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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.

≥ 2.1308255793978239Formalized record≥ 2.15Open frontier

9 provers on it17 of 18 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

4 provers on it7 of 7 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

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

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

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

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

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

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

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

References:

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

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

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

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

≤ 27Formalized record≤ 5Open frontier

38 provers on it13 of 15 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 $2$ .

≥ 2.1308255793978239Formalized record≥ 2.15Open frontier

9 provers on it17 of 18 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

4 provers on it7 of 7 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

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

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

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

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

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

References:

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

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

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

≤ 27Formalized record≤ 5Open frontier

38 provers on it13 of 15 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Completed

Economics Operations Research Optimization·Captain: mikedeng1

Climate Impact Investing 2: With Poisson Climate Uncertainty λ, If the Green Portfolio Is Green Then y*_λ Rises to 1 in λ, So Equilibrium Emissions Rise as Uncertainty GrowsResearch Paper

Motivation

Climate impact investing refers to investors who tilt their portfolios away from companies with large greenhouse-gas emissions. Its stated aim goes beyond aligning portfolios with values: by raising the cost of capital of polluting companies, it is meant to change what those companies do. De Angelis, Tankov and Zerbib (SSRN 3562534, published in Management Science, 2022, doi:10.1287/mnsc.2022.4472) build an equilibrium model in which this mechanism can be computed. Green investors penalize the expected climate externalities of companies. Companies choose emission schedules to maximize their market value net of abatement costs. Prices, portfolios and emissions are determined jointly.

Section 4 of the paper asks what happens when the green investors are uncertain about future climate externalities, for instance because news on climate risk or regulation arrives in rare large shocks. Its answer is that, in the most common market configuration, more uncertainty makes green investors hedge by holding more brown assets, which weakens the incentive to abate, so that equilibrium emissions rise as climate uncertainty grows. This mission formalizes that result and the chain of statements behind it.

Setting

Fix a date $t$ in a horizon $[0,T]$ ; every object below is evaluated at that date, and its subscript is dropped where harmless. There are $n$ companies, indexed by $i$ , with asset covariance matrix $\Sigma_t$ (positive definite). Green investors hold wealth share $\alpha\in(0,1)$ , regular investors the rest, with risk aversions $\gamma^g,\gamma^r>0$ .

Company $i$ emits at rate $\psi^i_t\ge 0$ and pays the marginal abatement cost $c^i_t\ge 0$ . The green investors attach to company $i$ the climate externality $\theta^i_t(\psi)=\kappa_{0,t}-\frac{\kappa_t}{2}\psi^2$ , the companies use $\theta^{c,i}_t(\psi)=\kappa^c_{0,t}-\frac{\kappa^c_t}{2}\psi^2$ , and all $\kappa$ 's are positive. With a rate of time preference $\rho>0$ , the time factors are $\beta^c_t=(e^{-\rho t}-e^{-\rho T})/(1-e^{-\rho T})$ and $\beta_t=(1-e^{-\rho t})/(1-e^{-\rho T})$ .

Climate uncertainty is a Poisson process of intensity $\lambda>0$ : a smaller $\lambda$ means rarer, larger shocks, i.e. more uncertainty. For a fixed externality vector $\theta$ , the equilibrium of Proposition 5 is described by a scalar $y_\lambda>0$ solving

y_\lambda=\exp\Big\{-\frac{\alpha\gamma^g}{\lambda}\Big(\mathbf 1^\top\theta+\frac{y_\lambda}{\gamma^r}\theta^\top\Sigma_t^{-1}\theta\Big)\Big\},

and the green portfolio is $N^{g,\lambda}=\alpha(\mathbf 1+\frac{y_\lambda}{\gamma^r}\Sigma_t^{-1}\theta)$ . Without uncertainty, $y=1$ and the portfolio is $N^g$ . When the companies take $y$ as given (the large-market assumption), company $i$ 's optimal emission rate is

\psi^i_t(y)=\frac{c^i_t}{\alpha\beta_t\kappa_t\,y+\beta^c_t\kappa^c_t},

and $y$ itself must then solve the fixed-point equation $y=\exp\{-\frac{\alpha\gamma^g}{\lambda}g^*(y)\}$ , where

g^*(y)=\mathbf 1^\top\theta_t(\psi_t(y))+\frac{y}{\gamma^r}\theta_t(\psi_t(y))^\top\Sigma_t^{-1}\theta_t(\psi_t(y)).

Its solution is $y^*_\lambda$ . Finally, $\psi^{*,0}_t=\psi_t(1)$ is the schedule without uncertainty.

Formalization targets

Goal: Proposition 8(iii)

Assume condition (24) at the date $t$ : $\sum_j(\Sigma_t^{-1})_{ij}\,y\,\theta^j_t(\psi^j_t(y))\ge-\gamma^r/2$ for all $i$ and all $y>0$ . Let $\lambda\mapsto y^*_\lambda$ solve the fixed-point equation. Then the sign of the green portfolio's externality without uncertainty decides the direction of the response:

\theta_t(\psi^{*,0}_t)^\top N^g_t>0\ \Longrightarrow\ y^*_\lambda\uparrow 1\ \text{strictly in }\lambda,\quad \lambda\mapsto\psi^{*,i}_t\ \text{nonincreasing},

the opposite if $\theta_t(\psi^{*,0}_t)^\top N^g_t<0$ , and $y^*_\lambda\equiv 1$ with unchanged emissions if it is $0$ .

Milestones

In the order of the attack:

Proposition 7: for fixed $y>0$ , $\psi^i_t(y)$ is the unique maximizer of the company's pointwise objective.
Proposition 8(i): the fixed-point equation has a positive solution for every $\lambda>0$ .
Under (24), $g^*$ is increasing.
Proposition 8(ii): under (24), the solution is unique.
$\psi^*_t(1)=\psi^{*,0}_t$ , and $\alpha g^*(1)=\theta_t(\psi^{*,0}_t)^\top N^g_t$ .
Equation (20) has exactly one positive solution $y_\lambda$ .
Proposition 6(ii)–(iv): the same three-way comparative static at a fixed schedule, for $y_\lambda$ , the green portfolio and the expected returns.

Companion statements cover Proposition 6(i) (the green portfolio's market risk increases in $\lambda$ ), Lemma 1, and the algebraic core of Proposition 5.

Significance

The result. Proposition 8(iii) is the paper's headline claim about uncertainty, and it is quoted in its abstract: uncertainty about future climate externalities, by making green investors hedge, can push companies to increase their emissions. The paper's Lemma 1 shows that the positive case, a green portfolio with positive climate externality, is the common one. The sign condition is therefore not a curiosity. It separates markets where uncertainty undermines impact investing from markets where it strengthens it.

Formalizing it. The proofs in the appendix are short and leave several steps implicit, and in two places the printed formulas do not match their own derivation (see the scope section). A machine-checked version settles the precise hypotheses under which the comparative statics hold. It also provides reusable statements about scalar fixed-point equations of the form $-\lambda\log y=G(y)$ with nondecreasing $G$ . No part of this paper has been formalized before.

Difficulty

At a fixed schedule, everything follows from the fact that $g$ is affine and nondecreasing in $y$ . Once the schedule is endogenous, $g^*$ is nonlinear and in general not monotone, so neither uniqueness nor monotonicity in $\lambda$ comes for free. The work lies in turning condition (24) into monotonicity of $g^*$ . That requires differentiating the quadratic form $\theta(\psi(y))^\top\Sigma^{-1}\theta(\psi(y))$ through the schedule and controlling its sign company by company. The limit $y^*_\lambda\to 1$ as $\lambda\to\infty$ needs bounds on $g^*$ along the solution path rather than the formal Taylor expansion in $1/\lambda$ the paper gives for (20).

Formalization scope

The stochastic layer is not formalized: Brownian cash-flow news, subjective measures built by Girsanov's theorem, CARA utility maximization, the Poisson process of climate shocks, and Propositions 1 and 5 as equilibrium statements. The statements start from the objects the proofs start from: the first-order and market-clearing system (45), and equations (20), (22) and (23). Everything is stated at one fixed date $t$ , with vectors in $\mathbb R^n$ indexed by Fin n, $\Sigma_t$ positive definite and $\Sigma_t^{-1}$ Mathlib's matrix inverse.

