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Economics Mechanism Design Operations Research·Captain: mikedeng1

Incomplete Contracts in Multi-period Procurement: An English Auction with a Take-It-or-Leave-It Offer Attains the Optimal Complete-Contract Buyer SurplusResearch Paper

Motivation

Public and private buyers often procure the same input over several periods from a pool of competing suppliers: road maintenance, defence supply, energy, and IT services are typical cases. Between periods, conditions change. Input prices, regulations or technology shift, and so does the cost of serving the contract. A buyer who can write a complete contract commits in period one to what she will buy and pay in every future state of the world. Real procurement contracts rarely do this. They are incomplete: the first-period contract fixes the first-period deal, and later terms are renegotiated once the new state is known.

Incompleteness is usually costly in mechanism design. A buyer who learns a supplier's cost in period one cannot commit not to use that information later. This is the ratchet effect (Laffont and Tirole 1988), and suppliers anticipate it by hiding their costs. V. Gretschko and M. Pollrich (Incomplete Contracts in Multi-period Procurement, Management Science 2021; held version: ZEW Discussion Paper No. 19-040, 2019, titled Incomplete Contracts in Dynamic Procurement) show that in a two-period procurement model this loss can be avoided entirely. An English auction reveals who has the lowest cost but not what that cost is. Followed by a single spot offer, it gives the buyer exactly the surplus of the optimal complete contract. Related results on auctions with limited commitment are due to Skreta (2015), Liu, Mierendorff, Shi and Zhong (2019), and Vartiainen (2013).

Setting

A buyer buys one unit in period 1 and $Q > 0$ units in period 2 from one of $n \ge 1$ suppliers, and values each unit at $R > 0$ . She can buy in period 2 only if she bought in period 1.

Costs. Supplier $i$ privately knows a first-period cost $c^i$ . The costs are independent, with density $f$ on $[\underline c, \bar c]$ , and $f$ is bounded away from zero there. The distribution function is $F(c) = \int_{\underline c}^c f$ . At the start of period 2 a publicly observed state $\omega \in \Omega$ is drawn from a probability measure $G$ , and the second-period cost of a supplier is $k(c, \omega)$ . Here $k(\cdot, \omega)$ is strictly increasing, with derivative $k_c \in [0, K]$ .

Virtual costs. The first- and second-period virtual costs are

\Psi_1(c) = c + \frac{F(c)}{f(c)}, \qquad \Psi_2(c,\omega) = k(c,\omega) + \frac{F(c)}{f(c)}\,k_c(c,\omega).

Assumption 1 requires both to be non-decreasing in $c$ .

Thresholds. The threshold $c^\star_2(\omega)$ is where $\Psi_2(\cdot,\omega)$ crosses $R$ . The threshold $c^\star_1$ is where $\Psi_1$ crosses $R + \int_\Omega Q\max\{0, R - \Psi_2(\cdot,\omega)\}\,dG(\omega)$ .

Complete contracts. A direct mechanism maps reported costs to a period-1 allocation $x_1$ , a state-contingent period-2 allocation $x_2(\cdot,\omega)$ and transfers $t$ . It is incentive compatible (IC) if truthful reporting is optimal for each supplier given truthful reports by the others. It is individually rational (IR) if each supplier's expected profit is non-negative. Its buyer surplus is the expected value $\mathbb E\big[\sum_i (R x^i_1 + \int_\Omega Q R x^i_2\,dG - t^i)\big]$ .

Incomplete contracts. In the English auction with reserve $r^\star = c^\star_1$ , the price falls from the reserve. The last remaining supplier wins at the price $p_1$ at which his final opponent quit. After observing $p_1$ and $\omega$ , the buyer makes the winner a take-it-or-leave-it offer $p_2$ , which he accepts iff it covers his cost $k(c,\omega)$ .

Formalization targets

Goal: Proposition 3

With reserve $r^\star = c^\star_1$ and the offer

p_2(\omega,p_1) = \min\{k(p_1,\omega),\, k(c^\star_2(\omega),\omega)\},

the following hold.

Truthful exit $\beta^E(c) = c$ is a best response for every supplier.
The offer is optimal for the buyer given her belief that the winner's cost has distribution $F(c)/F(p_1)$ on $[\underline c, p_1]$ .
The buyer's expected surplus in the English auction equals

\max\{\text{buyer surplus of } (x,t) : (x,t) \text{ admissible, IC, IR}\}.

Milestones

Lemma 1: the envelope formula for the profit of an IC mechanism.
Virtual surplus: the buyer surplus of an IC mechanism equals the expected virtual surplus minus the profits of the highest types.
Proposition 1: the mechanism that buys from the lowest-cost supplier when $c^i \le c^\star_1$ , and in period 2 when also $k(c^i,\omega) \le k(c^\star_2(\omega),\omega)$ , with the transfers of its proof, is IC, IR and optimal.
Lemma 2: the derivative of the expected second-period profit $\Phi$ .
Equation (9): the buyer's optimal offer.
Supplier sign condition: winning below one's cost is unprofitable.
English-auction profit: the profit $U(c)$ of type $c$ in the English auction.

Significance

The result says that, in this model, the commitment power of a complete contract is worth nothing to the buyer once she uses the right auction format. The English auction screens enough to identify the efficient supplier, and it conceals enough that the buyer cannot exploit what she has learned. This supports the observed use of open descending auctions followed by spot negotiation, and it contrasts with sealed-bid formats, whose bids reveal the winner's cost exactly. Example 2 of the paper shows that a standard-auction implementation with option contracts is not sequentially rational.

As far as a search of the Prove2Me catalog shows, none of these results has been formalized. The Mathlib-based library has one-period optimal auction and screening results (Myerson's optimal auction, envelope lemmas for single-period screening, sequential screening). It has nothing on two-period procurement, nothing combining an English auction with renegotiation, and nothing on the value of incomplete contracts. A formalization would check the optimal complete-contract benchmark, with its two-period envelope argument, and the equilibrium-plus-surplus claim of Proposition 3. It would also make explicit several conventions the paper leaves implicit, listed below.

Difficulty

The obvious argument for optimality, pointwise maximization of the virtual surplus, needs the envelope formula for all IC mechanisms. In this model a supplier's profit depends on a whole second-period schedule $\omega \mapsto x_2(\cdot,\omega)$ . The envelope therefore involves differentiating an integral over the state space under only measurability and boundedness assumptions, and every interim quantity is an integral over the other suppliers' costs.

For the English auction, "revenue equivalence" is not available off the shelf. The auction's period-2 payment depends on $\omega$ and on the first-period price, and the induced direct mechanism is not one of the direct mechanisms above with an $\omega$ -independent transfer. Equality of surplus must come from computing the auction's expected supplier profit (Lemma 2 and the profit formula $U(c)$ ) and matching it with the optimal mechanism's. The thresholds add a further difficulty. With merely non-decreasing virtual costs and a density that need not be continuous, equations (1) and (2) need not have solutions, and the crossing points must be handled at the boundary.

Formalization scope

Representation.

Everything is real-valued.
The environment is a structure Env Ω over an arbitrary measurable space Ω, with a Standing predicate for the assumptions of §2 and an Assumption1 predicate.
Cost profiles are Fin n → ℝ, with law the product density $\prod_i f(c^i)$ on $[\underline c,\bar c]^n$ . Interim quantities overwrite one coordinate (Function.update).
A direct mechanism is admissible when, on the cost cube, $x_1$ is a lottery, $x_2(\cdot,\omega) \ge 0$ with $\sum_i x^i_2 \le \sum_i x^i_1$ (period-2 purchase requires a period-1 purchase), $x_1$ and $x_2$ are measurable, and each transfer is integrable.

Conventions and additions, each disclosed in the items.

$n \ge 1$ ; $k(c,\cdot)$ and $k_c(c,\cdot)$ are measurable; and $k$ is bounded on $[\underline c,\bar c]\times\Omega$ , as the proof of Lemma 1 asserts.
$x_2$ is the unconditional probability of supplying in period 2. Where the paper multiplies $X_2$ by $X_1$ , the formulas are rewritten, and the allocation (4) carries the conjunct $c^i \le c^\star_1$ .
$c^\star_1$ and $c^\star_2(\omega)$ are crossing points, defined as suprema. They coincide with the paper's roots when these exist. They replace the page's " $\bar c$ otherwise", which is wrong when the virtual cost stays above $R$ .
The set $A(c)$ in the transfer rule is the production set $\Omega(c)$ , which the paper identifies with it in footnote 16.
Ties are broken by lowest index (probability zero).
The equilibrium clauses of Proposition 3 are stated in reduced form: a best-response sign condition for suppliers, and the buyer's problem (9).

Ruled out. A formalization that stated the goal as "the English auction's allocation equals Proposition 1's allocation", or that defined the English-auction surplus by a virtual-surplus formula, would be trivial. The goal compares the surplus computed from the auction's outcome (winner, price, offer, acceptance) with every admissible IC/IR mechanism, and asserts attainment.

Not formalized. Proposition 2 (standard auctions with option contracts), Examples 1–2, and the extensions in §5.

Infrastructure. A complete development needs:

an envelope theorem for families of Lipschitz functions;
differentiation under the integral sign against $G$ ;
joint measurability of Carathéodory functions;
Fubini on the cost cube;
order statistics of i.i.d. costs.

The envelope and order-statistic lemmas are reusable beyond this mission. Contributions of any of these, or of the milestones in any order, are welcome.

Selected references

V. Gretschko and M. Pollrich, Incomplete Contracts in Dynamic Procurement, ZEW Discussion Paper No. 19-040, 2019. https://ssrn.com/abstract=3466701 — published as Incomplete Contracts in Multi-period Procurement, Management Science, 2021. https://doi.org/10.1287/mnsc.2021.4123
J.-J. Laffont and J. Tirole, The Dynamics of Incentive Contracts, Econometrica 56(5), 1988. https://doi.org/10.2307/1913100
R. B. Myerson, Optimal Auction Design, Mathematics of Operations Research 6(1), 1981. https://doi.org/10.1287/moor.6.1.58
J. Riley and R. Zeckhauser, Optimal Selling Strategies: When to Haggle, When to Hold Firm, Quarterly Journal of Economics 98(2), 1983. https://doi.org/10.2307/1885625
V. Skreta, Optimal Auction Design under Non-Commitment, Journal of Economic Theory 159, 2015. https://doi.org/10.1016/j.jet.2015.07.002
Q. Liu, K. Mierendorff, X. Shi and W. Zhong, Auctions with Limited Commitment, American Economic Review 109(3), 2019. https://doi.org/10.1257/aer.20170387
H. Vartiainen, Auction Design without Commitment, Journal of the European Economic Association 11(2), 2013. https://doi.org/10.1111/jeea.12003

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Operations Research Probability Statistics+1·Captain: mikedeng1

Quickest Detection Problems for Ornstein–Uhlenbeck Processes: The Optimal Detection Time Is the First Time the Posterior Odds Exceed b(X), the Unique Solution of a Nonlinear Integral EquationResearch Paper

Motivation

A change in the dynamics of an observed process is often visible only after noisy evidence has accumulated. Stopping early raises a false alarm; waiting too long increases detection delay. The quickest detection problem asks for a rule that balances these losses when the change time is unobserved. Glover and Peskir study this tradeoff for an Ornstein–Uhlenbeck process whose mean-reversion rate changes at an unknown time. Their cases include a stationary process becoming Brownian motion, Brownian motion becoming stationary, and changes between recurrent and transient behavior. The same model is motivated by monitoring a spread in pairs trading, where a change in mean reversion affects when a position should be closed. Glover and Peskir, §§2–3.

The paper's outcome is an explicit form for the optimal detection rule, expressed through a boundary that solves a nonlinear integral equation. The boundary is a function of the current observation, so a fixed threshold on posterior probability alone does not describe the optimal rule. The source for this mission is the authors' 29-page Research Report No. 11, labelled 2017 and later revised; the report's own page and theorem numbers are used throughout. Glover and Peskir, Corollary 12.

Setting

Let $X_t$ be the real-valued process observed at time $t\ge0$ . Its starting point is $x\in\mathbb R$ . Before the unobserved disorder time $\theta$ , its drift is $-\beta_0X_t$ ; at and after $\theta$ , the drift is $-\beta_1X_t$ . The diffusion coefficient is one and both mean-reverting levels are zero. The rates $\beta_0,\beta_1$ may have either sign or be zero, but they differ. Brownian noise drives the process. The prior distribution places mass $\pi\in[0,1]$ at $\theta=0$ ; conditional on $\theta>0$ , the time is exponentially distributed with rate $\lambda>0$ and is independent of the noise. Glover and Peskir, §3, pp. 5–7.

A detection time $\tau$ may use the observed history of $X$ through the time at which it stops. With delay penalty $c>0$ , its Bayes risk is

R(\tau)=\mathsf P_\pi(\tau<\theta)+c\,\mathsf E_\pi(\tau-\theta)^+, \qquad V(\pi)=\inf_{\tau\in\mathcal T^X}R(\tau).

The first term charges a false alarm, while the second charges time spent after the change and before detection. The infimum ranges over stopping times of the natural filtration of $X$ , not over rules that can see $\theta$ . Glover and Peskir, (3.4), pp. 6–7.

Write $\Pi_t=\mathsf P_\pi(\theta\le t\mid\mathcal F_t^X)$ for the posterior probability of a change by time $t$ , and $\Phi_t=\Pi_t/(1-\Pi_t)$ for its odds when the denominator is nonzero. Under the no-change law $\mathsf P^\infty$ , the paper studies a second value function,

\widehat V(\varphi,x)=\inf_{\tau\in\mathcal T^X} \mathsf E^\infty_{\varphi,x} \left[\int_0^\tau e^{-\lambda t}\left(\Phi_t-\frac{\lambda}{c}\right)dt\right],

where $\varphi\ge0$ is the initial odds. The stopping set $D$ consists of the points $(\varphi,x)$ with $\widehat V(\varphi,x)=0$ . Its lower boundary is $b(x)=\inf\{\varphi\ge0:(\varphi,x)\in D\}$ . This defines $b$ from the optimization problem; the integral equation below characterizes it later. Glover and Peskir, (4.8), (7.2), (7.7).

Formalization targets

Optimal Bayes detection

The goal is Corollary 12: the original Bayes value is recovered from the no-change value, and the first time the posterior odds cross $b(X_t)$ is optimal. With $\delta=\beta_0-\beta_1$ and $\kappa=\beta_0+\beta_1$ , the paper writes the odds as a functional of the observed path:

E_t^x=(\lambda-\delta/2)t+\frac\delta2 \left(X_t^2-x^2+\kappa\int_0^tX_s^2\,ds\right),\qquad \Phi_t=e^{E_t^x}\left(\frac\pi{1-\pi}+\lambda\int_0^t e^{-E_s^x}\,ds\right).

The target identities are

V(\pi)=(1-\pi)\left[1+c\widehat V\left(\frac\pi{1-\pi},x\right)\right], \qquad \tau^*=\inf\{t\ge0:\Phi_t\ge b(X_t)\}.

Here $\widehat V$ is also given by the expectation form of (9.6), using the paper's identity (9.4). The value identity includes $\pi=1$ ; the displayed odds formula and the optimality clause are used for $\pi<1$ , where their denominator is positive. Glover and Peskir, (9.4), Corollary 12, pp. 24, 26–27.

Boundary characterization

Theorem 11 identifies $b$ as the unique continuous solution, among functions decreasing on $(-\infty,0]$ , increasing on $[0,\infty)$ , and bounded below by $\lambda/c$ , of

\int_0^\infty e^{-\lambda t}\mathsf E^\infty_{b(x),x} \left[\left(\Phi_t-\frac\lambda c\right) \mathbf1_{\{\Phi_t<b(X_t)\}}\right]dt=0 \quad\text{for every }x\in\mathbb R.

The milestone list also includes the measure-change identity, the optimality and almost-sure finiteness of first entry into $D$ , monotonicity of the value and boundary, finiteness and continuity of the boundary, and the value representation and stopping rule of Theorem 11. Glover and Peskir, §§4, 7, 9.

Significance

The result specifies how to choose a stopping time using only data available to the observer. It gives both the minimum expected cost and a state-dependent rule attaining it. The two-sided monotonicity of $b$ says that observations farther from zero require a different level of posterior evidence for stopping, a feature a constant-threshold rule would miss. The integral equation supplies an independent characterization of the optimal threshold. Glover and Peskir, Corollary 7 and Theorem 11.

The mathematical result is proved in the paper. This mission asks for machine-checked proofs of its formal statements; the draft statements typecheck, but their proofs are open. A completed development would connect Bayesian filtering, continuous-time optimal stopping, boundary regularity, and the integral characterization in one Lean environment. The pathwise OU and odds definitions, and the treatment of extended-valued stopping costs, can be reused in related detection problems.

Difficulty

The boundary's behavior is the central obstacle. The stopping value depends jointly on posterior odds and the observed state, with both coordinates driven by the same noise. Moving the initial state farther from zero changes the future observation path and the likelihood of disorder together. This prevents a simple pointwise comparison of the running costs from proving monotonicity of $\widehat V$ in $|x|$ . Establishing that monotonicity and then the finiteness and continuity of $b$ are substantial results of §7. Glover and Peskir, Proposition 6 and Propositions 8–9.

Theorem 11 adds the challenge of identifying a unique boundary from an integral equation while preserving the original optimal-stopping definition of that boundary. The paper states the uniqueness result and refers to earlier methods for some details. A formal proof must also account for stopping times that can be infinite and for expectations whose positive part may be unbounded. Glover and Peskir, Theorem 11 and its proof, pp. 24–26.

