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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.

≤ 16Formalized record

5 provers on it13 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.

≤ 23Formalized record≤ 5Open frontier

38 provers on it15 of 17 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.

≤ 16Formalized record

5 provers on it13 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.

≤ 23Formalized record≤ 5Open frontier

38 provers on it15 of 17 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Completed

Economics Operations Research·Captain: mikedeng1

Brand Spillover as a Marketing Strategy: The Weak-Brand Firm Should Commit to Forgo Brand Spillover if and only if the Supplier Would Otherwise Drop It or the Insourcing Cost Is LowResearch Paper

Motivation

Competing brands often buy their products from the same contract manufacturer (CM). A weaker brand can exploit this. The paper's opening example is the Chinese retailer Netease Yanxuan, which advertises that its goods use "the same materials, the same factories, and the same workers" as international brands such as Adidas, Coach and MUJI. A weak brand that discloses its common supplier in this way borrows part of the leader's reputation. This practice is called brand spillover.

The obvious view is that a weak brand should use any spillover it can get. Wu, Zhang and Zhou (SSRN 3891915, Management Science 2021) show that this view is incomplete. A supplier that serves both brands has to keep the strong brand willing to outsource. If the weak brand's spillover makes that too expensive, the supplier stops serving the weak brand, or it extracts the spillover's value through its prices. The paper's main result, Proposition 4, says when the weak brand is better off committing in advance not to use the spillover. This mission formalizes that result together with the equilibrium analysis it rests on.

Setting

Two firms sell substitutable products. Firm S has brand power $\theta_S = 1$ and Firm W has brand power $\theta_W = \theta < 1$ . With quantities $q_S, q_W$ the market-clearing prices are

p_S = 1 - q_S - q_W,\qquad p_W = \theta_W - q_S - q_W .

Each firm can produce in-house at unit cost $c > 0$ , or buy from the CM, whose own unit cost is normalized to $0$ . A sourcing structure is one of $(O,O)$ , $(O,I)$ , $(I,O)$ , $(I,I)$ : the first letter is Firm S's choice and the second Firm W's, with $O$ for outsourcing and $I$ for insourcing. Brand spillover can occur only under $(O,O)$ . There, if Firm W uses it, its brand power becomes $\theta + \alpha(1-\theta)$ , where $\alpha \in [0,1]$ is the level of brand spillover.

The game has four stages:

Firm W decides whether to commit to not using brand spillover. Committing is equivalent to setting $\alpha = 0$ .
The CM announces wholesale prices $w_S$ and $w_W$ .
The firms choose their sourcing simultaneously.
The firms compete in quantities.

At stage 4 the quantities are the first-order-condition solutions, and each firm's profit is the square of its quantity. At stage 2, for each structure, the CM maximizes its revenue $\sum w_i q_i$ over the outsourcing firms, subject to participation constraints. A constraint says that a firm does at least as well outsourcing at the offered price as deviating to insourcing. The CM then induces the structure with the highest optimal revenue. The standing assumption $\theta \ge \underline\theta = \max\{(2+7c)/7,\ (1+c)/2\}$ keeps Firm W in the market.

The thresholds used below are $c^{oi} = \tfrac13(2-\theta)$ , $c^{oo}_1 = \tfrac12(2\theta - 1 - 3\alpha(1-\theta))$ and $c^{oo}_2 = \tfrac12(1 + \alpha(1-\theta))$ .

Formalization targets

Goal: Proposition 4

Assume $\alpha > 0$ . Let $\Pi_W$ be Firm W's profit at an equilibrium without commitment, and $\Pi^0_W$ its profit at an equilibrium with commitment. If the structure without commitment is $(O,I)$ , then $\Pi_W \le \Pi^0_W$ . Otherwise

\Pi_W < \Pi^0_W \iff c < \theta - \tfrac12 - \alpha + \alpha\theta \iff \alpha < \frac{2\theta - 1 - 2c}{2(1-\theta)} .

Milestones

The quantity stage has a unique Nash equilibrium, with profits $q_i^2$ .
Lemma 1: the CM never induces $(I,I)$ or $(I,O)$ .
Lemma 2: under $(O,I)$ the CM's optimal price is $w^{oi}_S = \min\{c, \tfrac14(2-\theta+c)\}$ .
Lemma 3: under $(O,O)$ the CM's optimal prices are $w^{oo}_S = \min\{c - \tfrac12\alpha(1-\theta), \tfrac12\}$ , and $w^{oo}_W$ given by (2)–(3).
Appendix B: the equilibrium quantities and their signs.
Proposition 2(b): under $(O,O)$ , Firm W's profit decreases in $\alpha$ exactly when $\alpha < (2\theta-2c-1)/(3(1-\theta))$ .
Proposition 3: the equilibrium sourcing structure for every spillover level.
With commitment, the CM always induces $(O,O)$ . Firm W then earns $((2\theta-1-c)/3)^2$ or $((2\theta-1)/6)^2$ .

Significance

Proposition 4 splits the parameter space into three regions. In the first, the supplier would drop the weak brand: it induces $(O,I)$ , and Firm W pays the insourcing cost. In the second, the supplier captures the spillover through a higher wholesale price, and the weak brand loses ground to the strong one. In the third, the spillover pays. The comparative static behind the second region is Proposition 2(b): brand spillover can lower the profit of the firm it benefits. The analysis also identifies when the strong brand is hurt by the weak brand's spillover and should act to prevent it (paper, abstract and §5).

The paper proves the result by hand, and no formal proof exists. A complete development would give machine-checked proofs of:

a four-stage game with participation-constrained pricing;
the closed-form solution of two non-convex price programs;
an equilibrium comparison that holds for every selection when the CM is indifferent between structures.

Formalizing the statements has already found two slips in Proposition 3: a sign in $T_1$ , and a boundary point that the printed case split assigns to the wrong structure.

Difficulty

The CM's feasible sets are not convex. Each participation constraint compares two squared quantities, $q_{\text{out}}^2 \ge q_{\text{in}}^2$ , so it admits two branches: a price below a bound, or a price so high that the quantity changes sign. The paper's proofs treat only the first branch. A complete proof must rule out the other branches as optimal, so maximizing a concave quadratic is not enough.

The goal compares equilibrium outcomes, not formulas. Each side combines an optimal-price problem with a choice between structures. The no-commitment side changes regime along the curves $T_1$ , $T_2$ , $T_3$ , which contain square roots. Firm W's profit has a kink at $c = c^{oo}_1$ .

Formalization scope

Model and its general part. The model is one definitions module, BrandSpillover.Commitment.Model. The profit functions use the referenced general linear Cournot payoff SupplierAudit.Competition.cournotPayoff with substitution parameter $1$ . All quantities and prices are real numbers.
Quantities. The stage-4 quantities are the first-order-condition formulas on $\mathbb R$ and are not clamped at $0$ , as in the paper. Clamping them changes the CM's feasible set and breaks Lemma 3.
Prices. Wholesale prices range over $\mathbb R^2$ , without a sign restriction. The price for a firm that insources is ignored.
Feasibility. The feasible set of each program is the page's set of profit comparisons. It is never replaced by the linear bounds derived in the proofs.
Equilibrium. An equilibrium is a structure whose optimal CM revenue is at least that of every other structure, together with optimal prices for it. Ties are allowed, and the goal holds for every selection.
Standing assumptions. The standing assumptions are $c > 0$ , $\underline\theta \le \theta < 1$ and $\alpha\in[0,1]$ , with the paper's weak inequality $\theta \ge \underline\theta$ . The goal adds $\alpha > 0$ , from p. 17.
Corrections. $T_1$ uses $(1-\theta)^2$ where the page prints $(1+\theta)^2$ . Condition (i) of Proposition 3 includes $c = c^{oo}_1$ . The committed profit is stated for every admissible $c$ , including $c > \tfrac12$ , which the page omits.
Ruled-out shortcut. Substituting the closed-form prices of Lemmas 2–3 into the goal would turn it into a comparison of squares. The goal is therefore stated about equilibrium outcomes of the price programs.
Contributions. Proofs of any milestone are welcome. Lemma 3 and Proposition 3 are the hardest.

Selected references

X. Wu, F. Zhang, Y. Zhou, Brand Spillover as a Marketing Strategy, SSRN preprint 3891915, June 2021; Management Science, 2021. https://ssrn.com/abstract=3891915, https://doi.org/10.1287/mnsc.2021.4165
A. Arya, B. Mittendorf, D. E. M. Sappington, The make-or-buy decision in the presence of a rival: strategic outsourcing to a common supplier, Management Science 54(10), 2008. https://doi.org/10.1287/mnsc.1080.0874
N. Singh, X. Vives, Price and quantity competition in a differentiated duopoly, RAND Journal of Economics 15(4), 1984. https://doi.org/10.2307/2555525

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Completed

Economics Operations Research·Captain: mikedeng1

On Co-opetitive Supply Partnerships with End-Product Rivals: Information Asymmetry, Dual Sourcing and Supply Market Efficiency: The Equilibrium Wholesale Price and When the VI Buys InformationResearch Paper

Motivation

Firms often buy key components from the very rivals they compete with in the end market. The paper's examples: Lockheed Martin has sourced engine control systems and gearboxes from AgustaWestland, its competitor in national helicopter markets, while avoiding single sourcing from it in markets where it holds an information advantage; vertically integrated generic-drug makers such as Teva supply active ingredients to finishing-only rivals such as Apotex. Such co-opetitive supply relationships raise an informational problem studied in operations management: the buyer's order quantity reveals what the buyer knows about demand, and the supplier, who is also a competitor, can use that signal when setting her own output. The buyer may therefore distort his orders to hide his information, and the supplier may lower her wholesale price to make revealing it worth his while.

S. H. Jung and P. Kouvelis (SSRN 3555184, 2020; published in Manufacturing & Service Operations Management) analyse this trade-off in a stylized model with a single outside supplier as the buyer's alternative. Their main result, Proposition 4, gives the supplier's equilibrium wholesale price in closed form and identifies when she pays an information rent to learn demand and when she prices to keep the buyer from leaving.

Setting

A vertically integrated firm (VI, "she") with production cost normalized to $0$ and a downstream outsourcing firm (DO, "he") compete in quantities. The market-clearing price is $P = 1 - Q + \epsilon$ , where $Q$ is total output and the demand shock $\epsilon$ is $t$ or $-t$ ( $t > 0$ ), each with probability $1/2$ . Only the DO observes $\epsilon$ . Play proceeds in two stages.

The VI offers a unit wholesale price $w_I$ .
The DO orders $Q_{O,1}$ from the VI at $w_I$ and $Q_{O,2}$ from an outside supplier at the unit price $w_A$ . The VI sees $Q_{O,1}$ but not $Q_{O,2}$ , updates her belief about $\epsilon$ , and produces $Q_I$ .

In state $\epsilon$ the VI earns $(1 - (Q_{O,1} + Q_{O,2} + Q_I) + \epsilon)Q_I + w_I Q_{O,1}$ . The paper solves the quantity subgame in closed form. Below the separating/pooling threshold $\bar w_{sep/pool}$ of its equation (3) the DO's order reveals the state (separating quantities, Lemma 1); between $\bar w_{sep/pool}$ and the pooling cap $\bar w_I$ of equation (4) the DO orders the same amount in both states and tops up from the outside supplier in high demand (pooling quantities, Lemma 2); at or above $\bar w_I$ he buys only from the outside supplier (Proposition 1). Averaging the VI's state profit gives $\Pi^s_I(w_I)$ , $\Pi^p_I(w_I)$ and $\Pi^{n*}_I = (3 + 4w_A)^2/100$ , and the VI's objective is equation (6):

\Pi_I(w_I) = \begin{cases}\Pi^s_I(w_I) & w_I \le \bar w_{sep/pool},\\ \Pi^p_I(w_I) & \bar w_{sep/pool} < w_I < \bar w_I,\\ \Pi^{n*}_I & w_I \ge \bar w_I,\end{cases}\qquad w_I^* = \arg\max_{w_I \ge 0}\Pi_I(w_I).

The analysis is carried out under the paper's standing assumptions: $\underline t := 28(4 + 5\sqrt2)/1105 \le t$ and $\underline w := 126(17 - 3\sqrt{34})/(570\sqrt{34} - 15829) < w_A < 1/2 - 5t/7$ . The constant $\bar t \approx 0.4005$ is the nonzero solution of equation (7), $153(\sqrt{5134} - 122)\bar t + 125\sqrt{21\bar t(51\bar t + 52)} = 0$ .

Formalization targets

Goal: Proposition 4

For $\underline t \le t < \bar t$ there is $\bar w_A > 0$ solving $\Pi^s_I(\bar w_{sep/pool}) = \Pi^p_I(\bar w_I)$ such that, for every admissible $w_A$ ,

w_A > \bar w_A \;\Rightarrow\; w_I^* = \bar w_{sep/pool}, \qquad w_A \le \bar w_A \;\Rightarrow\; w_I^* = \bar w_I^{(-)},

and for $t \ge \bar t$ , $w_I^* = \bar w_I^{(-)}$ for every admissible $w_A$ . Here $w_I^* = \bar w_{sep/pool}$ means this price is nonnegative and the unique maximizer of $\Pi_I$ on $w_I \ge 0$ , and $w_I^* = \bar w_I^{(-)}$ means $\sup_{w_I \ge 0}\Pi_I = \Pi^p_I(\bar w_I)$ , approached as $w_I \uparrow \bar w_I$ . The cap itself is excluded; at the crossing $w_A=\bar w_A$ , the separating boundary can attain the same value.

Milestones

Equation (13): $\Pi^p_I$ is a concave quadratic, uniquely maximized over $\mathbb R$ at $(3703 - 4066t - 242w_A)/9204$ .
The proof of Lemma 3: the left limit of $\Pi_I - \Pi^{n*}_I$ at $\bar w_I$ in closed form.
Lemma 3: some $w_I \in [0, \bar w_I)$ gives $\Pi_I(w_I) > \Pi^{n*}_I$ .
Lemma 4: $\Pi^s_I$ is strictly increasing on the separating region.
For $t > 35/92$ , $\bar w_{sep/pool} - \bar{\bar w}_{equal}$ is strictly increasing in $w_A$ , where $\bar{\bar w}_{equal}$ (equation (16)) equates the last separating piece with $\Pi^p_I(\bar w_I)$ .
Its limit as $w_A \uparrow 1/2 - 5t/7$ equals $(153(\sqrt{5134} - 122)t + 125\sqrt{21t(51t + 52)})/5250$ .
The limit itself is positive if and only if $t < \bar t$ .
Lemma 5 (i): for $t < \bar t$ , thresholds $\underline w_A$ , $\bar w_A$ inside the range separate the separating and the pooling outcomes. The pooling value is approached by feasible wholesale prices if its maximizing endpoint is excluded.

Significance

Proposition 4 turns a two-stage signaling game into a two-parameter phase diagram. The VI pays an information rent, by pricing at $\bar w_{sep/pool}$ below the outside price, only when demand uncertainty is low ( $t < \bar t$ ) and the outside market is inefficient ( $w_A > \bar w_A$ ). Otherwise she prices just below the level at which the DO walks away, and the DO conceals his information by pooling and dual sourcing in high demand. The paper builds its welfare comparison (Proposition 5) and its numerical study of the value of the partnership and of information on this characterization.

The result is proved on paper but has no machine-checked proof. The work is explicit real analysis: a piecewise objective with square roots, branch thresholds and a jump, and an optimization over a half-line. The formal statement also records corrections the paper needs (see the scope section), so that a proof checks the intended theorem rather than its typeset version.

Difficulty

$\Pi_I$ is neither concave, monotone nor continuous: the separating piece itself has four sub-branches (the low-demand order hits zero, the DO starts dual sourcing), and $\Pi_I$ jumps down to $\Pi^{n*}_I$ at $\bar w_I$ . A first-order condition on one formula does not work. The comparison of the two candidates, $\Pi^s_I(\bar w_{sep/pool})$ against $\Pi^p_I(\bar w_I)$ , involves nested radicals in $t$ and $w_A$ , and its sign changes along a curve $w_A = \bar w_A(t)$ that has no closed form. The paper locates that curve only through the monotonicity of $\bar w_{sep/pool} - \bar{\bar w}_{equal}$ in $w_A$ and the sign of its boundary limit; establishing both requires polynomial inequalities over a two-dimensional region with irrational coefficients.

Formalization scope

Everything is in $\mathbb R$ ; the parameters are the two reals $t$ and $w_A$ , and the decision variable is $w_I$ . The two-point prior is written out as an average of the two states; no measure theory is used.

The quantity subgame is taken as given. Lemmas 1–2, Definition 1 and Proposition 3 (existence and selection of the Pareto-dominant separating or pooling equilibrium) are not formalized; their closed-form quantities enter (6) as definitions, transcribed as printed. The goal is the VI's first-stage problem over the displayed (6), which is exactly what Proposition 4 is about.
Equation (4) is corrected. The page typesets $\bar w_I$ with the radical over $34(7 - 10t - 14w_A)$ ; the radical belongs over $34$ alone, $\bar w_I = (245 - 350t - 70w_A - 6\sqrt{34}(7 - 10t - 14w_A))/420$ . The typeset form contradicts the paper's own Lemma 3 identity and Table 1, and makes Lemma 3 and Proposition 4 false in part of the region.
Standing assumptions $t \ge \underline t$ , $w_A > \underline w$ (exact value) and $w_A < 1/2 - 5t/7$ are used on the VI optimization statements, including Lemma 3, which the paper states just before introducing $t \ge \underline t$ . Algebraic items such as equation (13) use their stated domains.
$w_I^* = \bar w_I^{(-)}$ is a supremum with a left limit, never an arg max at $\bar w_I$ , where $\Pi_I = \Pi^{n*}_I$ . The domain $w_I \ge 0$ is kept: over all of $\mathbb R$ the pooling quadratic would dominate. Any formalization that builds the answer into the definition of $\Pi_I$ , or replaces the threshold $\bar w_A$ by one outside the parameter range, trivializes the result and is excluded.
Positive parts are max 0, and Real.sqrt radicands are nonnegative wherever they are used. Printed slips in the proofs (a $\bar t$ for $t$ in a limit formula, $\Pi^{S*}_I$ for $\Pi^{p*}_I$ ) are corrected in the Lean and disclosed.

Contributions welcome: proofs of the milestones, which are self-contained real-analysis statements; reusable lemmas on piecewise-defined maximization and one-sided limits of branch functions; and, separately, a formal treatment of the quantity-subgame equilibria of Lemmas 1–2.

Selected references

S. H. Jung, P. Kouvelis, On Co-opetitive Supply Partnerships with End-Product Rivals: Information Asymmetry, Dual Sourcing and Supply Market Efficiency, SSRN preprint 3555184, 2020; Manufacturing & Service Operations Management, 2022. https://ssrn.com/abstract=3555184, https://doi.org/10.1287/msom.2021.0982

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Economics Operations Research·Captain: mikedeng1

Quality Regulation on Two-Sided Platforms: Exclusion, Subsidization, and First-Party Applications: Exclusion Is a Special Case of Subsidization and Is Dominated by ItResearch Paper

Why platform quality rules matter

An application platform can increase average quality by excluding low-quality developers, or it can pay high-quality developers to join. Either choice changes the number of applications, the consumer audience, and the fees the platform can charge. Huang, Lyu, and Xu study these linked effects in a two-sided market with consumers and two developer types. Their SSRN paper, §5.1.1 compares exclusion with a targeted subsidy from the platform's perspective. The comparison matters when a platform must decide whether to restrict entry or use payments to shape the mix of applications.

The paper appeared in Management Science in 2021. This mission uses the 43-page SSRN posting, whose page and equation numbering are the source for every target below. Its online appendices are absent from the held posting, so several closed forms that the paper says are available there cannot be cited as printed appendix statements.

Market and decisions

A two-sided platform serves consumers and application developers. A fraction $\lambda$ of developers are high quality, with application quality $q_h>0$ ; the remaining fraction are low quality, with quality normalized to zero. Write $m$ for the consumer network size, $n$ for the total developer network size, and $n_h,n_l$ for the high- and low-quality developer counts. Their average quality is $\bar q=q_hn_h/n$ when $n>0$ .

Consumers receive base value $w$ , a cross-side benefit $\alpha_c n$ , and a benefit $\beta\bar qn$ from total application quality. The separate average-quality preference $\mu$ is fixed to zero in this mission, matching §5.1. Developer types receive cross-side benefits $\alpha_{dh}m$ and $\alpha_{dl}m$ , while congestion costs them $bn$ . Consumers and developers also face uniformly distributed differentiation costs on $[0,\theta_c]$ and $[0,\theta_d]$ . The platform chooses a developer access fee $p_d$ and a consumer access fee $p_c$ .

Under exclusion, only high-quality developers can enter, so $n=n_h$ and $\bar q^E=q_h$ . Under subsidization, low-quality developers may enter, and the platform pays each high-quality entrant a fixed amount $\gamma\ge0$ . The paper's demand equations determine the network sizes from the fees. The resulting profits are

\Pi^E=p_dn+p_cm-\eta mn,\qquad \Pi^S=p_dn+p_cm-\eta mn-\gamma n_h,

where $\eta mn$ is the platform's operating cost. The model defines an optimum by comparison with every feasible outcome of the same strategy. Thus the goal concerns the platform's choices and induced participation, rather than a comparison of two isolated algebraic formulas.

Formalization targets

The main target is a corrected form of Proposition 4, p. 26. For optimal, interior exclusion and subsidization outcomes in the nonnegative-subsidy regime,

\Pi^{S*}\ge\Pi^{E*},\qquad \bar q^{S*}\le q_h=\bar q^{E*},\qquad m^{S*}>m^{E*},\qquad n^{S*}>n^{E*}.

It also asks for a feasible subsidy outcome that reproduces the optimal exclusion outcome: no low-quality entrant, the same consumer and high-quality developer counts, and the same platform profit. The goal is quantified over outcomes that optimize the original fee-and-participation model.

Five milestones follow the paper's attack path: the reduced exclusion profit in §4.1, p. 12; the exclusion optimum in Proposition 1.1(1), pp. 12–13; the reduced subsidy profit in §4.2, p. 16; the optimal subsidy in Proposition 2.2(1), p. 17; and the replication claim in the discussion following Proposition 4, p. 26. Each milestone carries the statement from the SSRN pages and its page provenance.

What the comparison establishes

Profit dominance would mean that, within the specified parameter regime, a platform with the subsidy option can attain at least the exclusion profit. The quality comparison says this gain may come with a lower average quality, even while the consumer and total developer networks grow. The replication clause gives the strategy comparison a direct interpretation in terms of fees and participation. These are distinct claims: a larger total network need not contain more high-quality developers.

The result is already claimed in the paper, but the printed Proposition 4 contains statements that do not hold as written. A formal proof would establish the corrected result for the exact fee, subsidy, and demand model, and would clarify the required regime and interior conditions. The reusable outputs are the model definitions and the reduced-profit identities; other quality-regulation comparisons could build on them. No machine-checked proof of this proposition is claimed here. The draft Lean theorem is an open target.

Why the target is difficult

The fee choices affect participation on both sides, while quality affects consumer demand and developer payments affect the platform's net revenue. Comparing the two optima therefore requires more than observing that a subsidy can raise high-quality participation. In particular, an unconstrained algebraic optimum for $\gamma$ can be negative, but the actual subsidy strategy requires $\gamma\ge0$ . The paper's claim that exclusion is a special case also has to respect this constraint while keeping the same realized network sizes and profit.

The source's part (3) shows another difficulty: high-quality participation, total developer participation, and developer access fees can move in different directions. Treating those quantities as interchangeable would prove a different and sometimes false comparison.

Formalization scope and corrections

All parameters, fees, network sizes, and profits are real numbers. The Lean model uses the paper's untruncated linear demand equations at $\mu=0$ . Feasibility requires nonnegative consumer and developer counts; it also requires $\gamma\ge0$ , $n_l\ge0$ , and $n_h\ge0$ for subsidization. The high-quality count in consumer demand makes total quality equal to $q_hn_h$ . The composite parameters $\bar\alpha_d,\xi,\xi_1,\rho,t,\tau$ use the coefficients and signs printed in §§3–4.

The standing parameter condition includes $0<\lambda<1$ , $q_h,\theta_c,\theta_d,b,w>0$ , $\alpha_{dl}\le\alpha_{dh}$ , and $\beta\ge0$ ; it puts no extra sign on $\alpha_c$ or $\eta$ . The strict inequalities on $\lambda$ strengthen the source's $\lambda\in[0,1]$ because the exclusion inverse demand divides by $\lambda$ and the comparison requires both developer types. Separate strict-concavity conditions make explicit the paper's joint-concavity assumption and its use of unique optima. The main theorem assumes interior optimal participation and $\alpha_{dh}-\alpha_{dl}\le\beta q_h$ .

That last restriction is needed for the printed profit-dominance claim. For example, $\lambda=1/5$ , $q_h=1/2$ , $\theta_c=\theta_d=1$ , $\alpha_{dh}=1$ , $\alpha_{dl}=0$ , $\alpha_c=b=1/2$ , $\beta=\eta=0$ , and $w=1$ give $\Pi^{S*}=150/551<22/79=\Pi^{E*}$ . The paper's printed part (3) also claims $n_h^{S*}>n^{E*}$ and $p_d^{S*}>p_d^{E*}$ . At its Figure 3 parameters with $\lambda=1/10$ , $n_h^{S*}=247650/27398773<525/52868=n^{E*}$ . At $\lambda=1/4$ , $q_h=1$ , $\theta_c=1$ , $\theta_d=4$ , $\alpha_{dh}=1$ , $\alpha_{dl}=1/2$ , $\alpha_c=3/2$ , $b=2$ , $\beta=1/2$ , $\eta=0$ , and $w=1$ , the subsidy developer fee is $-64/199<-2/7$ , the exclusion fee. The goal therefore states larger total developer and consumer networks and leaves out the false fee comparison.

The goal cannot be discharged by defining profit as its reduced quadratic formula or by merely substituting the reported optimal closed forms: the fee-and-demand definitions and optimality over feasible outcomes remain part of the statement. The development needs real algebra, strict concavity of quadratic objectives, inverse-demand identities, and inequality reasoning at the feasible boundary. The equation identities and basic optimization lemmas are useful contributions beyond this comparison.

Selected references

P. Huang, G. Lyu, and Y. Xu, Quality Regulation on Two-Sided Platforms: Exclusion, Subsidization, and First-Party Applications, Management Science, 2021; SSRN posting 3948778, §§3–5, especially Proposition 4, p. 26 (PDF 27).

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Economics Mechanism Design Operations Research·Captain: mikedeng1

On the Futility of Dynamics in Robust Mechanism Design: Minimax Regret Equals T Times the Single-Round Minimax Regret and Repeating a Direct Static Mechanism Is OptimalResearch Paper

Motivation

A seller who faces the same buyer again and again, a firm that contracts repeatedly with a supplier, or a platform that allocates resources to a returning user can adapt its terms over time. The buyer's behaviour reveals information about their preferences, and mechanisms can link rounds through bundles, discounts and promised future utility. When the principal does not know the distribution of the agent's preferences, adapting looks attractive. This is the repeated-interaction setting of the paper.

Balseiro, Kim and Russo ask whether such dynamics help when the principal evaluates a mechanism by its worst-case regret over a set of possible distributions, the robust objective common in dynamic pricing and online learning. Their answer, under two conditions that hold for a broad class of games, is that dynamics are useless. The optimal minimax regret over $T$ rounds is exactly $T$ times the single-round minimax regret restricted to point-mass distributions, and repeating a single-round direct mechanism achieves it. This mission formalizes that reduction (Theorem 1 of the paper).

Setting

A game consists of a set of shocks $\Theta$ , a set of outcomes $\Omega$ with a no-interaction outcome $\emptyset$ , a principal utility $u(\theta,\omega)$ and an agent utility $v(\theta,\omega)$ , with $u(\theta,\emptyset)=v(\theta,\emptyset)=0$ . The agent's shocks $\theta_1,\dots,\theta_T$ are i.i.d. from a distribution $F$ that the agent learns in Round 0 and the principal never sees. The agent observes $\theta_t$ privately at the start of round $t$ .