The standing assumptions are hypotheses of the statements:

$\rho,T>0$ and $t\in[0,T]$ ;
$0<\alpha<1$ ;
$\gamma^r,\gamma^g>0$ ;
all $\kappa$ 's positive;
$c^i_t\ge0$ .

The relation $\alpha=\gamma^r/(\gamma^r+\gamma^g)$ is assumed only where it is used, in the core of Proposition 5. The solution maps $\lambda\mapsto y_\lambda$ and $\lambda\mapsto y^*_\lambda$ enter only through the hypothesis that each value solves its equation. A theorem that assumed monotonicity of $g^*$ , uniqueness, or the sign of $g^*(1)$ would trivialize the goal; the goal assumes only condition (24).

Disclosed corrections of the printed text:

(23) and $g^*$ . The page prints a factor $\alpha y/\gamma^r$ in the quadratic term. Footnote 25 derives (23) from (20), which has $y/\gamma^r$ , and only that version makes the sign test of (iii) agree with the sign of $g^*(1)$ . The formalization uses $y/\gamma^r$ .
(24). The derivative of the quadratic form on p. 72 omits the factor 2 that the symmetry of $\Sigma^{-1}$ produces. With the corrected $g^*$ , the argument needs the bound $-\gamma^r/2$ , not $-\gamma^r/\alpha$ . Whether uniqueness holds under the printed bound is not claimed.
Existence at $t=T$ . The proof of (i) uses $\beta^c_t>0$ , so existence is stated for $t\in[0,T)$ .
Asymptotic uniqueness. The last sentence of (ii), uniqueness as $\lambda\to\infty$ , has no proof in the paper and is not stated.

Two readings are fixed. "Monotonically increasing" for $y_\lambda$ is strict. The emission response is weak, because at $t=0$ the schedule does not depend on $y$ . "Reallocate towards brown assets" means that the green portfolio's climate externality $\theta^\top N^{g,\lambda}$ falls as $\lambda$ falls.

The development needs elementary real analysis: the intermediate value theorem on $(0,\infty)$ , monotonicity from derivatives, and limits at infinity. It also needs finite-dimensional linear algebra for positive definite matrices. The lemmas about $-\lambda\log y=G(y)$ are reusable beyond this mission. Proofs of the companion statements, which are algebraic, are welcome as warm-ups.

Selected references

T. De Angelis, P. Tankov, O. D. Zerbib, Climate Impact Investing, SSRN 3562534, version of June 25, 2021. https://ssrn.com/abstract=3562534
T. De Angelis, P. Tankov, O. D. Zerbib, Climate Impact Investing, Management Science, 2022. https://doi.org/10.1287/mnsc.2022.4472
L. Pástor, R. F. Stambaugh, L. A. Taylor, Sustainable investing in equilibrium, Journal of Financial Economics 142(2), 2021. https://doi.org/10.1016/j.jfineco.2020.12.011

13 thms3 active usersReviewed

Completed

Number Theory Operations Research Optimization·Captain: mikedeng1

Integer Factorization: Why Two-Item Joint Replenishment Is Hard 2: For Odd Composite M, Every Optimal Plan of an Explicit Two-Item Periodic JRP Instance Has q₂ = M and 1 < gcd(q₁, M) < MResearch Paper

Motivation

The joint replenishment problem (JRP) is a basic model of inventory coordination: several items are ordered from one supplier, each item pays its own ordering and holding costs, and an additional item-independent joint ordering cost rewards ordering the items at common times. Two accounting conventions are studied. In the periodic JRP (also called the general integer model; see the survey of Goyal and Satir, 1989) the joint cost is charged at every point of a common grid, whether or not an item is ordered there; in the aperiodic JRP it is charged only at actual order times.

For decades the literature produced heuristics and enumerative algorithms for the JRP without a polynomial-time exact algorithm and without evidence that none exists. Schulz and Telha (arXiv:2007.09045v1; published in Operations Research, 2024, as Integer Factorization: Why Two-Item Joint Replenishment Is Hard) showed that already the two-item problem is at least as hard as integer factorization, in both variants. For the periodic JRP the paper notes (p. 10): "No hardness results were previously known for the periodic JRP."

Timeline, as recorded in the paper (pp. 4–5):

2011: Schulz and Telha announce that the two-item aperiodic JRP is at least as hard as integer factorization (Theorem 1(a)).
2018: Cohen-Hillel and Yedidsion prove that the aperiodic JRP with an arbitrary number of items is strongly NP-hard.
2020: the preprint arXiv:2007.09045v1 adds the periodic case (Theorem 1(b), via Lemma 2, the subject of this mission) and, assuming the Riemann hypothesis, NP-hardness of the two-item periodic JRP under randomized reductions (Theorem 2).

Setting

An item with ordering cost $K \ge 0$ and holding cost $H \ge 0$ , replenished every $q$ time units, where $q$ is a positive integer, has the EOQ cost

\mathrm{EOQ}_{K,H}(q) = \frac{K}{q} + Hq .

In the two-item periodic JRP, item $i \in \{1,2\}$ has parameters $(K_i, H_i)$ and an integer period $q_i \ge 1$ , and a joint ordering cost $K_0 > 0$ is incurred every $\rho$ time units, where $\rho$ divides both periods. At an optimum $\rho = \gcd(q_1,q_2)$ , and the problem is to minimize objective (3) of the paper,

\mathrm{JRP}(q_1,q_2) = \mathrm{EOQ}_{K_1,H_1}(q_1) + \mathrm{EOQ}_{K_2,H_2}(q_2) + \frac{K_0}{\gcd(q_1,q_2)}, \qquad q_1,q_2 \in \mathbb{Z}^+ .

An optimal solution is a pair of positive integers whose value is at most the value of every pair of positive integers.

Given a natural number $M$ , the instance of Lemma 2 is

K_0 = M,\quad K_1 = 0,\quad H_1 = 1,\quad H_2 = M(M+1),\quad K_2 = M^3(M+1),

with objective written $\mathrm{JRP}_M$ and item 2's cost $\mathrm{EOQ}_2(q) = K_2/q + H_2 q$ . Fixing $q_2 = M$ leaves the single-variable problem (4):

\min\ \frac{M}{\gcd(q,M)} + q \quad \text{s.t. } q \in \mathbb{Z}^+ .

An odd composite number is an odd integer $M > 1$ that is not prime.

Formalization targets

Goal: Lemma 2 (p. 11)

For every odd composite $M$ : the coefficients of the instance above satisfy $K_0 > 0$ and are natural numbers at most $M^5$ ; the instance has an optimal solution; and every optimal solution $(q_1,q_2)$ satisfies

q_2 = M \quad\text{and}\quad 1 < \gcd(q_1, M) < M .

The goal does not say which non-trivial divisor $\gcd(q_1,M)$ is: optimal solutions need not be unique (for $M = 15$ both $q_1 = 3$ and $q_1 = 5$ are optimal).

Milestones (proof of Lemma 2, pp. 11–12)

For $q_2 \ne M$ : $\mathrm{EOQ}_2(q_2) \ge \mathrm{EOQ}_2(M \pm 1) = \mathrm{EOQ}_2(M) + \frac{H_2}{M\pm 1} \ge \mathrm{EOQ}_2(M) + M$ .
For $q_2 \ne M$ : $\mathrm{JRP}_M(q_1,q_2) > M + \mathrm{EOQ}_2(M) + 1 = \mathrm{JRP}_M(M,M)$ .
With $q_2 = M$ the instance reduces to problem (4): $\mathrm{JRP}_M(q_1,M) = \mathrm{EOQ}_2(M) + \frac{M}{\gcd(q_1,M)} + q_1$ .
An odd composite $M$ has a divisor $p$ with $3 \le p \le M/3$ .
For such $p$ : $\frac{M}{\gcd(p,M)} + p \le \frac{2M}{3}$ .
If $\gcd(q,M) = 1$ or $\gcd(q,M) = M$ , then $\frac{M}{\gcd(q,M)} + q \ge M$ .