Formalization scope

The Lean development fixes $x_0=x_1=0$ and $\sigma=1$ , and carries $\lambda>0$ , $c>0$ , and $\beta_0\ne\beta_1$ as standing conditions. It represents Brownian motion by a continuous, pointwise measurable version and uses the pathwise integration-by-parts form of the OU flow. The odds formula is the pathwise expression of (9.12), so no stochastic integral appears in the definitions. The disorder time is independent of the entire Brownian path, and decisions use the natural filtration of $X$ . Under the no-change law, the natural filtration of the Brownian driver is used; it agrees with that of the OU observation. Glover and Peskir, §§3, 5, 9.

Stopping times take values in $[0,\infty]$ . Bayes risk uses extended nonnegative reals; the no-change running cost is represented as a difference of two nonnegative integrals in extended reals. Its real value is used only where the paper proves finiteness. The equation and value representation use the expectation side of (9.4), rather than introducing a transition density. “Increasing” and “decreasing” mean weak monotonicity. The real infimum defining $b$ is zero if its fibre is empty; Proposition 8 is the separate assertion excluding that case. For $\pi=1$ , the value identity is retained but the stopping rule's division by $1-\pi$ is outside its domain. Glover and Peskir, §§4, 7, 9.

The model keeps $V(\pi)$ as the infimum of the original false-alarm and delay risk, $\widehat V$ as an infimum over observation stopping times, and $b$ as the lower edge of $\{\widehat V=0\}$ . Replacing any of these by the equation or representation being proved would trivialize the target. Contributions toward the filtering and measure-change link, optimal entry times, value monotonicity, boundary regularity, integral equation, and final Bayes-optimality theorem all advance the mission. Mathlib's Brownian-motion, filtration, stopping-time, measure, and integration libraries provide the base objects; the specific disorder and boundary constructions are developed here.

Selected references

Kristoffer Glover and Goran Peskir, Quickest Detection Problems for Ornstein–Uhlenbeck Processes, Research Report No. 11, Probability and Statistics Group, University of Manchester, 2017 (revised version; published in Mathematics of Operations Research, 2024). DOI: 10.1287/moor.2021.0186. This mission cites the 29-page research report's numbering.

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Economics Operations Research·Captain: mikedeng1

Inventory Sharing Under Service Competition: With Exogenous Inventories the Highest Non-Moot Transfer Price Is Optimal and Beats No Sharing, yet No Transfer Price Coordinates the RetailersResearch Paper

Why inventory sharing between competitors

Inventory sharing (also called lateral transshipment) lets a retailer that runs short buy leftover stock from another retailer instead of placing an expensive emergency order. Between retailers under one owner, or retailers that do not compete, pooling is known to raise profits, and a suitably chosen transfer price can make independent retailers act exactly as a centralized firm would (Rudi, Kapur & Pyke 2001; Hu, Duenyas & Kapuscinski 2007, Management Science 53(8)). Real sharing agreements, however, are often struck between stores that compete for the same customers, for instance dealers of the same brand competing on service quality. Guo and Jiang ask what changes when the retailers compete on service: which transfer price the retailers want, whether they want to share at all, and whether any transfer price still reproduces the centralized outcome.

Setting

Two retailers $i=1,2$ sell one product at price $p$ . Each holds pre-season inventory $I_i$ bought at unit cost $c$ ; unmet demand can be filled by a quick-response order at unit cost $\bar c$ , and a unit shipped between the retailers costs $\delta$ . The standing assumptions are

0\le c<\bar c\le p,\qquad k>0,\qquad \beta\ge0,\qquad 0\le\delta<\bar c .

Retailer $i$ chooses a service level $s_i\in\mathbb R$ at cost $k s_i^2$ , and faces demand $D_i=\varepsilon_i\,d_i(s_i,s_j)$ with

d_i(s_i,s_j)=\tfrac12+\beta(s_i-s_j),

where $(\varepsilon_1,\varepsilon_2)$ is a random vector on $[0,1]^2$ with an exchangeable joint law $\nu$ that has a density; $\mu=E[\varepsilon_1]$ . Service choices must keep both deterministic demands non-negative.

Without sharing, retailer $i$ earns $\pi^N_i=p\,E[\varepsilon_i d_i]-\bar c\,E[(\varepsilon_i d_i-I_i)^+]-ks_i^2-cI_i$ . Under a sharing agreement at transfer price $t$ , a retailer short of stock buys the rival's leftover at $t+\delta$ before ordering from the manufacturer; its profit $\pi_i(s_i,I_i\mid s_j,I_j)$ (equation (3) of the paper) adds the revenue $t\,E[(\min\{I_i-\varepsilon_i d_i,\varepsilon_j d_j-I_j\})^+]$ and subtracts the purchase cost $(t+\delta)\,E[(\min\{\varepsilon_i d_i-I_i,I_j-\varepsilon_j d_j\})^+]$ and the residual quick-response cost. Sharing is non-moot only for $t\in[0,\bar c-\delta]$ . With exogenous inventories $I_1=I_2=I\in[0,1]$ , a symmetric equilibrium is a profile $(s,s)$ from which no retailer gains by a feasible unilateral deviation. A centralized owner sets $s=0$ and earns $\pi^c(I,I)$ , given by (A.1) of the Online Appendix; each retailer's share is $\pi^c(I,I)/2$ .

The paper's closed forms are $\hat s^N(I)=\beta\bigl(p\mu-\bar c\int_{2I}^1\varepsilon g(\varepsilon)\,d\varepsilon\bigr)/(2k)$ without sharing and $\hat s(t,I)=\beta(p\mu-A-B)/(2k)$ with sharing, where $A$ and $B$ are expectations of piecewise-linear functions of $(\varepsilon_1,\varepsilon_2)$ over the regions in which a transfer happens.

Formalization targets

Goal: Proposition 2 (corrected)

Assume $p\mu-A-B>0$ at $t=\bar c-\delta$ . Then

\pi_i\bigl(s,I\mid s,I\bigr)\Big|_{t}\ \le\ \pi_i\bigl(s',I\mid s',I\bigr)\Big|_{\bar c-\delta},\qquad \pi^N_i(s^N,I\mid s^N)\ \le\ \pi_i\bigl(s',I\mid s',I\bigr)\Big|_{\bar c-\delta}

for every $t\in[0,\bar c-\delta]$ and all symmetric equilibria $(s,s)$ at $t$ , $(s',s')$ at $\bar c-\delta$ and $(s^N,s^N)$ without sharing; and if $\beta>0$ , every symmetric sharing equilibrium at every non-moot $t$ satisfies

\pi_i(s,I\mid s,I)<\tfrac12\pi^c(I,I),\qquad s\neq 0 .

Milestones

Lemma A.1(a)–(b) (the exchange identities); Lemma 2(a) (no-sharing equilibrium is $\hat s^N$ ); Lemma 3(a), uniqueness (a sharing equilibrium is $\hat s$ ); the closed forms (A.9) and (A.11) at symmetric profiles; Lemma 3(b) and 3(c) in $t$ ; Proposition 1(a) through the identity $\hat s(t,I)-\hat s^N(I)=(\bar c-2t-\delta)\beta W(I)/(2k)$ ; Proposition 1(c) for $t\ge(\bar c-\delta)/2$ ; and the identity $\pi_i(s,I\mid s,I)=\pi^c(I,I)/2-ks^2$ from the proof of Proposition 2.

Significance

The result separates the two roles a transfer price plays. Between non-competing retailers it only redistributes pooled profit; under service competition it also changes how hard the retailers compete, and the largest non-moot price softens competition most. Proposition 2 says the retailers therefore pick $\hat t^*=\bar c-\delta$ , always sign the agreement at that price, and still fall short of the centralized profit whenever $\beta>0$ : inventory sharing cannot by itself undo service competition. This contrasts with the coordination results for non-competing retailers cited above and with the paper's own Lemma 1 for $\beta=0$ .

The paper's proofs are informal calculus on iterated integrals against a density. A formal development makes explicit which steps hold for every exchangeable law and which need extra conditions. Two such conditions surfaced during planning: the sign of $\hat s(\bar c-\delta,I)$ , and the existence of equilibria. No part of the paper had been machine-checked before this mission.

Difficulty

The obvious argument differentiates $\hat\pi_i(t,I)$ in $t$ and finds $-2k\hat s\,\partial_t\hat s$ , which is positive only if $\hat s\ge0$ . The paper asserts " $\hat s(t,I)>0$ for any $t$ given that $\beta>0$ " without proof, and it is false: for i.i.d. bimodal demand with $p=1$ , $\bar c=0.9$ , $\delta=0$ , $c=0.3$ , $I=0.324$ , $\beta=0.1$ , $k=1$ , the closed-form service level turns negative before $\bar c-\delta$ , the profit peaks near $t\approx0.605$ , and at that price it equals the centralized half-profit. The goal is therefore stated under the guard $p\mu-A-B>0$ at $\bar c-\delta$ . A second difficulty is that equilibria are defined by global best responses, while the paper's closed forms come from first-order conditions. The uniqueness direction needs the differentiability of expectations of kinked payoffs, for which the density matters. The existence direction is not true in general and is not part of the mission.

Formalization scope

All quantities are real numbers. The law of $(\varepsilon_1,\varepsilon_2)$ is a probability measure $\nu$ on $\mathbb R^2$ with $\nu([0,1]^2)=1$ , $\nu\circ\mathrm{swap}^{-1}=\nu$ and $\nu\ll$ Lebesgue. Exchangeability comes from the Online Appendix ( $g_{12}(x,y)=g_{12}(y,x)$ ); the main text states only equal marginals. The paper's iterated integrals are expectations of region indicators; open and closed region boundaries agree up to null sets. Service levels carry no sign restriction. Deviations must keep both deterministic demands non-negative. The bound $\beta\le\bar\beta$ of (A.7) is not imposed because no formalized result uses it. "Optimal" means "a maximizer": at $I=0$ every price is optimal. "Prefer" is weak, following the paper's tie-break rule (1).

The goal quantifies over equilibria defined by best responses. Stating it about the closed forms $\hat s,\hat s^N$ in place of equilibria would trivialize it into the milestones, and is ruled out. Since the goal is universally quantified, a sanity file checks that its hypotheses are jointly satisfiable (uniform demand, Figure 2's parameters, $I=0$ ) and that equilibria exist at $\beta=0$ . Existence for $\beta>0$ is open in this formalization.

A complete development needs: change of variables under the coordinate swap; integrability of bounded piecewise-linear functions on $[0,1]^2$ ; differentiation under the integral sign for kinked payoffs; and first-order conditions for concave or locally concave functions on an interval. The exchange identities and the symmetric-profile closed forms can be reused for other symmetric two-location sharing models. Proofs of any milestone are welcome, and so is an existence result for $\beta>0$ under explicit conditions.

Selected references

X. Guo and B. Jiang, Inventory Sharing under Service Competition, Manufacturing & Service Operations Management, 2023; SSRN version of March 2, 2023. https://ssrn.com/abstract=3512712 , https://doi.org/10.1287/msom.2020.0584
N. Rudi, S. Kapur and D. F. Pyke, A Two-Location Inventory Model with Transshipment and Local Decision Making, Management Science 47(12):1668–1680, 2001. https://doi.org/10.1287/mnsc.47.12.1668.10236
X. Hu, I. Duenyas and R. Kapuscinski, Existence of Coordinating Transshipment Prices in a Two-Location Inventory Model, Management Science 53(8):1289–1302, 2007.

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Operations Research Partial Differential Equations Probability+1·Captain: mikedeng1

Exact Optimal Stopping for Multidimensional Linear Switching Diffusions: The Optimal Stopping Boundaries Are the Unique Admissible Solution of a Nonlinear Integral EquationResearch Paper

Motivation

Optimal stopping of a diffusion whose coefficients switch with an exogenous Markov chain is the standard model for decisions taken in a random environment: an investment or abandonment decision when the economy alternates between regimes, a quickest-detection problem whose drift is driven by a hidden chain, a maintenance decision whose degradation rate depends on an operating mode. For one-dimensional problems without switching the optimal rule is a threshold, and the threshold is characterized by a nonlinear integral equation (Peskir and Shiryaev, Optimal Stopping and Free-Boundary Problems, 2006). In two dimensions with regime switching, the stopping region on each layer of the chain is bounded by a curve, the curves differ from layer to layer, and neither the optimality of the first entry time nor a characterization of the curves follows from the classical theory.

Ernst and Mei (arXiv:2107.14441, Mathematics of Operations Research, 2023) study such a problem for a two-dimensional linear diffusion driven by a two-state chain. They show that the first entry time into the stopping region is optimal although the running cost need not be integrable on $[0,\infty)$ , derive the regularity of the value function, and characterize the optimal stopping boundaries as the unique solution of a coupled system of nonlinear integral equations. Section 6 of the paper applies the results to quickest detection of a Markovian drift.

Setting

Let $M=\{1,2\}$ be the regimes and let $\alpha$ be a continuous-time Markov chain on $M$ with generator $Q=(q_{\iota\kappa})$ . Let $W=(W^1,W^2)$ be a standard two-dimensional Brownian motion independent of $\alpha$ . For $x\in[0,\infty)^2$ the state $X=(X^1,X^2)$ solves

dX^i_t=\Big(a^i(\alpha_t)+\sum_{j=1}^2 b^{ij}(\alpha_t)X^j_t\Big)dt+\sigma^i(\alpha_t)X^i_t\,dW^i_t,\qquad X_0=x,\quad i=1,2,

with $\lambda(\iota)>0$ , $a^i(\iota)\ge0$ , $\sigma^i(\iota)\ne0$ and $b^{ij}(\iota)\ge0$ for $i\ne j$ (Assumption 2.1). The running cost $H(\iota,\cdot)$ is concave, locally Lipschitz up to the boundary, and increasing on $[0,\infty)^2$ , with $H(\iota,(z,0))\to\infty$ and $H(\iota,(0,z))\to\infty$ (Assumption 2.2), and $\underline H=\min_\iota H(\iota,(0,0))<0$ . The value function is

V(\iota,x)=\inf_{\tau}\mathsf E_{\iota,x}\int_0^{\tau}e^{-\int_0^t\lambda(\alpha_s)ds}H(\alpha_t,X_t)\,dt,

the infimum over stopping times of the augmented natural filtration of $(\alpha,X)$ . The continuation and stopping regions are $C_\iota=\{V(\iota,\cdot)<0\}$ and $D_\iota=\{V(\iota,\cdot)=0\}$ in $(0,\infty)^2$ , the optimal stopping boundary is $b_\iota(x_1)=\inf\{x_2>0:V(\iota,x_1,x_2)=0\}$ , and $\tau_D$ is the first entry time of $(\alpha,X)$ into $D$ . Assumption 2.3 asks that each one-dimensional process $d\tilde X^i_t=(a^i(\alpha_t)+b^{ii}(\alpha_t)\tilde X^i_t)dt+\sigma^i(\alpha_t)\tilde X^i_t\,dW^i_t$ be recurrent to $(c,\infty)$ for every $c>0$ . The truncated problem $V^N$ replaces $H$ by $H^N=\min(H,N)$ .

Formalization targets

Goal: Theorem 5.1

With a constant discount rate $\lambda\equiv\lambda_0>0$ , the family $\{b_\iota\}$ is the unique solution, among bounded continuous $c=(c_\iota)$ with $H(\iota,(z,c_\iota(z)))\ge0$ , of

\int_0^\infty\sum_{\kappa\in M}\int_0^\infty\int_0^{c_\kappa(y_1)}e^{-\lambda_0t}\,p_t\big(\iota,x_1,c_\iota(x_1);\kappa,y\big)H(\kappa,y)\,dy_2\,dy_1\,dt=0,\qquad x_1>0,

and $V(\iota,x)=\int_0^\infty\sum_\kappa\int_0^\infty\int_0^{b_\kappa(y_1)}e^{-\lambda_0t}p_t(\iota,x;\kappa,y)H(\kappa,y)\,dy_2\,dy_1\,dt$ .

Milestones

Proposition 2.4: the one-dimensional truncated problem stops above finite positive thresholds $\bar x^1_\iota$ .
Lemmas 3.2 and 3.3: entry times of the truncated stopping regions are finite, and a criterion for optimality of $\tau_D$ .
Lemmas 3.5 and 3.6, Proposition 3.7: $V$ is increasing, concave and continuous; $b_\iota$ is decreasing, convex, continuous and bounded.
Theorem 3.8: $V(\iota,x)=\mathsf E_{\iota,x}\int_0^{\tau_D}e^{-\int_0^t\lambda(\alpha_s)ds}H(\alpha_t,X_t)\,dt$ .
Lemmas 3.9 and 3.11, Proposition 3.13: $V\in C^2(C_\iota)$ solves $\mathbb LV-\lambda V=-H$ there, $V\in C^1((0,\infty)^2)$ , and its gradient is Lipschitz on $(1/K,K)^2$ .

Significance

Theorem 3.8 settles the optimality of the first entry time for a cost that is bounded below but not integrable, a case outside the standard criterion. The concavity and smooth-fit results give the free-boundary characterization of $V$ (Theorem 4.1 of the paper), and the integral equation turns the unknown curves $b_1,b_2$ into the solution of a system that can be discretized and solved numerically, as is done for one-dimensional problems.

None of these results is formalized anywhere. A complete development would give a machine-checked account of an optimal stopping problem for a multidimensional diffusion with regime switching, including the passage from the truncated costs $H^N$ to $H$ , the regularity theory, and the integral equation. The uniqueness part of Theorem 5.1 is stated in the paper, but its proof is omitted ("we will omit further details", p. 18); a formal proof would also settle that argument in the presence of transition regions.