A dynamic mechanism $A$ fixes report spaces $\mathcal M_0,\dots,\mathcal M_T$ , with a report QUIT in $\mathcal M_0$ . It also fixes decision rules $\omega_t=\pi_t(m_t,h_t,z_t)$ , which depend on the current report, the public history $h_t=(m_{0:t-1},\omega_{1:t-1})$ and a private uniform $z_t$ . Finally it fixes a recommended strategy $\sigma$ , with $m_0=\sigma_0(F,y_0)$ and $m_t=\sigma_t(\theta_t,h^+_t,y_t)$ . Here $h^+_t=(F,\theta_{1:t-1},m_{0:t-1},\omega_{1:t-1})$ is everything the agent knows and $y_t$ is the agent's private uniform. After QUIT every outcome is $\emptyset$ .

The principal utility and agent utility of a strategy $\tilde\sigma$ are the expected sums $\mathbb E[\sum_{t=1}^T u(\theta_t,\omega_t)]$ and $\mathbb E[\sum_{t=1}^T v(\theta_t,\omega_t)]$ . $A$ is incentive compatible (IC) if, for every distribution $F$ on $\Theta$ and every feasible strategy $\tilde\sigma$ , following $\sigma$ gives the agent at least as much as $\tilde\sigma$ . Because QUIT is a deviation, IC contains individual rationality. With $\mathcal A$ the set of IC mechanisms:

\mathrm{OPT}(F,T)=\sup_{A\in\mathcal A}\mathrm{PrincipalUtility}(A,F,T),\qquad \mathrm{Regret}(A,F,T)=\mathrm{OPT}(F,T)-\mathrm{PrincipalUtility}(A,F,T),

\mathrm{Regret}(T)=\inf_{A\in\mathcal A}\ \sup_{F\in\mathcal F}\mathrm{Regret}(A,F,T),

where $\mathcal F$ is the set of distributions the principal guards against.

A direct static mechanism $S^{\times T}$ repeats a single-round rule $\tilde\pi(m,z)$ . The agent reports a shock or PASS (which yields $\emptyset$ ), the rule ignores the history and the round, and the recommended strategy is truthful. $\mathcal S^{\times1}$ is the class of such single-round mechanisms that are IC. Assumption 1 says that $\delta_\theta\in\mathcal F$ for every $\theta$ . Assumption 2 (extreme-point convexity) says that $\mathrm{OPT}(F,T)\le\mathbb E_{\theta\sim F}[\mathrm{OPT}(\delta_\theta,T)]$ for $F\in\mathcal F$ .

Formalization targets

Goal: Theorem 1

Under Assumptions 1 and 2,

\mathrm{Regret}(T)=T\cdot\inf_{S\in\mathcal S^{\times1}}\sup_{\theta\in\Theta}\mathrm{Regret}(S,\delta_\theta,1).

Moreover, an $S$ that is within $\epsilon/T$ of the single-round infimum satisfies $\sup_{F\in\mathcal F}\mathrm{Regret}(S^{\times T},F,T)\le\mathrm{Regret}(T)+\epsilon$ . The multi-round problem has an optimal mechanism if and only if the single-round problem does. No constant is fixed: the statement identifies the value, its near-optimal solutions and the existence of optimal solutions.

Milestones

In proof order:

Proposition 13 (parts 1–2): $\mathrm{OPT}(F,T)\le T\,\bar u(F)$ and $\mathrm{OPT}(\delta_\theta,T)=T\,\bar u(\delta_\theta)$ , where $\bar u$ is the first-best single-round value.
Proposition 1: $\mathrm{OPT}(\delta_\theta,T)=T\cdot\mathrm{OPT}(\delta_\theta,1)$ .
Lemma 4: every IC mechanism yields $S(A)\in\mathcal S^{\times1}$ with T times its point-mass principal utility.
Lemma 2: the lower bound $\sup_F\mathrm{Regret}(A,F,T)\ge T\sup_\theta\mathrm{Regret}(S,\delta_\theta,1)$ .
Claim 1: replicating any round of a deviation as a single-round deviation.
Lemma 5: $S^{\times T}$ is IC, and its value is linear in $T$ .
Lemma 6: the single-round mixture inequality.
Lemma 3: the upper bound $\sup_F\mathrm{Regret}(S^{\times T},F,T)\le T\sup_\theta\mathrm{Regret}(S,\delta_\theta,1)$ .

Significance

Theorem 1 replaces a $T$ -round dynamic mechanism design problem, whose decision variables are history-dependent rules over arbitrary message spaces, by a single-round problem over direct mechanisms and point masses. The single-round problem is often a linear program or is solvable with Myersonian methods. The paper uses this to compute the minimax regret $T/e$ for repeated selling of one good (Proposition 3) and to characterize the optimal mechanism of a principal–agent contract model (Proposition 4). For these games it shows that adaptive learning schemes cannot beat a myopic, non-adaptive mechanism under the minimax-regret criterion.

The result is proved on paper. The Lean statements in this mission compile, but their proofs remain open. A complete development would give a machine-checked model of repeated principal–agent interaction with a fully strategic, forward-looking agent: arbitrary report spaces, private randomization, augmented histories, and IC against all history-dependent deviations. It would also check a revelation-principle-type reduction (Lemma 4) whose informal proof leaves measurability and integrability implicit.

Difficulty

The obvious argument restricts nature to point masses and then "repeats the best single-round mechanism". It fails twice. First, an optimal dynamic mechanism may be IC only because of its dynamics; the lower bound needs a single-round mechanism $S(A)$ whose IC follows from IC of $A$ against deviations that mimic another point mass (Lemma 4). Second, a repeated mechanism must be IC against deviations that condition on the whole past. Showing that one round of such a deviation is itself a single-round deviation (Claim 1) requires the agent to simulate past shocks and the principal's randomness from private noise. Behind both steps sit time-averaged and conditional outcome laws that must be realized as measurable rules of a single uniform.

Formalization scope

The Lean game (FutileDynamics.StaticOpt.Game) gives the shock and outcome spaces, utilities, and no-interaction outcome. The mechanism model (FutileDynamics.StaticOpt.Model) imports it and fixes the following conventions, all disclosed in the items:

$u$ and $v$ are jointly measurable and bounded, so every expectation is a genuine Bochner integral.
$\Theta$ is a standard Borel space and nonempty, and $0<T$ .
Decision rules are jointly measurable. A feasible strategy, recommended or deviating, is jointly measurable in all its arguments, the distribution $F$ (Giry σ-algebra) included.
Rounds are 0-indexed (k : Fin T is round $k+1$ ), and uniforms live on unitInterval.
Every report space is a standard Borel space, so the set of measurable (feasible) deviations is not an artefact of the designer's choice of σ-algebra.
The round report space $\Theta\cup\{\mathrm{PASS}\}$ is Θ × Bool: $(\theta,\mathrm{true})$ reports $\theta$ , and every $(\theta,\mathrm{false})$ is PASS and yields $\emptyset$ . CONTINUE/QUIT is true/false.
$\mathrm{OPT}$ , $\mathrm{Regret}(A,T)$ , $\mathrm{Regret}(T)$ and the single-round value are real suprema and infima over sets that are nonempty (the always- $\emptyset$ mechanism is IC) and bounded.
The paper's $\bar u$ (E-8) is printed as a supremum over $\mathcal S^{\times1}$ , but its proof drops IC. The Lean $\bar u$ ranges over all direct rules (corrected).

The statement admits several trivializing formalizations, and the model rules each out:

$\mathcal A$ restricted to direct or static mechanisms. Here OPT and $\mathrm{Regret}(T)$ quantify over every report-space family.
IC required only against $F\in\mathcal F$ , only against point masses, or only against truthful misreports. Here IC holds against every probability distribution and every feasible history-dependent, randomized deviation, QUIT included.
$\mathcal S^{\times1}$ defined by the IC/IR program (3). Here it is IC in the dynamic sense at horizon 1.

A complete development needs:

trajectories of the interaction as measurable functions of the noise;
Fubini and independence arguments for i.i.d. shocks;
measurable sampling from a distribution on a standard Borel space;
splitting one uniform into countably many.

These pieces are reusable for any repeated-game or dynamic-mechanism formalization. Proofs of the milestones, of the lower and upper bounds, and of auxiliary measurability lemmas are welcome.

Selected references

S. R. Balseiro, A. Kim, D. Russo, On the futility of dynamics in robust mechanism design, Operations Research, 2021; SSRN working paper 3449698 (version of October 5, 2020), the text formalized here. https://doi.org/10.1287/opre.2021.2122 · https://ssrn.com/abstract=3449698

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Algorithmic Game Theory Mechanism Design Operations Research·Captain: mikedeng1

Preference Elicitation for Participatory Budgeting: Randomized Threshold Approval Votes Achieve Distortion O(log² m)Research Paper

Motivation

In participatory budgeting, residents of a city vote on how to spend part of a public budget on a list of proposed projects, each with a known cost. The practice has been adopted by cities in North America and Europe, including New York City, Chicago, Madrid and Paris, the last two each spending more than 100 million euros through participatory budgets in 2017 (Cabannes, 2004; paper §1). Every deployment must choose a ballot: voters may be asked to approve some projects, to rank them, or to fill a knapsack within the budget. The ballot determines how much information about the residents' preferences reaches the aggregation step, and so how much welfare the chosen set of projects can guarantee.

The implicit utilitarian framework of Procaccia and Rosenschein (2006) makes this question quantitative. Voters are assumed to hold cardinal utilities, which the ballot reveals only partially; the quality of a ballot-plus-aggregation pair is measured by its distortion, the worst-case ratio between the best achievable welfare and the welfare it actually obtains. Benadè, Nath, Procaccia and Shah (Management Science, 2021; AAAI 2017 conference version) apply this lens to participatory budgeting and compare the input formats used in practice. Their main positive result is that threshold approval votes with a randomly drawn threshold reach distortion $O(\log^2 m)$ , against $m$ for a rule that uses no information at all and $\Omega(\sqrt m)$ for any deterministic threshold.

Setting

There are $n$ voters $N=[n]$ and a set $A$ of $m$ alternatives (projects). Alternative $a$ costs $c_a\in[0,1]$ , and the budget is $1$ . Write $c(S)=\sum_{a\in S}c_a$ , and let

\mathcal F_c=\{S\subseteq A:\ c(S)\le 1,\ c(T)>1\text{ for all }S\subsetneq T\subseteq A\}

be the family of inclusion-maximal budget-feasible sets.

Voter $i$ has a utility $v_i(a)\ge 0$ for each alternative, utilities are additive over sets, and they are normalized: $\sum_{a\in A}v_i(a)=1$ . The social welfare of a set is $\mathrm{sw}(S,\vec v)=\sum_{a\in S}\sum_{i\in N}v_i(a)$ .

A threshold approval vote at threshold $t$ is the set $\tau_i=\{a\in A: v_i(a)\ge t\}$ . A randomized threshold rule chooses a threshold $t$ at random and maps the resulting profile $\vec\rho(\vec v,t)=(\tau_1,\dots,\tau_n)$ to a (possibly random) set in $\mathcal F_c$ . In the formalization, a rule is a finite mixture of arms; each arm has a probability, a threshold, and a deterministic aggregation map from approval profiles to $\mathcal F_c$ (ThresholdRule), and its expected welfare is the weighted sum over arms (ThresholdRule.expWelfare). The rule never sees the utilities, only the approval sets. The distortion of a rule is bounded by $D$ when

\max_{T\in\mathcal F_c}\mathrm{sw}(T,\vec v)\ \le\ D\cdot\mathbb E[\mathrm{sw}(\text{rule},\vec v)]\quad\text{for every normalized profile }\vec v,

and the distortion associated with an input format is that of the best rule for it.

Formalization targets

Goal: Theorem 3.5

\exists\,C>0\ \ \forall n,\ \forall A\text{ with }m\ge 2,\ \forall c\in[0,1]^A:\ \exists\text{ rule }R\ \ \forall\vec v,\ \forall T\in\mathcal F_c:\quad \mathrm{sw}(T,\vec v)\le C\log_2^2 m\cdot\mathbb E[\mathrm{sw}(R,\vec v)].

The goal fixes only the shape $O(\log^2 m)$ , with an absolute constant chosen before $n$ , $A$ and the costs, and it covers every $m\ge 2$ .

Milestones

For $m=2^K$ , with $\ell_j=2^{j-1}/m^2$ , $u_j=2^j/m^2$ , $I_j=(\ell_j,u_j]$ , cost buckets $S_j=\{a:c_a\in I_j\}$ ( $S_0$ uses $I_0=[0,1/m^2]$ ) and counts $n^a_j=|\{i:v_i(a)\in I_j\}|$ . In the statements involving $S^*$ , it is an optimal member of $\mathcal F_c$ for the given utility profile (IsWelfareOptimal).

Observation 3.1: some rule has distortion at most $m$ .
Eq. (6): $\sum_{j\ge1}n^a_j\ell_j\le\mathrm{sw}(a,\vec v)\le n/m^2+\sum_{j\ge1}n^a_ju_j$ .
At most $1/u_j$ alternatives from $S_j$ are budget-feasible.
Eq. (7): the same sandwich for $\mathrm{sw}(S^*\cap S_k,\vec v)$ .
Eq. (8): a greedy top set $B_{j,k}\subseteq S_k$ at threshold $\ell_j$ has $\mathrm{sw}(B_{j,k},\vec v)\ge\frac14 n^*_{j,k}u_j$ .
Mechanism A's expected welfare is at least $(\mathrm{sw}(S^*\setminus S_0,\vec v)-n/m)/(16\log^2 m)$ .
Mechanism B: $S_0$ is feasible and covers $S^*\cap S_0$ .
The explicit form for $m=2^K$ : distortion at most $2+16\log^2 m$ .

Significance

The theorem separates randomized from deterministic elicitation: with a fixed threshold no aggregation rule beats $\Omega(\sqrt m)$ (Theorem 3.4 of the paper), while drawing the threshold at random brings the distortion to $O(\log^2 m)$ , close to the $\Omega(\log m/\log\log m)$ lower bound of Theorem 3.6. It is the paper's argument for threshold approval as a practical, low-burden ballot, and it supports the empirical comparison in the paper's §5 between input formats on data from real participatory budgeting elections.

The result is proved in the paper; no machine-checked version is known. A formalization yields a reusable layer for implicit utilitarian voting with budgets (feasible-set families, normalized additive profiles, randomized rules over partial ballots, distortion), on which the paper's other bounds (Theorem 3.3 for rankings, Theorems 3.4 and 3.6 for lower bounds) and the related literature on distortion can be stated. It also makes explicit the reading of distortion under a random threshold that the proof actually uses.

Difficulty

The obvious rule picks the threshold and then funds the most-approved projects greedily. It fails because one threshold cannot serve both high-utility and low-utility voters, and because popular projects may be expensive: approval counts say nothing about costs. Mixing over thresholds alone does not help either; the ratio for a single badly chosen threshold can be of order $m$ . What has to be shown is that one fixed mixture, chosen before any utilities are seen, recovers a $1/O(\log^2 m)$ share of the optimum simultaneously for every profile, despite rounding of utilities into intervals, budget constraints within each cost class, and the loss of everything below the smallest threshold.

Formalization scope

All objects live in PBElicit.RandThresh (definition file PBElicit.RandThresh.Setting). Voters are Fin n; alternatives are a Fintype with $m$ = Fintype.card A. Utilities are real, nonnegative and normalized (IsUtilityProfile); costs satisfy $0\le c_a\le 1$ , which the paper uses when it calls a single alternative budget-feasible. Rules must return sets of $\mathcal F_c$ (IsMaxFeasible); the paper's mechanisms return feasible sets, which are extended to maximal ones. The maximum over $\mathcal F_c$ is stated as "for every $T\in\mathcal F_c$ "; distortion bounds are multiplicative. Logarithms are base $2$ . The milestones take $m=2^K$ , as the paper's proof does; the goal does not.

Distortion is read as the optimum over the total expected welfare (threshold and aggregation together), which is what the proof of Theorem 3.5 bounds; the paper's displayed definition puts the expectation over the threshold outside the ratio. The interval bound (8) is formalized as its outer inequality, because the printed middle step fails when a utility equals a threshold exactly.

The rule is fixed before the utilities, sees only threshold approval votes, and must return an inclusion-maximal feasible set; choosing the rule after the profile (which makes distortion $1$ trivially) is excluded by the quantifier order, and the uniform rule of Observation 3.1 only gives distortion $m$ , so a $C\log^2 m$ bound is a genuine improvement.

Welcome contributions: proofs of the milestones, the reduction from general $m$ to the next power of $2$ , and further results of the paper stated on the same definitions.

Selected references

G. Benadè, S. Nath, A. D. Procaccia, N. Shah, Preference Elicitation for Participatory Budgeting, Management Science, 2021 (AAAI 2017 conference version). https://doi.org/10.1287/mnsc.2020.3666
A. D. Procaccia, J. S. Rosenschein, The Distortion of Cardinal Preferences in Voting, Cooperative Information Agents (CIA), LNCS 4149, pp. 317–331, 2006. https://doi.org/10.1007/11839354_23
I. Caragiannis, A. D. Procaccia, Voting Almost Maximizes Social Welfare Despite Limited Communication, Artificial Intelligence 175(9–10), pp. 1655–1671, 2011. https://doi.org/10.1016/j.artint.2011.03.005
C. Boutilier, I. Caragiannis, S. Haber, T. Lu, A. D. Procaccia, O. Sheffet, Optimal Social Choice Functions: A Utilitarian View, Artificial Intelligence 227, pp. 190–213, 2015. https://doi.org/10.1016/j.artint.2015.06.003
Y. Cabannes, Participatory Budgeting: A Significant Contribution to Participatory Democracy, Environment and Urbanization 16(1), pp. 27–46, 2004. https://doi.org/10.1177/095624780401600104

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Algorithmic Game Theory Combinatorics Operations Research·Captain: mikedeng1

Best of Both Worlds: Ex-Ante and Ex-Post Fairness in Resource Allocation: Recursive Probabilistic Serial Is Ex-Ante Envy-Free and Ex-Post Envy-Free up to One GoodResearch Paper

Motivation

Many allocation problems hand out indivisible goods without money: course seats, shifts, donated food, inherited items. An integral allocation of such goods can rarely be envy-free. With one good and two agents, someone always envies. The standard remedies are of two kinds. Randomization restores fairness in expectation: the probabilistic serial rule of Bogomolnaia and Moulin (J. Econ. Theory 2001) yields a fractional allocation that is envy-free and can be implemented as a lottery over integral allocations. Approximation restores fairness after the draw: envy-freeness up to one good (EF1), introduced by Lipton, Markakis, Mossel and Saberi (EC 2004) and named by Budish (J. Political Economy 2011), asks only that any envy disappear once a single good is removed from the envied bundle.

The two remedies pull apart. Giving all goods to one agent chosen uniformly at random is envy-free ex-ante, but after the draw one agent holds everything. Round-robin picking with a uniformly random order is EF1 after every draw, and proportional but not envy-free in expectation (Proposition 2 and Example 1 of the paper). Freeman, Shah and Vaish (arXiv:2005.14122, 2020; with Aziz, Operations Research 2024) ask whether one lottery can be fair in both senses at once. They show that it can, and that the answer depends on how the lottery is built, not only on which fractional allocation it implements.

Setting

There is a finite set $N$ of $n \ge 1$ agents and a finite set $M$ of $m$ goods. Agent $i$ has an additive valuation with values $v_{i,g} \ge 0$ . A fractional allocation is a matrix $X$ with $X_{i,g} \in [0,1]$ the share of good $g$ given to agent $i$ , and $v_i(X_i) = \sum_g X_{i,g} v_{i,g}$ . An integral allocation gives each agent a bundle $A_i \subseteq M$ , with bundles pairwise disjoint, and $v_i(A_i) = \sum_{g \in A_i} v_{i,g}$ . A randomized allocation is a finite lottery $\{(p^k, A^k)\}_k$ over integral allocations. It implements the fractional allocation $\sum_k p^k A^k$ .

$X$ is envy-free (EF) if $v_i(X_i) \ge v_i(X_h)$ for all $i, h$ . An integral allocation $A$ is EF1 if for all $i, h$ with $A_h \neq \emptyset$ some $j \in A_h$ satisfies $v_i(A_i) \ge v_i(A_h \setminus \{j\})$ . A lottery is ex-ante EF if the allocation it implements is EF. It is ex-post EF1 if every allocation it draws is EF1.

The eating protocol $\mathrm{Eating}(M', 1)$ runs for one unit of time on the goods $M'$ , each of mass one. At every moment each agent eats her favorite good among those not yet fully consumed, at rate one. Recursive Probabilistic Serial (Algorithm 1) proceeds in $\lceil m/n \rceil$ stages, starting from $M^1 = M$ . Stage $t$ computes $X^t = \mathrm{Eating}(M^t, 1)$ and decomposes it as $X^t = \sum_k w^{t,k} A^{t,k}$ by the generalized Birkhoff–von Neumann theorem, Theorem 1 of the paper. It then draws $B^t = A^{t,k}$ with probability $w^{t,k}$ and continues on $M^{t+1} = M^t \setminus \bigcup_i B^t_i$ . The final bundles are $B_i = \bigcup_t B^t_i$ .

Formalization targets

Goal: Theorem 2 (p. 11)

For every instance, every way of breaking ties between equally valued goods, and every decomposition rule satisfying the conclusions of Theorem 1,

\text{the lottery of Algorithm 1 is ex-ante envy-free and ex-post EF1.}

Milestones

Theorem 1 (p. 9). A matrix with entries in $[0,1]$ and row and column sums at most $1$ is a convex combination of $0/1$ allocations. Each of their row and column sums is the floor or the ceiling of the corresponding sum of $X$ .
One eating stage (§3.1, p. 9). $\mathrm{Eating}(M', 1)$ satisfies the hypotheses of Theorem 1 and is supported on $M'$ . If $|M'| \ge n$ , every agent eats a total mass of exactly one.
One item per stage (§3.1, p. 9). Every allocation of the decomposition uses only goods of $M'$ . If $|M'| \ge n$ , it gives each agent exactly one good.
Each eating stage is envy-free (proof of Theorem 2, p. 11).
Lemma 1 (p. 11). In iteration $t < \lceil m/n \rceil$ , each agent weakly prefers the good she receives to every good still unallocated.
Lemma 2 (p. 11). Every allocation returned by Algorithm 1 is EF1.
Algorithm 1 returns a randomized allocation (§3.1, p. 9).
The expectation recursion (proof of Theorem 2, p. 11; Algorithm 3, p. 33). The implemented allocation is the first eating stage plus the weighted sum of the allocations implemented on the remaining goods.

Significance

Theorem 2 shows that ex-ante envy-freeness and ex-post EF1 are compatible for additive goods. Algorithm 1 is a constructive witness. The paper builds on this in two directions. A polynomial-time variant (§3.2) computes such a lottery. An impossibility result (Theorem 3) shows that adding ex-post fractional Pareto optimality is too much. The paper also extends the approach to bads and mixed manna (§7), where the natural extension guarantees only ex-post EF2.

The result is proved on paper. No formalization of probabilistic serial, the eating protocol, EF1 of randomized allocations or the rectangular Birkhoff–von Neumann theorem with floor and ceiling bounds is known to us in any proof assistant. Mathlib has the square doubly stochastic case (doublyStochastic_eq_convexHull_permMatrix), and Prove2Me has it as FamousTheorems.birkhoff_von_neumann. This mission asks for a machine-checked proof of Theorem 2 about an explicit definition of the eating protocol and of Algorithm 1.

Difficulty

The proof is short on paper but hides two pieces of continuous-time bookkeeping. First, the eating protocol is a process in time. The stage statements need a description of it in which "agent $i$ ate a positive share of $g$ while $g^*$ was still available" can be stated and checked. Second, the ex-ante argument conditions on the history of earlier draws. Because the decomposition may have exponentially many branches, it has to be turned into an induction over the recursive lottery. The obvious shortcut, proving the result for "the" BvN decomposition of each stage, fails because the decomposition is not unique and the theorem must hold for all of them. Weight-zero terms of a decomposition can violate Lemma 1. They are never drawn, but any argument over the whole support must exclude them explicitly.

Formalization scope

Agents and goods are finite types with decidable equality. The good set $M$ is the whole type of goods, and $n \ge 1$ is [Nonempty N]. Values are real and non-negative. All quantities are in $\mathbb R$ , and $\lceil m/n \rceil$ is Nat.ceil of a real quotient. Integral allocations are bundle maps N → Finset G, and lotteries are finite lists of (probability, allocation) pairs.

The eating protocol is a definition computed event by event. Between events every agent eats a fixed good chosen by a choice rule. The next event comes when the time budget ends or a good being eaten is exhausted. The minimum runs only over goods actually being eaten, so no division by zero occurs. $|M| + 1$ events suffice, and later iterations change nothing. Algorithm 1 is a definition: a recursion over stages producing a lottery over stage histories.

The only free parameters of the goal are the tie-breaking rule, constrained to choose a most valuable available good, and the decomposition rule, constrained to satisfy Theorem 1's conclusions. The eating outcome, the lottery and its support are computed by the definitions, never assumed. A statement taking $X^t$ or the lottery as free variables with envy-freeness as a hypothesis would trivialize the goal and is not this mission.

Conventions that differ from the page:

Ex-post EF1 is over allocations drawn with positive probability.
The without-loss-of-generality assumption that every good is valued positively by some agent is omitted. The proof does not use it, so the goal is stronger.
Theorem 1 is stated as existence, without its running-time clause.
Algorithm 1's printed line 2 initializes $M^0$ , while the loop reads $M^1$ . Its line 8 is a union over all stages inside the loop. Both are read as the intended algorithm, consistent with Algorithm 3 and with the proof of Theorem 2.
Histories are indexed from $0$ .

Not included are Proposition 1, the §3.2 polynomial-time variant, the SD-EF remark after Theorem 2, and §§4–7 (Theorem 3, the group-fairness results, the MNW characterization, bads). A proof of Theorem 1 in this rectangular form, with the floor and ceiling bounds, would be reusable well beyond this mission, for example in other fair-division and assignment results. So would a clean theory of the eating protocol. Contributions to either are welcome.

Selected references

R. Freeman, N. Shah, R. Vaish, Best of Both Worlds: Ex-Ante and Ex-Post Fairness in Resource Allocation, arXiv:2005.14122v1, 2020. https://arxiv.org/abs/2005.14122 (journal version with H. Aziz, Operations Research, 2024).
A. Bogomolnaia, H. Moulin, A New Solution to the Random Assignment Problem, Journal of Economic Theory 100(2), 2001. https://doi.org/10.1006/jeth.2000.2710
R. J. Lipton, E. Markakis, E. Mossel, A. Saberi, On Approximately Fair Allocations of Indivisible Goods, ACM EC 2004. https://doi.org/10.1145/988772.988792
E. Budish, The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes, Journal of Political Economy 119(6), 2011. https://doi.org/10.1086/664613
E. Budish, Y.-K. Che, F. Kojima, P. Milgrom, Designing Random Allocation Mechanisms: Theory and Applications, American Economic Review 103(2), 2013. https://doi.org/10.1257/aer.103.2.585

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Operations Research Probability Stochastic Systems·Captain: mikedeng1

When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence: Every Exogenous Service–Patience Dependence Has an Equivalent Endogenous DependenceResearch Paper

Motivation

In call centers, hospitals and other service systems, customers who wait too long leave: each has a patience time, and abandons the queue once the wait in queue exceeds it. Empirical work has found that service times and waiting are dependent. Customers who waited longer often need longer (or shorter) service, and those who were served are a self-selected group: they had enough patience to wait. Two models of this dependence are in use.

In the exogenous model, each customer arrives carrying a pair (service time, patience time) with a joint density. The dependence is a property of the customer population.
In the endogenous model, the service time a customer requires depends on the delay that customer actually experiences in queue, and the patience is independent of everything else.

The two models suggest different statistical procedures. Estimating an exogenous dependence means fitting a two-sided censored bivariate distribution. Estimating an endogenous one means a univariate patience estimate (Kaplan–Meier) plus a conditional service-time distribution per delay. Wu, Bassamboo & Perry (2022) prove that, as far as the law of the queueing dynamics is concerned, every exogenous dependence is an endogenous one. The simpler estimation procedure therefore never loses generality.