Significance

The result. Lemma 2 says that an exact solver for the two-item periodic JRP, applied to an instance whose coefficients have size $O(\log M)$ , returns a pair from which a non-trivial divisor of $M$ is read off as $\gcd(q_1, M)$ . Repeating on the factors yields the full factorization, so a polynomial-time algorithm for the two-item periodic JRP would give a polynomial-time algorithm for integer factorization (Theorem 1(b), p. 12). Since the security of RSA-type cryptosystems rests on the belief that factoring is hard, the lemma explains why exact algorithms for even two items have not been found. Section 3 of the paper extends the same construction to a reduction from a range-divisor problem that is NP-complete under randomized reductions, assuming the Riemann hypothesis; that extension is a separate mission of this series.

Formalizing it. The result is proved in the paper, and the proof is short. To our knowledge it has no machine-checked proof. This mission produces a Lean statement of the reduction's correctness with the explicit instance, and its proof together with the intermediate steps of the paper's argument. One printed step needs a correction (milestone 4, see below), which a machine-checked proof makes explicit.

Difficulty

Two points need care. First, the claim concerns every optimal solution of a problem over all pairs of positive integers, so it needs both a global lower bound for every pair with $q_2 \ne M$ and the existence of an optimum, which the paper does not state: the objective must be shown to attain its infimum over an infinite discrete domain. Second, the step separating non-trivial from trivial greatest common divisors depends on a divisor of a specific size: the printed claim "a non-trivial divisor $p$ with $3 \le p < M/3$ " is false for $M = 9$ , whose only non-trivial divisor is $3 = M/3$ . The corrected bound $p \le M/3$ still gives the value $2M/3 < M$ that the argument needs, but only because $M \ge 9$ for every odd composite $M$ . A direct convexity argument for item 2 alone does not finish the proof: item 1's cost and the joint cost interact through $\gcd(q_1,q_2)$ , which is not convex in either variable.

Formalization scope

This mission formalizes the correctness of the reduction behind Theorem 1(b); polynomial running time is not formalized. Theorem 1(b) itself, the recursion that turns one divisor into a full factorization, primality testing, and the treatment of even $M$ (p. 10) are not stated.

Conventions committed to:

Periods are natural numbers cast to $\mathbb{R}$ , and every optimality notion and every milestone ranges over positive periods only, because in Lean $K/0 = 0$ and $\gcd(0, M) = M$ .
Costs are real numbers; the coefficients of the instance are natural-number expressions in $M$ .
Objective (3) is the model and is defined directly; the derivation $\rho = \gcd(q_1,q_2)$ (p. 10, with $B = 1$ ) is not formalized.
"Odd composite" is encoded as Odd M ∧ 1 < M ∧ ¬ M.Prime.
"The size of the coefficients is at most $5\log(M)$ " is stated as "every coefficient is at most $M^5$ ".

Disclosed differences from the printed lemma: the goal names the explicit instance of the proof's first sentence instead of "there is an instance" (a stronger statement); it adds the existence of an optimal solution, so that "every optimal solution satisfies …" is not vacuous; and milestone 4 uses $p \le M/3$ in place of the printed $p < M/3$ . Milestones 1–3 and 6 are stated for every $M$ with $M \ge 2$ (or every $M$ ), which contains the paper's case. A formalization with an existential instance, an unattained or empty set of optima, or periods allowed to be $0$ would make the goal trivial or wrong; all three are excluded.

Infrastructure: the definitions (EOQ cost, objective (3), optimality over positive integer pairs) are short and local to this mission; the same objects appear in the companion missions of this series under their own namespaces. Mathlib supplies Nat.gcd, Nat.minFac and Nat.minFac_sq_le_self. Proofs of any milestone are welcome independently; the existence of an optimum is the one ingredient not present among the milestones.

Selected references

A. S. Schulz and C. Telha, Integer factorization and Riemann's hypothesis: Why two-item joint replenishment is hard, arXiv preprint, 2020. https://arxiv.org/abs/2007.09045v1 (published as Integer Factorization: Why Two-Item Joint Replenishment Is Hard, Operations Research, 2024).
S. K. Goyal and A. T. Satir, Joint replenishment inventory control: Deterministic and stochastic models, European Journal of Operational Research 38(1):2–13, 1989. https://doi.org/10.1016/0377-2217(89)90463-3
T. Cohen-Hillel and L. Yedidsion, The periodic joint replenishment problem is strongly NP-hard, Mathematics of Operations Research 43(4):1269–1289, 2018. https://doi.org/10.1287/moor.2017.0904
F. W. Harris, How many parts to make at once, Factory, The Magazine of Management 10(2):135–136, 152, 1913; reprinted in Operations Research 38(6):947–950, 1990. https://doi.org/10.1287/opre.38.6.947
M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Annals of Mathematics 160(2):781–793, 2004. https://doi.org/10.4007/annals.2004.160.781

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Graph Theory Linear Optimization Operations Research·Captain: mikedeng1

Linear Programming and Community Detection 3: The Metric LP Relaxation Recovers the Planted Bisection When d_in − d_out ≥ n/4 Under Regular SandwichingResearch Paper

Motivation

The minimum bisection problem asks, for a graph $G=(V,E)$ on an even number $n$ of vertices, for a split of $V$ into two halves of size $n/2$ that cuts as few edges as possible. It is NP-hard, and it is also the maximum a posteriori estimator for the two-community stochastic block model, the standard random model of networks with community structure. A tractable surrogate is a convex relaxation. Semidefinite relaxations of minimum bisection are known to recover a planted community structure under near-optimal conditions; Del Pia, Khajavirad and Kunisky (arXiv:2006.03213, Mathematics of Operations Research, 2023) ask the same question for the cheaper and more classical linear programming relaxation, the metric relaxation obtained by outer-approximating the cut polytope by triangle inequalities.

This mission covers the deterministic, sufficient side of their analysis (§3.1 of the paper): explicit conditions on a graph with a planted bisection under which the LP relaxation returns exactly that bisection. These conditions are the input to the paper's probabilistic recovery theorem for the very dense block model (Theorem 6).

Setting

Let $V=\{1,\dots,n\}$ . A bisection is a partition $V_1,V_2$ of $V$ with $|V_1|=|V_2|=n/2$ . A fixed but unknown bisection is the planted bisection. Write $G_1=G[V_1]$ and $G_2=G[V_2]$ for the subgraphs induced on the two sides, and $G_0$ for the bipartite graph of the edges with one end on each side. Two degree parameters describe the instance: $d_{\mathrm{in}}$ , a degree inside the communities, and $d_{\mathrm{out}}$ , a degree across them. A graph is $d$ -regular if every vertex has exactly $d$ neighbours.

The cut vector $\bar x\in\{0,1\}^{\binom n2}$ of the planted bisection has $\bar x_{ij}=0$ for $i,j$ on the same side and $\bar x_{ij}=1$ otherwise. With $a_{ij}=1$ if $(i,j)\in E$ and $0$ otherwise, and the convention $x_{ji}=x_{ij}$ , the metric relaxation is

\text{(LP-P)}\qquad \min_x \sum_{1\le i<j\le n} a_{ij}x_{ij}\quad\text{s.t.}\quad \begin{aligned} &x_{ij}\le x_{ik}+x_{jk} && \text{distinct } i,j,k,\ i<j, &(1)\\ &x_{ij}+x_{ik}+x_{jk}\le 2 && 1\le i<j<k\le n, &(2)\\ &\textstyle\sum_{i<j}x_{ij}=n^2/4. && &(3) \end{aligned}

The LP recovers the planted bisection if $\bar x$ is an optimal solution and the only one.