Difficulty

The obvious route to Theorem 3.8 is the classical criterion that the first entry time into $\{V=0\}$ is optimal when $\mathsf E\int_0^\infty e^{-\Lambda_t}|H(\alpha_t,X_t)|\,dt<\infty$ . That integrability fails here, because $H$ grows without bound. Without it, the value need not be attained by any entry time a priori, and the entry time $\tau_D$ need not even be finite.

The second obstacle is the chain. On a layer $\iota$ , points of $C_\iota\cap D_\kappa$ form transition regions where the process can jump from continuation into stopping without crossing a boundary, so the regularity theory for a Dirichlet problem with one common boundary does not apply, and any characterization of the curves must account for boundaries that differ by layer.

Formalization scope

Regimes and coordinates are Fin 2, 0-based: regime 0 is the paper's state 1 and x 0 is $x_1$ . The model is a structure of coefficients; the probabilistic setting is one probability space carrying the chain, the Brownian motions, the flow $x\mapsto X^x$ of strong solutions of (2.2) for every $x\in[0,\infty)^2$ , the one-dimensional flows $\tilde X^i$ , and the filtrations. The SDEs are required in integral form with continuous paths; the stochastic integrals are the published relation EthierKurtz.HasBrownianItoIntegral. $P$ is complete, both initial regimes have positive probability, and $\mathsf P_{\iota,x}$ is $P$ conditioned on $\alpha_0=\iota$ , applied to $X^x$ . Stopping times take values in $[0,\infty]$ and the stopping cost is computed in extended reals from positive and negative parts, so a non-integrable cost is never silently replaced by $0$ . The standing $\underline H<0$ of p. 4 is a hypothesis of every statement.

The integrals against the transition density in (5.1)–(5.2) are written as expectations, $\int_0^\infty e^{-\lambda_0t}\,\mathsf E\big[H(\alpha_t,X_t)\mathbf 1\{X_t\text{ below the curve}\}\big]dt$ , with absolute convergence required; this equals the density form whenever the density exists. A formalization that drops the integrability requirement, or that quantifies over an abstract Markov family instead of the solution of (2.2), would make the uniqueness statement false or vacuous, and is ruled out. Admissible functions take values in $[0,\infty)$ , where $H$ is defined. Green regularity (used only in Lemma 3.3) is read in probability. $W^{2,\infty}((1/K,K)^2)$ is read, as on p. 11 of the paper, as Lipschitz continuity of the first partial derivatives on the square.

Two printed slips are corrected: Lemma 3.6's $b_\iota(z)<b^N_\iota(z)$ becomes $\le$ (both vanish for large $z$ ), and Proposition 3.7(4)'s undefined $\bar x^2_\iota$ becomes the threshold $\bar x^1_\iota$ of the one-dimensional problem.

A complete development needs Itô's formula for switching diffusions, the strong Markov property of $(\alpha,X)$ , comparison for the linear SDE, Snell envelope arguments in continuous time, and interior elliptic regularity. The chain and filtration definitions and the extended-real stopping value are reusable for other regime-switching stopping problems. Proofs of any milestone, of the existence of the setting, and of reusable lemmas on the two-state chain are welcome.

Selected references

P. A. Ernst, H. Mei, Exact optimal stopping for multidimensional linear switching diffusions, Mathematics of Operations Research, 2023. arXiv:2107.14441
G. Peskir, A. Shiryaev, Optimal Stopping and Free-Boundary Problems, Birkhäuser, 2006. doi:10.1007/978-3-7643-7390-0
G. G. Yin, C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, 2010. doi:10.1007/978-1-4419-1105-6
J. du Toit, G. Peskir, The trap of complacency in predicting the maximum, Annals of Probability 35, 2007. doi:10.1214/009117906000000638

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Algorithmic Game Theory Operations Research Optimization·Captain: mikedeng1

Characterizing and Computing the Set of Nash Equilibria via Vector Optimization: Shared Constraint Nash Equilibria Are Exactly the Pareto Optimal Points Under a Non-Convex Ordering ConeResearch Paper

Motivation

A Nash equilibrium is a feasible choice of strategies at which no player can lower her own cost by changing her strategy alone. A Pareto minimizer is a feasible point that cannot be improved under a specified comparison of vector outcomes. These ideas usually select different points: a change that helps several players at once is relevant to ordinary Pareto comparison, while Nash stability holds the other players' strategies fixed. Feinstein and Rudloff ask whether a suitable vector comparison can recover all equilibria, including when the equilibrium set has several points or is empty. Their answer for games with one shared feasibility constraint is Theorem 2.6 of the pinned preprint. The result addresses the full equilibrium set rather than the existence or computation of a single point.

The paper places this characterization alongside vector optimization methods that can compute solution sets for some structured games. It also extends the framework to generalized Nash games and vector-valued costs. This mission concentrates on the shared-constraint, scalar-cost statement on which those later sections build. The formal target is a known theorem from the paper, not a claim that the game always has an equilibrium or that a particular numerical algorithm finds one.

Setting

Let $I$ be a finite set of at least two players. Player $i$ chooses a strategy in a real linear space $\mathcal X_i$ ; the spaces may differ. A joint strategy $x$ is a member of $\prod_{i\in I}\mathcal X_i$ . Player $i$ has an arbitrary real-valued cost function $f_i$ on joint strategies. A set $\mathbb X\subseteq\prod_i\mathcal X_i$ specifies the shared constraint: only joint strategies in $\mathbb X$ are feasible. No continuity, convexity, boundedness, or nonemptiness condition is imposed on the costs or on $\mathbb X$ .

For a joint strategy $x^*$ , write $(x_i,x^*_{-i})$ for the strategy obtained by replacing only player $i$ 's coordinate. The set $\mathrm{NE}(f,\mathbb X)$ consists of feasible $x^*$ such that every feasible replacement obeys $f_i(x^*)\le f_i(x_i,x^*_{-i})$ . The feasibility condition applies to the whole replacement; it cannot be replaced by membership of $x_i$ in a coordinate projection of $\mathbb X$ . This is Definition 2.2 in the source.

For a vector space $Z$ and a cone $K\subseteq Z$ , define $a\le_K b$ when $b-a\in K$ . For a domain $D$ , feasible set $A\subseteq D$ , and objective $g:D\to Z$ , the paper calls $x^*$ a $K$ -minimizer if it is feasible and every feasible $x$ satisfying $g(x)\le_K g(x^*)$ also satisfies $g(x^*)\le_K g(x)$ . This permits equivalent outcomes and does not require a convex cone. The mission restricts comparisons to feasible points, matching Definition 2.3.

The objective for the game is $g(x)=(x,f(x))$ , where $f(x)=(f_i(x))_{i\in I}$ . Its target space is $\bigl(\prod_i\mathcal X_i\bigr)\times\mathbb R^I$ . For each player $i$ , the cone $C_i$ permits any change in strategy coordinate $i$ and in the other players' cost coordinates, requires zero change in every other strategy coordinate, and requires a nonnegative change in cost coordinate $i$ . The paper's ordering cone is $C=\{(u,v):\exists i,\ u_j=0\text{ for }j\ne i,\ v_i\ge0\}$ . Equation (7) gives $C_i$ as an explicit product; equation (6) gives $C$ directly. The resulting relation compares costs only while other players' strategies remain fixed.

Formalization targets

Player-cone structure

The paper states that $C=\bigcup_{i\in I}C_i$ and that each $C_i$ is a convex cone. It then characterizes comparisons by $C_i$ and $C$ in terms of fixed strategy coordinates and a player's cost. These are the first two milestones, from §2.2, pages 6–7 of the preprint.

Minimizer decomposition

Lemma 2.7 is the third milestone:

C\text{-argmin}\{(x,f(x)):x\in\mathbb X\} =\bigcap_{i\in I} C_i\text{-argmin}\{(x,f(x)):x\in\mathbb X\}.

Main equality

The goal is Theorem 2.6, an equality of complete solution sets:

\mathrm{NE}(f,\mathbb X) = C\text{-argmin}\{(x,f(x)):x\in\mathbb X\}.

The equality covers an empty equilibrium set as well as games with one, several, or infinitely many equilibria. It asserts neither existence nor uniqueness.

Significance

The theorem gives a vector optimization description of every stable joint strategy in a shared-constraint game. Because the objective retains the strategy vector, the cone can express which coordinates remain fixed during a unilateral deviation. The decomposition into player cones also identifies separate convex comparisons inside a potentially nonconvex overall preference. In settings where suitable vector optimization methods are available, the paper uses this characterization as a route toward finding full equilibrium sets; it does not claim a universal exact algorithm for arbitrary games §1 and §2.2.

Formalization adds a reusable generic cone-induced relation and vector-minimizer definition, plus a shared-constraint game interface that later results can import. It pins down the direction of comparison, the feasible deviation set, and the product-coordinate constraints in the cones. The source proves the goal and its supporting statements on paper. The local Prove2Me prior-art search found no published machine-checked statement with this exact shared-constraint model and cone; the mission asks solvers to supply proofs of the drafted statements.

Difficulty

Applying the ordinary coordinatewise order to the cost vector $f(x)$ compares outcomes from arbitrary simultaneous changes of strategy. It therefore does not encode the unilateral test in the Nash definition. Including the strategy vector in the objective and using $C$ makes the relevant changes comparable, but $C$ is a union of player cones and need not be convex. A proof must respect the paper's minimizer definition for this relation and establish the set equality in both directions. Restricting the theorem to smooth or convex games would miss the generality of Theorem 2.6.

Formalization scope

Lean represents $I$ by a finite type $\iota$ with decidable equality and states $2\le|I|$ , the paper's standing convention. Each $\mathcal X_i$ is a possibly different type carrying an additive commutative group and a real module structure. Joint strategies are dependent functions $\forall i,\mathcal X_i$ ; feasible deviations use Function.update. The shared constraint is an arbitrary set of joint strategies, and each cost is an arbitrary real-valued function. An empty feasible set remains permitted. The objective space is the product of joint strategies with real cost profiles, and $C_i$ is built as the product of the coordinate sets in equation (7). The condition $x_{-i}=0$ in equation (6) means that every coordinate other than $i$ equals its vector-space zero.

The generic vector-minimizer definition, the Nash equilibrium definition, and the two cones are independent. In particular, Nash equilibrium is defined directly by unilateral feasible deviations; $C$ is defined directly by the existential condition in (6); and vector minimization is generic in its cone, objective, and feasible domain. This rules out a definition that makes the main equality true by construction. The $|I|\ge2$ hypothesis is copied from the paper, although these particular equalities also hold with fewer players. Contributions to the cone comparison lemmas, the minimizer decomposition, and the full set equality are welcome. Later generalized Nash and vector-valued results, numerical examples, and Corollary 2.8 lie outside this mission.

Selected references

Zachary Feinstein and Birgit Rudloff, Characterizing and Computing the Set of Nash Equilibria via Vector Optimization, arXiv:2109.14932v5, 2022; published as an Operations Research technical note, 2024. Pinned preprint

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Algorithmic Game Theory Control Theory Probability·Captain: mikedeng1

A Probabilistic Approach to Extended Finite State Mean Field Games: With Lipschitz Rates and Costs and a Unique Hamiltonian Minimizer, a Nash Equilibrium in Closed-Loop Strategies ExistsResearch Paper

Motivation

Mean field games describe the limit of stochastic differential games with a large number of statistically identical players, each interacting with the others only through the empirical distribution of their states (Lasry and Lions, Mean field games, 2007; Huang, Malhamé and Caines, 2006). Models whose players move among finitely many states (an infection status, a cyber-security level, a choice among a few technologies) are used in epidemiology, botnet defense (Kolokoltsov and Bensoussan, 2016; reference [19] of the paper) and economics. For these finite state games the analytic approach of Gomes, Mohr and Souza (Continuous time finite state mean field games, 2013) characterizes equilibria by a coupled system of ordinary differential equations, and so restricts attention to Markovian controls and to interactions through the distribution of states only.

Carmona and Wang (arXiv:1808.07635, 2018; Mathematics of Operations Research, 2021) treat the finite state game in the weak formulation, by a change of measure on the canonical path space, following the diffusion case of Carmona and Lacker (A probabilistic weak formulation of mean field games and applications, 2015). This handles two extensions at once: strategies may be arbitrary closed-loop, path-dependent controls, and the rates and costs may depend on the distribution of the controls themselves (an extended mean field game).

Setting

There are $m$ states, identified with the basis vectors $e_1,\dots,e_m$ of $\mathbb R^m$ . The canonical space $\Omega$ consists of the càdlàg paths $[0,T]\to\{e_1,\dots,e_m\}$ that are continuous at $T$ ; $X_t(\omega)=\omega_t$ is the canonical process and $\mathbb F=(\mathcal F_t)$ its natural filtration. The reference measure $\mathbb P$ makes $X$ a continuous-time Markov chain with a given initial law $p^\circ$ and rate matrix $Q^0$ (diagonal entries $-(m-1)$ , off-diagonal entries $1$ ).

Controls take values in a compact set $A\subseteq\mathbb R^l$ . Write $\mathcal S$ for the probability simplex of $\mathbb R^m$ , $\mathcal P(A)$ for the probability measures on $A$ and $\mathcal W_1$ for the 1-Wasserstein distance on $\mathcal P(A)$ . A strategy $\alpha\in\mathbb A$ is an $\mathbb F$ -predictable process with values in $A$ . A mean field is a pair of measurable flows $p:[0,T]\to\mathcal S$ (states) and $\nu:[0,T]\to\mathcal P(A)$ (controls). Transition rates $q(t,i,j,\alpha,p,\nu)$ , a running cost $f(t,e_i,\alpha,p,\nu)$ and a terminal cost $g(e_i,p)$ are given.

A strategy and a mean field change $\mathbb P$ into $\mathbb Q^{(\alpha,p,\nu)}$ , whose density is the Doléans-Dade exponential of (5)–(7); under it $X$ jumps from $e_i$ to $e_j$ at rate $q(t,i,j,\alpha_t,p_t,\nu_t)$ . The cost is

J(\alpha,p,\nu)=\mathbb E^{\mathbb Q^{(\alpha,p,\nu)}}\Big[\int_0^Tf(t,X_t,\alpha_t,p_t,\nu_t)\,dt+g(X_T,p_T)\Big].

A Nash equilibrium (Definition 2.8) is a triple $(\alpha^*,p^*,\nu^*)$ such that $\alpha^*$ minimizes $J(\cdot,p^*,\nu^*)$ over $\mathbb A$ , and for every $t$ , $p^*_t$ is the law of $X_t$ and $\nu^*_t$ the law of $\alpha^*_t$ under $\mathbb Q^{(\alpha^*,p^*,\nu^*)}$ .

The reduced Hamiltonian is $H_i(t,z,\alpha,p,\nu)=f(t,e_i,\alpha,p,\nu)+\sum_{j\neq i}(z_j-z_i)(q(t,i,j,\alpha,p,\nu)-1)$ , and $\hat H_i=\inf_{\alpha\in A}H_i$ .

Formalization targets

Goal: Theorem 4.1

Under Assumption 2.2 (each $Q(t,\alpha,p,\nu)$ is a Q-matrix, off-diagonal rates lie in $(C_1,C_2)$ with $C_1>0$ , rates are Lipschitz in $(\alpha,p,\nu)$ ), Assumption 2.7 ( $f$ and $g$ Lipschitz) and Assumption 3.4 ( $H_i(t,z,\cdot,p,\nu)$ has a unique minimizer $\hat a_i(t,z,p)$ , independent of $\nu$ , measurable and Lipschitz in the sense of (21)),

\exists\,(\alpha^*,p^*,\nu^*)\ \text{a Nash equilibrium in the sense of Definition 2.8.}

Milestones

Lemma 3.7: $\hat H$ and $\hat a$ are Lipschitz in $z$ for the stochastic seminorm and in $(p,\nu)$ with factor $1+\|z\|$ .
Lemma 3.8: the cost of any strategy is $\mathbb E^{\mathbb P}[Y_0]$ for the BSDE (28).
Proposition 3.9: the BSDE (29) gives the value, and $\hat\alpha_t=\hat a(t,X_{t-},Z_t,p_t)$ is the $dt\otimes d\mathbb P$ -a.e. unique optimal control.
Proposition 4.2: the set $\mathcal P_0$ of measures with $L^2$ -bounded densities is convex and relatively compact for the J1-weak topology.
Lemma 4.7: $\mathcal R_0$ is convex and compact for the stable topology.
Lemma 4.8: randomized rates and Hamiltonian are Lipschitz in $\bar{\mathcal W}_1$ .
Proposition 4.10: for some $C_0>0$ , $\bar{\mathcal P}_0\times\mathcal R_0$ is stable for the fixed-point map $\Phi$ .
Lemma 4.15: $\int_0^T\mathcal W_1(\nu^{(n)}_t,\nu^{(0)}_t)\,dt\to0$ implies stable convergence of $\mathcal L(dt)\times\delta_{\nu^{(n)}_t}$ .

Significance

The theorem gives equilibria for finite state mean field games without the Markovian restriction and with interaction through the distribution of controls, in a setting where the analytic (ODE) approach does not apply. The same weak formulation supports the paper's uniqueness result under a Lasry–Lions monotonicity condition and the construction of approximate Nash equilibria for the $N$ -player game (Theorems 5.4 and 6.5), which are not part of this mission.