Setting

A service system has $n \ge 1$ identical servers. Customers arrive one at a time, at times $\kappa_1 < \kappa_2 < \cdots$ , and are served first come, first served. A customer who finds a free server starts service at once. Otherwise the customer waits, and abandons if the wait exceeds the customer's patience.

The offered wait $Z_i$ of customer $i$ is the time the customer would wait with infinite patience. With the sorting operator $\mathcal R$ (which sorts a vector of $\mathbb R^n$ in increasing order), $e_1 = (1,0,\dots,0)$ , $\mathbf 1 = (1,\dots,1)$ and interarrival times $\alpha_{i+1} = \kappa_{i+1} - \kappa_i$ , the offered-wait vectors follow the recursion

\mathbf Z_{i+1} = \mathcal R\big([\mathbf Z_i + \mathbb I_{\{\mathbf Z_i(1) \le T_i\}} S_i(\mathbf Z_i(1))\,e_1 - \alpha_{i+1}\mathbf 1]^+\big), \qquad Z_i = \mathbf Z_i(1). \tag{2}

Here $T_i$ is the patience of customer $i$ and $S_i(z)$ the service time it requires at offered wait $z$ .

The queue is described by the process $\mathbf X(t) = (Y(t), Q(t), U(t), V(t), B(t))$ , $t \ge 0$ . Its components are:

$Y(t)$ and $Q(t)$ , the numbers of customers in service and in queue;
$U(t) \in \mathbb R^n_+$ , which lists the idle servers as zeros, followed by the remaining service times in increasing order;
$V(t)$ , the elapsed waiting times of the customers in queue, in decreasing order and padded with zeros;
$B(t)$ , the time to the next arrival.
Exogenous dependence $f^{ex}$ . The pairs $(S^{ex}_i, T^{ex}_i)$ are i.i.d. with joint density $f^{ex}$ on $\mathbb R_+^2$ , and independent of the arrivals and the initial state. The service time does not depend on the wait. The patience marginal is $f^{ex}_T(y) = \int_0^\infty f^{ex}(x,y)\,dx$ .
Endogenous dependence $(f^{en}_T, \psi^{en})$ . The patience times $T^{en}_i$ are i.i.d. with density $f^{en}_T$ . Customer $i$ has a virtual service-time process $z \mapsto S^{en}_i(z)$ whose value at $z$ has density $\psi^{en}(\cdot, z)$ . These are independent across customers, of the patience and of the arrivals.

$\mathcal L^{ex}(A, n; f^{ex})$ and $\mathcal L^{en}(A, n; f^{en}_T, \psi^{en})$ denote the laws of the process $\mathbf X$ . The two dependencies are equivalent (Definition 1) if these laws agree for every arrival process $A$ and every $n$ , whenever the initial states agree in law.

Formalization targets

Goal: Theorem 1 (p. 3348)

For every joint density $f^{ex}$ on $\mathbb R_+^2$ , the pair

f^{en}_T(z) = f^{ex}_T(z), \qquad \psi^{en}(x,z) = \frac{\int_z^\infty f^{ex}(x,y)\,dy}{\int_z^\infty f^{ex}_T(y)\,dy} \tag{1}

is an endogenous dependence (a patience density and a conditional service-time density), and it is equivalent to $f^{ex}$ :

\mathcal L^{ex}(A, n; f^{ex}) = \mathcal L^{en}(A, n; f^{en}_T, \psi^{en}) \quad \text{for every } n \ge 1 \text{ and every arrival process } A.

Milestones

Formula (1) defines densities.
Lemma 1, sufficiency: the law of $\mathbf X^{en}$ is determined by $(f_T, \psi)$ , the arrivals and $n$ .
The pathwise coupling: two systems in which every customer has the same outcome at its offered wait have the same offered waits and the same path $\mathbf X$ .
The display after the proof: $\bar\Psi^{en}(x,z)\bar F^{en}_T(z) = \int_z^\infty\int_x^\infty f^{ex}(u,v)\,du\,dv$ , decreasing in $x$ and in $z$ .
The coupling kernel: the pair generated by steps (ii)–(iii) of the proof has joint density $f^{ex}$ , whatever the offered wait.
The law of $\mathbf X^{ex}$ is determined by $f^{ex}$ , the arrivals and $n$ .

Significance

The theorem shows that the systems with exogenous dependence form a subclass of those with endogenous dependence. Two consequences follow:

Any statement about the law of a queue with exogenous dependence (delay distribution, abandonment probability, occupancy) is a statement about some endogenous system.
Statistical analysis can be carried out with the endogenous model, a Kaplan–Meier patience estimate and binned conditional service-time distributions, without assuming the dependence is truly endogenous.

The formula for $\psi^{en}$ also explains the self-selection effect. It is the service-time density of a customer, conditioned on having patience exceeding the offered wait.

The paper's proof omits the verification step "for brevity". It also states the law-determination facts (milestones 2 and 6) without proof, and treats an initially empty system. A formal proof would supply the per-customer density computation, the measure-theoretic construction of the coupled system, and the measurability of the path map.

Difficulty

The obvious argument would compare the two systems customer by customer with the same random draws. It fails: in the exogenous system the service time and the patience are drawn together before the customer meets the queue, while in the endogenous system the service time is drawn after the offered wait is known. Equality holds only in law, through a coupling built on a new probability space, and the coupled pairs must be shown to be i.i.d. with density $f^{ex}$ even though they are constructed from the state of the queue.

Two technical points are not visible in the paper. First, the law of the whole path $t \mapsto \mathbf X(t)$ must be shown to be determined by the primitives. This requires measurability of a map built from sorting, counting and an infimum over arrival indices. Second, the conditional densities in (1) and in steps (ii)–(iii) are ratios whose denominators can vanish, so null sets need care.

Formalization scope

Everything lives in the namespace DelayDep.Equiv. The conventions are:

Paths and time. Customers are numbered from $0$ (customer $i$ is the paper's customer $i+1$ ). Time is NNReal. The law of $\mathbf X$ is the image measure on NNReal → State n with the product σ-algebra, which is the σ-algebra of the finite-dimensional distributions.
Densities. Densities are real-valued, nonnegative and measurable on $[0,\infty)$ or $[0,\infty)^2$ , and enter measures through withDensity.
Ties. Recursion (2)'s convention is used: a customer is served iff offered wait $\le$ patience. Ties are null events.
Arrivals and initial state. The arrivals are any measurable positive interarrival times whose sums tend to infinity. The initial state is an empty queue with arbitrary nonnegative residual service times. "Any arrival process $A$ with $\mathbf X^{ex}(0) \overset{d}{=} \mathbf X^{en}(0)$ " becomes "any joint law of interarrival times and initial residual service times, the same in both systems".
Assumption 1. Assumption 1 (differentiability of $\Psi^{en}$ ) is not imposed. The $\psi^{en}$ of (1) need not satisfy it, and Theorem 1 does not use it.
Tail guard. The endogenous service-time law is prescribed only at offered waits with positive patience tail. Elsewhere a customer abandons almost surely, and (1) is $0/0$ .

Every law equality includes the almost-everywhere measurability of both path maps as part of its conclusion. Without it, the equality could hold trivially through Lean's convention that the image of a measure under a non-measurable map is $0$ . For the same reason, the goal also asserts that (1) defines densities, so that the endogenous systems it quantifies over are not vacuous.

Out of scope are Theorem 2, the necessity half of Lemma 1, Example 1, Theorem A.1 and the estimation section §4. Useful contributions are:

a general measurability lemma for sample-path functionals of this kind;
the kernel computation of milestone 5;
the construction of the coupled system;
the law-determination lemmas, which apply to any queue driven by i.i.d. customer primitives.

Selected references

C. A. Wu, A. Bassamboo, O. Perry, When service times depend on customers' delays: A relationship between two models of dependence, Operations Research 70(6):3345–3354, 2022. https://doi.org/10.1287/opre.2021.2179

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Operations Research Probability Stochastic Systems·Captain: mikedeng1

Technical Note–A Simple Heuristic Policy for Stochastic Distribution Inventory Systems with Fixed Shipment Costs: An Upper Bound on the Long-Run Average Cost of Modified Echelon (r,Q) PoliciesResearch Paper

Motivation

A one-warehouse, multi-retailer (OWMR) distribution system is the basic model of a supply chain in which a central warehouse buys from an outside supplier and ships to several retail locations. When every shipment carries a fixed cost and a positive lead time, the optimal policy of the stochastic system is unknown, and the classical approach is to search over a structured class of policies and bound its cost. Exact evaluation of even a simple policy is hard, because the warehouse can run out of stock and then the retailers' replenishments interact.

Zhu, Chen, Hu and Yang (SSRN 2921743; Operations Research, 2021, doi:10.1287/opre.2021.2101) study modified echelon $(r,Q)$ policies for continuous-review OWMR systems with Poisson demand. Their central tool is an explicit upper bound on the long-run average cost of any such policy, written as the retailers' single-stage $(r,Q)$ costs, a warehouse term that depends on the retailer policies, and a fixed-cost allowance. Their heuristic policy, its worst-case performance guarantee and its asymptotic optimality (Theorems 2–4 of the paper) are all built on this bound. The single-stage cost functions follow Zheng (1992) and the echelon cost accounting follows Chen and Zheng (1994).

Setting

There are $N\ge1$ retailers, indexed $i=1,\dots,N$ , and one warehouse, indexed $0$ . Retailer $i$ faces Poisson demand of rate $\lambda_i>0$ , independent across retailers, so the warehouse sees Poisson demand of rate $\lambda_0=\sum_i\lambda_i$ . Installation $i$ has a constant lead time $L_i>0$ , a fixed cost $K_i>0$ per shipment, and an echelon holding cost rate $h_i>0$ ; unmet retailer demand is backlogged at rate $p_i>0$ . Write $\bar K=\max_iK_i$ .

A modified echelon $(r,Q)$ policy has reorder levels $r_i$ and order quantities $Q_i>0$ for every installation. When the echelon inventory position $IP_i$ of installation $i$ is at or below $r_i$ and the upstream installation has stock on hand, a shipment raises $IP_i$ as close as possible to $r_i+Q_i$ .

With $D_i\sim\mathrm{Poisson}(\lambda_iL_i)$ the lead-time demand of retailer $i$ , the single-stage holding-and-backlog rate and the single-stage $(r,Q)$ cost are

G_i(y)=\mathbb E\big[h_i(y-D_i)^++(h_0+p_i)(y-D_i)^-\big],\qquad C_i(r_i,Q_i)=\frac{\lambda_iK_i+\int_{r_i}^{r_i+Q_i}G_i(y)\,dy}{Q_i}.

With $G_i(w_i)=\sup_{r_i<z\le r_i+Q_i}G_i(z)$ , $M_j=\max\{G_j(w_j),C_j(r_j,Q_j)\}$ and $S_i=\sum_{j\ne i}(r_j+Q_j)$ , the envelope of Lemma 3 at warehouse echelon level $x$ is

\bar\Gamma_i(x)=\sum_{j\ne i}M_j+\begin{cases}M_i & x-S_i>r_i,\\ \max\{G_i(x-S_i),M_i\} & \text{otherwise},\end{cases}\qquad \bar\Gamma(x)=\max_i\bar\Gamma_i(x).

With $D_0\sim\mathrm{Poisson}(\lambda_0L_0)$ , set $\hat G=\bar\Gamma-\sum_iC_i$ , $\Lambda_0(y)=\mathbb E[h_0(y-D_0)+\hat G(y-D_0)]$ and

\hat C_0(r_0,Q_0)=\frac1{Q_0}\Big[\lambda_0K_0+\int_{r_0}^{r_0+Q_0}\Lambda_0(y)\,dy\Big].

The long-run average system cost is (1) of the paper,

C(r,Q)=\frac1{Q_0}\Big[\lambda_0K_0+\int_{r_0}^{r_0+Q_0}\mathbb E\big[h_0(y-D_0)+\Omega(y)\big]\,dy\Big],

where $\Omega(y)$ is the expected cost rate of all retailers when the warehouse inventory position is $y$ .

Formalization targets

Goal: Theorem 1 (p. 13)

C(r,Q)\le C^U(r,Q)\equiv\sum_{i=1}^NC_i(r_i,Q_i)+\hat C_0(r_0,Q_0)+\frac{\lambda_0\bar K}{Q_0}.

It is stated for every policy, with no assumption on the parameters beyond $Q_i,Q_0>0$ .

Milestones

App. A (3): the per-retailer bound: the charge of retailer $i$ is at most the own-retailer term of $\bar\Gamma_i(x)$ .
Lemma 3: the total retailer charge at a feasible state is at most $\bar\Gamma(x)$ .
App. A (8): $\hat\Gamma(x)\le\bar\Gamma(x)=\hat G(x)+\sum_iC_i$ .
App. A (9): $\Gamma_0(y)\le\Lambda_0(y)+\sum_iC_i$ .
App. A (10): $\frac1{Q_0}[\lambda_0K_0+\int_{r_0}^{r_0+Q_0}\Gamma_0]\le\hat C_0(r_0,Q_0)+\sum_iC_i$ .

Significance

The bound replaces the cost of an $(N+1)$ -installation stochastic system, which has no closed form, by computable retailer costs and a warehouse term. This is what makes the paper's heuristic policy computable and what its performance guarantees compare against: each retailer uses its single-stage optimal $(r_i^*,Q_i^*)$ , and the warehouse minimizes the remaining term. Bounds of this decomposition type are a standard device in multi-echelon inventory theory, and Lemma 3's envelope, which charges each retailer at the lowest inventory position the others can leave it, is the step specific to the distribution setting.

The result is proved in the paper's online appendix. It has not been formalized. A formal development adds to the platform a single-stage $(r,Q)$ cost with Poisson lead-time demand, the echelon envelope of Lemma 3 for general $N$ , and an explicit OWMR cost decomposition, all of which are reusable by later work on distribution systems, including the paper's own performance bounds.

Difficulty

The core is Lemma 3 for general $N$ . Bounding each retailer separately by its worst case $\max\{G_i(w_i),C_i\}$ fails, because a retailer below its reorder level can have a much larger cost rate $G_i(IP_i)$ ; the bound must use that the inventory positions are coupled through the warehouse echelon level. The printed proof is an induction on $N$ whose step needs convexity of a sum in which one term is only bounded by a convex function, so the order of bounding and convexity arguments matters. Convexity and continuity of $G_i$ (a Poisson mixture of piecewise-linear functions) must be established, and every expectation against the Poisson law needs an integrability argument, since the integrands grow linearly.

Formalization scope

All declarations live in the namespace DistribRQ.UpperBound. Retailers are indexed by Fin N with [NeZero N]; the warehouse data $L_0,K_0,h_0,r_0,Q_0$ are separate. The primitives are bundled in Primitives N (all rates and costs positive) and the policy in Policy N (real $r_i,r_0$ ; $Q_i,Q_0>0$ ), following the paper's continuous approximation (p. 7). Poisson expectations are Bochner integrals over $\mathbb N$ against poissonMeasure, evaluated at real arguments $y-n$ . $G_i(w_i)$ is the supremum of $G_i$ over $(r_i,r_i+Q_i]$ , since the paper's $\arg\max$ need not be attained on a half-open interval.

The sample-path model (cycles, shipment intervals, the paper's Lemmas 1–2) has no substrate, so the system is encoded as the paper's proof uses it:

$C(r,Q)$ is (1), with $\Omega$ any interval-integrable function satisfying the bound of p. 12, $\Omega(y)\le\lambda_0\bar K/Q_0+\mathbb E[\hat\Gamma(y-D_0)]$ ; this bound is where Lemmas 1–2 enter and is a hypothesis.
$\hat\Gamma(x)=\int\sum_ic_i(z_i)\,\kappa(x)(dz)$ for any Markov kernel $\kappa$ concentrated on the feasible states of p. A1 ( $z_j\le r_j+Q_j$ ; $\sum_jz_j\le x$ ; and $\sum_jz_j<x$ forces $z_i>r_i$ for all $i$ ), and any measurable, nonnegative admissible charges ( $c_i(z)\le\max\{G_i(z),C_i\}$ for $z>r_i$ , $c_i(z)\le G_i(z)$ for $z\le r_i$ ).
The theorem holds for every such $\kappa$ , $c$ and $\Omega$ , so the real system is an instance.

$\hat\Gamma$ is never a free function bounded by $\bar\Gamma$ , the feasible set keeps the clause for positive warehouse stock, and $G_i$ , $C_i$ , $\Lambda_0$ are computed from the Poisson law rather than taken as parameters; formalizations that relax any of these, or that rely on a junk value of sSup, an integral or a division, trivialize the statement and are not accepted.

Contributions welcome: convexity and continuity of $G_i$ , Poisson integrability lemmas, Lemma 3 for general $N$ , and the averaging steps of App. A (8)–(10).

Selected references

H. Zhu, Y. (F.) Chen, M. Hu, Y. Yang, Technical Note—A Simple Heuristic Policy for Stochastic Distribution Inventory Systems with Fixed Shipment Costs, Operations Research, 2021. SSRN copy of 2020-09-08: https://ssrn.com/abstract=2921743; doi:10.1287/opre.2021.2101.
Y.-S. Zheng, On properties of stochastic inventory systems, Management Science 38(1), 1992. https://doi.org/10.1287/mnsc.38.1.87
F. Chen, Y.-S. Zheng, Lower bounds for multi-echelon stochastic inventory systems, Management Science 40(11), 1994. https://doi.org/10.1287/mnsc.40.11.1426

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Operations Research Probability Statistics·Captain: mikedeng1

Optimal Sequential Multi-class Diagnosis: Under Exponential Tilting, Every Reachable Posterior Belief Is a Closed-Form Function of an r-Dimensional Diagnostic Statistic, r = rank(B)Research Paper

Motivation

In sequential multi-class diagnosis, a decision maker observes a system repeatedly, then chooses when to stop and which hidden class to report. Each new observation can improve the diagnosis, but waiting has a cost. A direct dynamic program uses the vector of posterior class probabilities as its state. With $N+1$ possible classes, that state lies in an $N$ -dimensional probability simplex. Wang's model asks when the observed sequence can instead be summarized by fewer coordinates, so that the same posterior can be recovered without retaining a full belief vector as an independent state Wang, §2–§3.

The paper considers class distributions related by exponential tilting. This relationship appears in common one-dimensional distribution families, including the normal, binomial, Poisson, and gamma examples collected in its Table 1 Wang, p. 8. The result targeted here is the belief reconstruction identity of Theorem 1. Later parts of the paper use the reduced statistic in optimality equations and study stopping regions; those results are outside this mission Wang, pp. 12–18.

Setting

The hidden class is $j\in\{0,\ldots,N\}$ . Before observing data, the decision maker has a prior belief $\theta=(\theta_0,\ldots,\theta_N)$ : every component is nonnegative and their sum is one. Conditional on class $j$ , observations $y_1,\ldots,y_t$ are independent and have the same density $f_j$ on the observation space $\mathcal Y$ . Class $0$ supplies a baseline density $f_0$ . The one-step Bayesian operator $\mathcal B$ multiplies each current class weight by that class's likelihood at the new observation and divides by the sum of all weighted likelihoods. Starting at $\Pi_0=\theta$ , the belief $\Pi_t$ is obtained by applying $\mathcal B$ once for each observation in arrival order Wang, pp. 6–7.

In the exponential tilting model, a transformed observation $h(y)\in\mathbb R^p$ and a parameter $\beta_j\in\mathbb R^p$ describe each nonbaseline density:

f_j(y)=f_0(y)\exp\{\beta_{j0}+\beta_j^{\mathsf T}h(y)\},\qquad \beta_{j0}=-\log\int_{\mathcal Y}\exp\{\beta_j^{\mathsf T}h(y)\}f_0(y)\,dy.

Here $j=1,\ldots,N$ . Place the $\beta_j^{\mathsf T}$ in the rows of the diagnostic parameter matrix $\mathbb B$ . Its rank decomposition is $\mathbb B=\mathbb R\mathbb P$ , where $\mathbb R$ has $N$ rows and $r$ linearly independent columns, while $\mathbb P$ has $r$ linearly independent rows and $p$ columns. Thus $r=\operatorname{rank}(\mathbb B)$ . The paper calls $\mathbb R$ the reconstruction matrix and $\mathbb P$ the projection matrix Wang, Definition 1, p. 8.

Formalization targets

Theorem 1: posterior reconstruction

For $t\ge1$ , define the diagnostic statistic $x_t=\mathbb P\big(\sum_{m=1}^t h(y_m)\big)/t\in\mathbb R^r$ . For classes $j=1,\ldots,N$ , the target identity is

\Pi_t(y_1,\ldots,y_t,\theta)_j =\mathcal T_j^t(x_t,\theta) =\frac{\theta_j\exp\{t(\beta_{j0}+e_j\mathbb R x_t)\}} {\theta_0+\sum_{i=1}^{N}\theta_i\exp\{t(\beta_{i0}+e_i\mathbb R x_t)\}}.

The baseline coordinate is $\mathcal T_0^t=1-\sum_{j=1}^{N}\mathcal T_j^t$ . The normalizer appearing in $\mathcal T^t$ is evaluated from the row $e_i\mathbb R\mathbb P$ , as in equation (3), and the theorem identifies it with the model normalizer through $\mathbb B=\mathbb R\mathbb P$ . The target also identifies the set of posteriors reachable after exactly $t$ observations with $\mathcal T^t(\Omega_t,\theta)$ , where $\Omega_t$ is the set of attainable diagnostic statistics. These statements give a precise reading of the paper's claim that reachable beliefs admit an $r$ -coordinate representation Wang, Theorem 1, p. 9; Definition 2, p. 10.

The goal additionally states that the map $x\mapsto\mathcal T^t(x,\theta)$ is injective on $\mathbb R^r$ when every prior component is positive. This is an explicit, qualified reading of the paper's phrase that the dimensionality is equal to $r$ . For discrete observations, the reachable set $\Omega_t$ can be finite, so the phrase cannot always mean its geometric dimension is literally $r$ . The five milestones follow the source's Bayes operator and the displayed identities in its proof of Theorem 1 Wang, p. ec1.

Significance

The formula determines every posterior component from $r$ real coordinates, the prior, and the observation count. When $r$ is smaller than $N$ , this supplies the representation used by the paper to reformulate a diagnosis problem in a smaller state space. The paper's examples show cases with one or two diagnostic coordinates; Proposition 1 then rewrites its optimality equations using those coordinates Wang, pp. 9–12. This mission formalizes the reconstruction claim itself, including the exact reachable-set statement, while leaving the decision costs and Bellman equations for later work.

A completed development would provide reusable definitions for a finite-class Bayes update and its sequential iterate, an exponential-tilting model over densities, and a rank-factorized reconstruction map. The result is proved in the cited paper. The Lean declarations in this proposal compile as statements, but their sorry bodies are open proof obligations; this mission seeks machine-checked proofs of them.

Difficulty

The main verification burden is connecting several expressions for the same posterior without quietly replacing one definition by another. A sequential posterior is an iterated one-step update; the paper's closed form is a normalized product of likelihoods; and equation (3) is an exponential expression in a projected sum of transformed observations. Each form has its own denominator, so the positivity conditions that make division meaningful must be stated. The rank claim also requires full-rank assumptions on both factors of $\mathbb B$ . For a prior with zero mass on a class, the posterior formula remains meaningful, but the injectivity assertion needs a separate full-support condition.

Formalization scope

Lean represents classes by Fin (N + 1), with class zero as the baseline. Row i : Fin N of the diagnostic matrix is class i.succ. Observations form a List in arrival order; $t$ is its length, and the posterior is the list fold of $\mathcal B$ . The observation space is a measurable type with a reference measure $\mu$ , covering continuous and discrete densities with one notation. This generalizes the paper's $\mathcal Y\subseteq\mathbb R$ ; no order or arithmetic operation on observations is used by Theorem 1. Integrals are real-valued, and $N=0$ and $p=0$ are permitted. The reconstruction identity is stated for $t\ge1$ , matching the paper and avoiding the division by zero in $x_t$ .

The standing assumptions say that every $f_j$ is measurable, $f_0$ is positive at every admitted observation, its integral is one, and each tilted baseline density is integrable. These make the normalizers and the Bayesian denominators meaningful. Positivity, normalization, and integrability are explicit refinements of the paper's use of densities and normalizing constants. The prior lies in the standard simplex; $\theta_0$ need not be positive for reconstruction. The additional full-support condition appears only in the injectivity clause. The paper's conditional independence assumption is represented by multiplying class likelihoods along the list.

The generic Bayesian update and its reachable set are separate from the paper-specific model. In particular, the posterior is an actual fold of the one-step update, and the class densities are given data constrained by exponential tilting. Defining the posterior directly as equation (3), or defining the densities to make the tilting identity true by construction, would erase the result being formalized. A complete proof needs finite-sum identities, exponential and logarithm facts, integrability facts, matrix rank and row identities, and the algebra of probability normalization. The generic Bayes definitions and list-level likelihood identity are reusable beyond this paper; contributions that establish those interfaces or close any source-derived milestone are welcome.

Selected references

J. Wang, Optimal Sequential Multi-class Diagnosis, Operations Research 70(5), 2022; arXiv:1506.08915v4. The mission cites the v4 preprint's pages and indices.

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Operations Research Probability Stochastic Systems·Captain: mikedeng1

Exploiting Random Lead Times for Significant Inventory Cost Savings: With Exponential Lead Times, the Generalized Base-Stock Policy's Scaled Stationary Inventory Level Converges to N(0, 1/(βγ))Research Paper

Motivation

In continuous-review inventory systems with backlogging, the classical constant base-stock (CBS) policy orders one unit whenever one unit of demand arrives, so that the inventory position (net inventory plus in-transit inventory) stays constant. Under CBS with Poisson demand of rate $r$ and i.i.d. lead times with mean $1/\beta$ , the in-transit inventory has a Poisson law with mean $r/\beta$ , and the steady-state net inventory level $Y^r(\infty)$ fluctuates on the scale $\sqrt r$ : $\mathbb E|Y^r(\infty)|/\sqrt r \to 2(2\pi\beta)^{-1/2}$ . Average holding and backlog costs therefore grow like $\sqrt r$ .

Stolyar and Wang (arXiv:1801.02646, Operations Research 2022) observe that when lead times are random and orders may cross, an inventory manager can exploit the randomness: by ordering more aggressively when the inventory level is low and less when it is high, the variance of the inventory level can be reduced. Their generalized base-stock (GBS) policy implements this with a single gain parameter $\gamma$ . The main theoretical result, Theorem 1, identifies the diffusion-scale steady-state law of the inventory level under GBS when lead times are exponential: it is normal with variance $1/(\beta\gamma)$ , a factor $\gamma$ below the CBS variance. With $\gamma = \gamma(r) \to \infty$ (Corollary 2), the scaled average cost $\mathbb E|Y^r(\infty)|/\sqrt r$ tends to $0$ , so GBS becomes infinitely better than CBS in the limit.

Setting

Demand units (customers) arrive as a Poisson process of rate $r$ . Supply units (items) are ordered by the policy and each arrives after an independent exponential lead time with mean $1/\beta$ . Let $Y$ be the net inventory level (positive or negative, backlog allowed) and $Z \ge 0$ the number of items in transit. Write $a\vee b=\max(a,b)$ , $a\wedge b=\min(a,b)$ , $a^+=a\vee 0$ , $a^-=(-a)^+$ .

The GBS policy has an integer gain $\gamma\ge 1$ , a truncation level $f>0$ and a centering $x_*$ . With $X_* = r/\beta$ it uses the target and the truncated target

X = X_* - \gamma(Y - x_*), \qquad T = [X\wedge(X_*+f)]\vee 0 .

At every arrival: a customer sets $Y := Y-1$ ; an item sets $Y := Y+1$ , $Z := Z-1$ . Then $T$ is recomputed and $A = \lceil T - Z\rceil\vee 0$ items are ordered, $Z := Z + A$ . Under exponential lead times $(Y,Z)$ is a continuous-time Markov chain on $\mathbb Z\times\mathbb Z_+$ with customer rate $r$ and item rate $\beta Z$ .