Formalization targets

Goal: Theorem 4 (p. 12)

Assume $G_1$ and $G_2$ contain $d_{\mathrm{in}}$ -regular subgraphs on the same node sets, and $G_0$ is a subgraph of a $d_{\mathrm{out}}$ -regular bipartite graph with the same bipartition. Then

d_{\mathrm{in}}-d_{\mathrm{out}}\ \ge\ \frac n4\quad\Longrightarrow\quad \text{(LP-P) recovers the planted bisection.}

Milestones

Proposition 1 (p. 7). If $G_1,G_2$ are $d_{\mathrm{in}}$ -regular with $d_{\mathrm{in}}\in\{1,\dots,n/2-1\}$ , $G_0$ is $d_{\mathrm{out}}$ -regular with $d_{\mathrm{out}}\in\{0,\dots,n/2\}$ , $n\ge4$ , and $d_{\mathrm{in}}-d_{\mathrm{out}}\ge n/4-1$ , then $\bar x$ is optimal for (LP-P).
Theorem 3 (pp. 9–10), Mangasarian's criterion: for an LP $\min\{c^\top x: Ax=b,\ Cx\ge d\}$ with primal optimum $\bar x$ and dual optimum $(\bar u,\bar v)$ , $\bar x$ is the unique optimum if and only if $Ax=0$ , $C_Kx=0$ , $C_Lx\ge0$ has no nonzero solution, where $K$ and $L$ are the active constraints with positive and zero multiplier.
Proposition 2 (p. 10). For regular $G_1,G_2,G_0$ as above, $d_{\mathrm{in}}-d_{\mathrm{out}}\ge n/4$ implies recovery.
Lemma 2 (p. 11). Recovery survives deleting crossing edges.
Lemma 3 (p. 11). Recovery survives adding edges inside $V_1$ or $V_2$ .

Significance

Theorem 4 is a deterministic certificate: any graph that can be sandwiched between a regular within-community graph and a regular across-community graph with a degree gap of $n/4$ is solved exactly by a polynomial-size LP. For regular instances the threshold is tight: for $8\mid n$ and any $d_{\mathrm{in}}-d_{\mathrm{out}}\le n/4-1$ in range, the paper's Theorem 2 constructs regular graphs on which even the exact minimum bisection fails to recover the planted bisection. The sandwich form is what makes the result usable for random graphs: the paper's Theorem 6 shows that in the very dense block model with constant edge probabilities $p>q$ and $p-q>1/2$ the sandwich hypotheses hold with high probability, so the LP recovers the communities.

None of these statements has, to our knowledge, a machine-checked proof. A complete development gives a verified dual certificate for the metric polytope, a verified version of Mangasarian's uniqueness criterion for linear programs in the general form $Ax=b$ , $Cx\ge d$ (useful well beyond this paper), and the monotonicity lemmas that transfer LP recovery between graphs.

Difficulty

Optimality of $\bar x$ (Proposition 1) needs an explicit dual solution of (LP-P), with one multiplier per triangle inequality, and the bookkeeping that verifies the dual equality constraint for every pair $i<j$ in six cases. Optimality alone is not recovery: the obvious argument, exhibiting a dual solution, says nothing about other optima, and the LP has many fractional feasible points. Uniqueness requires a strictly complementary structure and a separate argument that the linear system attached to the active constraints has only the zero solution; on the page this step uses Dirac's theorem on Hamiltonian cycles inside each community. Finally, the passage from regular to sandwiched graphs is not a perturbation statement: adding or deleting edges changes the objective, and the lemmas rely on the bounds $0\le x_{ij}\le 1$ , which are implied by the triangle inequalities rather than imposed.

Formalization scope

Vertices are Fin n (0-based), a graph is a SimpleGraph (Fin n), and a bisection is a finset V1 with 2 * V1.card = n; the other side is its complement. A point of $\mathbb R^{\binom n2}$ is an array x : Fin n → Fin n → ℝ that is symmetric with zero diagonal, and these two conditions are part of LP feasibility. Recovery is optimality of the cut vector together with equality of every optimal solution to it. Regularity of $G_1$ and $G_2$ with one common degree is "every vertex has exactly $d_{\mathrm{in}}$ neighbours on its own side"; regularity of $G_0$ is "every vertex has exactly $d_{\mathrm{out}}$ neighbours on the other side". Degree conditions are compared in $\mathbb R$ .

Committed readings:

Theorem 4 is posed for arbitrary integers $d_{\mathrm{in}},d_{\mathrm{out}}$ satisfying the sandwich hypotheses. The page names them the minimum and maximum degrees, but the paper's Theorem 6 applies Theorem 4 with other integers; the literal reading is a special case.
The two $d_{\mathrm{in}}$ -regular subgraphs of $G_1$ and $G_2$ are encoded as a single graph with no crossing edges.
Proposition 1 carries the standing assumptions of §3.1.1 ( $n\ge4$ and the degree ranges). Proposition 2 and Theorem 4 carry no range hypotheses: they are implied by the remaining hypotheses for $n\ge4$ , the hypotheses are unsatisfiable at $n=2$ , and the conclusion is trivial at $n=0$ .
Lemma 2's strict inclusion $E(\tilde G_0)\subset E(G_0)$ is posed as $\subseteq$ , since equality is trivial.
Theorem 3 renames the cost vector $p$ to $c$ and the number of variables $n$ to $N$ .

Two encodings would trivialize the mission and are ruled out: an array $x$ without symmetry and zero diagonal (its unconstrained entries make uniqueness false), and an LP with added box constraints $0\le x\le1$ (a different polytope). Dirac's theorem, used inside the proof of Proposition 2, is posed separately on the platform (AppliedComb.Graphs.dirac) and is not restated here. Contributions welcome: the dual certificate of Proposition 1, a general-purpose proof of Theorem 3 from LP duality, and the two monotonicity lemmas.

Selected references

A. Del Pia, A. Khajavirad, D. Kunisky, Linear Programming and Community Detection, Mathematics of Operations Research, 2023; arXiv:2006.03213v2. https://arxiv.org/abs/2006.03213
O. L. Mangasarian, Uniqueness of solution in linear programming, Linear Algebra and its Applications 25 (1979), 151–162. https://doi.org/10.1016/0024-3795(79)90014-4
G. A. Dirac, Some theorems on abstract graphs, Proceedings of the London Mathematical Society s3-2 (1952), 69–81. https://doi.org/10.1112/plms/s3-2.1.69
M. M. Deza, M. Laurent, Geometry of Cuts and Metrics, Springer, 1997. https://doi.org/10.1007/978-3-642-04295-9

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Algorithmic Game Theory Economics Operations Research+1·Captain: mikedeng1

Accelerability vs. Scalability: R&D Investment Under Financial Constraints and Competition II: Unconstrained R&D Race: Best Responses Rise on Accelerable Projects and a Symmetric Equilibrium ExistsResearch Paper

Why investment races matter

Two firms developing competing innovations face a decision that a single firm does not: a rival's discovery can end the value of continuing the project. If the first discovery wins the payoff, each firm chooses how much to spend per period while knowing that greater spending may change both its payoff on success and its speed of discovery. The investment decision therefore depends on what the rival chooses. Danmo Lin's model separates these two channels and asks when competition itself raises a firm's chosen R&D intensity Lin (2022).

This mission concerns the paper's race of two financially unconstrained firms. The result is a comparative statics claim about a firm's best response to its rival, together with existence of a symmetric pure-strategy equilibrium. The paper proves the result in Appendix A.6. The problem here is to give machine-checked statements and ultimately a proof for the same stochastic race, including its strategy ranges and first-arrival payoffs.

The two-firm race

Firm $i\in\{1,2\}$ chooses a fixed investment intensity $R_i$ , the amount it pays per unit time while the race continues. Its technology has parameters $(A_i,\beta_i,\eta_i,\gamma_i)$ . If it discovers first, its mean project payoff is $u_i(R_i)=A_iR_i^{\beta_i}$ . Its discovery time $\tau_i$ is exponentially distributed with rate $\lambda_i(R_i)=\eta_iR_i^{\gamma_i}$ . The discovery times are independent, represented by the product law $\mathcal E(\lambda_1,\lambda_2)$ , and the earlier discovery wins. Both firms stop paying at $\min\{\tau_1,\tau_2\}$ . The loser receives no discovery payoff. Future cash flows are discounted at rate $r>0$ Lin (2022), §3.1.

The technology conditions are $A_i,\eta_i>0$ , $0\le\beta_i,\gamma_i<1$ , and $\beta_i+\gamma_i<1$ . A project is scalable when its discovery payoff increases with investment, corresponding here to $\beta_i>0$ . It is accelerable when investment increases its discovery rate, corresponding to $\gamma_i>0$ . The model permits the two firms' parameters to differ. The lower intensity is $1$ ; the upper intensity is $\bar R_i=(A_i\eta_i)^{1/(1-\beta_i-\gamma_i)}$ . The standing condition $A_i\eta_i>1$ makes $[1,\bar R_i]$ a nondegenerate range in which investment takes place Lin (2022), (2), (7).