The result is proved in the paper; nothing here is open mathematics. No part of it is machine-checked. A formalization would add, beyond the theorem itself: the reference Markov chain on the canonical càdlàg space, Girsanov changes of measure for chains with bounded rates, backward SDEs driven by a Markov chain (Cohen and Elliott's theory), the J1 topology on finite-valued paths, and the stable topology of Jacod and Mémin. While drafting, Proposition 4.5 of the paper was found to be false as printed (see Formalization scope); the formalization records the corrected argument structure.

Difficulty

The obvious approach, a fixed point of $(p,\nu)\mapsto$ the laws under the best response, fails for lack of compactness: sets of measurable flows $[0,T]\to\mathcal S$ or $[0,T]\to\mathcal P(A)$ have no useful compact subsets. The paper replaces the state flow by a law on path space, where compactness comes from tightness in the Skorokhod topology, and the control flow by a measure on $[0,T]\times\mathcal P(A)$ with the stable topology, where compactness requires fixing the time marginal. Continuity of the fixed-point map then needs stability of Markov-chain BSDEs and $L^2$ control of Girsanov densities, both uniform in the mean field. In Lean, Schauder's fixed point theorem, Prokhorov's theorem on Skorokhod space, Girsanov's theorem for jump processes and BSDE theory are all absent from Mathlib.

Formalization scope

All declarations are in the namespace ExtFSMFG.Existence. Time is indexed by $\mathbb R_{\ge0}$ , and paths are frozen after $T$ (in bijection with the paper's paths on $[0,T]$ ); states are Fin m. The reference measure is a hypothesis, IsReference, fixing the finite-dimensional distributions through $e^{tQ^0}$ . $\mathbb Q^{(\alpha,p,\nu)}$ is defined by the closed form of $\mathcal E(L^{(\alpha,p,\nu)})_T$ (an exponential of the compensator times the product of the rates at the finitely many jumps), and stochastic integrals against the chain's martingale are pathwise. $\|p-p'\|$ is Euclidean. $\mathcal P(A)$ carries the weak topology and its Borel $\sigma$ -field; $\mathcal W_1$ and $\bar{\mathcal W}_1$ are infima over couplings. $\hat a$ appears as an argument of Assumption 3.4, which pins it down.

One hypothesis is added and disclosed: $q$ and $f$ are jointly measurable and $f$ is bounded (RegularData). The paper uses both (for (5), (12) and the BSDE drivers) without stating them. BSDE solutions have a predictable $Z$ (the paper writes "left-continuous"); $\bar{\mathcal W}_1$ has the marginal constraint that (40) omits; (42) uses the rates of (5).

The equilibrium is stated through Definition 2.8 itself, so a formalization in which the equilibrium is a fixed point of $\Phi$ , the strategy set is shrunk to Markov feedback controls, the density is a junk value, or optimality is phrased through an infimum that may default to $0$ , is ruled out: optimality is " $J(\alpha^*)\le J(\alpha)$ for every predictable $A$ -valued $\alpha$ ".

Proposition 4.5 (" $\mathcal R$ is compact, metrizable, and Polish") is not a milestone: as printed it is false, since $\delta_{(T/(n+1),\,m_0)}$ has no cluster point in $\mathcal R$ for the stable topology (test $g(t,m)=\mathbb 1_{S_k}(t)$ with $S_k=\{T/(n+1):n\ge k\}$ ). Only the compactness of $\mathcal R_0$ (Lemma 4.7) is used. Lemmas 3.1, 3.3, 4.3, Propositions 4.12, 4.14, 4.16 are further steps of the proof that may be added as milestones later. Contributions welcome: a proof of the sanity-level facts (the density integrates to $1$ ; $\mathcal W_1$ is a metric on $\mathcal P(A)$ ), Lemma 3.7 and Lemma 4.8, which are elementary, and Markov-chain BSDE infrastructure.

Selected references

R. Carmona, P. Wang, A Probabilistic Approach to Extended Finite State Mean Field Games, Mathematics of Operations Research, 2021. https://arxiv.org/abs/1808.07635 (v1 is the version formalized)
R. Carmona, D. Lacker, A probabilistic weak formulation of mean field games and applications, Annals of Applied Probability 25, 2015. https://doi.org/10.1214/14-AAP1020
S. N. Cohen, R. J. Elliott, Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions, Annals of Applied Probability 20, 2010. https://doi.org/10.1214/09-AAP619
D. A. Gomes, J. Mohr, R. R. Souza, Continuous time finite state mean field games, Applied Mathematics & Optimization 68, 2013. https://doi.org/10.1007/s00245-013-9202-8
J. Jacod, J. Mémin, Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité, Séminaire de Probabilités XV, Lecture Notes in Mathematics 850, 1981 (reference [17] of Carmona–Wang).
J.-M. Lasry, P.-L. Lions, Mean field games, Japanese Journal of Mathematics 2, 2007. https://doi.org/10.1007/s11537-007-0657-8

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Algorithmic Game Theory Combinatorics Linear Optimization+1·Captain: mikedeng1

Network Inspection for Detecting Strategic Attacks: The Equilibrium Detection Rate and Inspection Strategies Do Not Depend on the Attacker's Number of ResourcesResearch Paper

Motivation

Utility networks such as water pipelines, gas distribution systems and power grids are exposed to attacks on their physical components. An operator who cannot monitor every component must decide where to place a limited number of detectors, without knowing which components an adversary will target or how many of them. Dahan, Sela and Amin (arXiv:1705.00349, Operations Research 2022) model this as a two-player zero-sum game between a defender who places detectors and an attacker who chooses a set of components to disrupt. Their game sits in the line of security games and network interdiction models (Washburn and Wood 1995; Tambe 2011), with set cover and set packing as the combinatorial structure that governs its equilibria.

The practical question the paper answers is whether the defender needs to know the attacker's resources. The main result, Theorem 2, says it does not, within a range described by a set packing number of the network.

Setting

A detection model $\mathcal G=(\mathcal V,\mathcal E,\{\mathcal C_i,\ i\in\mathcal V\})$ consists of a finite set $\mathcal V$ of nodes, a finite set $\mathcal E$ of components, and for each node $i$ the set $\mathcal C_i\subseteq\mathcal E$ of components a detector at $i$ monitors. Every component is monitored from at least one node. For $S\subseteq\mathcal V$ let $\mathcal C_S=\bigcup_{i\in S}\mathcal C_i$ , and define the detection function

F(S,T)=|\mathcal C_S\cap T|,

the number of attacked components in $T$ that some detector of $S$ monitors.

The defender (P1) has $b_1$ detectors and chooses a positioning $S\in\mathcal A_1=\{S\subseteq\mathcal V:|S|\le b_1\}$ . The attacker (P2) has $b_2$ resources and chooses an attack plan $T\in\mathcal A_2=\{T\subseteq\mathcal E:|T|\le b_2\}$ . Both may randomize: $\sigma^1\in\Delta(\mathcal A_1)$ , $\sigma^2\in\Delta(\mathcal A_2)$ . The attacker's payoff is the expected number of undetected attacks

U(\sigma^1,\sigma^2)=\mathbb E_{(\sigma^1,\sigma^2)}\bigl[|T|-F(S,T)\bigr],

the defender receives $-U$ , and this zero-sum game is $\Gamma(b_1,b_2)$ . A Nash equilibrium is a profile $(\sigma^{1*},\sigma^{2*})$ with $U(\sigma^{1*},\sigma^2)\le U(\sigma^{1*},\sigma^{2*})\le U(\sigma^1,\sigma^{2*})$ for all $\sigma^1,\sigma^2$ ; the set of equilibria is $\Sigma(b_1,b_2)$ . The value $U^*(b_1,b_2)$ is $\min_{\sigma^1}\max_{T\in\mathcal A_2}U(\sigma^1,T)$ . The expected detection rate of a profile is

r(\sigma)=\mathbb E_\sigma\Bigl[\frac{F(S,T)}{|T|}\Bigr].

A set cover is a set of nodes monitoring every component; $n^*$ is the minimum size of a set cover (MSC). A set packing is a set of components no node monitors two of; $m^*$ is the maximum size of a set packing (MSP). The paper's case of interest is $b_1<n^*$ and $b_2<m^*$ : the defender cannot monitor everything, and the network is large compared with the attacker's resources.

Formalization targets

Goal: Theorem 2

For every $b_1$ there is $r^*_{b_1}\in[0,1]$ such that

\forall\, b_2 \text{ with } 1\le b_2<m^*,\ \ \forall\,\sigma^*\in\Sigma(b_1,b_2):\qquad r(\sigma^*)=r^*_{b_1},

and every inspection strategy that is part of an equilibrium of $\Gamma(b_1,1)$ is part of an equilibrium of every $\Gamma(b_1,b_2)$ with $1\le b_2<m^*$ . The goal fixes no value of $r^*_{b_1}$ : it asserts only that one number serves all admissible $b_2$ .

Milestones

Corollary EC.1: $F(S,T)\le\sum_{i\in S}F(\{i\},T)$ and $F(S,T)=\sum_{e\in T}F(S,\{e\})$ .
Lemma EC.2: $m^*\le n^*$ .
Lemma EC.3 (Lemma 1 in the body): the cyclic strategies $\sigma^1(S,b_1)$ , $\sigma^2(T,b_2)$ , uniform over $|S|$ (resp. $|T|$ ) cyclic windows, inspect every node of $S$ with probability $b_1/|S|$ and target every component of $T$ with probability $b_2/|T|$ , and satisfy $F(S,\{e\})\le\frac1{b_1}\sum_kF(S^k,\{e\})$ .
Proposition 1: for a minimal set cover $S'$ and a set packing $T'$ with $|T'|\ge b_2$ ,

\max\Bigl\{0,b_2\Bigl(1-\frac{b_1}{|T'|}\Bigr)\Bigr\}=\min_{S\in\mathcal A_1}U(S,\sigma^2(T',b_2))\le U^*(b_1,b_2)\le\max_{T\in\mathcal A_2}U(\sigma^1(S',b_1),T)=b_2\Bigl(1-\frac{b_1}{|S'|}\Bigr).

Theorem 1: for $b_1<n^*$ , $b_2<m^*$ , every equilibrium uses positionings of size exactly $b_1$ and attack plans of size exactly $b_2$ , and the equilibrium strategies are the optimal solutions of the linear programs $(\overline{\mathrm{LP}}_1)$ , $(\overline{\mathrm{LP}}_2)$ over $\bar{\mathcal A}_1=\{|S|=b_1\}$ , $\bar{\mathcal A}_2=\{|T|=b_2\}$ .
Lemma EC.4: for $b_2<m^*$ , every $\rho\in[0,1]^{\mathcal E}$ with $\sum_e\rho_e=b_2$ is the vector of attack probabilities of some $\sigma^2\in\Delta(\mathcal A_2)$ .
Proof of Theorem 2, step (i.a): for $b_2=1$ , some optimal solution of $(\overline{\mathrm{LP}}_2)$ targets every component with probability at most $1/m^*$ .
(18): for $b_1<n^*$ and $1\le b_2<m^*$ , $U^*(b_1,b_2)=b_2\,U^*(b_1,1)$ , i.e. $U^*(b_1,b_2)=(1-r^*_{b_1})b_2$ .

Significance

The result. Theorem 2 separates the defender's problem from the attacker's resources. The defender may solve the game for a single attack resource, $\Gamma(b_1,1)$ , a linear program over attack plans of size one, and use the resulting inspection strategy against any attacker with fewer than $m^*$ resources; the guaranteed detection rate is the same. Together with Proposition 2 of the paper, this reduces the problem of choosing the fewest detectors that achieve a target detection rate to games with $b_2=1$ , and so to linear programs whose size does not grow with the attacker's resources. The linearity (18) of the value in $b_2$ is the quantitative form of the statement.

Formalizing it. The result is proved in the paper; it has no machine-checked proof. A formalization fixes the boundary conventions the paper leaves implicit (the empty attack plan in the rate, $b_2\ge1$ ), supplies the existence of equilibria of finite zero-sum games that the paper takes for granted, and makes explicit the cases $b_1\in\{0,1\}$ of step (i.a), which the printed argument does not cover.

Difficulty

The central difficulty is step (i.a): an optimal attack strategy of $\Gamma(b_1,1)$ may concentrate its probability on a few components. Scaling its attack probabilities by $b_2$ can exceed $1$ , so it need not define a strategy of $\Gamma(b_1,b_2)$ . The required conclusion is stronger than the existence of an equilibrium: an optimal single-attack strategy must exist whose probability on each component is at most $1/m^*$ .

Theorem 1 also requires strict use of both players’ resources in equilibrium, even though their action sets permit smaller plans. Without that result, the expected detection rate cannot be identified from the game value by dividing by $b_2$ . The marginal-realization claim in Lemma EC.4 must work for every vector in the stated polytope, including its boundary.

Formalization scope

Everything is finite and there is no measure theory. The game is an instance of the published finite zero-sum game BlottoLP.Flow.ZeroSum: the attacker is the row player over the subtype $\{T:|T|\le b_2\}$ , the defender the column player over $\{S:|S|\le b_1\}$ , mixed strategies are nonnegative real weight functions summing to $1$ , and payoffs and the rate are finite double sums. An inspection strategy therefore never uses more than $b_1$ detectors. $U^*$ is the optimal value of (LP1); $n^*$ and $m^*$ are the infimum and supremum of the cover and packing sizes, well defined because every component is monitored from some node.

Conventions fixed in Lean: the ratio $F(S,\emptyset)/|\emptyset|$ is $1$ (an empty attack plan counts as detected), which the case $b_1\ge n^*$ of Theorem 2 requires; $b_2\ge1$ is assumed wherever $b_2$ is quantified, since at $b_2=0$ every profile is an equilibrium; Lemma EC.3 assumes $b_1,b_2\ge1$ ; cyclic windows are indexed from $0$ . The standing assumptions $b_1<n^*$ , $b_2<m^*$ of §3.3 are hypotheses of each statement that uses them.

A trivializing formalization is ruled out: the equilibrium predicate is exactly (5) over the correct strategy sets, and a sanity check exhibits an equilibrium of $\Gamma(1,1)$ in a network with $n^*=m^*=2$ . The extrema used in the statements have nonempty, bounded domains under their hypotheses, and the empty attack plan has an explicit rate convention. When $m^*\le1$ the goal is vacuous, as in the paper.

Infrastructure needed: the minimax theorem for finite zero-sum games, Farkas' lemma (on the platform as LinearOptimization.farkas_lemma), and finite sums over subsets. The coverage-function facts, the cyclic strategies, and the marginal-realization lemma are reusable for other security and inspection games. Proofs of any milestone, and alternative proofs of step (i.a), are welcome.

Selected references

M. Dahan, L. Sela, S. Amin, Network Inspection for Detecting Strategic Attacks, Operations Research 70(2), 2022, https://doi.org/10.1287/opre.2021.2180; preprint arXiv:1705.00349v4, 2020. https://arxiv.org/abs/1705.00349
A. Washburn, K. Wood, Two-Person Zero-Sum Games for Network Interdiction, Operations Research 43(2), 1995. https://doi.org/10.1287/opre.43.2.243
M. Tambe, Security and Game Theory: Algorithms, Deployed Systems, Lessons Learned, Cambridge University Press, 2011. https://doi.org/10.1017/CBO9780511973031
R. J. Lipton, E. Markakis, A. Mehta, Playing Large Games Using Simple Strategies, ACM EC 2003. https://doi.org/10.1145/779928.779933

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Economics Operations Research·Captain: mikedeng1

Bank Competition and Risk-Taking Under Market Integration: Integration Lowers Loan Rates and Risk-Taking in the Borrower-Heavy Market, and a Strong Bank-Customer Effect Raises Them in the OtherResearch Paper

Motivation

Whether more competition among banks makes the financial system safer or more fragile is a long-running question in banking theory and regulation. The competition-fragility view holds that competition erodes bank margins and charter values and so encourages risk-taking; the competition-stability view, formalized by Boyd and De Nicolò (2005), holds that competition lowers loan rates, which reduces the incentive of borrowers to choose risky projects. Empirical studies of deregulation episodes, such as the removal of interstate branching restrictions in the United States, have found evidence for both views.

Dam and Sengupta (2024) point out that such episodes do not merely add competitors to a single market: they integrate previously separate markets with different customer compositions. In their model integration changes both the number of competing banks and the ratio of borrowers to depositors each bank faces, and the second channel can reverse the conventional prediction in one of the integrating markets. This mission formalizes that result, Proposition 2 of the paper, together with the equilibrium characterizations it rests on.

Setting

There are two banking markets $j=1,2$ . Market $j$ has $a_j>0$ depositors, $b_j>0$ borrowers (entrepreneurs) and $n\ge1$ banks, and $\lambda>1$ is a model constant. A borrower chooses the risk $\theta\in[0,\lambda]$ of a unit-size project, which returns $\theta$ with probability $1-\theta/\lambda$ and $0$ otherwise. At loan rate $r$ the borrower maximizes $(1-\theta/\lambda)(\theta-r)$ ; a maximizer is a borrower-optimal risk.

Inverse deposit supply and inverse loan demand in market $j$ are $R_j=D_j/a_j$ and $r_j=\lambda-L_j/b_j$ , where $D_j$ and $L_j$ are aggregate deposits and loans. Banks are financed by deposits only, so $D_j=L_j$ . Each bank $i$ chooses a loan volume $L_{ij}\ge0$ to maximize its expected profit, display (2),

\frac{L_j}{2\lambda b_j}\Big(\lambda-\frac{L_j}{b_j}-\frac{L_j}{a_j}\Big)L_{ij},\qquad L_j=\sum_i L_{ij}.

A Cournot equilibrium is a profile of volumes from which no bank gains by changing its own volume. The segmented market equilibrium (SME) of market $j$ is the Cournot equilibrium of this game; its loan rate is $r_j$ and its borrower-optimal risk $\theta_j$ .