The systems are indexed by $r\to\infty$ , with $f = f(r)$ satisfying

f(r)/r\to 0, \qquad f(r)/\sqrt r\to\infty,

and $x_* = x_*(r)$ arbitrary. A stationary distribution $p$ of a countable chain with rates $q(s,s')$ and exit rates $q(s)$ is a probability mass function with finite mean exit rate satisfying global balance $\sum_s p(s)q(s,s') = p(s')q(s')$ .

Formalization targets

Goal: Theorem 1

For every $r>0$ some state is reachable from every state, for all large $r$ the chain has a unique stationary distribution, and for every choice of stationary distributions

\frac{Y^r(\infty)-x_*}{\sqrt r}\Rightarrow\mathcal N\big(0,(\beta\gamma)^{-1}\big),\qquad \mathbb E\Big[\frac{Y^r(\infty)-x_*}{\sqrt r}\Big]^{\pm}\to(2\pi\beta\gamma)^{-1/2}.

Milestones

In proof order: the stability part of Theorem 1; the conservation law $\beta\mathbb E Z^r(\infty)=r$ (14); the mean bound for the reflected random walk $Q_n$ that dominates the order gap (p. 16, EC.3); Lemma 4, $\mathbb E D^r(\infty)\le\mathbb E\tilde D^r(\infty)\le C$ ; the conservation law (19) and Lemma 5 for the artificial process, a birth–death chain $\hat Y^r$ that keeps $\hat Z = \lceil\hat T\rceil$ exactly; Corollary 6; the stochastic comparison Lemma 8, $\hat Y^r(\infty)\le_{st}Y^r(\infty)$ and $\hat T^r(\infty)\ge_{st}T^r(\infty)$ ; the target-gap bound (31); the transfer Lemma 9; and the reduction (36), $\mathbb E Y^r(\infty)/\sqrt r - \mathbb E\hat Y^r(\infty)/\sqrt r\to 0$ . For §4 milestones the centering is $x_*\equiv 0$ , as in the paper (p. 12).

Significance

The result. Theorem 1 quantifies the value of random lead times: the GBS policy divides the asymptotic variance of the steady-state inventory level by $\gamma$ , and the half-normal limits of $\mathbb E[\cdot]^\pm$ give the asymptotic holding and backlog costs, which is what makes the cost comparison with CBS (bounds (12)–(13) and Corollary 2) possible. The artificial-process comparison and Lemma 9 form a reusable template for transferring a diffusion limit from a tractable one-dimensional chain to a higher-dimensional controlled chain.

Formalizing it. The result is proved in the paper; no machine-checked version exists. The mission formalizes the GBS chain itself as explicit transition rates, so the theorem cannot be discharged by a model that only resembles the policy. The stability part is only sketched in the paper ("We omit further details of the stability proof", p. 13), so a complete formal proof also fills that gap. Lemma 5 is a stationary Halfin–Whitt-type limit for a birth–death chain with state-dependent up-rates, including uniform integrability of the scaled tails.

Difficulty

The two-dimensional chain has no product form, and its in-transit inventory does not track the target: after an item arrival $Y$ rises by one while the target $X$ falls by $\gamma$ , but only one item leaves transit, so $Z$ overshoots its target and the gap $D = Z - \lceil T\rceil$ is random. The natural argument, "the chain behaves like the birth–death chain with $Z = \lceil T\rceil$ ", fails on two counts: the steady-state gap must be bounded uniformly in $r$ (Lemma 4), although the rates of the chain grow with $r$ and $Z$ is unbounded on the state space, and the truncations of $T$ at $0$ and at $X_*+f$ produce error terms that must vanish after $1/\sqrt r$ scaling (Corollary 6). Stochastic order alone gives only an inequality between the two laws; closing it needs the mean identity (36), which rests on the exact conservation laws (14) and (19).

Formalization scope

All objects live in the namespace RandomLeadGBS.Diffusion. The state is $(Y,Z)\in\mathbb Z\times\mathbb N$ ; the demand rate $r$ is real and $r\to\infty$ along atTop. The parameters are $\beta>0$ , $\gamma\in\mathbb N$ with $\gamma\ge 1$ , $f:\mathbb R\to\mathbb R$ with $f>0$ and (10), and arbitrary $x_*:\mathbb R\to\mathbb R$ . The policy is encoded literally: target (4), truncated target (5), order step $Z + \mathrm{toNat}\lceil T - Z\rceil$ applied after every arrival; the item rate is $\beta Z$ and the cap is $X_*+f$ (not $X_*+\gamma x_*+f$ ).

Conventions committed to:

a stationary distribution is a probability solution of global balance with finite mean exit rate (the state space is unbounded in $Z$ );
expectations and probabilities are series over states, and every expectation bound also asserts the summability it needs;
weak convergence $\Rightarrow$ is stated by its definition, through every bounded continuous test function, against Mathlib's gaussianReal 0 (1/(βγ));
the paper's "irreducible" is replaced by "some state is reachable from every state", because states with $Z<\lceil T(Y)\rceil$ are never entered;
"uniformly in $r$ " and " $\sup_r$ " (Lemma 4, Corollary 6, (31)) are stated for all $r\ge r_0$ .

The goal concerns the two-dimensional GBS chain; stating it for the artificial chain, or only for an empty family of stationary laws, would trivialize it. The uniqueness clause and the existence clause of Lemma 5 rule out the vacuous reading.

A complete development needs a theory of stationary laws of countable continuous-time chains (existence via drift/Foster criteria, embedded-chain time change), birth–death stationary laws and their local normal asymptotics, and stochastic comparison of chains by coupling. The countable-chain stationarity layer and Lemma 9 are reusable beyond this mission. Proofs of any milestone, and alternative routes to the goal, are welcome. Propositions 3 and 7 (generic comparison results used for Lemma 4 and Lemma 8) are not posed as milestones.

Selected references

A. L. Stolyar and Q. Wang, Exploiting random lead times for significant inventory cost savings, Operations Research, 2022; preprint arXiv:1801.02646v6, 2021. https://arxiv.org/abs/1801.02646
S. M. Disney, A. Maltz, X. Wang and R. D. H. Warburton, Inventory management for stochastic lead times with order crossovers, European Journal of Operational Research 248(2), 473–486, 2016. https://doi.org/10.1016/j.ejor.2015.07.047
S. Halfin and W. Whitt, Heavy-traffic limits for queues with many exponential servers, Operations Research 29(3), 567–588, 1981. https://doi.org/10.1287/opre.29.3.567
T. Lindvall, Lectures on the Coupling Method, Dover, 1992.
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer, 1993. https://doi.org/10.1007/978-1-4471-3267-7

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Algorithmic Game Theory Economics Operations Research·Captain: mikedeng1

Implications of Price Matching in Supply Chain Negotiation: Under Symmetric Bargaining Power the Seller Prefers Price Matching to Simultaneous Negotiation and Both Buyers Prefer the ReverseResearch Paper

Motivation

A seller that supplies several competing buyers may negotiate a wholesale price with each of them independently and simultaneously (simultaneous negotiation, SN), or may commit to charge all of them the same price (price-matching negotiation, PM). PM is the bargaining counterpart of the most-favoured-customer clause, a contractual guarantee that a customer receives the best price the vendor gives anyone. Retailers such as Walmart and Costco demand such clauses from common suppliers, and buyers command them expecting to obtain the same price as their rivals (Shang and Cai, §1). Shang and Cai (MSOM, 2021; SSRN 3021563) ask who benefits from PM relative to SN when wholesale prices are the outcome of Nash bargaining between the seller and each buyer, and show that with symmetric buyers PM benefits the seller and hurts both buyers.

The model builds on the bilateral-bargaining framework of Horn and Wolinsky (1988), in which a common supplier negotiates simultaneously with competing downstream firms, and on the analysis of negotiated wholesale contracts by O'Brien and Shaffer (2005, RAND Journal of Economics 36, 573–595) and Dukes, Gal-Or and Srinivasan (2006).

Setting

A seller supplies an intermediate good to Buyer 1 and Buyer 2, who each turn one unit of it into one unit of a substitutable end product. With demand base $a > 0$ and substitutability $\gamma \in [0,1)$ , retail prices and quantities satisfy the reverse demand system $p_i = a - x_i - \gamma x_j$ , $j = 3 - i$ .

The game has two stages.

Retail stage. Given wholesale prices $(w_i, w_j)$ , the buyers set retail prices in a differentiated Bertrand game. Buyer $i$ earns $\pi_i = (p_i - w_i)x_i$ and the seller earns $\pi_s = w_i x_i + w_j x_j$ . Writing $\hat p_i(w_i,w_j)$ and $\hat x_i(w_i,w_j)$ for the equilibrium prices and quantities, the second-stage profits are $\pi_i(w_i,w_j)$ and $\pi_s(w_i,w_j)$ .
Negotiation stage. Wholesale prices are set by generalized Nash bargaining: Buyer $i$ has bargaining power $\theta_i \in (0,1)$ against the seller, who has $1-\theta_i$ . A bargain over a price $w \ge 0$ with buyer payoff $U(w)$ , seller payoff $V(w)$ and disagreement point $(0, d)$ selects the individually rational $w$ maximizing $U(w)^{\theta_i}(V(w) - d)^{1-\theta_i}$ .

If a negotiation fails, the seller can sell to the other buyer alone. Then Buyer $j$ faces demand $a - p_j$ , and the seller earns the monopoly profit $\pi^M_s(w_j)$ .

The two mechanisms are:

Simultaneous negotiation (SN). The seller bargains with both buyers at once. A pair $(w^{SN}_1, w^{SN}_2)$ is an equilibrium if each $w^{SN}_i$ solves dyad $i$ 's bargain with payoffs $\pi_i(\cdot, w^{SN}_j)$ , $\pi_s(\cdot, w^{SN}_j)$ and disagreement point $(0, \pi^M_s(w^{SN}_j))$ .
Price matching (PM). The seller bargains with one buyer first; the other buyer accepts the same price. If the first bargain fails, the seller bargains with the other buyer alone, reaching $\hat w_j$ . The PM price $w^{PM}$ solves the bargain with payoffs $\pi_i(w,w)$ , $\pi_s(w,w)$ and disagreement point $(0, \pi^M_s(\hat w_j))$ .

The Lean development uses this notation: pHat, xHat, piB ( $\pi_i$ ), piS ( $\pi_s$ ), piMb, piMs, IsNashBargain, IsSNEq, IsPMPrice.

Formalization targets

Goal: Theorem 1 (p. 10)

With symmetric power $\theta_1 = \theta_2 = \theta \in (0,1)$ , an SN equilibrium and a PM equilibrium price exist, and for every SN equilibrium $(w_1, w_2)$ and every PM price $w$

\pi_s(w_1,w_2) < \pi_s(w,w), \qquad \pi_1(w,w) < \pi_1(w_1,w_2), \qquad \pi_2(w,w) < \pi_2(w_2,w_1).

The goal is stated about the equilibrium predicates, not about the closed-form prices, so that it asserts the comparison of the two mechanisms as the paper defines them.

Milestones

The retail equilibrium $\hat p_i$ , $\hat x_i$ and the profit formulas (1)–(2), §3.1.1.
The single-channel monopoly price and profits, §3.1.2.
The single-channel bargain $\hat w_j = (1-\theta_j)a/2$ , with $\pi^M_s(\hat w_j) = (1-\theta_j^2)a^2/8$ , §3.2.2.
The SN equilibrium is unique and equals $w^{SN} = \dfrac{(1-\theta)(1-\gamma)(2+\gamma)a}{2(1-\gamma)(2+\gamma) + \gamma^2(1+\gamma)(2-\gamma)\theta}$ , §3.2.1.
The PM equilibrium price is unique and equals $w^{PM}_i = \tfrac a4\bigl(3 - \theta_i - \sqrt{1 - \gamma\theta_i + \gamma^2\theta_i + \theta_i^2 + (1+\gamma)(2-\gamma)\theta_i\theta_j^2}\bigr)$ , §3.2.2, for general $\theta_i, \theta_j$ .
$0 < w^{SN} < w^{PM} \le a/2$ under symmetric power, §3.2.2.
At a common price, $\pi_s(w,w)$ increases on $[0, a/2]$ and $\pi_i(w,w)$ decreases on $[0,a]$ (proof of Theorem 1, Online Supplement p. 1).

Significance

Theorem 1 separates the effect of the negotiation mechanism from any asymmetry between buyers: under symmetric power the buyers are interchangeable, SN gives them the same price, and still the mechanism alone moves surplus from the buyers to the seller. The paper's explanation is that price matching lowers a buyer's incentive to negotiate the price down, so the matched wholesale price is higher (p. 10). The paper builds its asymmetric-power, side-payment and spot-market results (§§4–6) on this comparison.

The result is proved in the paper; nothing here is open. Neither the result nor its ingredients have been formalized. The mission produces a machine-checked account of a two-stage bargaining-over-competition model: the Bertrand stage, the Nash-bargaining stage with endogenous disagreement points, and the closed-form equilibria. Two steps the paper only asserts become formal obligations: uniqueness of the SN equilibrium, and the claim that the closed-form PM price is the global maximizer of the Nash product.

Difficulty

The comparison step is short once the closed forms are known: by milestones 6 and 7 the PM price is higher, the seller's diagonal profit increases, and the buyers' decreases. The work is in the equilibrium characterizations.

In SN each dyad's disagreement payoff $\pi^M_s(w_j)$ depends on the other dyad's outcome, so an equilibrium is a fixed point of two coupled Nash-product maximizations. The paper asserts uniqueness. The best-response map is not a contraction for $\gamma$ near $1$ , so a contraction argument does not give uniqueness directly.
The Nash products involve real powers with exponents $\theta$ and $1-\theta$ . Showing that the first-order condition picks out the global maximizer over the individually rational set requires a concavity or single-peakedness argument, not just a derivative computation.
The PM closed form contains a square root. Comparing it with the rational function $w^{SN}$ is an inequality in three parameters $(\gamma, \theta, a)$ .

Formalization scope

All quantities are real numbers. The standing assumptions are $a > 0$ , $\gamma \in [0,1)$ and every bargaining power in $(0,1)$ . The sign $a > 0$ is added; the paper never states it.

Bargaining. A Nash bargaining solution maximizes $U^{\theta}(V-d)^{1-\theta}$ over the individually rational prices $\{w \ge 0 : U(w) \ge 0,\ V(w) \ge d\}$ . Lean's Real.rpow is the real power only on non-negative bases, and the individually rational set guarantees non-negative bases.
Feasible domain. Equilibrium wholesale prices are required to lie in $[0,a]$ , the paper's "entire feasible domain" (p. 9; Online Supplement p. 1); bargaining deviations range over all individually rational $w \ge 0$ .
Second-stage profits. $\pi_i$ and $\pi_s$ are defined from the demand system at $(\hat p_i,\hat p_j)$ , not by the closed forms (1)–(2), so milestone 1 is a genuine identity.
PM outcome. In PM the matched price applies to both buyers, and the seller negotiates first with Buyer 1. With symmetric power the choice of first buyer is immaterial.
Preference. "Prefers" is strict; $\theta \in (0,1)$ excludes the paper's equality cases.

Substituting the closed-form prices $w^{SN}$ , $w^{PM}$ into the profit comparison would turn Theorem 1 into the algebra of milestones 6–7. The goal is therefore stated for every equilibrium of each mechanism, together with existence of both equilibria, so neither the closed forms nor a vacuous equilibrium set can discharge it.

A complete development needs:

concavity of the Nash product along the individually rational interval (reusable for other one-dimensional Nash-bargaining models);
the Bertrand equilibrium of a linear differentiated duopoly;
real-analysis inequalities involving square roots.

Contributions to any milestone, or to reusable lemmas on maximizing $u^{\theta} v^{1-\theta}$ over an interval, are welcome.

Selected references

W. Shang and G. Cai, Implications of Price Matching in Supply Chain Negotiation, Manufacturing & Service Operations Management, 2021 (accepted manuscript, SSRN 3021563). https://ssrn.com/abstract=3021563 — DOI https://doi.org/10.1287/msom.2021.0992
H. Horn and A. Wolinsky, Bilateral Monopolies and Incentives for Merger, RAND Journal of Economics 19(3), 408–419, 1988. https://doi.org/10.2307/2555721
D. P. O'Brien and G. Shaffer, Bargaining, Bundling, and Clout: The Portfolio Effects of Horizontal Mergers, RAND Journal of Economics 36, 573–595, 2005.
A. Dukes, E. Gal-Or and K. Srinivasan, Channel Bargaining with Retailer Asymmetry, Journal of Marketing Research 43, 84–97, 2006. https://doi.org/10.1509/jmkr.43.1.84
E. Kalai, Nonsymmetric Nash Solutions and Replications of 2-Person Bargaining, International Journal of Game Theory 6(3), 129–133, 1977. https://doi.org/10.1007/BF01770215

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Algorithmic Game Theory Operations Research Theoretical Computer Science·Captain: mikedeng1

Dynamic Fair Resource Division: The Optimal Fairness Ratio of d-Uniform Control Vectors Is 1/((d+1)·ln((d+1)/d)), Attained by the Frugally Fair AlgorithmResearch Paper

Motivation

Shared computing infrastructure — a data-centre cluster, a cloud platform's memory — is divided among users who arrive over time. Two requirements pull against each other. Proportional fairness asks that, when $t$ users are present, each holds at least a $1/t$ fraction of the resource. Limited disruption asks that the system not take resource away from running jobs at every arrival, because preemption and reallocation are costly; production cluster managers such as Google's Borg impose an explicit cap on the number of task disruptions an update may cause (Verma et al., 2015, as quoted in Vardi–Psomas–Friedman 2022, pp. 945–946). If reallocation were free, the resource could be re-split evenly at every arrival; if fairness did not matter, the first user could keep everything. The question is how much fairness survives a hard cap on disruptions.

Earlier work on dynamic fair division (Walsh 2011, online cake cutting; Kash, Procaccia and Shah 2014) did not allow reallocation at all. Friedman, Psomas and Vardi (EC 2017) introduced control vectors and gave non-optimal algorithms. Vardi, Psomas and Friedman (Mathematics of Operations Research, 2022) give an optimal algorithm for every control vector once the target ratio is known, and compute the optimal ratio exactly for uniform disruption budgets. This mission formalizes that exact computation.

Setting

One unit of a homogeneous, divisible resource is shared. Users arrive one at a time and never leave (the paper treats arrivals and departures separately; §§3–5, which this mission covers, are arrivals only). Users are labelled by arrival order, and $X^t_k$ is the share of the $(k+1)$ -th user when $t$ users are present.

A control vector $\psi$ of length $n \in \{1,2,\dots\}\cup\{\infty\}$ fixes, for each arrival $1 \le t \le n$ , a budget $\psi[t]$ : when the $t$ -th user arrives, at most $\psi[t]$ of the $t-1$ users already present may have their share strictly reduced (disrupted). Increasing a share is free. The $d$ -uniform control vector has $\psi[t] = d$ for every $t$ , of any length, finite or infinite.

An allocation sequence $X = (X^1, X^2, \dots)$ for $\psi$ is feasible if at every time $1 \le t \le n$ the shares are nonnegative, sum to at most $1$ , and respect the disruption budget. Its fairness ratio is

\inf_{1 \le t \le n}\; t \cdot \min_{k<t} X^t_k ,

the smallest share relative to the proportional share $1/t$ , in the worst time step. The fairness ratio of the control vector, $\mathrm{FAIRNESS}(\psi)$ , is the supremum of this quantity over all feasible allocation sequences, and

\mathrm{FAIRNESS}(d\text{-uniform}) = \inf_{\psi\ d\text{-uniform}} \mathrm{FAIRNESS}(\psi).

The frugally fair algorithm $\mathrm{FFA}(\psi,\sigma)$ takes a target ratio $\sigma$ : when user $t$ arrives, it gives the newcomer $\sigma/t$ and cuts the $\psi[t]$ largest existing shares to $\sigma/t$ . It may allocate more than one unit; $S^t$ denotes its total allocation at time $t$ .

Formalization targets

Goal: Theorem 4

For every integer $d \ge 1$ ,

\mathrm{FAIRNESS}(d\text{-uniform}) \;=\; \sigma^*_d \;:=\; \frac{1}{(d+1)\ln\!\big(\frac{d+1}{d}\big)} .

The goal states this in three parts: the infinite $d$ -uniform vector admits one feasible allocation sequence with ratio at least $\sigma^*_d$ ; for every $\sigma > \sigma^*_d$ some finite $d$ -uniform vector admits no feasible allocation sequence with ratio $\sigma$ ; and the infimum over all lengths of $\mathrm{FAIRNESS}$ equals $\sigma^*_d$ . For $d = 1$ , $\sigma^*_1 = 1/(2\ln 2) \approx 0.721$ .

Milestones

Feasible runs of FFA yield feasible allocation sequences with the same shares and ratio $\sigma$ (§3, p. 949).
Theorem 1 (p. 950): if any algorithm achieves a nonnegative ratio $\sigma$ on $\psi$ , then $\mathrm{FFA}(\psi,\sigma)$ is feasible at every time.
Lemma 1 (p. 951): $\mathrm{FFA}(\psi,\sigma)$ is $\mathrm{FFA}(\psi,\sigma')$ scaled by $\sigma/\sigma'$ .
Proposition 2 (p. 952): $\mathrm{FAIRNESS}(\psi) \le \mathrm{FAIRNESS}(\psi')$ when $\psi'$ is a prefix of $\psi$ .
Lemma E.1 (p. 959): the explicit allocation of FFA on the infinite $d$ -uniform vector at time $t = \ell(d+1)+i$ .
$S^t$ is nondecreasing in $t$ on the $d$ -uniform vector (p. 959).
Display (E.1) (p. 959): $\lim_{\ell\to\infty} S^{\ell(d+1)} = \sigma(d+1)\ln\frac{d+1}{d}$ .

Significance

The theorem gives a closed-form price of limited disruption: with $d$ reallocations per arrival, a system can guarantee every user $\sigma^*_d/t$ of the resource forever, and no algorithm can guarantee more uniformly over all horizons. Since $\sigma^*_d = 1 - \frac{1}{2d} + O(d^{-2})$ , a modest budget already recovers most of proportional fairness, which is the quantitative content a system designer needs to choose $d$ . Combined with Theorem 1, the value also turns FFA into a concrete optimal algorithm for the infinite $d$ -uniform vector, running online with no knowledge of the horizon.

The result is proved in the paper; to the best of a search of the Prove2Me catalogue, nothing about control vectors, disruption-limited fair division or this constant is formalized there. The mission produces a machine-checked model of disruption-limited dynamic allocation, a formal optimality proof for a greedy online algorithm (Theorem 1) that is reusable for other control vectors (the paper's finite-vector algorithm and $c$ -gap bounds rest on the same theorem), and an explicit harmonic-sum asymptotic.

Difficulty

The impossibility half must rule out every algorithm, including ones that over-allocate early, raise shares opportunistically and choose whom to disrupt adaptively. The first idea, comparing total allocations time by time, fails: an arbitrary algorithm's allocation is not entrywise comparable with any fixed reference allocation, and a single well-chosen disruption can change which users are largest. The attainment half needs one allocation sequence for an unbounded horizon, so it cannot be obtained from finitely many feasibility checks; and the value $\sigma^*_d$ is not the outcome of any finite computation but a limit of harmonic-number differences. The paper's own computation of the one-step change in total allocation contains a slip (a missing factor $\ell$ ; the sign is unaffected), so that step cannot be transcribed as printed.

Formalization scope

Everything is real-valued and lives in DynFairDiv.Uniform. A control vector is ψ : ℕ → ℕ (index $t \ge 1$ ) with a horizon n : ℕ∞, where ⊤ is the infinite vector; uniformCV d is the constant vector. Allocation sequences are X : ℕ → ℕ → ℝ with users labelled by arrival order, which is equivalent to the paper's sorted vectors because feasibility uses only sums, fairness only minima, and a disruption is a strict decrease of a given user's share; the newcomer is never counted as disrupted. Fairness at least $\sigma$ is σ / t ≤ X t k for all $1 \le t \le n$ and $k < t$ . $\mathrm{FAIRNESS}(\psi)$ is a real sSup, genuine for $n \ge 1$ . FFA's state is a list of shares kept in nonincreasing order, so "the $\psi[t]$ largest shares" is the front of the list. Departures (the paper's §6) are not modelled.

Added hypotheses, each disclosed in its item: $d \ge 1$ throughout §5.1 (the paper's own assumption); $\sigma \ge 0$ for milestones 1, 2 and 6 (the paper's fairness ratios lie in $[0,1]$ ); $\sigma' \ne 0$ in Lemma 1; a prefix of length at least $1$ in Proposition 2. Milestone 1 identifies the resulting sequence's shares with FFA's shares at each time.

A trivializing reading is ruled out: the impossibility half quantifies over every feasible allocation sequence, not only FFA-like or sorted ones, and the attainment half asks for a single sequence on the infinite horizon rather than one per finite length.

A complete development needs basic facts about FFA's list (length, order, scaling) and the harmonic asymptotics already in Mathlib (Real.tendsto_harmonic_sub_log). The FFA model and Theorem 1 apply to arbitrary control vectors and are reusable for the paper's finite-vector and $c$ -gap results; contributions of auxiliary lemmas about FFA (length, sortedness, scaling) are welcome.

Selected references

S. Vardi, A. Psomas, E. Friedman, Dynamic Fair Resource Division, Mathematics of Operations Research 47(2):945–968, 2022. https://doi.org/10.1287/moor.2021.1155
E. Friedman, C.-A. Psomas, S. Vardi, Controlled Dynamic Fair Division, Proc. 18th ACM Conference on Economics and Computation (EC), 461–478, 2017.
I. A. Kash, A. D. Procaccia, N. Shah, No Agent Left Behind: Dynamic Fair Division of Multiple Resources, Journal of Artificial Intelligence Research 51:579–603, 2014.
T. Walsh, Online Cake Cutting, Algorithmic Decision Theory (ADT), Springer, 292–305, 2011.
A. Verma, L. Pedrosa, M. Korupolu, D. Oppenheimer, E. Tune, J. Wilkes, Large-Scale Cluster Management at Google with Borg, Proc. 10th European Conference on Computer Systems (EuroSys), 2015.

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Markov Chain Operations Research Stochastic Systems·Captain: mikedeng1

Smart Policies for Multisource Inventory Systems and General Tandem Queues with Order Tracking and Expediting: State Space and Product-Form Stationary Distribution under Policy-MResearch Paper

Why state-dependent sourcing matters

An inventory system can replenish demand from several stages of a supply process. When an order is placed, the manager chooses where it enters; when work finishes at one stage, the manager may send it past intermediate stages to speed delivery. These choices change the future congestion of every stage. A stationary occupancy distribution is needed to calculate how often stock is available, how often demand waits, and how often expediting is used. Song, Xiao, Zhang, and Zipkin study such policies through a tandem network whose arrival and routing decisions depend on the current occupancies. They identify Policy-M as one family with a product-form stationary distribution, despite this state dependence (Song et al., SSRN 3702068, §§3–4).

The paper's Theorem 1 is the target here. Its formula permits the system's long-run inventory cost to be evaluated by summing over states with known product weights. The same paper develops a distinct Policy-E family, but its theorem has a different routing rule and is outside this mission (Song et al., Theorems 1–2).

The tandem supply system

There are $K\ge1$ service nodes in series, numbered $1,\ldots,K$ . A state $n=(n_1,\ldots,n_K)\in\mathbb N^K$ records the number of units at each node. The segment occupancy $n[a,b]=\sum_{l=a}^{b}n_l$ counts units between nodes $a$ and $b$ , inclusive. Demand arrives at positive rate $\lambda$ . Node $k$ completes service at positive rate $\mu_k$ whenever $n_k>0$ . A completion at node $K$ leaves the supply system.

Policy parameters are thresholds $u_{ij}$ for $0\le i<j\le K$ , with values in $\mathbb N\cup\{\infty\}$ . For fixed origin $i$ , thresholds are nondecreasing as $j$ grows; for fixed destination $j$ , they are nonincreasing as $i$ grows. Every terminal threshold $u_{iK}$ equals $\infty$ . An arriving demand places a new unit at node $1$ when no prefix $n[1,k]$ reaches $u_{0k}$ ; otherwise it places the unit at one node beyond the largest such $k$ . A service completion at node $m<K$ follows the analogous Policy-M rule using downstream segments $n[m+1,k]$ and thresholds $u_{mk}$ . An empty set of reached thresholds sends the unit to $m+1$ (Song et al., (5), (7), §4).