Write $P_i(R_1,R_2)$ for Firm $i$ 's expected discounted payoff from the first-arrival race. Its best-response set $BR_i(R_{-i})$ contains every maximizer of $P_i$ over its own fixed interval $[1,\bar R_i]$ , holding the rival's intensity fixed. A pure Nash equilibrium is a pair $(R_1,R_2)$ in which each intensity belongs to that firm's best-response set. This set-valued definition keeps the claim meaningful when there is more than one maximizer.

Formalization targets

The first milestone identifies each expected payoff with the closed form in Lemma 2:

P_i(R_1,R_2)=V_i(R_1,R_2) =\frac{u_i(R_i)\lambda_i(R_i)-R_i}{r+\lambda_1(R_1)+\lambda_2(R_2)}.

The second milestone is the Appendix A.6 cross-partial identity. For Firm 1, the expression $V_1\,\partial^2_{R_1R_2}V_1-(\partial_{R_1}V_1)(\partial_{R_2}V_1)$ equals $\lambda'_2(u_1\lambda_1-R_1)^2\lambda'_1/(r+\lambda_1+\lambda_2)^4$ . It is strictly positive when both discovery rates rise with intensity and Firm 1's project surplus is nonzero Lin (2022), (A.27)–(A.28).

The goal is Proposition 3. When both projects are accelerable, higher rival investment orders the best-response sets upward, with strict movement for an old response above $1$ . When neither project is accelerable, the best-response sets are constant in the rival's intensity. For identical firms, the theorem asserts a symmetric pure Nash equilibrium:

\exists R\in[1,\bar R]\quad R\in BR_1(R)\cap BR_2(R).

Best-response sets are required to be nonempty for every positive rival intensity. The ordering statements therefore make substantive claims about actual choices Lin (2022), Proposition 3.

What the result establishes

The result distinguishes a change in the timing of innovation from a change in its eventual payoff. In an accelerable project, a rival who spends more discovers faster, and the firm's optimal investment response moves upward. If investment cannot affect discovery speed, a rival's intensity does not alter the firm's best-response set in this model. The equilibrium clause says the symmetric game has at least one profile at which neither firm gains by changing its own intensity. These are statements about competition's effect on project spending, rather than only about the numerical value of a firm Lin (2022), §3.1.

The mathematical result is proved in the paper; the mission seeks a Lean formalization of that known result. The expectation-to-value identity requires a reusable treatment of independent exponential discovery times and discounted spending stopped at the first arrival. The best-response statement then supplies a precise set-valued comparative statics target. A verified development of these ingredients could be reused for other innovation races and first-arrival games. This mission does not claim that such a machine-checked proof already exists.

Mathematical difficulty

The payoff is an expectation over two random times, with the prize awarded only to the earlier discovery and spending stopped at their minimum. The closed form must agree with that stochastic expression for each positive pair of intensities. The comparison of best responses involves all maximizers, including possible ties and boundary choices. A derivative of one selected optimizer is not a well-defined substitute when the optimizer is set-valued; moreover, strict response cannot be demanded at the lower endpoint. Equilibrium existence must hold for the same best-response correspondence that defines the competitive race. These points make a direct translation of the paper's derivative notation insufficient Lin (2022), Appendix A.5–A.6.

Formalization scope

Lean uses real intensities, real technology parameters, Real.rpow for the power functions, and ProbabilityTheory.expMeasure for each discovery time. Their product measure encodes independence. Firm 1's discovery time is the first coordinate throughout, including Firm 2's payoff. The two firms have separate technology records. A best response maximizes the expectation from (8) over the own interval $[1,\bar R_i]$ ; that interval never depends on the rival's action. The named closed form $V_1$ is separate from the expectation and appears in the Appendix A.6 milestone. The random payoff on discovery is represented by its mean $u_i(R_i)$ , as the paper's calculation uses only that mean. The other cash-flow component of the underlying firm value is independent of the R&D choice and is omitted from the race payoff.

The hypotheses $A_i\eta_i>1$ , $r>0$ , and positive intensities are explicit. The comparison allows any positive rival intensity, while each firm's own maximization uses (7). The paper's derivative claims about $R_i^*$ are posed as orderings of complete best-response sets; strictness is asserted for an old response above $1$ . Both firms must be accelerable for the increasing-response clause, and the symmetric equilibrium clause applies to identical technologies. The strict inequality in (A.28) includes the necessary nonzero-surplus condition, since its square vanishes at the upper endpoint. These choices and the printed endpoint issue are recorded in the mission's moderation notes.

The target excludes empty best-response sets, rival-dependent feasible ranges, payoffs defined by the closed form (9), an equilibrium asserted without mutual best-response membership, and comparisons that discard the strict interior part. A complete development will need integration and exponential-distribution facts, continuity and maximizer existence on compact intervals, ordered best-response sets, and a fixed-point argument for the symmetric correspondence. The first-arrival probability and integration results can be reused beyond this paper; contributions to those interfaces and to the two stated milestones are welcome.

Selected references

Danmo Lin, Accelerability vs. Scalability: R&D Investment Under Financial Constraints and Competition, Management Science, Articles in Advance, 2022. DOI: 10.1287/mnsc.2022.4503.

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Algorithmic Game Theory Graph Theory Linear Optimization+3·Captain: mikedeng1

Designing Approximately Optimal Search on Matching Platforms 1: Some Stationary Equilibrium of Directed Search Attains a Quarter of the First-Best WelfareResearch Paper

Motivation

Two-sided platforms for dating, freelancing and housing do more than list participants: they decide whom each participant is shown. When search is costly and agents leave the market over time, the platform's choice of assortments (which types of agents meet which, and how often) determines who matches, and agents respond strategically by deciding which matches to accept. A platform that ignores those responses can design meetings that agents then decline.

Immorlica, Lucier, Manshadi and Wei (arXiv:2102.09017, Management Science 2023) model this as directed search in a stationary flow economy and ask how much welfare a platform can secure when it controls only meetings, while agents retain the decision to accept. Computing the optimal assortments is NP-hard to approximate within 24/23 (their Theorem 4.2), so the relevant question is whether a constant fraction of the best conceivable welfare is reachable at all. Their Theorem 4.1 answers it with the factor 4, measured against the platform's first best, a benchmark in which the platform also dictates acceptance.

The mission formalizes the existence part of Theorem 4.1 and the chain of results in §4 of the paper that leads to it.

Setting

There are finitely many types on two sides, a set $\mathcal M$ and a set $\mathcal W$ ; write $\Theta=\mathcal M\cup\mathcal W$ and let $\theta'$ denote a generic type on the side opposite to $\theta$ . Agents of type $\theta$ enter at rate $\alpha_\theta>0$ and each agent leaves unmatched at rate $\delta>0$ . When an agent of type $m$ meets one of type $w$ , both observe one symmetric utility $u$ drawn from a law $F_{mw}$ on $\mathbb R$ with no atoms and finite mean; they match, and leave with utility $u$ each, if both accept.

The platform picks an assortment $\lambda_\theta(\theta')\ge0$ for each type: type $\theta$ meets type $\theta'$ at rate $\lambda_\theta(\theta')$ . Agents use thresholds $\tau_\theta$ and accept exactly when $u\ge\tau_\theta$ , so a pair matches when $u\ge\max(\tau_m,\tau_w)$ . With population masses $\eta_\theta$ and matching rates $\xi_\theta$ , the assortments induce a stationary equilibrium when, for every type,

\alpha_\theta=(\delta+\xi_\theta)\eta_\theta,\qquad \eta_m\lambda_m(w)=\eta_w\lambda_w(m),\qquad \sum_{\theta'}\lambda_\theta(\theta')\le1,

\xi_\theta=\sum_{\theta'}\lambda_\theta(\theta')\int_{\max(\tau_\theta,\tau_{\theta'})}^\infty dF_{\theta\theta'},\qquad \tau_\theta=\frac{\sum_{\theta'}\lambda_\theta(\theta')\int_{\max(\tau_\theta,\tau_{\theta'})}^\infty u\,dF_{\theta\theta'}}{\delta+\xi_\theta}.