The integrated market has $a_1+a_2$ depositors, $b_1+b_2$ borrowers and $2n$ banks, with profit (7) of the same form. Its Cournot equilibrium is the integrated market equilibrium (IME), with loan rate $r^*$ and risk $\theta^*$ . The customer composition of a market is the ratio of borrowers to depositors, $\xi_j=b_j/a_j$ and $\xi^*=(b_1+b_2)/(a_1+a_2)$ . Market 2 is labelled as the borrower-heavy one, $\xi_1<\xi_2$ .

The comparison is organized by the two terms of display (11):

C=-\lambda\cdot\frac{1}{1+\xi^*}\Big(\frac{2n+1}{2n+2}-\frac{n+1}{n+2}\Big),\qquad B_j=\lambda\cdot\frac{n+1}{n+2}\Big(\frac{1}{1+\xi_j}-\frac{1}{1+\xi^*}\Big),

the bank-competitor effect (more banks) and the bank-customer effect with respect to market $j$ (a different customer composition).

Formalization targets

Goal: Proposition 2

If $\xi_1<\xi_2$ , then for all Cournot equilibria of the three markets and all borrower-optimal risks at their loan rates,

\text{(a)}\quad r^*<r_2,\ \ \theta^*<\theta_2;\qquad \text{(b-i)}\quad B_1<-C\ \Rightarrow\ r^*<r_1,\ \theta^*<\theta_1;\qquad \text{(b-ii)}\quad B_1>-C\ \Rightarrow\ r^*>r_1,\ \theta^*>\theta_1 .

Milestones

Equation (1): the borrower-optimal risk is unique and equals $\theta(r)=(\lambda+r)/2$ .
Lemma 6(a): with marginal cost $c$ the Cournot game has a unique equilibrium, symmetric, with deposit rate, loan rate and risk (A.4)–(A.6).
Lemma 1: the SME, with (3)–(5).
Lemma 2: the IME, with (8)–(10), and $\xi_1<\xi^*<\xi_2$ .
Display (11): $r^*-r_j=C+B_j$ for $j=1,2$ .
The bank-competitor effect is negative, $C<0$ .
The bank-customer effects have opposite signs, $B_2<0<B_1$ .

Significance

Proposition 2 shows that a single deregulation episode can lower loan rates and borrower risk-taking in one integrating market while raising them in the other, even though the number of competing banks doubles. Its consequence for empirical work is that the competition-fragility and competition-stability views can both be consistent with the same episode, depending on each market's initial customer composition; it also separates within-market competition (captured by $C$ , which reproduces the classical Boyd–De Nicolò prediction) from across-market competition. Corollary 1 and the bank-merger analysis of §3 build on the same equilibria.

The result is proved in the paper; to our knowledge it has no machine-checked proof. The formal work this mission adds is the Cournot layer: the paper's proof of Lemma 6 argues only through the first-order condition, whereas the formal statements require that the closed-form profile is a global best response for every bank and that no other equilibrium, symmetric or not, exists.

Difficulty

The algebra of (11) and of the sign comparisons is elementary once the equilibrium rates are known. The substantive step is the equilibrium characterization. A bank's profit is a cubic in its own volume, so the first-order condition is necessary only at an interior choice and is not sufficient. A complete argument must rule out corner equilibria in which some banks lend nothing, show that every equilibrium is symmetric, and show that the stationary point is a global maximum over all nonnegative volumes. The integrated market has $2n$ banks indexed by Fin (2 * n), so the cast from $2n$ to the real number $2n$ enters every closed form.

Formalization scope

All quantities are real numbers, the number of banks is a natural number, and banks are indexed by Fin n (segmented) and Fin (2 * n) (integrated). A Cournot equilibrium is the published feasible pure-strategy Nash equilibrium CarbonDoubleCount.Planner.IsPureNash, with feasible set $[0,\infty)$ for each bank and payoff given by display (2)/(7). The loan and deposit rates are functions of the aggregate equilibrium volume. A borrower-optimal risk is a maximizer of $(1-\theta/\lambda)(\theta-r)$ over $[0,\lambda]$ ; it is never defined by a closed form. The standing assumptions $\lambda>1$ , $a_j,b_j>0$ , $n\ge1$ of p. 6 are bundled in one predicate, and $\xi_1<\xi_2$ is a separate hypothesis.

Added or chosen conventions: the volumes are nonnegative; equation (1) is stated for $-\lambda\le r\le\lambda$ ; Lemma 6(a) assumes $c<\lambda$ and is stated for $m\ge1$ banks (the appendix prints $n\ge2$ , while Lemma 1 needs $n\ge1$ ); "sufficiently weak/strong" in Proposition 2 is the strict comparison of $B_1$ with $-C$ , and the tie $B_1=-C$ is not covered, as on the page.

A trivializing formalization would define $r_j$ , $r^*$ , $\theta_j$ , $\theta^*$ by the closed forms (4), (5), (9), (10), which reduces the goal to arithmetic; the goal instead quantifies over all Cournot equilibria and all borrower-optimal risks, and the weak/strong conditions are stated through the defined effect terms, not through $r^*$ and $r_1$ . All hypotheses are satisfiable: the parameters $\lambda=3/2$ , $n=3$ , $a_1=a_2=4$ , $b_1=1$ , $b_2=6/5$ satisfy the premise of (b-i), and $\lambda=2$ , $n=1$ , $a_1=a_2=1$ , $b_1=1$ , $b_2=2$ that of (b-ii).

Contributions welcome: proofs of the milestones, in particular a reusable characterization of symmetric Cournot equilibria for cubic profits of this form, which also covers the merger analysis of §3.

Selected references

K. Dam and R. Sengupta, Bank competition and risk-taking under market integration, Management Science, 2024 (doi:10.1287/mnsc.2022.03495); cited here from Federal Reserve Bank of Kansas City Research Working Paper RWP 20-21, updated October 2024. https://doi.org/10.18651/RWP2020-21
J. H. Boyd and G. De Nicolò, The theory of bank risk taking and competition revisited, Journal of Finance 60(3), 2005. https://doi.org/10.1111/j.1540-6261.2005.00763.x
J. E. Stiglitz and A. Weiss, Credit rationing in markets with imperfect information, American Economic Review 71(3), 1981. https://www.jstor.org/stable/1802787
F. Allen and D. Gale, Competition and financial stability, Journal of Money, Credit and Banking 36(3), 2004. https://doi.org/10.1353/mcb.2004.0038

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Algorithmic Game Theory Dynamic Programming Economics·Captain: mikedeng1

Asset Dissemination through Dealer Markets: The Strategies Built from the Collected Demand Function Form a Rank-Free EquilibriumResearch Paper

Motivation

Many assets, such as corporate and municipal bonds, securitized products and new issues, trade over the counter: there is no central exchange, and an asset reaches its final buyers through a sequence of bilateral trades between dealers. Empirical work on these markets documents intermediation chains, in which a block of securities passes through several dealers, each of whom keeps part of it for her own customers and sells the rest onwards: in US municipal bonds, Li and Schürhoff (2019) find that chains account for 82% of the trades that start with a customer selling to a dealer, and in registered securitizations Hollifield, Neklyudov and Spatt (2017) find that about 27% of customer sales give rise to a sequence involving more than one dealer (figures as reported by Colliard and Demange, p. 1). Colliard and Demange (SSRN 3244321, forthcoming in Management Science) build a model of this process in which the length of the chain, the prices and the quantities are all equilibrium outcomes, and the friction that makes the chain finite is asymmetric information about how much each dealer's customers want to buy.

The equilibrium of that model rests on one object, the collected demand $\Omega$ , defined by a functional equation. This mission formalizes the existence and properties of $\Omega$ and the theorem that the strategies built from it form an equilibrium of the dealer game.

Setting

An asset pays a sure return $\rho>0$ per unit. Dealers $D_1,D_2,\dots$ arrive in sequence. Each is available to trade with probability $q\in(0,1)$ , independently. Dealer $D_n$ 's customers buy up to $v^C_n$ units at price $\rho$ ; the customer demand $\omega_n=\rho v^C_n$ is private information of $D_n$ and is drawn i.i.d. from a distribution $G$ on $\mathbb R_+$ with finite mean. Write $H(\omega)=\Pr(\tilde\omega\ge\omega)$ . $G$ may have atoms. Dealers borrow cash at the funding spread $r>0$ .

A dealer of type $\omega$ who accepts a take-it-or-leave-it offer $(p,v)$ (volume $v$ at unit price $p$ ) sells to her customers and has financing needs $y=\max(pv-\omega,0)$ , which she covers by borrowing or by making a new offer $(p_n,v_n)$ to the next dealer, with $0\le v_n\le v$ . If the next dealer accepts with probability $\Phi(p_n,v_n)$ , her expected profit (5) is

\pi=(\rho-p)v-ry+q\,\Phi(p_n,v_n)\bigl[r\min(p_nv_n,y)-(\rho-p_n)v_n\bigr].

A rank-free equilibrium (Definition 1) is a strategy (an acceptance rule $A(p,v,\omega)$ and a new-offer rule $(P,V)$ ) that does not depend on the dealer's position in the chain, such that $\Phi$ is the acceptance probability induced by $A$ , the new offer maximizes $\pi$ over all feasible offers, and a dealer accepts exactly when this maximal profit is non-negative.

The collected demand is the function $\Omega$ on $\mathbb R_+$ solving

\Omega(0)=0,\qquad \Omega(y)=\sup_{0\le\omega\le y} qH(\omega)\bigl(\omega+\Omega(y-\omega)\bigr)\quad(y>0).\tag{9}

The maximizers form the set $\mathcal T(y)$ of targeted values (10); a selection is a function $T$ with $T(y)\in\mathcal T(y)$ for all $y\ge0$ .

Formalization targets

Goal: Theorem 2

For $\Omega$ bounded, continuous and solving (9), and any selection $T$ , put $\pi^*(p,v,y)=(\rho-p)v-ry+r\Omega(y)$ . The strategy "accept $(p,v)$ iff $\pi^*(p,v,y)\ge0$ ; on acceptance with $y>0$ offer $(P(y),V(y))$ with

P(y)V(y)=y,\qquad (\rho-P(y))V(y)-ry+r\bigl(T(y)+\Omega(y-T(y))\bigr)=0\tag{14}

(and no offer when $y=0$ )" is a rank-free equilibrium. Moreover $P(y)\le\rho$ , $0\le V(y)\le v-\omega/\rho$ , the offer's expected transaction payoff is $r\Omega(y)$ , and the dealer's profit is $\pi^*(p,v,y)$ .

Milestones

$\Psi(\beta)(y)=\max_{0\le\omega\le y}qH(\omega)(\omega+\beta(y-\omega))$ maps bounded continuous non-negative $\beta$ with $\beta(0)=0$ to such functions and is a $q$ -contraction in sup-norm (proof of Theorem 1).
Theorem 1: a unique bounded continuous solution of (9) exists; the supremum in (9) is reached; $\Omega$ is $q$ -Lipschitz and non-decreasing; hence $\Omega(z)\le qz$ and $z\mapsto z-\Omega(z)$ is non-negative and increasing.
Lemma A.2: for $f$ non-decreasing and $q$ -Lipschitz on $\mathbb R_+$ , $\min(pv,y)+f(pv-\omega)\le pv+f(y-\omega)$ whenever $\omega\le pv$ , $\omega\le y$ .
The three points of the proof of Theorem 2: the offer is feasible; no offer yields a transaction payoff above $r\Omega(y)$ ; the offer $(P(y),V(y))$ is accepted exactly by types $\omega\ge T(y)$ and achieves $r\Omega(y)$ .

Significance

Theorem 2 is the existence result on which the rest of the paper is built. The comparative statics of dissemination length, customer purchases and prices along the chain (Propositions 1–4, Implications 1–6) are all statements about this equilibrium. It shows that the infinite-horizon screening problem of each dealer, against a chain of unknown length, reduces to the one-dimensional equation (9), and that the reduction is exact: the expected transaction payoff from the new offer is $r\Omega(y)$ .

The paper proves Theorem 2 in Appendix A; Lemma A.2 is proved in an Online Appendix that is not part of the source. To the knowledge of this mission, none of these results has a machine-checked proof. The mission's output is a checked existence-and-optimality argument for a dynamic screening game with a discontinuous survival function, together with the fixed-point theory of the operator $\Psi$ , which is not an expectation operator and does not fall under standard Bellman-operator results.

Difficulty

The operator $\Psi$ takes a supremum over the moving interval $[0,y]$ of a product with the survival function $H$ , which jumps downward at every atom of $G$ . Standard dynamic-programming theorems assume a continuous kernel or a finite state space; here attainment of the supremum and continuity of $\Psi(\beta)$ rely on $H$ being left-continuous and non-increasing, and on $\omega+\beta(y-\omega)\ge0$ . In the equilibrium part, the natural first step, verifying optimality only against the strategy's own offer or against offers accepted with a fixed probability, does not work: a deviation changes the set of types that accept it, so the acceptance probability must be recomputed from the acceptance rule for every candidate offer. The deviation can change both its unit margin and its probability of acceptance, making a uniform payoff bound the central difficulty.

Formalization scope

All objects live in the namespace DealerDissem.RankFree, in one definitions file. Parameters $\rho,r,q$ are bundled with $\rho>0$ , $r>0$ , $0<q<1$ . $G$ is a Measure ℝ that is a probability measure, gives no mass to $(-\infty,0)$ and has an integrable identity; $H(\omega)$ is the measure of $[\omega,\infty)$ (closed, so acceptance probabilities are exact at atoms). $\Omega$ , $T$ , $P$ , $V$ are functions ℝ → ℝ; every hypothesis and conclusion about them is restricted to $[0,\infty)$ , and uniqueness in Theorem 1 is equality on $[0,\infty)$ . Suprema in (9) and (46) range over $0\le\omega\le y$ . Acceptance rules are predicates $A(p,v,\omega)$ , and $\Phi$ is computed as the $G$ -measure of the acceptance set. $V(y)$ is (14) solved for $V$ (the paper's (16)), and $P(y)=y/V(y)$ ; at $y=0$ , $V=0$ and Lean's value $P(0)=0/0=0$ is irrelevant.

Definition 1 (ii) is encoded as: after acceptance the strategy's offer is feasible and maximizes (5) over every feasible offer; and acceptance holds iff some feasible offer gives a non-negative profit. The equilibrium predicate requires measurable acceptance events for feasible offers, so the quantity called a probability in Definition 1 (i) has that meaning. Added hypotheses, each where the printed sentence is false otherwise: $v>0$ in "this case surely occurs for $p>\rho$ "; $y>0$ for the bound $0\le V(y)\le v-\omega/\rho$ ; non-negative $\beta$ in the contraction milestone.

A trivializing formalization is ruled out: $\Phi$ is never a free parameter, optimality is against all feasible offers rather than the strategy's own, $\Omega$ must solve (9) at every $y>0$ , and $T(y)$ must be a maximizer, not merely a point of $[0,y]$ .

A complete development needs suprema over compact intervals of upper semicontinuous functions, the Banach fixed-point theorem on bounded continuous functions on $\mathbb R_+$ , and Markov's inequality for $\omega H(\omega)\le\mathbb E\tilde\omega$ . The fixed-point machinery for sup-over-interval operators is reusable for other sequential search and screening models. Proofs of any milestone are welcome, as is a proof of Theorem 2 from the milestones.

Selected references

J.-E. Colliard and G. Demange, Asset Dissemination through Dealer Markets, Management Science (forthcoming; authors' version of September 17, 2020), SSRN 3244321. https://ssrn.com/abstract=3244321 — published version: https://doi.org/10.1287/mnsc.2020.3779
D. Li and N. Schürhoff, Dealer Networks, The Journal of Finance 74(1), 91–144, 2019. https://doi.org/10.1111/jofi.12728
B. Hollifield, A. Neklyudov and C. Spatt, Bid-Ask Spreads, Trading Networks, and the Pricing of Securitizations, The Review of Financial Studies 30(9), 2017. https://doi.org/10.1093/rfs/hhx027
N. L. Stokey and R. E. Lucas, with E. C. Prescott, Recursive Methods in Economic Dynamics, Harvard University Press, 1989 (Theorem 3.3, Blackwell's sufficient conditions). https://www.hup.harvard.edu/books/9780674750968

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Algorithmic Game Theory Mechanism Design Operations Research·Captain: mikedeng1

The Design of Optimal Pay-as-Bid Procurement Mechanisms: Every Optimal Equitable Assortment Mechanism Is Matched by a k-Soft-Reserve Rule in the Symmetric Two-Point Cost ModelResearch Paper

Motivation

Public procurement agencies and group-buying platforms increasingly run framework agreements: suppliers post prices, and the agency then decides which suppliers' products appear in the catalogue from which buyers choose. Chile's ChileCompra framework agreements and drug formularies work this way. The agency faces a trade-off. Showing many products serves buyers with heterogeneous tastes. Showing few products makes suppliers compete harder on price.

Saban and Weintraub (2021, Operations Research; SSRN 3453144) formulated this as a mechanism design problem and solved it when the agency may use arbitrary payments. In practice, payments are pay-as-bid: a supplier included in the catalogue is paid the price it posted. In addition, the selection rule must be anonymous (it may depend on prices, not on identities) and monotone (a price cut never hurts the supplier who makes it). Choi, Saban and Weintraub (2022; SSRN 3785023) ask which selection rule maximizes expected consumer surplus under these constraints. Their answer, for symmetric suppliers with two possible cost levels, is a simple three-parameter rule: the k-soft reserve.