Assumption A imposes two inequalities on these thresholds: a triangle inequality $u_{kj}\le u_{ki}+u_{ij}$ for $k<i<j$ , and supermodularity $u_{ij}+u_{kl}\le u_{kj}+u_{il}$ for $k<i<j<l$ . The indices may include the demand source $0$ . The admissible state set $S_M(u)$ consists of all occupancies whose segments obey $n[k+1,l]\le u_{kl}$ for every $0\le k<l\le K$ (Song et al., Assumption A and (11)).

Formalization targets

The first part of Theorem 1 identifies the states reachable from the empty system through positive-rate Policy-M transitions:

\operatorname{Reach}_{M}(0)=S_M(u) =\{n\in\mathbb N^K:n[k+1,l]\le u_{kl}\text{ for }0\le k<l\le K\}.

For $\lambda<\mu_K$ and finite $u_{0i}$ whenever $1\le i<K$ , define $w(n)=\prod_{k=1}^{K}(\lambda/\mu_k)^{n_k}$ and $G(u)=\sum_{n\in S_M(u)}w(n)$ . The second part gives a normalized product law:

0<G(u)<\infty,\qquad \pi_M(n)=\frac{\mathbf1_{S_M(u)}(n)}{G(u)} \prod_{k=1}^{K}\left(\frac{\lambda}{\mu_k}\right)^{n_k}.

The mission also requires $\pi_M$ to have total mass one and to satisfy global balance for the rates generated by the stated routing rules. Its milestones are the closure claims (A-2) and (A-5), equality of the Appendix proof sets in (A-30), the indicator identity (A-25), and the local balance equations (A-12)–(A-15) (Song et al., Theorem 1 and Appendix A).

What the result gives

The formula describes the joint occupancy of all service nodes from $K$ rates and the policy thresholds. Once its normalizer is known, probabilities of stock levels and congestion events can be summed over $S_M(u)$ , supplying the inputs to the paper's long-run cost calculation (9). The state-space result determines which occupancies contribute to those sums (Song et al., (9), Theorem 1).

Theorem 1 is proved in the paper. This mission asks for machine-checked proofs of its formal statements, not a new queueing result. A complete development would provide reusable finite-segment reasoning, routing-set identities, and countable-state balance calculations. The declarations staged here are open Lean statements, so they do not yet constitute machine-checked proofs.

The obstacle in the balance equations

With fixed routing, a proposed product weight can be compared across neighboring states using only the node rates. Here the route of a newly arriving or completed unit changes with several segment occupancies. An incoming transition to a state may have skipped nodes, and the condition that selected its destination must hold in the predecessor state. Checking only the product ratio therefore leaves out the routing indicators. The source's local equations (A-12)–(A-15) and its proof-set identity (A-25) make those indicators explicit. The paper also gives an example showing that state-dependent routing does not yield product form for every policy (Song et al., Example 1 and Appendix A).

Formalization scope

Lean represents $n$ as a function from Fin K to natural numbers, while the functions occ and seg retain the paper's one-based node numbers. The threshold array takes values in extended natural numbers so $u_{iK}=\infty$ is representable; a strict comparison with a threshold expresses the paper's “at most $u-1$ ” condition. Both routing rules specify the destination when no threshold is reached. A service transition is enabled only when its origin node is occupied. The theorem states $K\ge1$ , which is implicit in the paper's numbered-node model, along with positive arrival and service rates.

The source defines $\pi$ as the limiting law of a continuous-time Markov chain and invokes known ergodicity. This Lean development encodes its stationary claim by a summable, normalized product weight and the global balance equations for explicit Policy-M rates. It does not formalize the CTMC limit or uniqueness of that limit. The normalizer is a countable sum; the goal asserts summability and $G(u)>0$ , so a default value for a divergent sum or division by zero cannot certify the claim. The state space is constructed from $u$ and proved reachable rather than supplied as a hypothesis. Empty-node departures are guarded to avoid natural-number subtraction at zero. The special $K=1$ case remains in the goal; the milestone combining four separate local equations assumes $K\ge2$ because its node-1 and node- $K$ formulas must refer to distinct nodes.

The staged definition supplies segments, thresholds, routing, transition rates, product weights, and the source's Appendix sets. Contributions may prove the listed milestones, build general lemmas about segment sums and reachability, or establish summability and the passage from local to global balance. These supporting tools can be reused for other countable-state queueing models.

Selected references

J.-S. Song, L. Xiao, H. Zhang, and P. Zipkin, Smart Policies for Multisource Inventory Systems and General Tandem Queues with Order Tracking and Expediting, SSRN 3702068, version of September 28, 2020; subsequently published in Operations Research, DOI 10.1287/opre.2021.2124. Source preprint.

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Operations Research Optimization Theoretical Computer Science·Captain: mikedeng1

Dynamic Resource Allocation in the Cloud with Near-Optimal Efficiency: The Multiplicative Weight Algorithm Nearly Matches the Offline Dynamic Optimum's Work While Approximately Satisfying Every SLAResearch Paper

Motivation

Cloud platforms divide a shared physical resource, such as a pool of processors, among many users at once. Each user is promised a service level agreement (SLA): a fraction $\beta(i)$ of the resource that it should effectively receive whenever it has work to do. The operator wants two things that pull against each other: to keep the resource busy (total work done should be large) and to honour every SLA. The difficulty is informational. The operator does not see future demand, and in the setting of Perez-Salazar, Menache, Singh and Toriello (arXiv:1809.02688) it does not even see current demand: at each step it only learns which users have a non-empty queue.

The paper answers with a multiplicative weight rule. Multiplicative weight methods go back to online learning (Freund and Schapire, 1997; survey: Arora, Hazan and Kale, 2012) and can be read as entropic mirror descent (Beck and Teboulle, 2003). Their classical guarantees compare an online learner with the best fixed decision in hindsight. This paper compares the algorithm with the best time-varying offline allocation instead, and proves the comparison directly, without the standard regret analysis.

Setting

There are $N \ge 2$ users and discrete time steps $t = 1, 2, \dots$ . Each user $i$ has an SLA $\beta(i) \ge 0$ , with $\sum_i \beta(i) \le 1$ . Loads $L_t(i) \ge 0$ arrive online. User $i$ keeps a queue $Q_t(i)$ of unfinished work. At each step:

the decision maker observes the active users $A_t = \{i : Q_t(i) > 0\}$ ;
it chooses an allocation $h_t$ with $h_t(i) \ge 0$ and $\sum_i h_t(i) \le 1$ ;
the load $L_t$ arrives, user $i$ completes $w_t(i) = \min\{h_t(i),\, L_t(i) + Q_t(i)\}$ , and $Q_{t+1}(i) = \max\{0,\, L_t(i) + Q_t(i) - h_t(i)\}$ .

The total work of an allocation sequence $h$ up to time $t$ is $\mathrm{work}_h(t) = \sum_i \sum_{\tau \le t} w_\tau(i)$ .

Algorithm 1 has parameters $0 < \varepsilon \le 1/10$ and $0 < \eta < 1/3$ , and sets $\lambda = \varepsilon^2/(8N)$ . It keeps $h_t$ in the truncated simplex $\Delta_\varepsilon = \{x : \sum_i x(i) = 1,\ x(i) \ge \varepsilon/N\}$ , starting anywhere in $\Delta_\varepsilon$ . At step $t$ it assigns the gain $g_t(i) = 1 + \lambda$ to active users below their SLA ( $A^1_t$ : $h_t(i) < \beta(i)$ ), $g_t(i) = 1$ to the other active users ( $A^2_t$ ) and $g_t(i) = 0$ to inactive users. It then sets $\widehat h_{t+1}(i) = h_t(i)e^{\eta g_t(i)}$ and takes $h_{t+1} = \pi_{\Delta_\varepsilon}(\widehat h_{t+1})$ , the KL projection: the minimizer over $\Delta_\varepsilon$ of $\sum_i x(i)\ln(x(i)/\widehat h_{t+1}(i))$ .

Formalization targets

Goal: Theorem 1 with the explicit constants of p. 10

Let $\beta(i) \ge 2\varepsilon/N$ for all $i$ and let $h$ be any run of Algorithm 1. For every offline allocation sequence $h^*$ and every time $t$ ,

\mathrm{work}_{\mathrm{alg}}(t) \ge (1 - 4\varepsilon)\,\mathrm{work}_{h^*}(t) - 2000\,\frac{N}{\varepsilon^2\eta}\ln\frac{N}{\varepsilon},

and for every user $i$ , every comparator $h'$ with $0 \le h'_t(i) \le \beta(i)$ and every $t$ ,

\sum_{\tau \le t} w_\tau(i) \ge (1 - 2\varepsilon)\sum_{\tau \le t} w'_\tau(i) - \beta(i)\,\tilde p, \qquad \tilde p = \frac{32N^2}{\varepsilon^3\eta}\ln\frac{N}{\varepsilon}.

Intermediate targets (milestones)

Theorem 3: against every sequence of $(1-4\varepsilon)$ -allocations, $\mathrm{work}_{h^*}(t) - \mathrm{work}_{\mathrm{alg}}(t) \le \frac{8N}{\varepsilon^2\eta}\ln\frac{N}{\varepsilon} + 2$ at every $t$ .
Theorem 4: if $\beta(j) > e^{\eta(1+\lambda)}\varepsilon/N$ for all $j$ , each user's work by time $t + \lceil\tilde p\rceil$ is at least what the constant share $(1-2\varepsilon)\beta(i)$ completes by time $t$ .
Lemma 1 and Lemma 2: per-step growth factors $1 + c$ , $1 - \varepsilon c$ and $1 + c'$ of a busy user's allocation, with $c = \varepsilon\eta/(4N)$ and $c' = \varepsilon\eta\lambda/(2N)$ .
Proposition 1(b): the closed form of the KL projection onto $\Delta_\varepsilon$ .
Proposition 2 (weak-duality half), Claims 3.1 and 3.2 (first conclusion): the amortization behind Theorem 3.
The scaling step of p. 10: scaling an allocation by $\kappa \le 1$ keeps at least a $\kappa$ fraction of its work, and work is monotone in the allocation.
Existence and uniqueness of the KL projection, hence of the run from each start.

Significance

The result. Theorem 1 is a bicriteria guarantee against a dynamic benchmark. The work benchmark may change its allocation arbitrarily over time with full knowledge of all loads, and the algorithm, which sees only which queues are non-empty, loses only an additive amount that does not grow with time, at the price of a $4\varepsilon$ fraction of capacity. At the same time every user receives, up to a fixed delay, at least the work of a dedicated $(1-2\varepsilon)$ fraction of its SLA. The paper also shows (arXiv:1809.02688, Theorem 2) that against the unscaled offline optimum every deterministic online algorithm loses $\Omega(\sqrt T)$ work on some load sequence, so some form of slack is unavoidable.

Formalizing it. The result is proved on paper; no machine-checked proof exists. A formalization would check the constant bookkeeping, where the printed text has slips (an $\varepsilon$ versus $\varepsilon/4$ in Theorem 1, a real block length $\tilde s$ used as an integer, an off-by-one in Claim 3.1). It would also produce a reusable deterministic queueing model and a verified closed form of the KL projection onto a truncated simplex.

Difficulty

The obvious first idea, the regret analysis of mirror descent, does not apply. Regret bounds compare with a fixed decision, but here the benchmark changes over time. The algorithm's gains also depend on its own past decisions through the queues, so they are not an exogenous sequence of losses. The paper notes (p. 10) that its guarantees do not follow from the standard analysis of mirror descent. A second obstacle is the feedback: a busy user is not the same as a user that uses its whole allocation. A user whose queue empties during a step completes less than its allocation, so utilization and work done can differ. Finally, the constants matter, because the goal states them explicitly. The printed derivation mixes $\varepsilon$ and $\varepsilon/4$ , uses the real number $\tilde s$ as a block length, and is off by one step when it identifies the work done with the arrived load.

Formalization scope

Model. Everything is real-valued. Time is 0-based: Lean's step $t$ is the paper's step $t + 1$ , the queue at Lean time $t$ is the paper's $Q_{t+1}$ , and $\sum_{\tau=1}^{t}$ is a sum over the first $t$ steps. Queues start empty, a convention the paper uses but does not state. Loads form an infinite sequence; the horizon $T$ is not a parameter and every bound holds at every $t$ .
Runs. The KL projection is a predicate for positive inputs, the domain where the logarithm is defined; Algorithm 1 always projects a positive vector. A run is any sequence satisfying the recursion. The gains are computed from the queues the run itself produces before step $t$ , so $h_{t+1}$ never uses the load of step $t$ or later. A milestone proves that each start in $\Delta_\varepsilon$ has exactly one run, and a sorry-free sanity check exhibits one. Statements about runs are therefore not vacuous.
Benchmarks. "The optimal offline sequence" is replaced by every allocation sequence with $\sum_i h^*_t(i) \le 1$ (or $\le 1 - 4\varepsilon$ ), which is equivalent and avoids a supremum. The SLA comparator ranges over every sequence with $h'_t(i) \le \beta(i)$ , not only the constant one.
Disclosed readings and corrections. Part 1 of the goal uses the $(1 - 4\varepsilon)$ form of p. 10 for the same $\varepsilon$ , not the headline's $(1-\varepsilon)$ . Theorem 3 carries a $+2$ for integer blocks. Claim 3.1 is stated in its corrected 0-based form. Theorem 4 assumes the SLA bound strictly. $0 < \eta < 1/3$ (Algorithm 1's input) is assumed throughout. $\sum_i \beta(i) \le 1$ replaces the $= 1$ used in the proof of Lemma 2.
Ruled out. The goal is stated for every run of the algorithm, with the active sets read from the run's own queues, against every offline sequence. A formalization that fixes a closed-form allocation, decouples the queues from the allocations, or restricts the benchmark to static allocations does not count.
Needed and reusable. The deterministic single-user queue with its work and monotonicity facts, KKT conditions for an entropic objective over a box-constrained simplex, and elementary exponential and logarithm inequalities. The queue model and the projection closed form are reusable beyond this mission. Contributions to any milestone are welcome.

Selected references

S. Perez-Salazar, I. Menache, M. Singh, A. Toriello, Dynamic Resource Allocation in the Cloud with Near-Optimal Efficiency, arXiv:1809.02688v3, 2021 (Operations Research, 2022). https://arxiv.org/abs/1809.02688
Y. Freund, R. E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, Journal of Computer and System Sciences 55(1), 1997. https://doi.org/10.1006/jcss.1997.1504
S. Arora, E. Hazan, S. Kale, The multiplicative weights update method: a meta-algorithm and applications, Theory of Computing 8, 2012. https://doi.org/10.4086/toc.2012.v008a006
A. Beck, M. Teboulle, Mirror descent and nonlinear projected subgradient methods for convex optimization, Operations Research Letters 31(3), 2003. https://doi.org/10.1016/S0167-6377(02)00231-6

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Economics Operations Research Probability·Captain: mikedeng1

Costly Interpretation of Asset Prices: With Costly Price Interpretation There Is a Unique Linear Financial Market Equilibrium, and Its Price Carries Endogenous NoiseResearch Paper

Motivation

A central question of financial economics is how much of the information dispersed among traders ends up in market prices. The workhorse model is the noisy rational expectations equilibrium (REE): investors with private signals trade a risky asset, the price aggregates their information, and each investor learns from the price as well as from her own signal (Grossman and Stiglitz 1980; Hellwig 1980). In that tradition the price is a free public signal: every investor reads off from it exactly what it contains.

Vives and Yang (SSRN 2763813; published with Mondria as Mondria, Vives and Yang 2021) drop that assumption. Extracting information from a price takes effort, so each investor observes the price signal only through her own interpretation error, and can buy sophistication to reduce that error. The interpretation errors share a common component, and the paper's first result is that this common error re-enters the price. Even with no exogenous noise traders, costly interpretation makes the price noisy. This mission formalizes that first result: the existence, uniqueness and closed form of the linear financial-market equilibrium (Proposition 1, with the remark $a_u > 0$ that follows it).

Setting

A risky asset pays $\tilde V = \tilde v + \tilde\xi$ , with independent $\tilde v \sim N(0,\tau_v^{-1})$ (learnable) and $\tilde\xi \sim N(0,\tau_\xi^{-1})$ (unlearnable). A continuum $i\in[0,1]$ of investors have CARA utility with risk aversion $\gamma$ . Investor $i$ observes:

a private signal $\tilde s_i = \tilde v + \tilde\varepsilon_i$ , with $\tilde\varepsilon_i \sim N(0,\tau_\varepsilon^{-1})$ ;
a personalized price signal $\tilde s_{p,i} = \tilde s_p + \tilde u + \tilde e_i$ , where $\tilde s_p$ is the information in the price, $\tilde u \sim N(0,\tau_u^{-1})$ is a common interpretation error and $\tilde e_i \sim N(0,\tau_e^{-1})$ an idiosyncratic one;
an interpretation signal $\tilde z_i = \tilde u + \tilde e_i + \tilde\eta_i$ , with $\tilde\eta_i \sim N(0,\tau_\eta^{-1})$ , where $\tau_\eta$ is the common sophistication level.

All shocks are independent and centred, and all precisions and $\gamma$ are positive. A linear price function is $\tilde p = a_v\tilde v + a_u\tilde u$ ; when $a_v\ne 0$ it is equivalent to the signal $\tilde s_p = \tilde v + \alpha\tilde u$ with $\alpha = a_u/a_v$ . Each investor demands $D_i = (E(\tilde V\mid \tilde s_i,\tilde s_{p,i},\tilde z_i) - \tilde p)/(\gamma\,\mathrm{Var}(\tilde V\mid \tilde s_i,\tilde s_{p,i},\tilde z_i))$ , computed by Bayes' rule from the jointly Gaussian signals. A linear equilibrium is a pair $(a_v,a_u)$ , $a_v\neq0$ , such that aggregate demand $\int_0^1 D_i\,di$ is zero for every realization of $(\tilde v,\tilde u)$ .

The paper's closed forms use the precision of the combined price information,

\tau_p = \frac{\tau_e\tau_u + (\tau_e+\tau_u)\tau_\eta}{\tau_u + \tau_e(\alpha+1)^2 + \alpha^2\tau_\eta},

and the cubic

f(\alpha) = (\tau_e\tau_\varepsilon + \tau_\varepsilon\tau_\eta)\alpha^3 + 2\tau_e\tau_\varepsilon\alpha^2 + (\tau_e\tau_\varepsilon + \tau_e\tau_\eta + \tau_u\tau_\varepsilon)\alpha - \tau_e\tau_u.

Formalization targets

Goal: Proposition 1 and the noise remark

There is exactly one linear equilibrium $(a_v,a_u)$ . In it

a_v = \frac{\tau_\varepsilon+\tau_p}{\tau_v+\tau_\varepsilon+\tau_p},\qquad a_u = \frac{\tau_p}{\tau_v+\tau_\varepsilon+\tau_p}\cdot\frac{\tau_u(\tau_e+\alpha\tau_e+\alpha\tau_\eta)}{\tau_e\tau_u+\tau_e\tau_\eta+\tau_u\tau_\eta},

with $\tau_p$ evaluated at $\alpha = a_u/a_v$ ; $\alpha$ lies in $(0,\bar\alpha)$ , $\bar\alpha = \tau_e\tau_u/(\tau_e\tau_\varepsilon+\tau_e\tau_\eta+\tau_u\tau_\varepsilon)$ , and is the unique positive root of $f$ ; and $a_u > 0$ .

Milestones, in the order of the paper's proof

The Bayes coefficients $(\beta_s,\beta_p,\beta_z)$ of $E(\tilde v\mid\cdot)$ in closed form (proof of Proposition 1, p. 55).
Equation (17): $\mathrm{Var}(\tilde V\mid\cdot) = (\tau_v+\tau_\varepsilon+\tau_p)^{-1} + \tau_\xi^{-1}$ (p. 18).
Market clearing holds iff $a_v = \beta_s+\beta_p$ and $a_u = \beta_p(\alpha+1)+\beta_z$ (p. 56).
These sums equal the closed forms of $a_v$ , $a_u$ (p. 56).
$\alpha\,a_v(\alpha) = a_u(\alpha)$ iff $f(\alpha)=0$ (p. 56).
$f(0) < 0 < f(\bar\alpha)$ , hence a root in $(0,\bar\alpha)$ (p. 56).
The discriminant of $f$ is the printed 25-term expression $\Delta<0$ , hence $f$ has one real root (p. 56).

Significance

Proposition 1 is the foundation of the rest of the paper. Every comparative static on price informativeness, volatility, return predictability, disagreement and trading volume (Propositions 2–3), and the analysis of the sophistication-acquisition game (Propositions 4–6), is computed from the closed forms of $a_v$ , $a_u$ and the root $\alpha$ . The remark $a_u>0$ is the paper's mechanism: the common interpretation error is a source of price noise generated by the traders themselves, which vanishes only in the limit of full sophistication (Corollary 1).

The result is proved in the paper; nothing here is open. To our knowledge no part of it, nor any noisy-REE equilibrium with Bayesian price learning, has been machine-checked. The mission produces a formal account of the equilibrium construction of the noisy REE literature, from Gaussian conditioning to market clearing to a polynomial fixed point. The same pattern (conjecture a linear price, condition, match coefficients, solve for the fixed point) recurs throughout that literature.

Difficulty

Each step is elementary, but each is a large symbolic computation. Inverting the $3\times3$ covariance matrix and simplifying $\Sigma^{-1}c$ to the printed fractions involves rational functions in seven variables. The discriminant of $f$ is a polynomial of degree ten with 25 terms, and its sign has to be established for all positive parameters.

The conceptual pitfall is uniqueness. Showing that $f$ has one positive root is not enough: a linear equilibrium may a priori have $\alpha = a_u/a_v \le 0$ , so uniqueness among all linear equilibria needs $f$ to have one real root, which is what the discriminant gives. The fixed-point condition $\alpha = a_u(\alpha)/a_v(\alpha)$ also needs $a_v(\alpha)\neq0$ at every candidate, which has to be proved rather than assumed.

Formalization scope

All quantities are real. A parameter structure carries $\tau_v,\tau_\xi,\tau_\varepsilon,\tau_e,\tau_u,\tau_\eta,\gamma$ ; the standing assumption is that all seven are positive, and nothing else is assumed. The model is encoded in reduced form:

Conditioning. The conditional mean coefficients are the linear projection $\Sigma(\alpha)^{-1}c$ , with $\Sigma(\alpha)$ the covariance matrix of $(\tilde s_i,\tilde s_{p,i},\tilde z_i)$ and $c$ their covariance with $\tilde v$ . The conditional variance is $\tau_v^{-1} - c^\top\Sigma^{-1}c + \tau_\xi^{-1}$ . No probability space is built.
Market clearing. It is stated for every $(\tilde v,\tilde u)\in\mathbb R^2$ rather than almost surely, which is equivalent for linear functions of a nondegenerate Gaussian pair. Aggregate demand is the demand at the cross-sectional mean $(\tilde v,\ \tilde v+(\alpha+1)\tilde u,\ \tilde u)$ of the signals, the usual continuum convention.
The equilibrium predicate. It requires $a_v\neq0$ , as on p. 17 of the paper.
Equation (4). As printed, it attaches the law $N(0,\tau_\eta^{-1})$ to $\tilde z_i$ ; (5) and (15) show it belongs to $\tilde\eta_i$ , which is the reading used.

The equilibrium predicate is defined from Bayesian demands and market clearing only. Defining it by the closed forms of $a_v$ , $a_u$ , or as " $\alpha$ is a root of (18)", would make Proposition 1 a statement about a cubic and is ruled out. Likewise the Bayes coefficients are $\Sigma^{-1}c$ , never the printed fractions.

A complete development needs $3\times3$ matrix inversion over $\mathbb R$ (Mathlib's Matrix.inv), rational-function simplification, Mathlib's Cubic and its discriminant, and the intermediate value theorem. A reusable lemma "a real cubic with negative discriminant has exactly one real root" would serve beyond this mission. Proofs of any milestone, and alternative proofs of uniqueness, are welcome.

Selected references

X. Vives and L. Yang, Costly Interpretation of Asset Prices, SSRN working paper 2763813, September 2018 version (the version cited throughout). https://ssrn.com/abstract=2763813
J. Mondria, X. Vives and L. Yang, Costly Interpretation of Asset Prices, Management Science, 2021. https://doi.org/10.1287/mnsc.2020.3871
S. J. Grossman and J. E. Stiglitz, On the Impossibility of Informationally Efficient Markets, American Economic Review 70(3), 1980. https://www.jstor.org/stable/1805228
M. F. Hellwig, On the Aggregation of Information in Competitive Markets, Journal of Economic Theory 22(3), 1980. https://doi.org/10.1016/0022-0531(80)90065-5

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Operations Research Optimization Probability·Captain: mikedeng1

Stochastic Knapsack Revisited: The Service Level Perspective: Anticipative Policies Need at Most 2Δ More Capacity Than Adaptive Policies to Meet Type-I Service LevelsResearch Paper

Motivation

A central planner holds a single pool of $Q$ units of a product and must supply $N$ customers whose demands $\tilde d_1, \dots, \tilde d_N$ are random. Each customer $i$ asks to be fully served, i.e. to receive her entire demand, with probability at least a target $\alpha_i$ . This Type-I service level (the persistency value of the stochastic knapsack) is the standard requirement in supply-chain allocation and in humanitarian logistics. The planner's question is how much capacity the targets require.

The answer depends on when the planner learns the demands. Lyu, Chou, Teo, Zheng and Zhong (SSRN 3674463, 2021) compare three information regimes:

a responsive policy sees every demand before allocating;
an adaptive policy learns a customer's demand only once it has decided to serve her, and may choose the next customer in light of what it has seen (Dean, Goemans and Vondrák, 2008);
an anticipative policy fixes a priority order before any demand is realized (Alptekinoğlu, Banerjee, Paul and Jain, M&SOM 15(1), 2013).

Anticipative rules are the ones used in practice, because they are simple to announce and to audit. The paper's main result quantifies what they cost in capacity compared with adaptive rules.

Timeline. Alptekinoğlu et al. (2013) introduced capacity pooling with differentiated service probabilities under anticipative priority rules. Dean, Goemans and Vondrák (2008) bounded the adaptivity gap of the stochastic knapsack for revenue maximization by a constant factor. Zhong, Zheng, Chou and Teo (Management Science 64(4), 2018) treated the Type-II (fill-rate) version, in which partial service counts in proportion. They showed that its achievable region is a polymatroid $\Gamma(Q)$ and that anticipative policies are optimal there. Lyu et al. (2021) use this Type-II system to compare the Type-I classes.

Setting

Customers are $i \in \{1, \dots, N\}$ . The demands are random variables on a probability space $(\Omega, \mathcal F, \mathbf P)$ . An allocation run with capacity $Q \ge 0$ serves customers one at a time. A chosen customer is fully served if her demand does not exceed the remaining capacity, and her demand is then deducted. Otherwise she is not fully served, and the run stops.

A deterministic adaptive policy chooses the next customer as a function of the customers chosen so far and their realized demands.
A deterministic anticipative policy is a priority order $\sigma$ , fixed in advance.
A randomized policy in either class is a finite mixture $\sum_k w_k A_k$ of deterministic ones, drawn independently of the demands.

The optimal Type-I capacities are

Q^{\mathrm{Adap}}_I = \inf\{Q \ge 0 : \text{some randomized adaptive policy has } \mathbf P(i \text{ fully served}) \ge \alpha_i\ \forall i\},

and $Q^{\mathrm{Anti}}_I$ is defined in the same way with anticipative policies. These are the optimal values of problem (CPP-A) of the paper.

On the Type-II side, $\Gamma(Q)$ is the set of $\alpha$ with $\sum_{i \in U} \alpha_i \mathbf E[\tilde d_i] \le \mathbf E[\min(Q, \sum_{i\in U} \tilde d_i)]$ for every $U \subseteq \{1,\dots,N\}$ . $Q_{II}$ is the least $Q \ge 0$ with $\alpha \in \Gamma(Q)$ . For a priority order $\sigma$ , the greedy fill $x^*_i(\sigma, \tilde d, Q) = \min\{\tilde d_i, \max(0, Q - \sum_{j \text{ ahead of } i} \tilde d_j)\}$ solves the fractional knapsack (CPP-F).