These are constraints (6a)–(6f) of the paper; the last one, (6e), says each threshold equals the type's expected utility, which is the agents' best response. The welfare is the flow of realized utility,

\mathrm{SW}=\sum_{m}\sum_{w}\big(\eta_m\lambda_m(w)+\eta_w\lambda_w(m)\big)\int_{\max(\tau_m,\tau_w)}^\infty u\,dF_{mw}.

The first-best relaxation (§4.1) drops (6e) and replaces $\max(\tau_m,\tau_w)$ by a free pairwise threshold $\tau_{mw}$ : the platform chooses both the meetings and which matches happen. Its welfare $\mathrm{FB}$ is the same expression with $\tau_{mw}$ .

Formalization targets

Goal: Theorem 4.1

For every market satisfying the standing assumptions there are assortments, thresholds, rates and masses satisfying (6a)–(6f) such that, for every first-best feasible candidate $Q$ ,

\mathrm{FB}(Q)\le 4\,\mathrm{SW}.

Milestones

Proposition 4.4. The first best equals the linear program $\max\,2\sum_{m,w}\rho_{mw}\beta_{mw}$ subject to $\beta\ge0$ , $\sum_w\beta_{mw}\le\alpha_m$ , $\sum_m\beta_{mw}\le\alpha_w$ , where $\rho_{mw}=\max_{\tau\ge0}\int_\tau^\infty u\,dF_{mw}/(\delta+\int_\tau^\infty dF_{mw})$ .
Lemma 4.5. The maximizer of $A(\tau)=\int_\tau^\infty u\,dF/(\delta+\int_\tau^\infty dF)$ over $\tau\ge0$ is a fixed point of $A$ , and $A$ increases before it and decreases after it.
Lemma 4.6. The LP has an optimal solution whose support is a forest on $\Theta$ .
Lemma 4.7. An edge-weighted forest keeps half its weight on a union of vertex-disjoint stars.
Lemma 4.9. In a star-shaped market (one side has a single type) the LP is a fractional knapsack with a greedy optimum.
§4.2.2 remark, Lemmas 4.10–4.12. The idealized thresholds $\hat\tau_{w_i}$ exist, coincide with or lie below $\rho_{mw_i}$ , and bound the equilibrium thresholds from above and the pair thresholds $\max(\tau_m,\tau_{w_i})$ from below.
Proposition 4.8. In a star-shaped market some induced stationary equilibrium satisfies $\mathrm{FB}(Q)\le2\,\mathrm{SW}$ for every first-best feasible $Q$ .

Significance

Theorem 4.1 is a price-of-anarchy statement for platform design: restricting the platform to meetings, and letting agents choose acceptance selfishly, costs at most a factor 4 against a platform that dictates everything. Since every equilibrium candidate is also a first-best candidate with the same welfare, it gives a 4-approximation to the optimal directed search as well. The construction behind it has a structure that is itself informative: the equilibrium consists of disjoint submarkets in each of which one side has a single type.

The results are proved in the paper; none of them has a machine-checked proof in Lean or another proof assistant on record. Formalizing them yields a checked model of stationary search markets with threshold play, a checked reduction of the first best to a generalized assignment LP, and two graph lemmas (forest support of LP optima, the star-forest halving) that recur in rounding arguments for generalized assignment and budgeted allocation.

Difficulty

The first best is easy to reach once Proposition 4.4 linearizes it; the obstacle is incentives. Turning a first-best solution into an equilibrium by keeping its meeting rates fails, because agents raise or lower their thresholds relative to the pairwise thresholds the platform would choose, and constraint (6e) couples every type's threshold to all thresholds on the opposite side through $\max(\tau_m,\tau_w)$ . The paper therefore has to control equilibrium thresholds without computing them, which it does only in star-shaped markets, and it needs a case analysis there (Case 1 and Subcases 2(a), 2(b) of Proposition 4.8) in which the meetings of the first best are sometimes discarded. Each of the reductions loses a factor: 2 in the passage from a forest to stars and 2 inside each star.

Formalization scope

The sides are separate finite types M and W with one law F m w per pair, serving both sides. The standing assumptions are $\delta>0$ ; $\alpha_\theta>0$ for every type; each $F_{mw}$ a probability measure with no atoms (NullSingletonClass, the paper's "continuous distribution"); and an integrable identity under each $F_{mw}$ . The positivity of $\alpha_\theta$ and the integrability are added: the paper uses both implicitly. Tails are taken over $[a,\infty)$ . LP feasibility includes $\beta\ge0$ , which the paper leaves implicit; Lemma 4.6 also assumes nonnegative capacities. $\rho_{mw}$ is the supremum of $A$ over $[0,\infty)$ , never defined as a fixed point.

"Approximation of the first-best optimum" is stated against every first-best feasible candidate, so no supremum appears. An induced stationary equilibrium always includes the best-response constraint (6e); a formalization that drops it, or that lets the equilibrium use pairwise thresholds, makes the goal compare the first best with itself and is ruled out. With an empty side both welfares vanish and the goal holds, as in the paper.

The "polynomial-time" part of Theorem 4.1 is not posed: the paper's computational model (oracle access to the laws, footnote 15) is not specified precisely enough to formalize. The source is the arXiv v2 preprint (18 Aug 2021); the Management Science version may number results differently. Proofs of any milestone, auxiliary lemmas about tail integrals of atomless laws, and LP vertex arguments are all welcome.

Selected references

N. Immorlica, B. Lucier, V. Manshadi, A. Wei, Designing Approximately Optimal Search on Matching Platforms, Management Science, 2023; preprint arXiv:2102.09017v2. https://arxiv.org/abs/2102.09017
J. K. Lenstra, D. B. Shmoys, É. Tardos, Approximation algorithms for scheduling unrelated parallel machines, Mathematical Programming 46, 1990. https://doi.org/10.1007/BF01585745
D. Chakrabarty, G. Goel, On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP, SIAM Journal on Computing 39(6), 2010. https://doi.org/10.1137/080735503
S. Banerjee, S. Gollapudi, K. Kollias, K. Munagala, Segmenting two-sided markets, WWW 2017. https://doi.org/10.1145/3038912.3052578

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Algorithmic Game Theory Economics·Captain: mikedeng1

Online Advertising, Data Sharing, and Consumer Control 3: Under Topics-Style Category Tracking Both Advertisers Bid on ADX1 Exactly When the Inaccuracy τ Is Below an Explicit Threshold τ̃ > 1Research Paper

Motivation

Online display advertising is sold through ad exchanges, and much of its value comes from targeting: showing an ad to consumers known to be interested in the advertised product. For two decades the main tracking technology was the third-party cookie, under which an exchange can follow a consumer only after an advertiser agrees to share the data from its own website. Google's proposed replacement for Chrome, the Topics system, instead lets the browser itself assign consumers to interest categories. Advertisers no longer control that data, and the categories are coarse, so some ads reach consumers who are not interested.

Johnson, Jungbauer and Preuss (SSRN 3898798, Nov. 2022; published in Management Science, 2024, doi:10.1287/mnsc.2022.03385) study how data sharing and consumer control shape competition between ad exchanges. Their §3 adds a Topics-style category system to the base model and asks when advertisers use it, and how the exchange that runs it profits from its accuracy. This mission formalizes that section.

Setting

There are two advertisers $A_1, A_2$ and two exchanges. ADX1 runs the Topics system. It reaches all $1+\eta_2$ consumers interested in the product category, but reaching a mass one of them takes $\tau>1$ ad impressions; $\tau$ is the system's inaccuracy. ADX2 offers retargeting through another technology. An advertiser bidding there reaches a fraction $\sigma$ of the $1-\alpha$ consumers who recently visited its own website. An impression costs $w$ . A consumer reached by a single advertiser is worth the monopoly profit $M$ , and one reached by both is worth the competition profit $C$ . The parameters satisfy

0<\alpha<\tfrac12,\qquad 0<w<C<M<2C,\qquad 0<\varphi<1,\qquad \eta_2>0,\qquad \tau>1,\qquad 0\le\sigma<1,

where $\varphi$ is the share of the ad price an exchange keeps.