Setting

There are $n$ suppliers $N=\{1,\dots,n\}$ . Each supplier's unit cost is $\theta_L$ with probability $f_L$ and $\theta_H>\theta_L>0$ with probability $f_H=1-f_L$ , independently, with $0<f_L<1$ .

Demand. Buyers are a single representative consumer with consumer surplus

CS(x,p)=c'x-\tfrac12x'Dx-p'x,\qquad D=\Gamma^{-1},\ c=\Gamma^{-1}\alpha,

where $x$ is the vector of market shares and $p$ the vector of prices. In the symmetric affine model (Model 2), $\alpha=(a,\dots,a)$ and $\Gamma$ has $r_o>0$ on the diagonal and $-r_c\le0$ off it, and $\Gamma$ is positive definite. Given a nonempty assortment $Q\subseteq N$ and prices $p\ge0$ , the demand $d(Q,p)$ maximizes $CS(\cdot,p)$ over $x\ge0$ with $\sum_ix_i=1$ and $x_i=0$ for $i\notin Q$ (Problem (1)). Total demand is one: there is no outside option.

Mechanisms. An assortment function $q:\mathbb R^n_+\to\{0,1\}^n$ includes supplier $i$ when $q_i(b)=1$ . A reserve price $r\ge\theta_H$ forces $q_i(b)=0$ whenever $b_i>r$ . The rule is equitable if it is anonymous, $q(P_\pi b)=P_\pi q(b)$ , and monotone: lowering $b_i$ weakly raises $q_i$ and weakly lowers every other $q_j$ . The allocation is $x^q(b)=d(q(b),b)$ .

The k-soft-reserve rule $q^{k,s,r}$ excludes supplier $i$ if $b_i>r$ , or if $s<b_i\le r$ and at least $k$ suppliers price at most $s$ ; otherwise it includes $i$ . With $k=n$ it is the no-competition rule, which includes every supplier priced at most $r$ .

Problem (8). Suppliers play a symmetric Bayes–Nash pricing strategy $(b_L,b_H)$ . The agency chooses $(q,r,b_L,b_H)$ to maximize

ECS(q,b_L,b_H)=\mathbb E_\theta\big[CS(x^q(\beta(\theta)),\beta(\theta))\big]

subject to: neither type gains by deviating to any price in $\mathbb R_+$ (BN-L, BN-H); $b_L\ge\theta_L$ and $b_H\ge\theta_H$ (IR); $q$ equitable (AM); and the reserve constraints RP(1), RP(2).

Formalization targets

Goal: Proposition 1 (p. 17)

For every optimal solution $(q^*,r^*,b^*_L,b^*_H)$ of Problem (8) there is $k\in\{1,\dots,n\}$ such that

(q^{k,b^*_L,b^*_H},\ b^*_H,\ b^*_L,\ b^*_H)\ \text{is optimal for Problem (8)}\quad\text{and}\quad ECS(q^{k,b^*_L,b^*_H},b^*_L,b^*_H)=ECS(q^*,b^*_L,b^*_H).

The goal does not fix the optimal $k$ , $b^*_L$ or $b^*_H$ ; those depend on all the parameters, and the paper computes them numerically.

Milestones, in attack order

Problem (1) is well posed (p. 9): for nonempty $Q$ and $p\ge0$ the demand exists and is unique.
Lemma 1 (p. A1): an equitable rule includes everyone at equal prices $\le r$ , and always includes a lowest-priced supplier with price $\le r$ .
Lemma 2 (p. A1): Model 2 is equivalent to the model with $\alpha=0$ and $\Gamma=\mathrm{diag}(1/\delta)$ , $\delta=1/(r_o+r_c)$ . Demands coincide and surpluses differ by a constant on the simplex.
Allocation properties (remark closing App. A.1, p. A3): under an equitable rule, allocations are permutation-equivariant and rise when one's own price falls.
Assortment monotonicity of demand (step (9) of Lemma 4's proof, p. A3): enlarging the assortment weakly lowers each included supplier's demand.
Lemma 3 (p. A2): $b^*_L\le b^*_H$ at every optimum.
Lemma 4 (p. A2): the goal in the reduced model $\alpha=0$ , $\Gamma=\mathrm{diag}(1/\delta)$ .

Significance

The result. Problem (8) is an optimization over all equitable assortment functions, an infinite-dimensional space of $\{0,1\}^n$ -valued maps, coupled to equilibrium constraints. The problem is a mixed-integer and non-convex program (p. 14). Proposition 1 collapses it to a search over three numbers, $(k,s,r)$ . The paper's numerical study of the best soft-reserve rule, and its comparison with the centralized benchmark of Saban and Weintraub (2021) (Proposition 3), both rest on this reduction. The companion Proposition 2 ( $r^*=\theta_H$ ) narrows the search further.

Formalizing it. The result is proved in the paper; this mission produces the first machine-checked version. The proof is short but informal at several points that a formalization must make precise. The interim utility is written by anonymity in a binomial form that presupposes anonymity of the demand. Two mechanisms are claimed "identical in equilibrium" when one of them keeps suppliers with zero demand that the other excludes. Lemma 2's displayed inverse of $\Gamma$ swaps its diagonal and off-diagonal entries. No proof exists in Lean or any other proof assistant.

Difficulty

The obvious argument copies $q^*$ on the equilibrium path and makes it as harsh as possible off the path. The difficulty is that off-path behaviour is exactly what sustains equilibrium. One must show that replacing $q^*$ by a $k$ -soft reserve never makes a deviation more attractive, for every deviation price and every number of low-type opponents. This requires comparing demands under two different assortments at the same prices. That comparison holds only because demand in this model is monotone in the assortment and in prices. Neither property is automatic for a general quadratic program; both rest on the symmetric structure of $\Gamma$ . The choice of $k$ is the minimal number of low-type suppliers at which high types receive zero demand, and showing that it is well defined uses monotonicity again.

Formalization scope

All objects live in the namespace PayAsBid.SoftReserve and are defined in one definition file. Suppliers are Fin n (0-indexed), prices are real vectors, types are Bool (true is $\theta_L$ ), and inclusion is Bool with false < true. Demand is a chosen maximizer of Problem (1), and 0 when Problem (1) has no feasible point. Expected values are finite sums over the $2^n$ type profiles. The interim utility (5) sums over full profiles with the deviator's own price replaced; this equals the paper's sum over opponents' types.

Conventions and additions to the page, each disclosed in the item where it occurs:

Standing assumptions: $n\ge1$ , $r_o>0$ , $r_c\ge0$ , $\Gamma$ positive definite, $0<\theta_L<\theta_H$ , $0<f_L<1$ .
RP(2) is $r\ge\theta_H$ , since $\min_i\max\Theta_i=\theta_H$ .
Individual rationality is Problem (8)'s $b_L\ge\theta_L$ , $b_H\ge\theta_H$ .
Domain of Problem (1): feasibility requires the assortment to be nonempty whenever some price is at most $r$ ("total demand adds up to one", p. 8) and on every equilibrium profile ((Demand) of Problem (6)).
$k$ ranges over $1,\dots,n$ , and Definition 5's $s<r$ is not imposed: when $b^*_L=b^*_H$ every soft reserve is the no-competition rule, which p. 17 lists as the second optimal form.

The domain conditions rule out a trivializing reading. Without them, a mechanism that excluded everybody on some profile would earn the junk surplus $CS(0,p)=0$ , which exceeds every genuine surplus in the reduced model, and the goal would fail. The hypothesis of the goal is optimality, not feasibility, as on the page. A companion verification file proves that an optimum exists: for $n=1$ the no-competition rule with $r=b_L=b_H=\theta_H$ is optimal. The hypothesis is therefore not vacuous.

A complete development needs:

existence and uniqueness for strictly concave quadratic programs on a face of the simplex;
the Sherman–Morrison inverse of $\Gamma$ ;
monotone comparative statics of the simplex-constrained demand (water-filling);
finite-sum manipulations over type profiles.

The demand lemmas (milestones 1, 3, 5) are reusable for any assortment problem with affine demand and no outside option. Contributions are welcome on every milestone; Lemmas 1 and 3 are self-contained, and milestones 4–5 are the analytic core.

Selected references

J. Choi, D. Saban, G. Weintraub, The Design of Optimal Pay-as-Bid Procurement Mechanisms, Manufacturing & Service Operations Management, 2022. SSRN preprint 3785023 (version of 2022-12-08), https://ssrn.com/abstract=3785023; https://doi.org/10.1287/msom.2022.1180
D. Saban, G. Weintraub, Procurement Mechanisms for Assortments of Differentiated Products, Operations Research 69(3):795–820, 2021. https://ssrn.com/abstract=3453144
A. Farahat, G. Perakis, A Nonnegative Extension of the Affine Demand Function and Equilibrium Analysis for Multiproduct Price Competition, Operations Research Letters 38(4):280–286, 2010.

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Operations Research Probability Stochastic Systems·Captain: mikedeng1

ε-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations: The Last Record-Breaking Dyadic Level Has Finite Exponential Moments and Bounds the Uniform ErrorResearch Paper

Motivation

Fractional Brownian motion (fBM) is the standard Gaussian model of long-range dependence and rough paths. It appears in network traffic, finance (rough volatility), hydrology and queueing, and it drives a class of stochastic differential equations whose solutions are defined pathwise by Young integration. Simulating such models needs sample paths of fBM, and an error guarantee in the strongest sense: a path that lies within a prescribed distance $\varepsilon$ of the true one, with probability one, on a finite computer. This guarantee is $\varepsilon$ -strong simulation.

The concept was introduced for Brownian motion by Beskos, Peluchetti and Roberts (Bernoulli 2012) and extended to reflected Brownian motion by Blanchet and Chen (AAP 2015) and to multidimensional Brownian-driven SDEs by Blanchet, Chen and Dong (AAP 2017), whose "record-breaker" strategy this paper adopts. For $H\ne 1/2$ the coefficients of Lévy's midpoint-displacement expansion are correlated and the earlier arguments do not apply. Chen, Dong and Ni (arXiv:1902.07824, 2019; Mathematics of Operations Research, 2021) extend the method to fBM of every Hurst index. The central fact is that, along the dyadic midpoint-displacement expansion, only finitely many levels "break a record", the last such level has finite exponential moments, and beyond it the remaining error is controlled deterministically. This mission formalizes that fact.

Setting

Fix a probability space $(\Omega,\mathcal F,\mathbb P)$ and a Hurst index $H\in(0,1)$ . A fractional Brownian motion on $[0,1]$ is a family of random variables $\{B(t)\}_{t\in[0,1]}$ that is a centered Gaussian process with covariance (1.1)

r(s,t)=\mathbb E[B(s)B(t)]=\tfrac12\big(|s|^{2H}+|t|^{2H}-|s-t|^{2H}\big).

At $H=1/2$ this is $\min(s,t)$ , Brownian motion.

The dyadic grid of level $n$ is $D_n=\{t^n_i=i/2^n:\ i=0,\dots,2^n\}$ . The dyadic discretization $B_n$ is the piecewise-linear interpolation of the values $B(t^n_0),\dots,B(t^n_{2^n})$ . Passing from level $k-1$ to level $k$ adds the midpoints $t^k_{2j+1}$ , and the midpoint displacement at the new point is $a^k_j-b^k_j$ , where (3.1)

a^k_j=B(t^k_{2j+1}),\qquad b^k_j=\tfrac12\big(B(t^{k-1}_j)+B(t^{k-1}_{j+1})\big),\qquad j=0,\dots,2^{k-1}-1 .

Fix $\delta\in(0,H)$ and $\rho>0$ , and put $\ell_k=2^{-(H-\delta)k}$ . A record is broken at level $k\ge1$ if (3.11)

\max_{0\le j\le 2^{k-1}-1}|a^k_j-b^k_j|\ \ge\ \rho\,\ell_k ,

and $N=\sup\{k\ge1:\text{a record is broken at level }k\}$ is the last record level ( $\sup\emptyset=0$ , and $N=\infty$ if records occur at infinitely many levels). Theorem 3.1 also uses $K(\nu)=\sup\{n\ge1:4\sqrt n>\nu\,2^{\delta n}\}$ (3.2).

Formalization targets

Goal: Theorem 3.6 (p. 12)

Assume the paths of $B$ are almost surely continuous on $[0,1]$ . For every $\delta\in(0,H)$ , $\rho>0$ and $t>0$ ,

\mathbb E\big[e^{tN}\big]<\infty,

and almost surely, for every $n>N$ ,

\sup_{s\in[0,1]}|B(s)-B_n(s)|\ \le\ \frac{\rho\cdot2^{-(H-\delta)(n+1)}}{1-2^{-(H-\delta)}} .

Milestones

Sup-norm identity (display after (3.1), p. 8): $\|B_k-B_{k-1}\|_\infty=\max_j|a^k_j-b^k_j|$ , for every path.
Lemma 3.2 (p. 9): the conditional variance of $a^k_j$ given the values on $D_{k-1}$ is at most $2\cdot2^{-2kH}$ .
Borell–TIS inequality (finite form, cited at (3.6), p. 9): for a centered Gaussian vector with coordinate variances $\le\sigma^2$ , $\mathbb P(\max_i|X_i|-\mathbb E\max_i|X_i|>u)\le e^{-u^2/(2\sigma^2)}$ .
Lemma 3.3 (p. 10): $\mathbb E[\max_{i\le n}|X_i|]\le2\sqrt{2\log(2n)}\,\sigma$ for centered Gaussians with variances $\le\sigma^2$ .
Lemma 3.4 (p. 11): $\operatorname{Var}(c^k_j-b^k_j)\le2\cdot2^{-2kH}$ , with $c^k_j=\mathbb E[a^k_j\mid B^H_{k-1}]$ .
Theorem 3.1 (p. 8): for $\nu,\nu^*>0$ , $\rho=2(\nu+\nu^*)$ and $k>K(\nu)$ ,

\mathbb P\Big(\max_j|a^k_j-b^k_j|\ge\rho\ell_k\Big)\le2\exp\{-\nu^{*2}\cdot2^{2k\delta-2}\}.

$N<\infty$ almost surely (proof of Theorem 3.5, pp. 11–12).
Theorem 3.5 (p. 11): $B_n$ converges almost surely, uniformly on $[0,1]$ , to a continuous fBM that is a modification of $B$ .

A companion statement after the goal records the truncation rule of Algorithm 1 (p. 14): with $N(\varepsilon)=\max\{N,\lceil\log_2(\rho(\varepsilon(1-2^{-(H-\delta)}))^{-1})/(H-\delta)\rceil\}$ , $\|B-B_{N(\varepsilon)}\|_\infty\le\varepsilon$ almost surely.

Significance

The result. Theorem 3.6 supplies both conditions of the record-breaker scheme: the last record level is finite with a light tail (so an algorithm that searches for it terminates with finite expected effort), and beyond it the truncation error is bounded by an explicit, deterministic quantity. Together they give an exact $\varepsilon$ -accurate path in finite time, the basis of the paper's simulation algorithm for fBM and, through the Hölder-norm analogue (Theorem 3.8), of its Euler scheme for fBM-driven Young SDEs and its unbiased multilevel Monte Carlo estimators. Theorem 3.5 is the almost-sure validity of Lévy's midpoint-displacement series for fBM with correlated coefficients.

Formalizing it. The results are proved in the paper; none of them is machine-checked. The mission produces a formal model of fBM on $[0,1]$ , its dyadic discretizations and the record-breaker level, together with two general Gaussian tools that are absent from Mathlib: the Borell–TIS inequality for finitely many correlated Gaussians and an upper bound on the expected maximum of Gaussians. Both are reusable well beyond this paper (suprema of Gaussian processes, chaining, high-dimensional statistics).

Difficulty

For Brownian motion the midpoint displacements are independent of the coarser levels and of each other, and every step is a computation with independent normal variables. For $H\ne1/2$ that independence is lost: given the values on $D_{k-1}$ , the new value $a^k_j$ is neither centered at $b^k_j$ nor independent of the other new values, and the bounds must hold for a maximum over $2^{k-1}$ correlated coordinates with constants uniform in $k$ . Two ingredients carry the weight, and Mathlib has neither:

Gaussian concentration of the maximum. The Borell–TIS inequality for correlated Gaussians, and the Gaussian log-Sobolev or Poincaré machinery that usually proves it.
Gaussian conditioning. For $H\ne1/2$ the values at the new midpoints depend on all values at the previous level, not only on the two neighbours, so the conditional variance and the bias of Lemmas 3.2 and 3.4 are statements about Gaussian conditional expectations, for which Mathlib has no regression formula.

Beyond these, Theorem 3.5 requires constructing the limit process and identifying its finite-dimensional laws, and the exponential moment in Theorem 3.6 requires a tail for $N$ that decays faster than any exponential.