Standing assumptions (§3.3): the demands are independent and positive, and almost surely $\max_i \tilde d_i \le \Delta$ for a constant $\Delta > 0$ . The targets satisfy $0 \le \alpha_i \le 1$ .

Formalization targets

Goal: Theorem 5

Q^{\mathrm{Anti}}_I - Q^{\mathrm{Adap}}_I \le 2\Delta.

The bound is uniform in $N$ and in the joint law of the demands, subject only to the standing assumptions.

The two halves: Theorems 3 and 4

Q^{\mathrm{Anti}}_I \le Q_{II} + \Delta, \qquad Q^{\mathrm{Adap}}_I \ge Q_{II} - \Delta.

Milestones inside the proofs

Achievability of $\Gamma(Q)$ in fill rate by randomized priority orders (p. 21, from Zhong et al. 2018).
Structure of the greedy fill: it is fractional for at most one customer, and positive exactly when the demand ahead is below $Q$ (p. 17).
$\mathbf E[x^*_i]/\mathbf E[\tilde d_i] \le \mathbf P(x^*_i > 0)$ (p. 17).
Theorem 3.
The rounded adaptive policy $A'$ , which also serves the one customer the run turns away, satisfies $\mathbf E[\sum_{k\in U}\tilde d_k y^{A'}_k] \ge \sum_{k \in U} \mathbf E[\tilde d_k]\alpha_k$ (p. 37).
Adaptive feasibility of $Q$ implies $\alpha \in \Gamma(Q + \Delta)$ (p. 37). The page writes $Q^{\mathrm{Adpt}}_I$ , a misprint for $Q^{\mathrm{Adap}}_I$ .
Theorem 4.

Significance

The result. Theorem 5 says that the information an adaptive planner gains costs at most two maximal demands of capacity, independently of the number of customers. Anticipative priority rules are therefore near-optimal for Type-I service whenever individual demands are small relative to the pool. This contrasts with the responsive regime: Example 1 of the paper shows that $Q^{\mathrm{Adap}}_I$ can exceed $Q^{\mathrm{Resp}}_I$ by an unbounded factor. In Theorems 3–4 the Type-II capacity $Q_{II}$ serves as a common yardstick, so each policy class is compared with an explicitly computable polymatroid quantity.

Formalizing it. The theorem is proved on paper. To our knowledge none of it is machine-checked. A formal development has to make precise what the paper leaves informal: what a randomized adaptive policy is, why the event "customer $i$ is reached" is independent of $\tilde d_i$ , and why the infima defining the optimal capacities are over non-empty sets. The milestones on the greedy fill and the polymatroid $\Gamma(Q)$ can be reused for other fill-rate and persistency problems.

Difficulty

The obvious comparison fails because the two classes face different constraints. A Type-I requirement is an all-or-nothing event, so a deterministic priority order cannot hedge. With three identical unit-demand customers and targets $1/3$ , any fixed order needs $Q = 3$ , while $Q_{II} = 1$ . Randomization over orders is therefore essential on the anticipative side, and the link between a Type-I requirement and the Type-II region $\Gamma(Q)$ is not a statement about any single order. On the adaptive side, the decisions depend on revealed demands. Whether customer $k$ is fully served is correlated with $\tilde d_k$ , so a Type-I service probability does not translate directly into a bound on the expected demand served, which is what $\Gamma(Q)$ constrains. Conditioning arguments that ignore this correlation give false inequalities.

Formalization scope

Demands are a vector d : Fin N → Ω → ℝ on a probability space. Probabilities are μ.real, expectations are Bochner integrals. The standing assumptions are bundled in Standing μ d Δ α: measurable, iIndepFun, almost surely positive, almost surely ≤ Δ, 0 < Δ, and α i ∈ [0, 1].
A run has at most $N$ steps. A demand equal to the remaining capacity is served. Every run, anticipative ones included, stops at the first customer not fully served.
Adaptive policies are functions of the chosen list and the demand vector that depend only on the chosen customers' demands, with measurable decision events. Anticipative policies are permutations, embedded as adaptive policies that ignore the demands.
Randomized policies are finite mixtures. Service levels are probabilities, not the long-run scenario frequencies of p. 11.
$Q^{\mathrm{Adap}}_I$ , $Q^{\mathrm{Anti}}_I$ and $Q_{II}$ are real infima. Under the standing assumptions every feasible set is non-empty ( $Q = N\Delta$ works) and bounded below by $0$ , so no value is a junk default. $Q_{II}$ uses the subset form $\Gamma(Q)$ , which p. 17 shows equals the Type-II problem (CPP-B).
The greedy fill is stated for a priority order $\sigma$ rather than for weights $\theta'$ . It equals $x^*(\theta', \tilde d, Q)$ for every $\theta'$ whose decreasing order, ties broken by index, is $\sigma$ .
Two encodings would trivialize the goal, and both are excluded. Restricting to deterministic priority orders makes Theorem 3 false. Letting adaptive runs skip an oversized customer and continue enlarges the adaptive class beyond the paper's.

The infrastructure the proofs need: measurability of runs, independence of functions of disjoint coordinates, continuity of $Q \mapsto \mathbf E[\min(Q, S)]$ (so that $Q_{II}$ is attained), and the greedy description of a polymatroid's base polytope. Proofs of any milestone are welcome. So are alternative proofs of milestone 1 that avoid the general polymatroid theorem.

Selected references

G. Lyu, M. C. Chou, C.-P. Teo, Z. Zheng, Y. Zhong, Stochastic Knapsack Revisited: The Service Level Perspective, SSRN working paper 3674463, 2021. https://ssrn.com/abstract=3674463
Y. Zhong, Z. Zheng, M. C. Chou, C.-P. Teo, Resource Pooling and Allocation Policies to Deliver Differentiated Service, Management Science 64(4), 1555–1573, 2018.
A. Alptekinoğlu, A. Banerjee, A. Paul, N. Jain, Inventory Pooling to Deliver Differentiated Service, Manufacturing & Service Operations Management 15(1), 33–44, 2013.
B. C. Dean, M. X. Goemans, J. Vondrák, Approximating the Stochastic Knapsack Problem: The Benefit of Adaptivity, Mathematics of Operations Research 33(4), 945–964, 2008. https://doi.org/10.1287/moor.1080.0330

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Machine Learning Operations Research Optimization·Captain: mikedeng1

Risk Guarantees for End-to-End Prediction and Optimization Processes: The Excess Squared-Loss Risk Bounds the Squared Excess Optimality-Gap Risk, Scaled by the Squared Diameter of the Feasible SetResearch Paper

Motivation

Many operational decisions are optimization problems whose cost vector is unknown when the decision is made but can be predicted from observed features: shortest paths under predicted travel times, portfolios under predicted returns, production plans under predicted prices. The common practice, predict-then-optimize, first fits a regression model $g$ mapping features $w$ to a cost estimate $g(w)$ and then solves the optimization problem with $g(w)$ in place of the true cost $c$ . What the decision maker cares about is not the prediction error but the optimality gap of the resulting decision under the true cost. Elmachtoub and Grigas (arXiv:1710.08005, Management Science 2022) introduced the SPO+ surrogate to train predictors against that gap directly, and established its Fisher consistency.

Ho-Nguyen and Kılınç-Karzan (arXiv:2012.15046, Management Science 2022) ask which surrogate losses give quantitative guarantees: when does a small excess surrogate risk force a small excess optimality-gap risk, uniformly over a class of data distributions? Their answer adapts the calibration theory of Steinwart (Constructive Approximation 26, 2007) to the end-to-end setting, and its most striking consequence concerns the ordinary squared loss.

Setting

Let $X \subset \mathbb R^m$ be a non-empty convex compact set and $f$ a convex function, continuous on $X$ . For a cost vector $c$ the decision maker solves

\min_{x \in X}\ \{ f(x) + c^\top x \}.

An argmin selection is a fixed map $x^* : \mathbb R^m \to X$ with $x^*(d)$ a minimizer of $f(x) + d^\top x$ over $X$ for every $d$ . Using a prediction $d$ instead of the true cost $c$ incurs the true loss

L(d,c) = f(x^*(d)) + c^\top x^*(d) - \min_{x \in X}\{f(x) + c^\top x\} \ \ge 0 .

Features and costs form a random pair $(w,c) \in W \times \mathbb R^m$ with joint law $\mathbb P$ . For a loss $\ell$ and a measurable predictor $g : W \to \mathbb R^m$ , the risk is $R_\ell(g,\mathbb P) = \mathbb E[\ell(g(w),c)]$ and the Bayes risk is $R_\ell(\mathbb P) = \inf_g R_\ell(g,\mathbb P)$ over measurable $g$ ; for $\ell = L$ these are written $R(g,\mathbb P)$ and $R(\mathbb P)$ . The squared loss is $\ell_{LS}(d,c) = \|d - c\|_2^2$ .

Given a set $C \subset \mathbb R^m$ , the class $\mathcal P$ consists of the laws for which every conditional law $\mathbb P[\cdot \mid w]$ is concentrated on $C$ , is square integrable, and has mean $\mathbb E[c \mid w] \in \mathrm{Conv}(C)$ . Assumption 2 asks that $C$ be bounded. The constants are $B_X = \max_{x,x' \in X}\|x - x'\|_2$ , $B_f = \max_{x,x' \in X}\{f(x) - f(x')\}$ and $B_C = \max_{c \in C}\|c\|_2$ .

A loss $\ell$ is $\mathcal P$ -uniformly calibrated if for every $\epsilon > 0$ there is $\delta > 0$ such that, for every law in $\mathcal P$ , every $w$ and every $d$ , a conditional excess $\ell$ -risk $\mathbb E[\ell(d,c)\mid w] - \inf_{d'}\mathbb E[\ell(d',c)\mid w]$ below $\delta$ forces a conditional excess true risk below $\epsilon$ . The largest such $\delta$ is the calibration function $\delta_\ell(\epsilon;\mathcal P)$ .

Formalization targets

Goal: Corollary 3 (p. 22)

For every law $\mathbb P \in \mathcal P$ with $\mathbb E\|c\|_2^2 < \infty$ and every measurable predictor $g$ ,

\big(R(g,\mathbb P) - R(\mathbb P)\big)^2 \ \le\ B_X^2\,\big(R_{\ell_{LS}}(g,\mathbb P) - R_{\ell_{LS}}(\mathbb P)\big).

Milestones

Lemma 2 (p. 10): $g^*(w) = \mathbb E[c \mid w]$ attains $R(\mathbb P)$ , and $R(\mathbb P) = \mathbb E[\min_{d'}\mathbb E[L(d',c)\mid w]]$ .
Remark 5 (p. 19): the conditional excess true risk equals $f(x^*(d)) - f(x^*(\bar c)) + \bar c^\top(x^*(d) - x^*(\bar c))$ with $\bar c = \mathbb E[c\mid w]$ , and is at most $B_f + B_C B_X$ .
Lemma 3 (p. 20): uniform calibration holds iff $\delta_\ell(\epsilon;\mathcal P) > 0$ for all $\epsilon > 0$ .
Lemma 4 (p. 21): the alternative form (14) of $\delta_\ell$ .
Lemma 5 (p. 21): $\|d - \bar c\|_2 \ge \max\{0, f(x) - f(x') + \bar c^\top(x - x')\}/\|x - x'\|_2$ when $x^*(d) = x \ne x' = x^*(\bar c)$ .
Lemma 6 (p. 22): $\delta_{\ell_{LS}}(\epsilon;\mathcal P)$ is the purely geometric infimum of $\|d - \bar c\|_2^2$ .
Theorem 4 (p. 22): $\ell_{LS}$ is $\mathcal P$ -uniformly calibrated, with $\delta_{\ell_{LS}}(\epsilon;\mathcal P) \ge \epsilon^2/B_X^2 > 0$ .
Theorem 3 (p. 20): for any uniformly calibrated $\ell$ , $\delta^{**}_\ell(R(g,\mathbb P) - R(\mathbb P)) \le R_\ell(g,\mathbb P) - R_\ell(\mathbb P)$ , with $\delta^{**}_\ell$ the convex minorant of $\delta_\ell$ on $(0, B_f + B_C B_X]$ .

The goal is stated with the constant $B_X^2$ and no other problem data: it is independent of $f$ , of $C$ beyond boundedness, and of the law of the data.

Significance

The corollary says that a predictor trained by ordinary least squares, without any information about $f$ or $X$ , has excess optimality-gap risk at most $B_X\sqrt{R_{\ell_{LS}}(g,\mathbb P) - R_{\ell_{LS}}(\mathbb P)}$ . Since the excess squared-loss risk separates over coordinates, coordinate-wise regression bounds suffice for end-to-end guarantees (Remark 7). Any learning-theoretic rate for least-squares regression therefore transfers to a rate for the decision problem, up to a square root. Theorem 3 gives the same transfer for any uniformly calibrated surrogate, and is the template for the paper's further results on separable symmetric losses.

The results are proved in the paper; none of them has a machine-checked proof. Formalizing them requires a measure-theoretic treatment of conditional risks through kernels, a Lean version of Steinwart's convexified calibration inequality in this setting, and the geometric bound of Lemma 5. Each milestone is a self-contained target.

Difficulty

The true loss $L(d,c)$ is neither convex nor continuous in $d$ (Lemma 1): the decision $x^*(d)$ jumps between faces of $X$ . A pointwise comparison between $\|d - c\|_2^2$ and $L(d,c)$ fails, since a prediction close to $c$ can still select a far-away vertex when $c$ lies near a tie. The comparison only holds after conditioning on $w$ and averaging over $c$ , and it must hold uniformly over all conditional laws in $\mathcal P$ , not for a fixed distribution. The step from conditional comparisons to the integrated risks is where the uniform bound $B_f + B_C B_X$ on the conditional excess true risk and the convention at $\epsilon = 0$ in $\delta^{**}$ matter; Bayes risks are infima over predictors, so this step also needs the conditional minimizers to assemble into a measurable predictor.

Formalization scope

Vectors live in EuclideanSpace ℝ (Fin m). The standing setting (Standing X f) bundles convexity, compactness and non-emptiness of $X$ , convexity of $f$ on $X$ and continuity of $f$ on $X$ ; IsArgminSelection X f xstar is the selection property. A joint law is μ ⊗ₘ κ, with μ the law of $w$ and κ a Markov kernel giving $\mathbb P[\cdot \mid w]$ ; $\mathbb E[c \mid w]$ is the Bochner integral against κ w. Risks are $[0,\infty]$ -valued lower integrals of ENNReal.ofReal of the non-negative losses, Bayes risks are infima over measurable predictors, and excess risks are truncated differences. The class $\mathcal P$ is lsClass C, a set of conditional laws concentrated on $C`; the calibration function `deltaL` and `UniformlyCalibrated` range over such a set, which matches the page's "for all$ \mathbb P \in \mathcal P $,$ w \in W $" whenever$ W $is non-empty.$ B_X$ is Metric.diam X.

Added or pinned conventions:

measurability of xstar (the paper uses Lebesgue measurability, Lemma EC.4) and of predictors;
$\mathbb E\|c\|_2^2 < \infty$ in the goal, explicit though implied by support on bounded $C$ , to make the extended-real Bayes-risk subtraction finite;
$1/B_X^2$ multiplied out in the goal; Theorem 4 carries $B_X>0$ for its printed quotient;
the conclusion of (12) read conditionally on $w$ ;
$\delta^{**}$ constrained by $h'(0) \le 0$ (the convention $\delta_\ell(0) = 0$ of p. 19);
in Theorem 3, the identity $R_\ell(\mathbb P) = \mathbb E[\inf_{d'}\mathbb E[\ell(d',c)\mid w]]$ and finiteness of $R_\ell(\mathbb P)$ .

The Bayes risks are genuine infima over measurable predictors and the conditional excess risks are defined generically. Neither is defined by the closed form that Lemma 2 or Lemma 6 establishes, so neither lemma holds by definition.

Needed infrastructure: lower integrals against μ ⊗ₘ κ and their disintegration, measurability of the conditional mean and of the optimal value $c \mapsto \min_x\{f(x) + c^\top x\}$ , Jensen's inequality for real convex functions on a bounded interval, and the distance from a point to a half-space. The bias–variance decomposition for laws on $\mathbb R^m$ and the half-space distance formula are reusable beyond this mission. Proofs of any milestone are welcome, as are alternative arguments for the goal that bypass Theorem 3.

Selected references

N. Ho-Nguyen and F. Kılınç-Karzan, Risk guarantees for end-to-end prediction and optimization processes, Management Science 68(12), 2022. arXiv:2012.15046v1. https://arxiv.org/abs/2012.15046
A. N. Elmachtoub and P. Grigas, Smart "predict, then optimize", Management Science 68(1), 2022. https://arxiv.org/abs/1710.08005
I. Steinwart, How to compare different loss functions and their risks, Constructive Approximation 26, 2007. https://doi.org/10.1007/s00365-006-0662-3

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Algorithmic Game Theory Linear Optimization Operations Research·Captain: mikedeng1

Consensus Halving for Sets of Items: For n Agents with Additive Utilities a Consensus Halving That Cuts at Most min{n, m} of the m Items Always ExistsResearch Paper

Motivation

Consensus halving asks for a division of a resource into two parts that every one of $n$ agents values equally, so that a group with heterogeneous preferences can agree that the two parts are of the same worth. The classical setting is a divisible interval (a cake or a necklace), where Simmons and Su (2003), building on Hobby and Rice (1965), showed that $n$ cuts always suffice. The question has a direct application to fair division: once a consensus halving is found, two parties can each take a part without either side of a group of stakeholders preferring the other part. Since Filos-Ratsikas and Goldberg (2018) proved that approximate consensus halving on an interval is PPA-complete, the problem has also become a central example in the complexity theory of total search problems.

Goldberg, Hollender, Igarashi, Manurangsi and Suksompong (arXiv:2007.06754, Mathematics of Operations Research, 2022) study the same question for a set of items with no spatial order: houses, children's custody days, or a portfolio of assets, where an item may be shared fractionally but each shared item is a cost. Their first result, for additive utilities, is the subject of this mission.

Setting

There is a set of $n$ agents and a set of $m$ items. Agent $i$ values item $j$ at $u_i(j)$ , and §2 of the paper assumes $u_i(j)\ge 0$ .

A fractional set of items holds a fraction $x_j\in[0,1]$ of each item $j$ ; it is written as the vector $x=(x_1,\dots,x_m)$ .
Utilities are additive: agent $i$ values the fractional set $x$ at $u_i(x)=\sum_j x_j\,u_i(j)$ .
A partition into two fractional sets $M_1,M_2$ gives every item fractions summing to exactly $1$ , so it is determined by $M_1=x$ : the set $M_2$ holds $1-x_j$ of item $j$ .
A consensus halving (Definition 2.1) is such a partition with $u_i(M_1)=u_i(M_2)$ for every agent $i$ , that is,

\sum_j x_j\,u_i(j)=\sum_j (1-x_j)\,u_i(j)\qquad\text{for all } i .

Equivalently (display (1) of the paper), $\sum_j x_j\,u_i(j)=\tfrac12\sum_j u_i(j)$ for all $i$ .

Item $j$ is cut if a positive fraction of it lies in each part, i.e. $0<x_j<1$ .

In the Lean development these are util, IsFractional, IsConsensusHalving, IsCut and numCuts in the namespace ConsHalving.Additive.

Formalization targets

Goal: Theorem 2.2 (existence form)

For $n$ agents with nonnegative additive utilities over $m$ items there is a consensus halving with at most $\min\{n,m\}$ cut items:

\exists\,x\in[0,1]^m:\quad \sum_j x_j\,u_i(j)=\sum_j(1-x_j)\,u_i(j)\ \ \forall i,\qquad \#\{j:0<x_j<1\}\le\min\{n,m\}.

The paper states:

Theorem 2.2. For n agents with additive utilities, there exists a polynomial-time algorithm that computes a consensus halving with at most min{n, m} cuts.

The mission formalizes the existence of the algorithm's output. The polynomial-time clause is not formalized: the paper fixes no machine model, and its complexity argument is a reference to Gaussian elimination.

Milestones

The milestones follow the steps of the paper's proof (p. 5), which runs an iterative procedure starting from $x\equiv\tfrac12$ :

display (1): the equal-value condition is the linear system $\sum_j x_j u_i(j)=\tfrac12\sum_j u_i(j)$ ;
the case $n\ge m$ : halving every item gives a consensus halving with $m=\min\{n,m\}$ cuts;
the combination $(1-\gamma)x+\gamma y$ of two solutions of (1) solves (1);
the bounds of Step 3(c): the new point stays in $[0,1]^m$ and the minimizing item $j^*$ becomes uncut;
one pass of Step 3 preserves the invariants (consensus halving, every frozen item uncut) and freezes a new item;
if the current point is the unique solution of the system $S\cup T$ , then $|T|\ge m-n$ .

Significance

The result. Splitting every item in half is always a consensus halving, but it cuts all $m$ items. Theorem 2.2 shows that, however many items there are, sharing at most as many items as there are agents is enough. This matches the $n$ -cut bound of Hobby–Rice and Simmons–Su for the interval, and the paper shows it is tight: Theorem 2.5 proves that when the utilities are drawn independently from non-atomic distributions, almost surely every consensus halving cuts at least $\min\{n,m\}$ items. The paper generalizes it to consensus $k$ -splitting (Theorem 5.3, at most $(k-1)\min\{n,m\}$ cuts).

Formalizing it. The theorem is proved in the paper; no machine-checked version exists on Prove2Me or, to our knowledge, elsewhere. The mission produces a checked consensus-halving model for additive utilities over unordered items and a checked proof of the $\min\{n,m\}$ bound, together with a reusable fact about linear systems: a system of $n+|T|$ equations in $m$ unknowns with a unique solution has $n+|T|\ge m$ .

Difficulty

The obvious approach is to note that the consensus halvings form a polytope in $[0,1]^m$ cut out by $n$ equations, and that a vertex of that polytope has at most $n$ fractional coordinates. The difficulty is to make "vertex" precise and nonempty in Lean: one must show that the polytope (nonempty, since it contains $x\equiv\tfrac12$ ) has a point where enough box constraints are tight, which needs either an extreme-point argument for compact convex sets or the paper's explicit procedure. The procedure in turn needs a termination argument: each pass freezes one new coordinate at $0$ or $1$ , and when no further move exists, a dimension count on the solution set of $S\cup T$ is required. The step coefficient $\gamma_{j^*}$ may exceed $1$ , so the "convex combination" of the paper is an affine one, and the bound $s\in[0,1]^m$ comes from the minimality of $\gamma_{j^*}$ , not from convexity.

Formalization scope

Agents are Fin n, items Fin m (the paper's $[n]$ , $[m]$ , indexed from $0$ ). The bounds $n,m\ge1$ are not assumed; every statement remains true for $n=0$ or $m=0$ .
Utilities are u : Fin n → Fin m → ℝ. Nonnegativity, a standing assumption of §2, is a hypothesis of the goal only; the milestones hold for real utilities of any sign, as the paper remarks on p. 6 for combinations of goods and chores.
A partition is encoded by $x$ alone, with $M_2$ holding $1-x_j$ ; IsConsensusHalving includes $x\in[0,1]^m$ . Without that bound every solution of (1) would count, and the cut count would be meaningless, so a formalization dropping it is ruled out.
A cut is $0<x_j\wedge 0<1-x_j$ , not $x_j\ne0$ . numCuts is the cardinality of the set of cut items.
gamma x y j is the $\gamma_j$ of Step 3(a). At $y_j=x_j$ it takes the junk value $0$ , and every statement using it quantifies over $y_j\ne x_j$ only and assumes $y_{j^*}\ne x_{j^*}$ , so the junk value never enters.
In the termination milestone the current point is assumed to solve $S$ , so that the uniqueness hypothesis cannot hold vacuously for an infeasible system. The inequality $|T|\ge m-n$ is written $m\le n+|T|$ .
Mathlib supplies the needed linear algebra (finite-dimensional kernels, rank–nullity). The step lemmas carry the loop invariants as explicit hypotheses rather than formalizing the while loop itself. Solvers may close the goal along the paper's procedure, through the milestones, or by any other argument, such as an extreme-point argument.

Selected references

P. W. Goldberg, A. Hollender, A. Igarashi, P. Manurangsi, W. Suksompong, Consensus Halving for Sets of Items, Mathematics of Operations Research, 2022; preprint arXiv:2007.06754v1, 2020. https://arxiv.org/abs/2007.06754
C. R. Hobby, J. R. Rice, A moment problem in L1 approximation, Proceedings of the American Mathematical Society 16(4):665–670, 1965. https://doi.org/10.2307/2033900
F. W. Simmons, F. E. Su, Consensus-halving via theorems of Borsuk–Ulam and Tucker, Mathematical Social Sciences 45(1), 2003. https://doi.org/10.1016/S0165-4896(02)00087-2
A. Filos-Ratsikas, P. W. Goldberg, Consensus halving is PPA-complete, Proceedings of the 50th ACM Symposium on Theory of Computing (STOC), 51–64, 2018. https://doi.org/10.1145/3188745.3188880

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Combinatorics Graph Theory Operations Research·Captain: mikedeng1

Hypergraph k-Cut for Fixed k in Deterministic Polynomial Time: A Maximal Minimum k-Partition's First Part Is the Source-Maximal Minimum (S, T)-Terminal Cut for Some |S| = 2k − 2Research Paper

Why this question matters

The hypergraph k-cut problem asks for a least-cost way to split the vertices of a hypergraph into $k$ nonempty groups. An edge of a hypergraph may contain more than two vertices, so the graph version does not capture the behavior of every input. For fixed $k$ , randomized polynomial-time algorithms for hypergraph k-cut were known before the deterministic result of Chandrasekaran and Chekuri. Their paper identifies a structural property of optimal partitions that makes one of their deterministic approaches possible. This mission isolates that property as a mathematical theorem about finite hypergraphs.

The same paper explains why the distinction between hypergraph k-cut and partitioning by a general symmetric submodular set function matters. The hypergraph cut function is symmetric and submodular, but the cost of a k-partition counts each crossing hyperedge once; it is generally different from the sum of the cut values of the parts. The paper also gives a counterexample to a proposed extension of its uncrossing theorem to general submodular functions. The target here therefore uses the actual hyperedge crossing cost throughout. Chandrasekaran and Chekuri, §§1.1 and 3.

Finite hypergraphs and terminal cuts

Let $G=(V,E)$ be a finite unit-cost hypergraph. A hyperedge is a subset of the vertex set $V$ , and $E$ may contain parallel copies. For a vertex set $A$ , its cut value $d(A)$ is the number of edge copies meeting both $A$ and its complement $\overline A=V\setminus A$ . Thus empty edges, singleton edges, and edges lying entirely on one side contribute zero. This is the paper's convention after it reduces to unit costs in §1.1. Source, pp. 3 and 7.

An ordered k-partition $P=(V_1,\ldots,V_k)$ has $k$ nonempty, pairwise disjoint parts whose union is $V$ . Its cost counts each hyperedge once if that edge meets at least two distinct parts. A minimum k-partition has least cost among all such ordered partitions. It is maximal minimum when there is no other minimum k-partition whose first part strictly contains $V_1$ . The order matters because the theorem singles out $V_1$ . Source, p. 4.

For disjoint vertex sets $(S,T)$ , an $(S,T)$ -terminal cut is a source set $U$ with $S\subseteq U\subseteq V\setminus T$ . A minimum terminal cut has the smallest $d(U)$ among these sets. The source-maximal minimum terminal cut is the minimum cut whose source set contains the source set of every other minimum $(S,T)$ -terminal cut. This condition identifies one set, even when several minimum cuts have the same value. Source, p. 3.

Formalization targets

The structural theorem

For $k\ge 2$ , let $P=(V_1,\ldots,V_k)$ be a maximal minimum k-partition with $|V_1|\ge 2k-2$ . For every $T\subseteq\overline{V_1}$ that meets every $V_j$ with $j\ge 2$ , the goal is

\exists S\subseteq V_1,\quad |S|=2k-2 \quad\text{and}\quad V_1\text{ is the source-maximal minimum }(S,T)\text{-terminal cut}.