The advertisers bid simultaneously, each on one exchange. Write $\pi(x,y)$ for the profit of an advertiser on exchange $x$ when its rival is on $y$ (p. 17):

\pi(1,1)=(1+\eta_2)(C-\tau w),\qquad \pi(2,2)=\sigma[\alpha M+(1-2\alpha)C-(1-\alpha)w],

\pi(1,2)=(1+\eta_2)M-\sigma(1-\alpha)(M-C)-(1+\eta_2)\tau w,\qquad \pi(2,1)=\sigma(1-\alpha)(C-w).

An equilibrium is a pure Nash equilibrium of this symmetric $2\times2$ game. ADX1's revenue is $\varphi w$ per impression sold on it, and an advertiser bidding on ADX1 buys $\tau(1+\eta_2)$ impressions. The proof of the main result uses the thresholds

w_1=\frac{(1+\eta_2)-\sigma(1-\alpha)}{\tau(1+\eta_2)-\sigma(1-\alpha)}\,C,\qquad w_2=\frac{[(1+\eta_2)-\sigma]M+\alpha\sigma C}{\tau(1+\eta_2)-\sigma(1-\alpha)},

and the accuracy threshold

\tilde\tau=\frac{(1+\eta_2)-\sigma(1-\alpha)}{w(1+\eta_2)}\,C+\frac{\sigma(1-\alpha)}{1+\eta_2}.

In Lean these objects are Params, Params.Standing, payoff, IsNE, revenue, w₁, w₂ and tauTilde, in the namespace AdDataSharing.Topics.

Formalization targets

Goal: Proposition 5 (threshold corrected)

\tilde\tau>1\quad\text{and}\quad\Big[(1,1)\text{ is the unique equilibrium}\iff \tau<\tilde\tau\Big].

Bidding on ADX1 means cross-targeting the rival's visitors, which Topics cannot avoid. The goal therefore says that both advertisers cross-target exactly when the system is not too inaccurate, and that this region is never empty.

Milestones

$(1,1)$ is an equilibrium $\iff w\le w_1\iff \tau\le\tilde\tau$ (proof of Proposition 5, p. 29).
$(2,2)$ is an equilibrium $\iff w\ge w_2$ (pp. 29–30).
Each asymmetric profile is an equilibrium $\iff w_1\le w\le w_2$ (p. 30).
$w_1<w_2$ (p. 30).
$\tilde\tau>1$ (p. 30).
Proposition 7 (i): for $\tau<\tau'<\tilde\tau$ , ADX1's equilibrium revenue at $\tau'$ strictly exceeds that at $\tau$ (p. 19).
Proposition 7 (ii): $\tilde\tau$ is strictly decreasing in $\sigma$ (p. 19).
Corollary 2: above some $\bar\tau$ the game has an equilibrium, and ADX1's revenue in every equilibrium is strictly below its revenue in every equilibrium of the base game of §1 without Topics (p. 19).

Significance

Proposition 5 is the entry point of the paper's Topics analysis. It identifies when a category-based system is actually used, and it implies that removing advertisers' property rights over their data leads to more information being shared about consumers' purchase interests. Proposition 7 adds an incentive result: inside the region of use, the exchange running Topics gains from making it less accurate, because inaccuracy sells more impressions. A weaker rival exchange widens that region. Corollary 2 adds a regulatory point: a privacy rule that sets a high minimum inaccuracy can lead the exchange not to introduce the system at all. Corollary 2 compares against ADX1's equilibrium revenue in the base game of §1, which mission 1 of this series formalizes; its definitions are shared with this mission.

The results are proved in the paper, and none of them has a machine-checked proof. Formalizing them also corrects the printed statement. Display (6) on p. 17 omits a factor $1/(1+\eta_2)$ , and with the printed threshold the equivalence of Proposition 5 is false. The mission states the threshold that the page's own deviation condition gives.

Difficulty

The algebra is elementary, and the care lies in the equilibrium bookkeeping. "In the equilibrium both advertisers bid on ADX1" is a uniqueness claim. At $\tau=\tilde\tau$ the profile $(1,1)$ is still an equilibrium, but the asymmetric profiles are equilibria too. A proof that only checks deviations from $(1,1)$ therefore gets the wrong boundary. A proof also has to rule out every other profile, using $w_1<w_2$ and the signs of the denominators. Each threshold is a quotient, and its denominator $\tau(1+\eta_2)-\sigma(1-\alpha)$ is positive only because $\tau>1>\sigma$ . Proposition 7 (i) compares two different games, and Corollary 2 needs equilibrium existence together with a threshold that is uniform in the profile.

Formalization scope

All quantities are real. The game is the platform's general two-player pure Nash equilibrium CompetingChains.Cournot.IsPureNash, which this mission references, applied with payoffs $\pi(x,y)$ and $\pi(y,x)$ . There is no contracting stage and no equilibrium selection, as footnote 22 of the paper states. Deviation inequalities are weak, so the exact conditions read $w\le w_1$ and $w\ge w_2$ where the proof writes strict inequalities as sufficient conditions.

Three assumptions are added relative to §3, and each is disclosed. $C>w$ is used throughout the paper and stated on p. 21. $\sigma\ge0$ is the reading of "a fraction $\sigma<1$ ". $\eta_2>0$ comes from the base model, where it reads $\eta_2>\eta_1>0$ . The inequalities $2C>M$ and $\varphi<1$ are kept as paper assumptions; they are inert here. Divisions outside the standing assumptions return Lean's junk value $0$ , and every theorem assumes the standing assumptions.

Several formalizations would trivialize the targets, and none is used. "Both advertisers bid on ADX1" is defined through IsNE and uniqueness, not as an inequality on payoff numbers. The printed threshold (6) is never posed. Statements quantifying over all equilibria also assert that an equilibrium exists, so they cannot hold vacuously. All definitions are local to this mission except the referenced Nash equilibrium notion. Contributions are welcome on any milestone, and a reusable lemma characterizing pure equilibria of $2\times2$ symmetric games would serve all of them.

Selected references

J. P. Johnson, T. Jungbauer, M. Preuss, Online Advertising, Data Sharing, and Consumer Control, SSRN working paper 3898798, version of November 2, 2022. https://ssrn.com/abstract=3898798
J. P. Johnson, T. Jungbauer, M. Preuss, Online Advertising, Data Sharing, and Consumer Control, Management Science, 2024. https://doi.org/10.1287/mnsc.2022.03385

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Operations Research Optimization Probability·Captain: mikedeng1

Lightning Network Economics: Channels II: A Payment Channel Beats On-Chain Settlement Only If the On-Chain Fee Exceeds an Explicit ThresholdResearch Paper

Motivation

Every Bitcoin transaction recorded on the blockchain pays a fee, and the blockchain processes only a few transactions per second. The Lightning Network is a layer-two protocol that moves repeated payments off chain: two parties lock funds in a payment channel, exchange payments by updating the channel's balance, and touch the blockchain only to open, close or reset the channel. Whether a channel is worth opening is an economic question. Locked funds forgo interest, and a channel whose balance is exhausted must be reset on chain at a cost.

Guasoni, Huberman and Shikhelman (Management Science, 2023) model this trade-off with Poisson payment streams and continuous discounting, and obtain closed-form channel costs (their Theorem 1). Their Theorem 2 turns those costs into necessary conditions: a channel can be cheaper than settling every payment on chain only if the on-chain fee exceeds an explicit threshold depending on the reset cost, the payment rate and the interest rate. This mission formalizes Theorem 2, its two proofs, and the on-chain benchmark (Lemma 1) against which every channel is measured.

Setting

Node 1 pays node 2 one unit at the jumps of a Poisson process with rate $\lambda_1$ , and node 2 pays node 1 at the jumps of an independent Poisson process with rate $\lambda_2$ . The discount rate $r>0$ is the opportunity cost of capital per unit of time, $B$ is the cost of resetting a channel, and $C$ is the cost of one on-chain transaction. Node $i$ commits the amount $l_i$ to the channel.

On-chain settlement. If every payment of a stream with rate $\lambda$ is settled on chain, the expected discounted cost is $\mathbb E\big[\sum_{n\ge1}e^{-r\tau_n}C\big]=C\lambda/r$ , where $\tau_n$ is the $n$ -th arrival time (Lemma 1). Two streams of rate $\lambda$ cost $2C\lambda/r$ .