Formalization scope

Model. $B:\mathbb R\to\Omega\to\mathbb R$ , of which only the times in $[0,1]$ are constrained. The predicate "fBM with Hurst index $H$ " requires each $B(t)$ to be measurable, $\{B(t)\}_{t\in[0,1]}$ to be a Gaussian process (Mathlib's IsGaussianProcess), $\mathbb E[B(t)]=0$ and $\operatorname{Cov}(B(s),B(t))=r(s,t)$ exactly as in (1.1). Everywhere $0<H<1$ and $0<\delta<H$ .
Continuity. The paper fixes the continuous modification of fBM once (p. 5). The goal and the companion assume almost-sure continuity of the paths on $[0,1]$ ; Theorem 3.5 assumes nothing about paths and constructs the continuous limit.
The last record level is valued in $\mathbb N\cup\{\infty\}$ , and $e^{tN}$ in $[0,\infty]$ with $e^{t\cdot\infty}=\infty$ . A natural-number supremum, which Lean sets to $0$ on unbounded sets, would make the exponential moment trivially finite on exactly the bad paths; this encoding rules that out.
$\rho$ . Theorem 3.1 has $\rho=2(\nu+\nu^*)$ ; the goal admits every $\rho>0$ , which is the same family.
Conditioning on $B^H_{k-1}$ is conditioning on the $\sigma$ -algebra generated by the values on $D_{k-1}$ ; Lemma 3.2 uses Mathlib's conditional variance (an almost-sure statement), and Lemma 3.4 states the diagonal of $\Sigma^{(k)}$ as $\operatorname{Var}(c^k_j-b^k_j)$ , which is how p. 11 describes it.
Sup norms are stated pointwise for every $s\in[0,1]$ , or as an attained maximum, never as a real supremum.
Printed constants are kept: $2\cdot2^{-2kH}$ (Lemmas 3.2, 3.4), $2\sqrt{2\log(2n)}\,\sigma$ (Lemma 3.3, natural logarithm), $2\exp\{-\nu^{*2}2^{2k\delta-2}\}$ (Theorem 3.1). The paper's display (3.6) contains a sign slip in the exponent; the Borell–TIS milestone states the inequality the citation names.

A formalization in which $N$ is defined as the level beyond which the error bound already holds, or in which the absence of records beyond $N$ is assumed, would make Theorem 3.6 definitional; $N$ here is computed from the path as the supremum of the record set.

Out of scope: the algorithms that sample $N$ (Algorithms 2–5, Theorems 4.6–4.8), the Hölder-norm results (Lemma 3.7, Theorem 3.8) and the SDE section. Contributions of general Gaussian infrastructure (Gaussian conditioning, Borell–TIS, expected-maximum bounds) are especially welcome.

Selected references

Y. Chen, J. Dong, H. Ni, ε-Strong simulation of fractional Brownian motion and related stochastic differential equations, arXiv:1902.07824v1, 2019; Mathematics of Operations Research, 2021. https://arxiv.org/abs/1902.07824
J. Blanchet, X. Chen, Steady-state simulation for reflected Brownian motion and related networks, Annals of Applied Probability 25(6), 2015. https://doi.org/10.1214/14-AAP1072
A. Beskos, S. Peluchetti, G. Roberts, ε-Strong simulation of the Brownian path, Bernoulli 18(4), 2012. https://doi.org/10.3150/11-BEJ383
J. Blanchet, X. Chen, J. Dong, ε-Strong simulation for multidimensional stochastic differential equations via rough path analysis, Annals of Applied Probability 27(1), 2017. https://doi.org/10.1214/16-AAP1214
R. J. Adler, J. E. Taylor, Random Fields and Geometry, Springer, 2007 (Borell–TIS inequality, Theorem 2.1.1). https://doi.org/10.1007/978-0-387-48116-6
B. B. Mandelbrot, J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review 10(4), 1968. https://doi.org/10.1137/1010093

10 thms1 active userReviewed

Algorithmic Game Theory Economics Mechanism Design+1·Captain: mikedeng1

Optimal Budget Aggregation with Star-Shaped Preference Domains: With Leontief Utilities, the Nash Product Rule Is the Only Continuous Group-Strategyproof Mechanism Satisfying Core Fair ShareResearch Paper

Motivation

In budget aggregation, a group has to split a fixed public budget among several projects, and each member proposes her own ideal split. A rule is needed to turn the proposals into one split. The question was studied for $\ell_1$ preferences by Freeman, Pennock, Peters and Wortman Vaughan (2021), who asked whether strategyproofness, efficiency and proportionality can be combined; the paper formalized here answers no for $n,m\ge 3$ . With $\ell_1$ preferences an agent cares only about the total distance from her ideal split. Many budgets are not like that: when projects are complementary, generous funding of one project does not make up for starving another. Leontief utilities model this: an agent's benefit is set by the project that gets the smallest fraction of what she proposed for it.

Brandt, Greger, Segal-Halevi and Suksompong (arXiv:2402.15904v3) show that, with Leontief utilities, the rule that maximizes the product of the agents' utilities is the only continuous rule that resists coalitional manipulation and gives every coalition its proportional share of decision power. The rule and several of its properties come from the same authors' work on donor coordination, Coordinating charitable donations with Leontief preferences (JET, 2025). The fairness axiom is the core fair share notion of Fain, Goel and Munagala (2016) and Aziz, Bogomolnaia and Moulin (2020), adapted to distributions.

Setting

There are $n\ge 1$ agents $N=[n]$ and $m\ge 1$ alternatives (projects) $M=[m]$ . A distribution is a vector $q=(q_1,\dots,q_m)$ with $q_j\ge 0$ and $\sum_j q_j=1$ ; the set of distributions is the simplex $\Delta^m$ . Each agent $i$ reports a peak $p_i\in\Delta^m$ , her ideal distribution. A profile $P=(p_{i,j})$ collects the peaks, and a mechanism is a map $f:(\Delta^m)^n\to\Delta^m$ .

Let $M_i=\{j:p_{i,j}>0\}$ ; it is nonempty because $p_i$ sums to one. Agent $i$ 's Leontief utility from a distribution $q$ is

u_i(q)=\min_{j\in M_i}\frac{q_j}{p_{i,j}},

which lies in $[0,1]$ and equals $1$ exactly at $q=p_i$ . The minimizers form the set $T_{q,i}$ of agent $i$ 's critical alternatives, and for a group $G\subseteq N$ , $T_{q,G}=\bigcup_{i\in G}T_{q,i}$ .

The axioms are as follows.

$q$ is efficient if no distribution makes every agent weakly better off and some agent strictly better off.
A nonempty coalition $N'\subseteq N$ blocks $q$ if some $q'\in\Delta^m$ makes every member of $N'$ strictly better off than at $q$ whenever the coalition's share $|N'|/n$ is spent on $q'$ , whatever the rest is spent on: for every $q''\in\Delta^m$ and every $i\in N'$ , $u_i\big(\tfrac{|N'|}{n}q'+(1-\tfrac{|N'|}{n})q''\big)>u_i(q)$ . The distribution $q$ has core fair share if no coalition blocks it.
A mechanism is group-strategyproof if, whenever a coalition $N'$ changes only its own reports, either some member is strictly worse off or all members are exactly as well off, with utilities measured at the true peaks.
A mechanism is continuous if nearby profiles give nearby outcomes.

The Nash product rule is

NASH(P)=\operatorname*{arg\,max}_{q\in\Delta^m}\ \prod_{i\in N}u_i(q),

and the paper notes, citing the donor coordination paper, that the maximizer is unique.

Formalization targets

Goal: Theorem 3 (p. 16)

For every mechanism $f$ ,

\big[f \text{ continuous, group-strategyproof, with core fair share at every profile}\big]\iff\big[\forall P:\ f(P)=NASH(P)\big].

Both directions are part of the goal. " $\Leftarrow$ " says NASH has the three properties, and " $\Rightarrow$ " says no other mechanism has them.

Milestones

The milestones are the paper's own results, in the order the proof uses them:

the Nash maximizer exists and is unique (§6);
core fair share implies efficiency (Proposition 3);
core fair share gives every agent utility at least $1/n$ (Lemma 2);
under core fair share, $q_j=0$ exactly when no agent values $j$ (Lemma 4);
efficient distributions with equal utility vectors coincide (Lemma 7);
moving every peak to its normalized restriction to the critical set, the profile $P^*$ , changes neither the outcome nor the critical sets (Lemma 9);
$q$ maximizes the Nash product if and only if $q$ splits into shares of $1/n$ , each supported on its agent's critical alternatives (Lemma 6);
Lemmas 10 and 11, which give $f(P^*)=NASH(P^*)$ and $NASH(P^*)=NASH(P)$ , and the uniqueness half of Theorem 3 assembled from Lemmas 9–11;
the existence half: Lemma 1 (efficiency through critical sets), Lemma 3 (a sufficient condition for core fair share), Proposition 2 (NASH has core fair share), Lemma 8 ( $q_j\le\max_i p_{i,j}$ for efficient $q$ ), Proposition 1 (NASH is continuous), and the efficiency and group-strategyproofness of NASH (§6).

Five of these results are cited by the paper from Brandt et al. (2025) rather than proved in it: uniqueness of NASH, Lemma 1, Lemma 6, Lemma 7, and the efficiency and group-strategyproofness of NASH.

Significance

Theorem 3 characterizes a mechanism by its axioms. A designer who asks for continuity, resistance to coalitional misreports, and proportional coalition power gets exactly one mechanism with Leontief utilities, so no further choice is needed. This contrasts with the paper's impossibility results for $\ell_1$ and $\ell_\infty$ preferences (Theorems 1 and 2), where efficiency, strategyproofness and proportionality cannot hold together for $n,m\ge 3$ . Core fair share is stronger than both efficiency (Proposition 3) and proportionality (Proposition 4), so the positive result covers both.

The theorem is proved on paper. No machine-checked proof of it, or of the cited results from the donor coordination paper, is known. The mission provides a definition layer for simplex-valued Leontief utilities, critical alternatives, coalition blocking with shares and group-strategyproofness. It states the characterization and every intermediate result its proof cites, so proofs can be built bottom-up, including the cited results that the paper does not prove.

Difficulty

Group-strategyproofness constrains only utilities, while the theorem identifies whole distributions. Turning one into the other needs efficiency (Lemma 7), which is why Proposition 3 sits early in the chain. The key step, Lemma 9, compares $f$ at $P$ and at $P^*$ . These are profiles at which several agents change their reports simultaneously, and continuity together with group-strategyproofness has to rule out a change of outcome. Agent-by-agent arguments do not suffice, because each single deviation can move the critical sets of the other agents. Continuity of NASH (Proposition 1) is also not immediate: as a peak coordinate tends to $0$ , that alternative leaves $M_i$ and the minimum defining $u_i$ changes form.

Formalization scope

Agents and alternatives are Fin n and Fin m, indexed from $0$ , with [NeZero n] [NeZero m] for the paper's convention that $n$ and $m$ are positive integers. Distributions are elements of Mathlib's stdSimplex ℝ (Fin m), profiles are functions Fin n → stdSimplex ℝ (Fin m), and mechanisms map profiles to distributions. All values are real, including the shares $|N'|/n$ and $1/n$ . Continuity is Continuous for the subtype topology, which is the topology of the paper's $\ell_1$ definition (Definition 7).

The formalization makes these commitments.

The Leontief minimum is a finite minimum over the positive support $M_i$ only, so division by zero never produces a spurious ratio of $0$ .
Core fair share ranges over nonempty coalitions. Read literally, Definition 3 lets the empty coalition block every distribution vacuously, which would make the axiom unsatisfiable and Theorem 3 empty. Every coalition used in the paper is nonempty.
The blocking condition keeps the order $\exists q'\,\forall q''\,\forall i\in N'$ .
Group-strategyproofness evaluates both outcomes at the true peaks.
NASH is the predicate " $q$ maximizes the Nash product over $\Delta^m$ ", not a chosen function. Its uniqueness is a milestone, so " $f=NASH$ " means " $f(P)$ is a maximizer for every $P$ ".
The profile $P^*$ of Lemma 9 is the page's construction from $P$ and $f(P)$ , not an arbitrary profile with the same critical sets.

These choices rule out the trivializing readings: an unsatisfiable core axiom, an identically zero utility, NASH as an arbitrary mechanism, and a one-directional goal presented as a characterization.

A complete development needs finite-minimum and product inequalities on the simplex, a compactness argument for existence of the maximizer, strict log-concavity arguments for its uniqueness, and the decomposition (flow) certificate of Lemma 6. The Leontief and core definitions can be reused in work on donor coordination, fair mixing and participatory budgeting. Contributions on any milestone are welcome, especially formal proofs of the results the paper cites from Brandt et al. (2025).

Selected references

F. Brandt, M. Greger, E. Segal-Halevi, W. Suksompong, Optimal Budget Aggregation with Star-Shaped Preference Domains, Mathematics of Operations Research, 2025; pinned version arXiv:2402.15904v3.
F. Brandt, M. Greger, E. Segal-Halevi, W. Suksompong, Coordinating charitable donations with Leontief preferences, Journal of Economic Theory 230:106096, 2025. doi:10.1016/j.jet.2025.106096
R. Freeman, D. M. Pennock, D. Peters, J. Wortman Vaughan, Truthful aggregation of budget proposals, Journal of Economic Theory 193:105234, 2021. arXiv:1905.00457
B. Fain, A. Goel, K. Munagala, The core of the participatory budgeting problem, WINE 2016, pp. 384–399. arXiv:1610.03474
H. Aziz, A. Bogomolnaia, H. Moulin, Fair mixing: the case of dichotomous preferences, ACM Transactions on Economics and Computation 8(4):18, 2020.

18 thms1 active userReviewed

Algorithmic Game Theory Linear algebra Operations Research·Captain: mikedeng1

Shapley–Snow Kernels, Multiparameter Eigenvalue Problems, and Stochastic Games: Every Discounted Value Is a Rank-Drop Root of a Kernel-Reduced Pencil, Algebraic of Degree at Most Its RankResearch Paper

Motivation

Zero-sum stochastic games model two opponents who interact over a finite set of states, each state being a matrix game whose outcome also decides the next state. Since Shapley (1953), every such game with discount factor $\lambda\in(0,1]$ has a value $v_\lambda^k$ at each initial state $k$ , characterised as the fixed point of a contraction. The value is the basic quantity of the theory, both for the discounted game and for its limits as $\lambda\to0$ , and computing it, or describing what kind of number it is, is a long-standing question in game theory and operations research.

A short history of the algebraic side of the question:

1950: Shapley and Snow describe the extreme optimal strategies of a matrix game through square sub-matrices and their cofactors.
1953: Shapley proves existence of $v_\lambda$ as the unique fixed point of the Shapley operator.
1976: Bewley and Kohlberg use Tarski–Seidenberg to show that $\lambda\mapsto v_\lambda$ is semi-algebraic and admits a Puiseux expansion near $0$ ; the degree of the algebraic equation is not controlled.
2018: Attia and Oliu-Barton (A solution for stochastic games, arXiv e-print, reference [2] of the paper) characterise $v^k_\lambda$ as the unique zero of $w\mapsto\mathrm{val}((-1)^n(\Delta^k-w\Delta^0))$ , with $n+1$ auxiliary matrices built from a Kronecker determinant.
2021: the present paper (arXiv:1810.08798v2, Mathematics of Operations Research 46(3)) refines this with Shapley–Snow kernels and the theory of multiparameter eigenvalue problems, and obtains an explicit univariate polynomial with $v^k_\lambda$ as a root, of degree bounded by a rank.

Setting

A stochastic game has states $K=\{1,\dots,n\}$ , finite nonempty action sets $I_k$ , $J_k$ at state $k$ , a payoff $g(k,i,j)\in\mathbb R$ , transition probabilities $q(\ell\mid k,i,j)\ge0$ with $\sum_\ell q(\ell\mid k,i,j)=1$ , and a discount factor $\lambda\in(0,1]$ .

A real matrix $A$ is a matrix game: the row player picks a mixed strategy $x$ , the column player a mixed strategy $y$ , and the row player receives ${}^txAy$ . Its value $\mathrm{val}(A)$ is the number that each player can guarantee. For $z\in\mathbb R^n$ , the local game $\mathcal G^k(\lambda,z)$ has entries $\lambda g(k,i,j)+(1-\lambda)\sum_\ell q(\ell\mid k,i,j)z^\ell$ , and the discounted value $v_\lambda\in\mathbb R^n$ is the unique solution of $\mathrm{val}(\mathcal G^k(\lambda,v_\lambda))=v^k_\lambda$ for all $k$ .

The data array $D(\lambda)$ is the $n\times(n+1)$ array of matrices $M^k_0=\lambda G^k$ , $M^k_k=(1-\lambda)Q^k_k-U$ and $M^k_\ell=(1-\lambda)Q^k_\ell$ for $\ell\neq k$ , where $G^k=(g(k,i,j))$ , $Q^k_\ell=(q(\ell\mid k,i,j))$ and $U$ is a matrix of ones. The Kronecker determinant of an $n\times n$ array of matrices replaces products of scalars by Kronecker products; entrywise, its entry at rows $(i_1,\dots,i_n)$ and columns $(j_1,\dots,j_n)$ is $\det\big(A_{k\ell}^{i_kj_k}\big)_{k,\ell}$ . The auxiliary matrices are $\Delta^\ell=(-1)^\ell\det_\otimes D^{(\ell)}(\lambda)$ , where $D^{(\ell)}$ deletes column $\ell$ , for $0\le\ell\le n$ .

A Shapley–Snow kernel of a matrix game $G$ is a square sub-matrix $\dot G$ with $S(\mathrm{co}(\dot G))\neq0$ whose cofactor strategies $\mathrm{co}(\dot G)\mathbf 1/S(\mathrm{co}(\dot G))$ and ${}^t\mathrm{co}(\dot G)\mathbf 1/S(\mathrm{co}(\dot G))$ , completed by zeros, are optimal in $G$ ; $\mathrm{co}$ is the cofactor matrix and $S$ the sum of entries. Choosing a kernel $\dot I_k\times\dot J_k$ of every local game $\mathcal G^k(\lambda,v_\lambda)$ and restricting each $M^k_\ell$ to it gives the reduced array $\dot D(\lambda)$ , with square auxiliary matrices $\dot\Delta^0,\dots,\dot\Delta^n$ .