This is Theorem 1.2, p. 4. The size is an equality, and the set $S$ may depend on $T$ . Source maximality is part of the conclusion.

Supporting statements

The milestone list includes symmetry and submodularity of $d$ , uniqueness of the source-maximal minimum terminal cut, the counting statements of §3 (Proposition 3.1, the identity $d(Y_1\cup W)+d(Y_2\cup W)=\sigma(Y_1,Y_2,W,Z)$ ; Lemma 3.1, the one-step refinement bound on $\sigma$ ; Lemma 3.2, the $\sigma$ bound for uncrossed minimum terminal cuts; Lemma 3.3, the selection of $k-1$ parts), and the containment and extremal claims used in §4 (Lemma 4.1, Claims 4.1 and 4.2). The principal intermediate target is Theorem 3.1: under its specified family of minimum terminal cuts, it produces a k-partition $P$ whose last part strictly contains a given complement $\overline U$ , and whose cost is at most half of every pair sum $d(A_i)+d(A_j)$ . These are separate theorem statements with the same finite hypergraph model.

What the result provides

Theorem 1.2 says that a part of a suitably chosen optimum partition can be specified by a bounded-size source set and a sink set meeting the other parts. It gives a finite certificate for recovering that part via a minimum terminal-cut problem. Unlike a statement about one specially chosen low-cut-value part, it applies to the first part of any maximal minimum k-partition that meets the size condition. The paper uses this theorem to derive its first deterministic polynomial-time algorithm. Chandrasekaran and Chekuri, Theorem 2.1 and §2.

This mission formalizes the known structural result, not a new conjecture. The Lean statements are compiled with proof placeholders; the mathematical proofs remain work for solvers. A complete development would produce reusable definitions and lemmas for hypergraph cut values, minimum terminal cuts, uncrossing, and counting edges crossing a partition. These objects are also useful for checking related claims about graph cuts and constrained partitions.

Where the proof is difficult

For a graph edge, crossing a partition has only two endpoints to track. A hyperedge may meet several parts at once, and the relationship between the sum of part cut values and the cost of the whole partition changes with that pattern. A direct submodularity argument does not control all the needed costs. Section 3 introduces an auxiliary count $\sigma$ that distinguishes edge patterns, and Theorem 3.1 combines minimum terminal cuts with a partition cost bound. Section 4 then uses that bound to rule out a failed choice of the source set in Theorem 1.2. The paper explicitly shows that the corresponding uncrossing statement fails for arbitrary symmetric submodular functions. Source, §§3–4, especially Remark 3.2.

Formalization scope

Vertices have a finite Lean type, vertex subsets are Finset V, and hyperedges form a Multiset (Finset V). The multiset preserves parallel copies under unit costs. Partitions are functions Fin k → Finset V; their parts are required to be nonempty, disjoint, and exhaustive, with P 0 representing $V_1$ . Complements are relative to all vertices. All counts are natural numbers. The theorem assumes $k\ge2$ , so $2k-2$ is the ordinary nontruncated value. It counts a crossing edge when that edge meets at least two parts, following the explicit mathematical test in the paper's definition.

The source-maximal predicate requires both minimum cut value and containment of every other minimum terminal source set. Neither a merely minimum cut nor an arbitrary maximal minimum cut is accepted as the target. The cost and d functions count edge copies, including duplicate hyperedges, so replacing the multiset by a set would change the problem. The published predicates for symmetric and submodular real-valued set functions are reused after casting the cut counts to real numbers.

Theorem 1.1, the polynomial-time algorithm claim, is outside this mission. It would need a precise machine and running-time model; an unqualified existential algorithm claim would also be satisfied by exhaustive search. The algorithms and running-time theorems of §§2 and 6, and the distinct structural results of §§5–6, are not represented here. Contributions toward proofs of the source-maximal cut theorem, the §3 counting estimates, and the §4 containment argument are within scope.

Selected references

Karthekeyan Chandrasekaran and Chandra Chekuri, Hypergraph k-cut for fixed k in deterministic polynomial time, arXiv:2009.12442v1, 2020; published in Mathematics of Operations Research, 2022. arXiv.

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Economics Mechanism Design Operations Research·Captain: mikedeng1

Mechanism Design for Managing Hidden Rebates and Inflated Quotes of a Procurement Service Provider: Paying a Fee Equal to the Hidden Rebate Is the Optimal Truthful Contract for Small RebatesResearch Paper

Motivation

Retailers that source apparel and other consumer goods from contract manufacturers in developing countries often work through a procurement service provider, an intermediary that solicits quotes, negotiates and manages production on the retailer's behalf. A documented problem in this industry is the hidden rebate: some manufacturers pay the intermediary a covert "referral commission" whenever they win an order. The intermediary then has a reason to favour those manufacturers, and, because the retailer sees quotes only through the intermediary, it can inflate the quotes of manufacturers that pay nothing. The retailer ends up buying from a more expensive source.

Fan, Chen and Tang (SSRN 3514358; Manufacturing & Service Operations Management, 2020) ask what a retailer can do about this with contracts alone, without audits or penalties, when it can verify a quote only with the manufacturer it finally selects. Their answer for the base model is an explicit contract: a selection rule that handicaps the rebate-paying manufacturer by the size of the rebate, together with a service fee equal to the rebate. The paper relates to the literature on middlemen and corruption in procurement (Burguet and Che 2004; Burguet and Perry 2007) and on mechanism design with partial verification, where Strausz (2016) showed that the revelation principle survives when only the selected party's report can be checked.

Setting

A retailer must buy one order from one of two manufacturers. Manufacturer $S$ is known to be ethical; manufacturer $W$ pays the intermediary a fixed hidden rebate $r \ge 0$ if and only if $W$ wins. The manufacturers' prices $x_S, x_W \ge 0$ are independent draws from a common distribution $\mu$ with density $g$ and survival function $\bar G(t) = \mu((t,\infty))$ . Because $W$ builds the rebate into its quote, the true quotes are

q = (q_S, q_W) = (x_S,\ x_W + r).

The intermediary observes $q$ and reports $z = (z_S, z_W)$ to the retailer. A deterministic contract $\langle A, F\rangle$ is a selection rule $A(z) \in \{S, W\}$ and a non-negative service fee $F(z)$ . After choosing $A(z)$ , the retailer checks the selected manufacturer's quote: if $z_{A(z)} = q_{A(z)}$ , the intermediary is paid $F(z)$ and, if $W$ was selected, also collects $r$ from $W$ ; otherwise it receives nothing. Its ex post payoff (1) is

\pi(z; \langle A,F\rangle) = \big(r\,\mathbf 1_{A(z)=W} + F(z)\big)\,\mathbf 1_{z_{A(z)} = q_{A(z)}}.

A contract is incentive compatible (13) when truthful reporting $z = q$ maximizes this payoff for every pair of true prices; ties are resolved in favour of truth-telling. Under truthful reporting the retailer pays $q_{A(q)} + F(q)$ , and its problem (14) is to minimize the expected value of this cost over incentive-compatible contracts.

The threshold of footnote 15 is

\bar r = \min\Big\{ \inf_{a>0,\,T\ge a} \frac{E\big((x_S-x_W)\mathbf 1_{x_S>x_W>T}\big) + E\big((x_W-x_S)\mathbf 1_{T>x_W>x_S>a}\big)}{\bar G(T)\,\bar G(a)},\ \inf_{T\ge 0} \Big[E\big(x_W \mathbf 1_{x_W<T}\big) + E(x_S)\,\bar G(T) - E\min\{x_S,x_W\}\Big] \Big\}.

In the Lean development these objects are quote, payoff, IsIC, Feasible, cost, contractStar, Gbar and rbar, in the namespace HiddenRebate.Fixed.

Formalization targets

Goal: Theorem 1 (p. 12)

Let $\langle A^*, F^*\rangle$ be the contract (15):

A^*(z) = \begin{cases} S & z_S \le z_W - r\\ W & z_S > z_W - r\end{cases},\qquad F^*(z) = \begin{cases} r & A^*(z) = S\\ 0 & \text{otherwise.}\end{cases}

If $0 \le r < \bar r$ , then $\langle A^*, F^*\rangle$ is incentive compatible, its expected cost is finite, and

E\,C(q;\langle A^*,F^*\rangle) \le E\,C(q;\langle A,F\rangle)

for every admissible incentive-compatible deterministic contract $\langle A,F\rangle$ .

Milestones

§5.1, p. 12. Under (15) the intermediary collects $r$ in every state, so (15) is incentive compatible.
§5.1, p. 12. In cost coordinates, $A^*(x_S, x_W + r) = S$ if and only if $x_S \le x_W$ .
(19), p. 13. $E\,C(q;\langle A^*,F^*\rangle) = E[\min\{x_S,x_W\} + r]$ .

Significance

Theorem 1 gives the retailer a contract that can be written down and implemented: it removes quote inflation entirely, selects the cheaper manufacturer in every state (no distortion), and lowers the retailer's expected cost from $E(x_W) + r$ under "lowest quote wins" to $E\min\{x_S, x_W\} + r$ (Corollaries 1 and 2 of the paper). The price of honesty is exactly the rebate, paid as a fee. The optimality half says that no other deterministic incentive-compatible contract does better as long as the rebate is below $\bar r$ .

The paper's proof of Theorem 1, including the characterization of incentive-compatible contracts and the optimal contract for $r > \bar r$ , is in an appendix that is not part of the preprint held for this mission. Formalizing the theorem therefore means reconstructing the optimality argument, not transcribing it. No machine-checked proof of this result, or of any result of the paper, is known.

Difficulty

Incentive compatibility of (15) is a short case check. The difficulty is optimality: the class of competitors is every pair of measurable maps on $\mathbb R^2$ satisfying (13), and the obvious comparison with the first-best cost $E\min\{x_S,x_W\}$ fails, because a competitor can pay less than $r$ in some states. One has to understand which contracts are incentive compatible under partial verification, where the intermediary can freely misreport the quote of the manufacturer that is not selected, and show that every such contract costs at least $E\min\{x_S,x_W\} + r$ when $r < \bar r$ . The threshold $\bar r$ is an infimum over a two-parameter family, so the comparison must hold uniformly over that family.

Formalization scope

Prices live in $\mathbb R$ with $\mu((-\infty,0)) = 0$ ; $\mu$ is a probability measure absolutely continuous with respect to Lebesgue measure (the density $g$ ); the joint law of $(x_S, x_W)$ is $\mu \otimes \mu$ . Reports range over all of $\mathbb R^2$ , and (13) is required for every pair $x_S, x_W \ge 0$ and every report, pointwise. A contract is a pair A : ℝ × ℝ → Supplier, F : ℝ × ℝ → ℝ with $F \ge 0$ , $\{A = S\}$ measurable and $F$ measurable. The cost (14) is the Bochner integral over $(x_S, x_W)$ of $q_{A(q)} + F(q)$ at $q = (x_S, x_W + r)$ .

Conventions and additions relative to the page, all disclosed in the item statements:

$E[x_S] < \infty$ and $r \ge 0$ are assumed;
competitors are required to have an integrable cost (a non-integrable non-negative cost is infinite and cannot beat (15)), and integrability of (15)'s cost is part of the conclusion;
each "min" in footnote 15 is an infimum, and pairs with $\bar G(T)\bar G(a) = 0$ are excluded from the first one;
footnote 15 is stated as printed: its second term is not divided by $\bar G(T)$ . This can only make $\bar r$ smaller, so the hypothesis is at least as strong as the evidently intended one;
the reduction from general mechanisms to deterministic direct ones (Strausz 2016) is cited by the paper and is not part of the statement.

The goal compares the cost of (15) with each competitor's cost directly; it is not stated with $E\min\{x_S, x_W\} + r$ substituted for the left side, and it is not reduced to incentive compatibility alone, which is milestone 1. For iid Exp(1) prices $\bar r = 1/2 - 1/e \approx 0.132$ , so the hypothesis $r < \bar r$ is satisfiable; for prices with bounded support $\bar r = 0$ and the theorem is vacuous, as on the page.

Needed infrastructure: product measures and Bochner integrals on $\mathbb R^2$ (in Mathlib), and a characterization of incentive-compatible contracts under partial verification, which would be reusable for other models in which only the selected agent is audited. Contributions of intermediate lemmas, such as lower bounds on the cost of incentive-compatible contracts, are welcome as linked theorems.

Selected references

X. Fan, Y.-J. Chen, C. S. Tang, Mechanism design for managing hidden rebates and inflated quotes of a procurement service provider, Manufacturing & Service Operations Management, 2020; preprint SSRN 3514358. https://ssrn.com/abstract=3514358
R. Strausz, Mechanism design with partially verifiable information, working paper, Humboldt University of Berlin, 2016 (as cited in the paper).
R. Burguet, Y.-K. Che, Competitive procurement with corruption, RAND Journal of Economics 35(1), 50–68, 2004.
R. Burguet, M. Perry, Bribery and favoritism by auctioneers in sealed-bid auctions, The B.E. Journal of Theoretical Economics 7(1), 1–27, 2007.

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Algorithmic Game Theory Economics Operations Research·Captain: mikedeng1

Optimal Price/Advertising Menus for Two-Sided Media Platforms: In Duopoly with Multi-Item Menus the Unique Equilibrium Is a Symmetric Two-Item Menu on Which Both Consumer Types Multi-HomeResearch Paper

Motivation

Ad-supported media and digital services (streaming music and video, news sites, social platforms) sell to two sides at once. Consumers value the content, advertisers value the consumers' attention, and consumers usually regard advertising as a nuisance whose cost differs from one person to the next. Many such platforms answer this with menu pricing: a free tier with ads next to a paid tier with few or no ads, so that consumers sort themselves by how much they mind advertising.

DeValve and Pekeč (SSRN 2715531; Operations Research 70(3), 2022, doi:10.1287/opre.2021.2230) characterize revenue-maximizing price/advertising menus when the platform does not observe a consumer's advertising disutility, first for a monopoly and then for a duopoly in which consumers decide endogenously whether to use one platform (single-homing), both (multi-homing) or neither. Earlier models of competing ad-supported platforms either fix homing behaviour through Hotelling competition (Anderson and Coate 2005; Armstrong 2006) or allow multi-homing with homogeneous ad disutility (Ambrus, Calvano and Reisinger 2016). This mission targets the paper's duopoly result, Proposition 3: the equilibrium with menus exists, is unique, and is symmetric, with both consumer types multi-homing.

Setting

A representative consumer has base value $v$ for content and is of type $\theta\in\{L,H\}$ with advertising disutility $d^\theta$ , $0<d^L<d^H$ . Type $H$ occurs with probability $r\in(0,1)$ ; write $r^H=r$ , $r^L=1-r$ . Two platforms $i=1,2$ each offer a menu of two items $(s,a)$ , a subscription price $s\in\mathbb R$ (negative prices allowed) and an advertising quantity $a\ge 0$ ; the two items of a menu may coincide.

A consumer who single-homes on platform $i$ with item $(s_i,a_i)$ and consumes $x_i\ge0$ gets $v x_i-\tfrac12x_i^2-(d^\theta a_i+s_i)$ . A consumer who multi-homes with items $(s_1,a_1)$ , $(s_2,a_2)$ gets

\sum_{i=1}^2\big(vx_i-\tfrac12x_i^2-(d^\theta a_i+s_i)\big)-\gamma x_1x_2,

where $\gamma\in(0,1)$ measures substitutability between the platforms. The utility of a choice is the supremum over nonnegative consumption; abstaining gives $0$ . Write $c^\theta(s,a)=d^\theta a+s$ for the cost of an item to type $\theta$ .

Advertisers pay a per-type market-clearing price $p=w-\sum_i a_i$ , the sum over the platforms the consumer uses, and a platform used with item $(s_i,a_i)$ earns $s_i+p\,a_i$ . Each platform maximizes its expected revenue over the two types.

An equilibrium is a menu profile with consumer choices such that: each type maximizes utility; on ties she multi-homes if multi-homing is optimal and participates if participating is optimal; and no platform can raise its expected revenue by switching to another admissible menu, when the consumers respond optimally under the same tie rules, the deviator may select the response most favourable to it, and a type that uses the rival platform before and after the deviation keeps her item there. The paper uses this concept in its proofs without naming it. The mission writes it out.

Let $\rho=(1-\gamma)/(1+\gamma)$ and $\alpha=\big(w-d^H-(1-r)(d^H-d^L)/r\big)^+$ .

Formalization targets

Goal: Proposition 3

With

(s^L,a^L)=\Big(\tfrac{\rho v^2}{2}-d^La^L-(d^H-d^L)a^H,\ \tfrac{w-d^L}{3}\Big),\qquad(s^H,a^H)=\Big(\tfrac{\rho v^2}{2}-d^Ha^H,\ \tfrac{\alpha}{3}\Big),

the profile in which both platforms offer $\{(s^L,a^L),(s^H,a^H)\}$ and each type $\theta$ multi-homes on $(s^\theta,a^\theta)$ is an equilibrium; the two items are distinct; in every equilibrium each menu is exactly these two items and each type multi-homes on her item; and each platform's expected revenue is

\frac{\rho v^2}{2}+\frac{(1-r)(w-d^L)^2+r\alpha^2}{9}.

Milestones

Lemma 6 A–E (p. 45): the multi-homing utility $v^2/(1+\gamma)-\sum_i c^\theta_i$ , the single-homing utility $\tfrac12v^2-c^\theta_i$ , the participation threshold $(1-\gamma)v^2/(2(1+\gamma))$ for multi- and single-homing, and cost minimization within each platform's menu.
Lemma 8 (p. 47): in every equilibrium both types multi-home.
Consumer choices (pp. 52–53): at the stated menus, multi-homing on the designed items is an admissible response for each type.
Platform best response (pp. 53–56): against the stated menu, no deviation earns more than the stated revenue.
Best response (22)/(24) (pp. 56–58): in every equilibrium, each platform's chosen ad quantities and prices are the explicit best response to the rival's.
Fixed point (pp. 58–59): the best-response equations (22) have the unique solution $a^L_1=a^L_2=(w-d^L)/3$ , $a^H_1=a^H_2=\alpha/3$ .

Significance

The proposition describes how competition shapes menus. With two platforms, consumers of both types use both platforms, the menus are symmetric, and the low-disutility type is given a price discount while the high-disutility type gets an advertising discount. When $\alpha=0$ the menu is a "free use with ads / pay for no ads" menu. The paper builds on this characterization: Corollary 1 compares monopoly and duopoly prices by type, and Proposition 4 shows the value of offering a menu instead of a single item can exceed $2$ under competition.

No machine-checked version of these results exists. A formal proof checks a uniqueness argument that runs over every combination of homing behaviour and every admissible deviation, and it fixes an equilibrium concept that the paper uses without defining. The model file is reusable for the paper's other duopoly results (Lemma 3, Proposition 4, Corollary 1) and for related two-sided pricing models with quadratic utility.

Difficulty

The consumer has nine possible choices per type, and a platform's deviation changes which of them is optimal. A best response is therefore not one optimization problem but a family indexed by homing behaviour, and the ad price $w-\sum_ia_i$ couples the two platforms through every type that multi-homes. Reducing the deviation problem to a single program requires showing that no deviation profitably moves a type from multi-homing to single-homing, which uses the rival's menu and the tie rules. The uniqueness part needs every non-multi-homing configuration ruled out separately (Lemma 8) and then a fixed-point argument for a piecewise-linear best-response map with a pooling regime. Tie-breaking matters: without the multi-homing tie rule or the status-quo rule on the rival platform, the stated profile is not an equilibrium.

Formalization scope

Lean represents platforms and item slots by Fin 2 (the paper's platforms 1, 2 are 0, 1), menus by Fin 2 → ℝ × ℝ with entries (s, a), and all quantities in ℝ. A menu is admissible when every ad quantity is nonnegative; prices are unrestricted. Utilities are suprema over nonnegative consumption, which are finite because $\gamma<1$ . They are not defined by the closed forms of Lemma 6. The equilibrium predicate IsEquilibrium encodes the four rules above: consumer optimality, multi-homing and participation on ties, arbitrary admissible deviations with the deviator's most favourable admissible response, and the status-quo rule on the rival platform. The strategy space is two-item menus, as in the paper after Lemma 7. Uniqueness is uniqueness of the chosen items and of each menu as a set, up to which slot holds which item.

Two hypotheses are added to the paper's standing assumptions and disclosed: $v>0$ , and $w>d^L$ , without which the stated $a^L$ is negative. The condition $w>d^H$ is not assumed; the case $\alpha=0$ is part of the result.

The goal is not posed as "each menu solves the platform's program (21) given the other". That would assume the reduction from the game to the program, which is the content of Lemma 8 and of the platform-best-response step. The goal quantifies over all equilibria of the game itself. Existence is a conjunct of the goal, so the uniqueness and revenue clauses cannot hold vacuously.

Needed: the model file (provided), elementary calculus of concave quadratics for Lemma 6, and case analysis over finite choice sets. Welcome contributions are proofs of the Lemma 6 parts, the fixed-point algebra, the consumer-choice step, and the two platform best-response steps.

Selected references

L. DeValve, S. Pekeč, Optimal Price/Advertising Menus for Two-Sided Media Platforms, SSRN 2715531 (version of 2021-04-15); published in Operations Research 70(3), 2022. https://ssrn.com/abstract=2715531, https://doi.org/10.1287/opre.2021.2230
A. Ambrus, E. Calvano, M. Reisinger, Either or Both Competition: A "Two-Sided" Theory of Advertising with Overlapping Viewerships, American Economic Journal: Microeconomics 8(3), 2016. https://doi.org/10.1257/mic.20150019
S. P. Anderson, S. Coate, Market Provision of Broadcasting: A Welfare Analysis, Review of Economic Studies 72(4), 2005. https://doi.org/10.1111/0034-6527.00356
M. Armstrong, Competition in Two-Sided Markets, RAND Journal of Economics 37(3), 2006. https://doi.org/10.1111/j.1756-2171.2006.tb00037.x
J.-C. Rochet, J. Tirole, Platform Competition in Two-Sided Markets, Journal of the European Economic Association 1(4), 2003. https://doi.org/10.1162/154247603322493212

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Economics Operations Research Probability·Captain: mikedeng1

Pricing of Debt and Equity in a Financial Network with Comonotonic Endowments: Closed-Form Default Probabilities and Expected Wealth, Debt Payments and Equity under Bankruptcy CostsResearch Paper

Motivation

Banks hold claims on one another. When one bank cannot pay, its creditors receive less, which can push them into default in turn. Clearing models compute the payments and wealths that are consistent across the whole network. Regulators and researchers use them to measure systemic risk and to price interbank debt. The basic model is that of Eisenberg and Noe (2001), doi:10.1287/mnsc.47.2.236.9835. Rogers and Veraart (2013), doi:10.1287/mnsc.1120.1569, added bankruptcy costs: a defaulting bank recovers only part of its assets.

The value of a bank's debt or equity is an expectation over random endowments. For general endowments that expectation requires partitioning the endowment space into up to $2^n$ default regions (Gouriéroux, Héam and Monfort 2012), and with bankruptcy costs those regions need not be convex. Banerjee and Feinstein, Pricing of debt and equity in a financial network with comonotonic endowments (Operations Research 2022; arXiv:1810.01372), observe that when all endowments are driven by one factor (they are comonotonic), banks default in a fixed order as the factor falls. Default probabilities and the prices of debt and equity then have closed forms.

Setting

There are $n$ banks and an outside society node. Bank $i$ owes a total of $\bar p_i > 0$ ; the fraction $\pi_{ij} \ge 0$ of it is owed to bank $j$ ( $\pi_{ii} = 0$ ), and the fraction $1 - \sum_j \pi_{ij} > 0$ to society (Assumption 2.1). The matrix $\Pi = (\pi_{ij})$ is the relative liabilities matrix. Banks start with endowments $x \in \mathbb R^n_+$ . A bank with negative wealth $V_i < 0$ defaults and keeps a fraction $\alpha_x \in [0,1]$ of its external assets and $\alpha_L \in [0,1]$ of its interbank assets.

With $V^- = \max(-V, 0)$ componentwise, the clearing process is

\Psi^*_i(V) = \big(\mathbb I_{\{V_i\ge0\}} + \mathbb I_{\{V_i<0\}}\alpha_x\big)x_i + \big(\mathbb I_{\{V_i\ge0\}} + \mathbb I_{\{V_i<0\}}\alpha_L\big)\sum_{j}\pi_{ji}(\bar p_j - V_j^-) - \bar p_i ,

and a variant $\Psi$ , which replaces $\bar p_j - V_j^-$ by its positive part, is used in the appendix. The greatest clearing wealth $V(x)$ is the largest real solution of $V = \Psi^*(V)$ . Bank $i$ 's payment is $p_i(x) = \bar p_i - V_i(x)^-$ and its equity is $E_i(x) = V_i(x)^+$ .

If the set of defaulting banks has indicator $z \in \{0,1\}^n$ , the clearing equations are linear. With $\operatorname{diag}(z)$ the diagonal 0–1 matrix of $z$ ,

\Delta(z) = \big(I - (I - (1-\alpha_L)\operatorname{diag}(z))\Pi^\top\operatorname{diag}(z)\big)^{-1}\big(I - (1-\alpha_x)\operatorname{diag}(z)\big),

\delta(z) = \big(I - (I - (1-\alpha_L)\operatorname{diag}(z))\Pi^\top\operatorname{diag}(z)\big)^{-1}\big[I - (I - (1-\alpha_L)\operatorname{diag}(z))\Pi^\top\big]\bar p .

Comonotonic endowments are $X = f(q)$ for a nonnegative random factor $q$ and a nondecreasing map $f : \mathbb R_+ \to \mathbb R^n_+$ (Assumption 3.2). The solvency threshold of bank $i$ is $q^*_i = \inf\{q \ge 0 \mid V_i(f(q)) \ge 0\}$ (Definition 3.4). Banks are numbered so that $q^*_1 \ge \dots \ge q^*_n$ , with $q^*_0 = \infty$ and $q^*_{n+1} = 0$ (Assumption 3.5). On $q \in [q^*_{k+1}, q^*_k)$ exactly banks $1,\dots,k$ default; $\Delta_k$ and $\delta_k$ are $\Delta$ and $\delta$ at that default set, with $\Delta_0 = I$ and $\delta_0 = (I - \Pi^\top)\bar p$ .

Formalization targets

Goal: Theorem 3.6 (p. 13), with right-continuous $f$

With $T_k = \Delta_k\,\mathbb E[f(q)\mathbb I_{\{q\in[q^*_{k+1},q^*_k)\}}] - \delta_k\,\mathbb P(q\in[q^*_{k+1},q^*_k))$ , for every bank $i$ :

\mathbb P(V_i(X)<0) = \mathbb P(q<q^*_i), \qquad \mathbb E[V_i(X)] = e_i^\top\sum_{k=0}^n T_k,

\mathbb E[p_i(X)] = \bar p_i + e_i^\top\sum_{k=i}^n T_k, \qquad \mathbb E[E_i(X)] = e_i^\top\sum_{k=0}^{i-1}T_k .

The goal is the theorem as printed, with one added hypothesis (see Formalization scope).

Milestones

Proposition A.1 (p. 34): $V = \Psi(V)$ has a greatest and a least solution, and every solution lies in $[-\bar p,\; x + \Pi^\top\bar p - \bar p]$ .
Proposition A.3 (p. 34): the greatest solution of $V = \Psi(V)$ is the greatest real fixed point of $\Psi^*$ , so it is $V(x)$ .
Proposition A.6 (p. 36): $V$ , $p$ and $E$ are nondecreasing in $x$ .
Corollary A.4 (p. 35): the fictitious default algorithm stops at $V(x)$ , and every matrix it inverts is nonsingular.
§3.1 display (p. 11): $V(x) = \Delta(\mathbb I_{\{V(x)<0\}})x - \delta(\mathbb I_{\{V(x)<0\}})$ for every $x \in \mathbb R^n_+$ .

Significance

Theorem 3.6 turns the valuation of every bank's debt and equity in a network with bankruptcy costs into $n+1$ one-dimensional integrals of the factor, replacing the $2^n$ default regions of the general case. The default probabilities depend only on the thresholds $q^*$ , which in turn depend only on the map $f$ and not on the law of $q$ . The same formulas price debt and equity in the paper's numerical study of the European banking system, and they are the starting point for the comonotonic bounds of Lemma 4.1, which are not part of this mission.

The paper gives a one-sentence proof of Theorem 3.6 and proves its appendix results largely by citation (Tarski, Milgrom–Roberts, Rogers–Veraart, input–output theory). No part of it has been machine-checked. A formalization supplies the missing steps: the identification of the two clearing maps, the attainment of the thresholds, nonsingularity of the linear systems, and the decomposition of the expectations. It also produces, as a reusable by-product, the Rogers–Veraart clearing model with its greatest fixed point, which other financial-network results can build on.

Difficulty

The obvious argument is: banks default in order, on each interval of $q$ the wealth is affine in $f(q)$ , so integrate piece by piece. Two steps of it fail as stated.

First, the threshold $q^*_i$ is an infimum, and the paper treats it as attained ("the minimal value such that firm $i$ is solvent"). With bankruptcy costs the wealth $V(f(q))$ jumps when a bank defaults, even for continuous $f$ , so attainment needs an argument about the greatest fixed point under decreasing endowments. Without right-continuity of $f$ it is false.

Second, the greatest fixed point of $\Psi^*$ is not reached by Tarski's theorem on $[-\infty, \cdot]$ , because $\Psi^*$ is not monotone below $-\bar p$ . That every real fixed point lies above $-\bar p$ uses Assumption 2.1. Nonsingularity of $I - (I-(1-\alpha_L)\Lambda)\Pi^\top\Lambda$ also rests on Assumption 2.1. Matching the default set $\{1,\dots,k\}$ to the interval $[q^*_{k+1},q^*_k)$ then combines monotonicity (Proposition A.6) with the ordering of Assumption 3.5.

Formalization scope

Model. The theorem statements assume $n>0$ , as in §2 of the paper. Banks are Fin n; bank i is the paper's bank $i+1$ . The data are $\Pi$ , $\bar p$ , $\alpha_x$ , $\alpha_L$ (structure Network), with the standing assumptions bundled in Network.Standing. $V(x)$ is the supremum of the real fixed points of $\Psi^*$ in the lattice $\mathbb R^n$ ; payments and equity follow Definition 2.2. Default sets are finite sets of banks. Inverses are Matrix.inv.
Thresholds are valued in $[0,\infty]$ : a bank that is never solvent has $q^*_i = \infty$ (the paper writes $q^* \in \mathbb R^n_+$ , but its own convention $q^*_0 = \infty$ requires the extended value).
Added hypothesis: $f$ is right-continuous on $[0,\infty)$ . As printed, Theorem 3.6 fails: with $n=1$ , $\Pi=0$ , $\bar p=1$ , $f=0$ on $[0,1]$ and $f=2$ afterwards, and $q \in \{1,2\}$ with probability $\tfrac12$ each, $\mathbb P(V_1<0) = \tfrac12$ but $\mathbb P(q<q^*_1) = 0$ .
Typo: Corollary A.4 (iv) prints $\Pi^\top\Lambda V^{(k)}$ where $\alpha_L\Pi^\top\Lambda V^{(k)}$ is meant in the default bracket; the closed form on its second line is used.
Index shift: the paper's sums over $k = i,\dots,n$ and $k = 0,\dots,i-1$ become $k \in \{i+1,\dots,n\}$ and $k \in \{0,\dots,i\}$ for the 0-based bank i.
Endowments are $X = f(q)$ literally, with $q$ measurable, $q \ge 0$ almost surely, and each $f_j(q)$ integrable.
Ruled out. $V$ is never defined by the piecewise formula, and $q^*$ is never a free parameter. Both are computed from the clearing map, and only their ordering (Assumption 3.5) is assumed. Integrability and Assumption 2.1 are present wherever an integral or an inverse appears, so neither can collapse to Lean's junk value $0$ .
Infrastructure. The development needs Knaster–Tarski on boxes in $\mathbb R^n$ , monotone comparative statics of greatest fixed points, invertibility of $I - A$ for nonnegative $A$ with column sums below $1$ , and decomposition of Bochner integrals over finitely many disjoint events. The clearing model and the greatest-fixed-point lemmas are reusable for other network models. Proofs of any milestone, and alternative routes to the goal, are welcome.

Selected references

T. Banerjee, Z. Feinstein, Pricing of debt and equity in a financial network with comonotonic endowments, Operations Research, 2022; preprint arXiv:1810.01372v5. https://arxiv.org/abs/1810.01372
L. Eisenberg, T. H. Noe, Systemic risk in financial systems, Management Science 47(2), 2001. https://doi.org/10.1287/mnsc.47.2.236.9835
L. C. G. Rogers, L. A. M. Veraart, Failure and rescue in an interbank network, Management Science 59(4), 2013. https://doi.org/10.1287/mnsc.1120.1569
C. Gouriéroux, J.-C. Héam, A. Monfort, Bilateral exposures and systemic solvency risk, Canadian Journal of Economics 45(4):1273–1309, 2012.
P. Milgrom, J. Roberts, Comparing equilibria, American Economic Review 84(3), 1994.

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Linear algebra Operations Research Optimization·Captain: mikedeng1

Market Efficient Portfolios in a Systemic Economy: f-Efficient Bank Holdings Exist, Attain Mean Squared Deviation (bᵀv)²/(1ᵀG⁻¹1), and Are Unique if and only if There Are Exactly Two BanksResearch Paper

Motivation

When banks are hit by losses they sell assets to restore their leverage targets, and these fire sales depress prices for every other holder of the same assets. The resulting deviation of market capitalization from its fundamental value is a systemic externality: it depends not on any single bank's portfolio but on how the stock of each asset is distributed across banks. Awiszus, Capponi and Weber (arXiv:2003.10121, Operations Research 2022) ask which distribution of a fixed stock of assets among banks with fixed budgets minimizes this externality, as measured by the expected squared deviation of market capitalization from fundamentals. The question matters to macroprudential regulators, who can influence portfolio overlap among banks, and it connects to the literature on overlapping portfolios and price-mediated contagion (Greenwood, Landier and Thesmar, 2015; Capponi and Larsson, 2015).

The paper's answer is explicit: the optimal mean squared deviation has a closed form, the optimum is computed by a two-step "aggregate first, then allocate" procedure, and the optimal allocation is unique exactly when there are two banks.

Setting

There are $K$ assets and $N$ banks. A holding matrix $Q\in\mathbb R^{K\times N}$ records in entry $Q_{ki}$ the holding of asset $k$ by bank $i$ , so column $i$ is bank $i$ 's portfolio. Initial prices are normalized to $1$ .

The vector $q\in\mathbb R^K$ gives the total initial holdings of each asset in the banking sector, and $b\in\mathbb R^N$ gives the banks' budgets. The feasible holding matrices are

\mathscr D(q,b) = \{Q\in\mathbb R^{K\times N} \mid Q\mathbf 1_N = q,\ \mathbf 1_K^\top Q = b^\top\}.

Price shocks $Z_1,\dots,Z_K$ have means $\mu\in\mathbb R^K$ and are uncorrelated with variances $\sigma_k^2$ , $\sigma_k>0$ (Assumption 2). Their second-moment matrix is

G = \mu\mu^\top + \mathrm{Diag}(\sigma^2).

The systemic significance vector $v\in\mathbb R^N$ (the paper's (3.1)) summarizes each bank's leverage targeting, liquidation strategy and the assets' illiquidity. To first order, the deviation of market capitalization from fundamentals is $(Qv)^\top Z$ , and its second moment is the mean squared deviation

\mathrm{MSD}(Q) = (Qv)^\top G\,(Qv).

A holding matrix $Q^*\in\mathscr D(q,b)$ is f-efficient (Definition 1) if it minimizes $\mathrm{MSD}$ over $\mathscr D(q,b)$ .
Assumption 3: $v_i\neq v_j$ for some $i,j$ , and $b^\top v\neq 0$ .
The aggregate problem (4.3) minimizes $y^\top G y$ over $\mathscr D_y = \{y\in\mathbb R^K \mid \mathbf 1_K^\top y = b^\top v\}$ . With $z = G^{-1}\mathbf 1_K$ , put $y^* = \frac{b^\top v}{\mathbf 1_K^\top z}\,z$ .

Formalization targets

Goal: Theorem 4.1

Let $N,K\ge 2$ . Then

\exists\,Q^*\ \text{f-efficient with}\quad \mathrm{MSD}(Q^*) = \frac{(b^\top v)^2}{\mathbf 1_K^\top G^{-1}\mathbf 1_K},

the f-efficient holding matrix is unique if and only if $N=2$ , and in that case it equals

Q^{N=2} = \frac{1}{v_2-v_1}\big(\,v_2 q - y^*\;\;\; y^* - v_1 q\,\big)\in\mathbb R^{K\times 2}.

Milestones

$G$ is positive definite, $G^{-1}$ has the Sherman–Morrison form, and $\mathbf 1_K^\top G^{-1}\mathbf 1_K\neq 0$ (proof of Lemma 4.1).
Lemma 4.1: $y^*$ is the unique solution of the aggregate problem (4.3).
Proposition 4.2: for $N\ge2$ , $K\ge1$ there is $Q^*\in\mathscr D(q,b)$ with $Q^*v=y^*$ .
Proposition 4.1: $Q^*\in\mathscr D(q,b)$ is f-efficient if and only if $Q^*v$ solves (4.3).
Remark EC.1.1: $Q^*$ is f-efficient if and only if $F\,\mathrm{vec}(Q^*) = (b,q,y^*)$ for the $(2K+N)\times KN$ matrix $F$ of (EC.1.12).
The value computation $\mathrm{MSD}(Q^*) = (y^*)^\top G y^* = (b^\top v)^2/\mathbf 1_K^\top G^{-1}\mathbf 1_K$ (proof of Theorem 4.1 a).
$\mathrm{rank}(F) = 2K+N-2$ and $\mathrm{null}(F) = (N-2)(K-1)$ (proof of Theorem 4.1 b).

Significance

The result itself. Theorem 4.1 gives the smallest achievable mean squared deviation of market capitalization from fundamentals for a given asset supply, budget profile and systemic significance profile, as an explicit function of the shock moments. It is the benchmark against which the paper measures the inefficiency of observed and diversified holdings (Section 4.2 shows that full diversification is f-efficient only in a degenerate case) and drives the comparative statics of Section 5. Part b) says that with three or more banks efficiency leaves an affine family of optimal allocations of dimension $(N-2)(K-1)$ , so a regulator has residual freedom to pursue other objectives at no cost in market efficiency.

Formalizing it. The result is proved in the paper; no machine-checked version is known. The formal development delivers the theorem together with a reusable treatment of minimizing a positive definite quadratic form over a hyperplane, the existence part of the transportation-type linear system $Q\mathbf 1_N=q$ , $\mathbf 1_K^\top Q=b^\top$ , $Qv=y$ , and an exact rank computation for the structured matrix $F$ . The uniqueness characterization in part b) depends on that rank computation, which the paper carries out by Gaussian elimination on block matrices.

Difficulty

The minimization over $\mathscr D(q,b)$ is a $KN$ -dimensional quadratic program whose objective is degenerate: it depends on $Q$ only through the $K$ -vector $Qv$ . Strict convexity therefore cannot give uniqueness, since the objective is constant along every direction in which $Qv$ and the constraints do not change, and the obvious argument for part b) fails. Existence is not automatic either: $\mathscr D(q,b)$ is unbounded, and one must show that every point of the hyperplane $\mathscr D_y$ is attained as $Qv$ by a feasible $Q$ , which fails when $v$ is constant. The uniqueness statement is a dimension count: it needs the exact rank of the $(2K+N)\times KN$ constraint matrix $F$ , whose rows carry exactly two linear dependencies under Assumption 3, so that the solution set has dimension $(N-2)(K-1)$ , which is positive as soon as $N\ge3$ and $K\ge2$ .

Formalization scope

Assets are Fin K, banks are Fin N, all quantities are real; holding matrices are Matrix (Fin K) (Fin N) ℝ. The paper's banks 1 and 2 are the indices 0 and 1.
$G^{-1}$ is Mathlib's Matrix.inv, which returns $0$ on singular matrices. Every statement that mentions $G^{-1}$ , $z$ or $y^*$ assumes $\sigma_k>0$ for all $k$ (Assumption 2), and every statement that divides by $\mathbf 1_K^\top z$ assumes $K\ge1$ .
The systemic significance $v$ is a free parameter in $\mathbb R^N$ subject to Assumption 3, not derived from the primitives by (3.1). Every result of Section 4 holds for each such $v$ , and $v$ does not depend on $Q$ .
Added hypothesis: $\sum_k q_k=\sum_i b_i$ appears on the statements that need $\mathscr D(q,b)\neq\emptyset$ (Proposition 4.2 and Theorem 4.1). The paper derives it from feasibility on p. 11; without it part a) is false.
Statements that use only the first half of Assumption 3 ( $v$ not constant) assume only that half. The goal carries Assumption 3 in full.
f-efficiency is an argmin. It is the minimization over $\mathscr D(q,b)$ of Definition 1, and $\mathrm{MSD}$ is the quadratic form $(Qv)^\top G(Qv)$ . Defining f-efficiency as $Qv=y^*$ , or defining $\mathrm{MSD}$ through $y^*$ , would remove the optimization from the goal and is ruled out.
$\mathrm{vec}(Q)$ is indexed by pairs (asset, bank) instead of stacked columns, and $\mathrm{null}(F)$ is the Module.finrank of the kernel of $x\mapsto Fx$ .
$Q^{N=2}$ is defined for $v\in\mathbb R^2$ ; the goal transports it along $N=2$ with Fin.cast.

The development needs the Sherman–Morrison identity for a diagonal-plus-rank-one matrix, the minimizer of a positive definite quadratic form under one linear constraint, explicit constructions in $\mathscr D(q,b)$ , and a rank computation. Contributions of reusable lemmas for these, and of proofs of the milestones in any order, are welcome.

Selected references

K. Awiszus, A. Capponi, S. Weber, Market Efficient Portfolios in a Systemic Economy, Operations Research 70(2), 2022. Preprint arXiv:2003.10121v3. https://arxiv.org/abs/2003.10121
R. Greenwood, A. Landier, D. Thesmar, Vulnerable Banks, Journal of Financial Economics 115(3), 2015. https://doi.org/10.1016/j.jfineco.2014.11.006
A. Capponi, M. Larsson, Price Contagion through Balance Sheet Linkages, Review of Asset Pricing Studies 5(2), 2015. https://doi.org/10.1093/rapstu/rav006
D. S. Bernstein, Matrix Mathematics, Princeton University Press, 2005.

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Operations Research Probability Statistics·Captain: mikedeng1

A General Framework to Compare Announcement Accuracy: Static vs. LES-Based Announcement: The Static Announcement Has No Larger MSE if and only if the Correlation Is at Most 1/2Research Paper

Why delay announcements matter

A customer who is told that service will be delayed can use the announced wait to decide whether to stay, leave, or plan around the delay. A service operator needs to know whether a changing, customer-specific announcement actually predicts the eventual wait more accurately than a fixed announcement. Bassamboo and Ibrahim studied this question for the last-entered-service (LES) announcement, which reports the wait of the last customer to enter service, and for a static announcement based on the mean wait of delayed customers. Their empirical examples show that neither method wins uniformly across queueing regimes; the comparison depends on how the announcement and the eventual wait move together (Bassamboo and Ibrahim, 2021, §§3–4 of the submitted manuscript).

The mission isolates the paper's general comparison criterion. Its statement needs no particular arrival process, service discipline, or queue length. This makes the result a reusable mathematical test for predictors in other service systems as well as for LES. The paper itself notes that the criterion applies irrespective of the specific queueing model, subject to the moment conditions under which its mean-squared errors and correlations are defined (submitted manuscript, §4.2, p. 14).

Random waits and predictions

Let $W_D$ be the real-valued waiting time of a delayed customer in steady state. Operationally, the paper takes the virtual waiting time $W_\infty$ and conditions on $W_\infty>0$ : $W_D$ has the conditional distribution of a positive wait. Let $P$ be a real-valued predictor issued to that customer. To compare a prediction with the outcome, the pair $(P,W_D)$ must be jointly distributed; their separate marginal distributions cannot determine the correlation. Both are represented as functions on one probability space in Lean.

The static announcement, denoted $\operatorname{EA}$ , always reports $\mathbb E[W_D]$ . A changing predictor can instead respond to the system's current state. The paper's mean-squared error of $P$ is the expectation of its squared discrepancy from the outcome:

\operatorname{MSE}(P)=\mathbb E[(W_D-P)^2].

Write $\operatorname{Var}(X)$ and $\operatorname{Cov}(X,Y)$ for variance and covariance. When both variances are positive, the Pearson correlation is $r[X,Y]=\operatorname{Cov}(X,Y)/\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}$ . The coefficient of variation of the delay is $C_{W_D}=\sqrt{\operatorname{Var}(W_D)}/\mathbb E[W_D]$ . These are the quantities in §4.2 and equation (3) of the submitted manuscript (published article; mission citations use the submitted manuscript's page and equation numbers).

Formalization targets

The first milestone establishes the error of the static predictor from its definition:

\operatorname{MSE}(\operatorname{EA})=\operatorname{Var}(W_D).

The next two milestones give the expanded error identity from the proof of Proposition 1 and its normalized form, equation (4). Suppose $\mathbb E[P]=\gamma\mathbb E[W_D]$ and $\operatorname{Var}(P)=\beta\operatorname{Var}(W_D)$ with $\gamma,\beta\ge0$ . Proposition 1 states

\operatorname{MSE}(P)=\operatorname{Var}(W_D) \left(\beta+1+\left(\frac{\gamma-1}{C_{W_D}}\right)^2 -2r[P,W_D]\sqrt\beta\right).

The goal is Corollary 1, equation (5). If $P$ has the same mean and variance as $W_D$ , it compares the two predictors at the exact threshold $1/2$ :

\operatorname{MSE}(\operatorname{EA})\le\operatorname{MSE}(P) \quad\Longleftrightarrow\quad r[P,W_D]\le\frac12.

Both inequalities are non-strict. The corollary is stated for a general predictor $P$ ; applying it to LES requires a separate analysis of the LES wait and its correlation with the arriving customer's wait. The submitted manuscript supplies that model-specific analysis later in §5 (Bassamboo and Ibrahim, 2021).

What the criterion gives

Under the corollary's equal-mean and equal-variance assumptions, the comparison of two expected squared errors reduces to one correlation. A predictor with correlation at most $1/2$ offers no mean-squared-error improvement over the static announcement; a predictor above that threshold does. The equality case is included. Proposition 1 also records how a mean shift or a variance change modifies this comparison, which is useful when a proposed announcement is biased or more variable than the delay.

The paper proves these identities. This mission asks for machine-checked proofs of their statements in Lean; the draft theorems themselves are open proof obligations. A completed development would provide reusable definitions of predictor MSE, the constant mean announcement, Pearson correlation, and coefficient of variation, together with exact interfaces for later queueing results. It does not assert that the paper's LES limit theorems are already formalized.

Where the proof work lies

The key mathematical work is passing from the expectation of a squared difference to the variance and covariance terms while keeping the two random variables on the same probability space. In Lean this requires the integrability consequences of square integrability, the relation between covariance and the centered product, and careful handling of the square-root normalization. Merely replacing MSE and correlation by unrelated real parameters would erase that work and yield an elementary numerical equivalence. The proposed definitions are therefore the expectations and covariance stated in the paper, and the goal quantifies over actual random variables.

The boundary $\beta=0$ also needs attention: the predictor has zero variance and ordinary Pearson correlation has a zero denominator. The term involving correlation is multiplied by $\sqrt\beta$ , so the displayed identity still has a meaningful value under the Lean convention described below. Proposition 1's other denominator, $C_{W_D}$ , requires a positive delay mean and variance in the intended setting.

Formalization scope

Lean uses an arbitrary measurable space with a probability measure and real-valued functions $W_D,P$ on that space. It does not construct the underlying queue; §4.2 makes the comparison at the level of the joint law of the delay and prediction. Each theorem explicitly assumes $W_D,P\in L^2$ where both appear. This supplies finite second moments for the paper's variances and MSE and avoids Lean's default value for a nonintegrable real integral. The EA identity only needs $W_D\in L^2$ .

Positive variance of $W_D$ is explicit in Proposition 1, its expansion, and the corollary. It gives the intended domain of correlation in the equal-variance corollary and prevents equation (4) from failing under division by zero. Proposition 1 additionally assumes $\mathbb E[W_D]>0$ , as expected for a positive conditional wait, so $C_{W_D}>0$ . No condition $\beta>0$ is imposed: at $\beta=0$ , Lean evaluates the zero-denominator correlation as zero and the entire correlation term vanishes. The error is defined with $(W_D-P)^2$ as in equation (3), although the proof writes the reversed difference. Contributions toward the integral identities, covariance normalization, and the final inequality are welcome; the four definitions may also serve other prediction-error formalizations.

Selected references

A. Bassamboo and R. Ibrahim, A General Framework to Compare Announcement Accuracy: Static vs. LES-Based Announcement, Management Science, 2021. DOI: 10.1287/mnsc.2020.3722. Mission theorem numbers and pages refer to the authors' submitted manuscript, §§3.1 and 4.2, pp. 7 and 12–14.

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Algorithmic Game Theory Economics Operations Research·Captain: mikedeng1

Multiregional Oligopoly with Capacity Constraints: The Capacity-Constrained Cournot Game Has a Unique Equilibrium, Implementing the Unique Fixed Point of the Marginal-Profit Best ResponseResearch Paper

Motivation

Firms that serve several geographic markets must decide both how much to produce and where to sell it. A capacity limit couples those regional decisions: an extra unit sent to one region cannot also be sent to another. Different transport costs and demand curves make a collection of independent single-region Cournot models inadequate. Alsabah, Bernard, Capponi, Iyengar and Sethuraman study whether this coupled quantity game has a determinate outcome. Their 2020 author version proves a unique equilibrium allocation and characterizes it through firms' marginal profits. This mission concerns that result, Theorem 4.5 on p. 12 of that version.

Setting

There are firms $i=1,\ldots,N$ and regions $r=1,\ldots,R$ . Firm $i$ sells quantity $q_r^i\ge0$ in region $r$ . Its total output is $q_{tot}^i=\sum_r q_r^i$ , constrained by capacity $\kappa^i$ . The total sales in region $r$ are $q_r^{tot}=\sum_i q_r^i$ . The inverse demand function $p_r(q)$ gives the price when total sales are $q$ . Firm $i$ pays production cost $a^i$ per unit and transport cost $t_r^i$ per unit sent to region $r$ . Its profit at allocation $Q=(q_r^i)$ is

\pi^i(Q)=\sum_r\bigl(p_r(q_r^{tot})-t_r^i\bigr)q_r^i-a^i q_{tot}^i.

A strategy for firm $i$ is any nonnegative regional allocation whose total is at most $\kappa^i$ . A Nash equilibrium is a feasible profile at which no firm can gain by changing its entire regional allocation while the others hold theirs fixed. This is the feasible unilateral-deviation definition in the paper's Definition 3.1, p. 8.

The marginal profit in region $r$ is

g_r^i(Q)=p'_r(q_r^{tot})q_r^i+p_r(q_r^{tot})-t_r^i-a^i.

For a vector $\rho=(\rho^i)$ of marginal-profit levels, condition (2) says $g_r^i(Q)\le\rho^i$ and $q_r^i(g_r^i(Q)-\rho^i)=0$ for every firm and region. Proposition 4.2 gives a unique nonnegative allocation satisfying these conditions, denoted $\hat Q(\rho)$ , when $\rho$ lies in the box $[0,\bar w]^N$ , where $w_r=p_r(0)$ and $\bar w=\max_r w_r$ . The reduced-form operator $\Phi$ assigns firm $i$ zero marginal profit if its output at $\rho^i=0$ is below capacity. Otherwise it assigns the marginal-profit level at which its output under $\hat Q$ equals $\kappa^i$ . This is the two-case definition on p. 11; it depends on the other firms' marginal-profit levels through $\hat Q$ .

Formalization targets

Fixed point of the reduced-form operator

Lemma 4.3 asserts that $\Phi$ has exactly one fixed point in the marginal-profit box:

\exists!\rho^*\in[0,\bar w]^N:\quad\Phi(\rho^*)=\rho^*.

Unique equilibrium allocation

Theorem 4.5 identifies every equilibrium of the original quantity game with the allocation at that fixed point:

Q\text{ is a Nash equilibrium}\quad\Longleftrightarrow\quad Q=\hat Q(\rho^*).

The goal includes existence of $\rho^*$ and the equivalence, so it asserts both existence and uniqueness of the equilibrium allocation. Its milestones follow the paper's supporting statements: existence of a pure equilibrium, marginal-profit conditions, existence and uniqueness of $\hat Q$ , continuity and comparative statics, fixed points of $\Phi$ , and the link from an equilibrium to a fixed point. The Algorithm 4.1 computation clause of Proposition 4.2 and the iteration-convergence clause of Lemma C.5 are identified as partial milestones; their full printed text is retained in the milestone provenance.

Significance

The result establishes that capacities, regional demand, and unequal shipping costs still lead to a single predicted sales allocation under the paper's regularity conditions. It also gives a common language for comparing the original game with a smaller game in marginal-profit vectors. Subsequent results in the paper analyze changes in costs, capacities, prices, and consumer surplus relative to this equilibrium Alsabah et al., §§5–6; those later comparative-statics claims are outside this mission.

The mathematical result is proved in the cited paper. The goal here is a machine-checked Lean development of its model, its fixed-point characterization, and the existence and uniqueness theorem. The current proposal contains definitions and theorem statements, with proofs open. The pure-Nash relation is a published reusable definition; the market model, complementarity allocation, and operator are new interfaces that could also support later formalizations of the paper's comparative-statics results.

Difficulty

An equilibrium of the quantity game satisfies the marginal-profit conditions, but those conditions alone describe candidate allocations and do not replace feasible unilateral deviations. Conversely, a fixed point of $\Phi$ does not become an equilibrium merely from the fixed-point equation: the capacity-constrained regional payoff problem still has to be connected to the allocation $\hat Q$ . Uniqueness also requires control of how all firms' regional sales respond when one marginal-profit coordinate changes. Lemma C.4 records weak and strict changes, including boundary cases where a firm stops selling in a region. The existence lemma invokes a general pure-strategy equilibrium argument; the statement is kept even though that infrastructure is substantial in Lean.

Formalization scope

Lean uses Fin N and Fin R for zero-based firm and region indices. An allocation has type Fin N → Fin R → ℝ, with the firm coordinate first, the transpose of the matrix orientation in the paper. Finite sums give firm and region totals. The inverse demands are $C^2$ on $\mathbb R$ , strictly decreasing and concave on nonnegative quantities; deriv is therefore the ordinary derivative at the quantities used here. The paper phrases Assumption 1 for direct demand $d_r$ on $[0,w_r]$ and then uses regularity of its inverse $p_r$ . The formal model makes that inverse-demand regularity an explicit assumption. It also states $w_r=p_r(0)\ge0$ , as implicit in the interval $[0,w_r]$ , positive capacities, and nonnegative production and transport costs. Those sign assumptions are needed for the box and the two branches of $\Phi$ to have the stated meaning. No positivity of prices at all quantities or finite choke quantity is assumed.

IsEquilibrium uses the published CarbonDoubleCount.Planner.IsPureNash relation, testing every feasible full strategy deviation. Qhat is chosen as the unique solution of condition (2), and Phi is computed from that allocation and the market capacities. On the stated box, the proposal's milestones establish that the selection and inverse branches denote the paper's objects. The intended theorem excludes encodings that choose an active set freely, pass $\Phi$ or $\rho^*$ as arbitrary parameters, or substitute first-order conditions or coordinatewise deviations for Nash equilibrium. The definition layer and results on continuity, monotonicity, and fixed points are reusable parts of the development. Contributions that prove these stated targets or provide supporting analysis of the firm's concave payoff problem are within scope.

Selected references

H. Alsabah, B. Bernard, A. Capponi, G. Iyengar and J. Sethuraman, Multiregional Oligopoly with Capacity Constraints, SSRN 3280688, author version dated June 6, 2020; published in Management Science, 2021. SSRN author version.

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