Formalization targets

Goal: Theorem 3.4.2

For every state $k$ and every choice of kernels:

\text{(i) } v^k_\lambda \text{ is the unique } w\in\mathbb R \text{ with } \mathrm{val}(\dot\Delta^k-w\dot\Delta^0)=0;\qquad \text{(ii) } \mathrm{rank}(\dot\Delta^k-v^k_\lambda\dot\Delta^0)<\max_{w\in\mathbb R}\mathrm{rank}(\dot\Delta^k-w\dot\Delta^0);

(iii) some nonzero polynomial $P^k$ of degree at most $\mathrm{rank}(\dot\Delta^0)$ , the determinant of a square sub-matrix of $\Delta^k-w\Delta^0$ , vanishes at $v^k_\lambda$ ; (iv) if all entries of $D(\lambda)$ are rational, then

[\mathbb Q(v^k_\lambda):\mathbb Q]\le\mathrm{rank}(\dot\Delta^0).

Milestones

In attack order: Shapley's fixed-point theorem (§2.1); the sign of $\Delta^0$ (Lemma 2.2.3) and unique solvability of $\mathrm{val}(A-wB)=0$ (Remark 2.2.4); the Shapley–Snow theorem (Theorem 2.3.1) and existence of kernels (§3.1); the cofactor lemmas B1–B3 and Proposition 2.3.4 on the kernel of $\dot G-v\dot U$ ; the rank-drop criterion (Lemma C1), the tensor kernel vector (Lemma C2) and Proposition 2.4.4; Proposition 3.2.1 placing $v_\lambda$ in the reduced solution sets; Proposition 3.4.1, the characterising polynomial.

Significance

The result. Theorem 3.4.2 turns the computation of a discounted value into univariate root finding: $v^k_\lambda$ is a root of one of finitely many explicit polynomials, read off from $D(\lambda)$ , of degree bounded by $\mathrm{rank}(\dot\Delta^0)\le\prod_k\min(|I_k|,|J_k|)$ . Part (iv) gives an explicit bound on the algebraic degree of the value of a rational game, a quantity that the semi-algebraic approach leaves uncontrolled. The same polynomials yield, in §4 of the paper, the convergence of $v_\lambda$ as $\lambda\to0$ and a bound on its rate. These are not part of this mission.

Formalizing it. The theorem is proved in the paper; no part of it is machine-checked. A complete development formalizes the value of a matrix game on arbitrary finite action sets, Shapley's fixed point, the Shapley–Snow description of extreme optimal strategies, Kronecker determinants of arrays of matrices, and the singular multiparameter-eigenvalue lemmas of Muhič–Plestenjak and Atkinson. Each of these is reusable outside stochastic games.

Difficulty

The obvious route is Cramer's rule for the multiparameter system $\det(M^k_0+z^1M^k_1+\dots+z^nM^k_n)=0$ : when $\Delta^0$ is invertible, the solutions are the eigenvalues of $\Delta^k(\Delta^0)^{-1}$ . Here $\Delta^0$ , and still more $\dot\Delta^0$ , is in general singular, and the polynomials $\det(\dot\Delta^k-w\dot\Delta^0)$ may vanish identically, so the determinant equation carries no information. The value must instead be located through a rank drop of the pencil. Proving the rank drop requires nonzero left and right kernel vectors with ${}^tx\dot\Delta^0y\neq0$ , which come from the sign of $\Delta^0$ (Lemma 2.2.3) and from nonnegative kernel strategies of the Shapley–Snow kernels transported through Kronecker products. Neither ingredient is available off the shelf.

A second obstacle is the paper's degree argument: it quotes the degree of $\det(A+wB)$ as equal to $\mathrm{rank}(B)$ , which is false in general; only the inequality holds and is posed.

Formalization scope

Model. States are Fin n with $n\ge1$ . The action sets I k, J k are finite, nonempty and may depend on the state, as in §4, p. 20, where the paper extends its results "word for word" to that case. The transition probabilities are nonnegative and sum to one, and 0 < lam ≤ 1.
Value. The value of a matrix game is the relation IsMatVal A w: optimal strategies of both players guarantee $w$ . No supremum or infimum over strategy sets is taken. $v_\lambda$ is any $v$ with IsShapleyFixed g q lam v; the milestone shapley_fixed_point gives existence and uniqueness, so the goal is not vacuous.
Kernels. A kernel is given by embeddings Fin s ↪ I, Fin s ↪ J. IsSSK requires $S(\mathrm{co}(\dot G))\neq0$ and optimality of the completed cofactor strategies. The paper also requires the pair to be basic, but by Theorem 2.3.1 this is equivalent.
Linear algebra. co M is the transposed adjugate. The Kronecker determinant is defined through its entry formula (K5), and rows and columns of $\Delta^\ell$ are indexed by tuples instead of lexicographic integers. Column $0$ of the array is the paper's $\ell=0$ , and the paper's $\Delta^k$ is Delta … k.succ. " $\max_w\mathrm{rank}$ " means "some real $w$ gives larger rank". The sets $S^M,S^\Delta$ accept complex vectors in Lemma C2, as in §2.4; $S^R,T^M$ are used over $\mathbb R^n$ in the game results. Proposition 3.4.1 takes its polynomial from a determinant of a reduced-pencil submatrix, as the proof constructs.
Conventions. Part (i) is the printed form, without the $(-1)^n$ of Result 1 (i); both hold for the square reduced matrices. Part (iv) is IsAlgebraic ℚ v ∧ (minpoly ℚ v).natDegree ≤ rank.

The goal ranges over Shapley–Snow kernels, not over arbitrary square sub-matrices, and $E^k$ is built from the full $\Delta^k-w\Delta^0$ with only its degree cap taken from $\dot\Delta^0$ . A formalization in which kernels are arbitrary square sub-matrices, or in which $E^k$ is built from $\dot\Delta$ , states a different theorem.

Contributions are welcome at every level. In particular: the value of a matrix game on Fintype index types with existence and uniqueness, Shapley's contraction, the Shapley–Snow theorem, the Kronecker determinant and its column-alternation property (K3), and the bound $\deg\det(A+wB)\le\mathrm{rank}(B)$ .

Selected references

L. Attia, M. Oliu-Barton, Shapley–Snow kernels, multiparameter eigenvalue problems and stochastic games, Mathematics of Operations Research 46(3), 2021; arXiv:1810.08798v2. https://arxiv.org/abs/1810.08798v2
L. Attia, M. Oliu-Barton, A solution for stochastic games, arXiv e-print, 2018 (reference [2] of the paper).
L. S. Shapley, Stochastic games, PNAS 39(10), 1953. https://doi.org/10.1073/pnas.39.10.1095
L. S. Shapley, R. N. Snow, Basic solutions of discrete games, Contributions to the Theory of Games I (Annals of Mathematics Studies 24), 1950, pp. 27–35.
T. Bewley, E. Kohlberg, The asymptotic theory of stochastic games, Mathematics of Operations Research 1(3), 1976. https://doi.org/10.1287/moor.1.3.197
A. Muhič, B. Plestenjak, On the singular two-parameter eigenvalue problem, Electronic Journal of Linear Algebra 18, 2009, pp. 420–437.
F. V. Atkinson, Multiparameter Eigenvalue Problems, vol. 1, Academic Press, 1972.
J. W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997 (cited for the degree of $\det(A+wB)$ ).

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Actuarial Science Operations Research Probability·Captain: mikedeng1

Endogenous Inverse Demand Functions: Necessary and Sufficient Conditions for a Unique Clearing Price of a Liquidated ClaimResearch Paper

Motivation

When a bank or fund in distress must sell a large position, the price it receives depends on how much it sells: the market's capacity to absorb the sale is finite. Models of fire sales and systemic risk (for instance Cifuentes, Shin and Ferrucci and Amini, Filipović and Minca) encode this through an inverse demand function, the price per unit as a function of the quantity liquidated. In most of this literature the inverse demand function is postulated exogenously, typically linear or exponential.

Bichuch and Feinstein (arXiv:2012.08002v3, 2022; Operations Research) derive the inverse demand function endogenously, from the utility-maximising behaviour of the market participants who buy the liquidated assets. Their starting point is the classical risk-exchange equilibrium of Bühlmann (1980), whose pricing measure generalises the Esscher premium of actuarial science. The price of a liquidated claim then becomes the solution of a fixed-point equation, and the first question the paper answers is when that equation has exactly one solution, so that the market produces a well-defined clearing price.

Setting

Fix a probability space $(\Omega,\mathcal F,\mathbb P)$ . The market participants' utilities share a domain $\mathbb D$ , either the real line $\mathbb R$ or the half line $\mathbb R_{++}=(0,\infty)$ . The aggregate endowment $\mathcal X$ is an essentially bounded random variable with $\mathcal X\in\mathbb D$ almost surely and $\operatorname{ess\,inf}\mathcal X\in\mathbb D$ (Assumption 2.1).

The market is summarised by one function $R:\mathbb D\to\mathbb R$ , the integral of the absolute risk aversion of the harmonic representative agent. Assumption 3.1 requires $R$ to be strictly increasing, differentiable and concave. If $\mathbb D=\mathbb R_{++}$ , $R(0)=\lim_{z\searrow0}R(z)$ in the broad sense, possibly $-\infty$ .

A claim $Z\in L^\infty$ is liquidated. Its price $v$ must satisfy $v=H_Z(v)$ , where

H_Z(v)=\frac{\mathbb E\big[Z\exp(-R(\mathcal X+Z-v))\big]}{\mathbb E\big[\exp(-R(\mathcal X+Z-v))\big]}.

For $\mathbb D=\mathbb R$ the domain $\operatorname{dom}H_Z$ is all of $\mathbb R$ ; for $\mathbb D=\mathbb R_{++}$ it consists of the $v$ with $\mathbb P(\mathcal X+Z\ge v)=1$ at which the denominator is finite. The set of clearing prices is

V(Z)=\{v\in\operatorname{dom}H_Z \mid v=H_Z(v)\}.

Write $e=\operatorname{ess\,inf}[\mathcal X+Z]$ for the worst-case aggregate wealth after the claim is added, and $\Theta_Z(v)=\mathbb E[(Z-v)\exp(-R(\mathcal X+Z-v))]$ . The uniqueness conditions of Theorem 3.3 are: (a) $Z$ and $\mathcal X+Z$ are comonotonic; (b) $R$ is linear; (c) $z\in\mathbb R_{++}\mapsto z\exp(-R(\mathcal X+z))$ is almost surely non-decreasing; (d) the same map is almost surely concave.

Formalization targets

Goal: Corollary 3.4

Let $\mathbb D=\mathbb R_{++}$ and assume any of (a)–(d). Then

|V(Z)|=1\iff\Big(e\in\operatorname{dom}H_Z,\ H_Z(e)\le e\Big)\ \text{or}\ \Big(e\notin\operatorname{dom}H_Z,\ \liminf_{v\nearrow e}H_Z(v)<e\Big),

and every clearing price satisfies $v\le e$ . Both directions of the equivalence, both branches of the case split, and the bound are part of the goal.

Milestones

Proposition D.1: monotonicity or concavity of $z\mapsto z\exp(-R(x+z))$ on $\mathbb R_{++}$ extends to $\mathbb D-x$ .
Continuity of $H_Z$ on $\operatorname{dom}H_Z$ , and the bounds $\operatorname{ess\,inf}Z\le H_Z(v)\le\operatorname{ess\,sup}Z\vee e$ (proof of Theorem 3.2).
Theorem 3.2: $V(Z)\ne\emptyset$ if $\mathbb D=\mathbb R$ , or if $\mathbb D=\mathbb R_{++}$ and $H_Z(v_0)\le v_0$ for some $v_0\in\operatorname{dom}H_Z$ , which holds under either branch of the goal's right-hand side; $V(Z)$ is compact.
Under (a) or (b), $H_Z$ is non-increasing; $v=H_Z(v)$ iff $\Theta_Z(v)=0$ ; under (c) or (d), $\Theta_Z<0$ beyond any root (proof of Theorem 3.3).
Theorem 3.3: under any of (a)–(d), $|V(Z)|\le1$ .
Corollary 3.5: if $\mathbb D=\mathbb R_{++}$ and $\mathbb E[\exp(-R(\mathcal X+Z-e))]=\infty$ , then $V(Z)\ne\emptyset$ .
The necessity step of Corollary 3.4: if $e\in\operatorname{dom}H_Z$ and $H_Z(e)>e$ , then $\Theta_Z>0$ on $[\operatorname{ess\,inf}Z,e]$ .

Significance

Corollary 3.4 turns the existence of a unique clearing price into a check of one number: the value of $H_Z$ at $e$ , or its limit inferior from the left when the denominator blows up there. It is the basis for the rest of the paper: the pricing function $V$ , its extension $\bar V$ to claims without a clearing price, and the endogenous inverse demand functions $f^q$ and $\bar f^q$ of a liquidated portfolio $q$ (Lemmas 3.9–3.10) are defined through the unique solution of (3.1). Without uniqueness the inverse demand function is not a function. The exponential- and power-utility special cases of Section 4 rely on the same statements.

The results are proved in the paper, with the proofs in its Online Appendix D. They have no machine-checked proof. Formalizing them produces a verified existence-and-uniqueness theory for Bühlmann-type pricing fixed points on both utility domains. It also records precisely how the broad-sense convention $R(0)=-\infty$ and an infinite denominator enter, which the printed definitions leave implicit.

Difficulty

$H_Z$ is a ratio of two expectations that both depend on the price $v$ . Continuity together with the bounds gives existence on $\mathbb R$ almost directly, but uniqueness does not follow from continuity: Example C.1 of the paper exhibits three clearing prices. Each uniqueness condition controls a different object. Conditions (a) and (b) control the sign of a covariance-type double integral, so that $H_Z$ is non-increasing. Conditions (c) and (d) control the shape of $\Theta_Z$ , and do so only for positive arguments, so Proposition D.1 is needed to reach the negative arguments $Z-v$ that actually occur.

On $\mathbb R_{++}$ the domain of $H_Z$ ends at $e$ , and $e$ may or may not belong to it. When $\mathbb E[\exp(-R(\mathcal X+Z-e))]=\infty$ , $H_Z(e)$ has no value and only the left limit inferior is available. Both sides of that limit, an expectation that becomes infinite and a ratio whose numerator and denominator diverge together, must be handled by monotone convergence and truncation. The naive approach of evaluating $H_Z$ at $e$ as a real number fails exactly in this branch.

Formalization scope

Everything is defined in the namespace EndoInvDemand.Clearing. The probability space is a measure P with IsProbabilityMeasure P. Only the aggregate $\mathcal X$ is modelled, since $H_Z$ sees the individual endowments only through their sum. Assumption 2.1 is stated for it with MemLp X ⊤ P. The domain is a two-valued flag Dom (real, pos). $R$ is a function ℝ → ℝ of which only the restriction to $\mathbb D$ is constrained and read.

The weight $\exp(-R(y))$ is valued in ℝ≥0∞. For $\mathbb D=\mathbb R_{++}$ its value at $y=0$ is $\sup_{z>0}\exp(-R(z))$ , which encodes $R(0)$ in the broad sense and is $+\infty$ exactly when $R(0)=-\infty$ . The denominator of $H_Z$ is a lower Lebesgue integral and may be $+\infty$ ; the numerator and $\Theta_Z$ are Bochner integrals.

The printed $\operatorname{dom}H_Z$ for $\mathbb R_{++}$ asks the denominator to be " $>0$ ", which always holds. The formalization asks it to be finite, the reading the proof of Corollary 3.5 uses, because the ratio has no value otherwise. "Linear" means affine on $\mathbb D$ . Comonotonicity is the $\mathbb P\otimes\mathbb P$ -almost-everywhere form. $|V(Z)|=1$ is ∃!, $|V(Z)|\le1$ is Set.Subsingleton, and $\liminf_{v\nearrow e}$ is Filter.liminf along 𝓝[<] e. The paper's derivative statements $H_Z'\le0$ and $\partial_v\Theta_Z<0$ are formalized as the monotonicity statements they are used for.

Neither side of the goal's equivalence can be satisfied by a degenerate encoding. $V(Z)$ is the fixed-point set of the explicit ratio $H_Z$ on its explicit domain; $H_Z$ never takes a junk value on that domain, because the denominator is finite and positive there; and $H_Z$ is bounded to the left of $e$ , so the limit inferior is genuine. Both branches of the case split occur: $R=\log$ puts $e$ outside the domain and $R(z)=z$ puts it inside.

A complete development needs dominated and monotone convergence for parametric integrals, the intermediate value theorem, and a Chebyshev-type covariance inequality for comonotonic variables under $\mathbb P\otimes\mathbb P$ . The covariance lemma and the treatment of $[0,\infty]$ -valued weights with a broad-sense boundary value are reusable beyond this mission. Proofs of any milestone are welcome, including ones that route around the paper's derivative computations.

Selected references

M. Bichuch and Z. Feinstein, Endogenous Inverse Demand Functions, arXiv:2012.08002v3, 2022; Operations Research. https://arxiv.org/abs/2012.08002
H. Bühlmann, An economic premium principle, ASTIN Bulletin 11(1), 1980. https://doi.org/10.1017/s0515036100006619
K. K. Aase, Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization, ASTIN Bulletin 23(2), 1993. https://doi.org/10.2143/ast.23.2.2005091
H. Amini, D. Filipović and A. Minca, Uniqueness of equilibrium in a payment system with liquidation costs, Operations Research Letters 44(1), 2016. https://doi.org/10.1016/j.orl.2015.10.005
R. Cifuentes, H. S. Shin and G. Ferrucci, Liquidity risk and contagion, Journal of the European Economic Association 3(2–3), 2005. https://doi.org/10.1162/jeea.2005.3.2-3.556

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