Unidirectional channel. If only one node pays, at rate $\lambda$ , a channel of size $m$ costs (Theorem 1(i), equation (1))

\mathrm{uniCost}(m)=m+B\Big(\big(\tfrac{r+\lambda}{\lambda}\big)^{m}-1\Big)^{-1}.

Bidirectional channel. With

\alpha_\pm=\frac{\lambda_1+\lambda_2+r\pm\sqrt{(\lambda_2-\lambda_1)^2+r^2+2r(\lambda_2+\lambda_1)}}{2\lambda_2}

(equation (4)), a channel with sizes $l_1,l_2$ costs $\mathrm{biCost}(l_1,l_2)$ , the expression (3) in $\alpha_\pm^{l_1}$ , $\alpha_\pm^{l_2}$ and $\alpha_\pm^{l_1+l_2}$ . In the symmetric case $\lambda_1=\lambda_2=\lambda$ it simplifies, with $\alpha=\alpha_+$ , to

l_1+l_2+\frac{B(\alpha^{l_1}+\alpha^{l_2})}{(\alpha^{l_1}-1)(\alpha^{l_2}-1)}.

A channel costs less than on-chain transactions if, for some size, its cost is strictly below the on-chain cost of the streams it carries.

Formalization targets

Goal: Theorem 2

For $B,r,\lambda>0$ and any fee $C$ :

if $\mathrm{uniCost}(m)<\lambda C/r$ for some $m>0$ , then

C>\log\Big(1+B\log\big(1+\tfrac r\lambda\big)\Big);\tag{5}

if $\mathrm{biCost}(l_1,l_2)<2C\lambda/r$ for some $l_1,l_2>0$ with $\lambda_1=\lambda_2=\lambda$ , then

C>3\Big(\frac{B\kappa^4}{4}\Big)^{1/3}-\frac B{12}\kappa^2,\qquad \kappa=\log\frac{r+2\lambda+\sqrt{r(4\lambda+r)}}{2\lambda}.\tag{6}

Milestones

Lemma 1: the on-chain cost $C\lambda/r$ .
Part (i) chain: the change of variable $\kappa=\log(1+r/\lambda)$ that turns the excess cost into $F(m)=C/(1-e^\kappa)+B/(e^{\kappa m}-1)+m$ ; the unique minimizer $\hat m=\cosh^{-1}(B\kappa/2+1)/\kappa$ and the minimum value; inequality (16); the sufficient condition (17)–(20) for on-chain settlement to be strictly cheaper; Theorem 2(i).
Part (ii) chain: the symmetric cost and $\lambda/r=\alpha/(\alpha-1)^2$ ; the reduction (21) to equal sizes; the hyperbolic form and the bounds $1/x^2-1/3<\operatorname{csch}^2x<1/x^2$ of (22); the minimization (24) of the lower bound (23); Theorem 2(ii).
Expansions: the right-hand sides of (5) and (6) equal $Br/\lambda+o(r)$ and $3(Br^2/(4\lambda^2))^{1/3}-Br/(12\lambda)+o(r)$ as $r\to0^+$ .

Significance

Theorem 2 gives the qualitative conclusion of the paper in quantitative form: channels pay off only when on-chain fees are high, reset costs low, payment rates high, or interest rates low. Read as a contrapositive, it says that below the thresholds, settling on chain is at least as cheap for every channel size, so no optimization over sizes is needed to rule a channel out. The expansions show the two thresholds scale differently in the interest rate, linearly for a unidirectional channel and as $r^{2/3}$ for a symmetric bidirectional one. The paper's subsequent asymptotic results (Theorem 3) rely on the same reductions.

The theorem is proved in the paper; nothing here is open. To our knowledge none of it is formalized. A formalization adds a machine-checked verification of a proof that contains several informal steps: a minimizer of a two-variable function is assumed to exist, two inequalities are stated non-strictly where strict ones are needed, and one sentence states an "if and only if" with $\le$ where $<$ is meant. Lemma 1 also requires the discounted sum over a Poisson process to be integrable, which the paper's renewal argument takes for granted.

Difficulty

The bidirectional cost (3) is a ratio of differences of real powers of two algebraic numbers, and its minimum over two sizes has no closed form, so the threshold cannot be read off an explicit optimum. The paper replaces the optimum by reductions whose rigorous versions are the substance of the work: the passage from (3) to the symmetric form, the reduction of a two-variable minimization to the diagonal (asserted in the paper through a Hessian computation and the existence of a minimizer, neither of which is carried out), and a lower bound on $\operatorname{csch}^2$ that holds on all of $(0,\infty)$ and is not in Mathlib. The obvious approach of bounding the cost at its minimizer fails for the same reason it fails on paper: no minimizer is given. For part (i), the minimizer involves $\cosh^{-1}$ , and the minimal value still depends on $C$ through two terms, so a condition on $C$ alone does not follow by rearrangement. Lemma 1 requires an exchange of expectation and infinite sum over a Poisson process, which the paper's renewal argument assumes.

Formalization scope

Costs as closed forms for real sizes. uniCost and biCost are the paper's expressions (1) and (3) with real sizes and real powers (Real.rpow). Theorem 2's proofs minimize over real sizes, and the hypotheses quantify over real sizes, so the statements imply their versions for natural sizes. The probabilistic derivation of (1) and (3) from the balance random walk is the subject of the sibling mission Channels I.
Explicit on-chain costs. The comparisons are with $\lambda C/r$ and $2C\lambda/r$ , written out; there is no opaque on-chain cost function.
Strict and existential. "Costs less" is strict and asserted for some size; the conclusions on $C$ are strict. No sign is imposed on $C$ ; $B,r,\lambda>0$ .
Lemma 1 uses the published definition QueueingFundamentals.Foundations.IsExpInterarrivals of i.i.d. exponential interarrival times; the conclusion asserts integrability as well as the value $C\lambda/r$ .
Corrections recorded in the statements. (17)–(20) is stated with the strict conclusion $F>0$ that (17) gives, for all $m>0$ ; (21) is stated in midpoint form, $H(m,m)\le H(l_1,l_2)$ for $m=(l_1+l_2)/2$ , which requires no minimizer.
Two different $\kappa$ . In part (i) $\kappa=\log(1+r/\lambda)$ ; in part (ii) $\kappa=\log\alpha_+$ . They are never identified.
Asymptotics are little-o along $r\to0^+$ (𝓝[>] 0) with $B$ and $\lambda$ fixed.

Ruled out as trivializing: fixing $B$ , $r$ or $\lambda$ ; restricting to natural sizes without saying so; an existentially chosen $\kappa$ ; weakening either strict inequality; or an on-chain cost under which the comparison is vacuous.

Needed infrastructure: real-power calculus and convexity on $(0,\infty)$ , $\cosh^{-1}$ and $\sinh$ bounds, little-o expansions of $\log$ and $\cosh^{-1}$ near their base points, and the Laplace transform of Gamma laws for Lemma 1. The csch inequality (22) and Lemma 1 are reusable beyond this mission. Contributions to any milestone are welcome independently.

Selected references

P. Guasoni, G. Huberman, C. Shikhelman, Lightning Network Economics: Channels, Management Science, Articles in Advance, published online Nov. 6, 2023, pp. 1–14. https://doi.org/10.1287/mnsc.2022.01664
J. Poon, T. Dryja, The Bitcoin Lightning Network: Scalable Off-Chain Instant Payments, technical report, 2016. https://lightning.network/lightning-network-paper.pdf
G. Huberman, J. D. Leshno, C. Moallemi, Monopoly without a Monopolist: An Economic Analysis of the Bitcoin Payment System, Review of Economic Studies 88(6), 2021. https://doi.org/10.1093/restud/rdab014

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Algorithmic Game Theory Economics Operations Research·Captain: mikedeng1

Sharing Sequentially Triggered Losses: Automated Conflict Resolution Through Smart Contracts 3: The Midpoint Fixed-Fraction Rule Is the Shapley Value and the Nucleolus of the Liability GameResearch Paper

Motivation

When one breached contract forces the next party to breach its own contract, a single failure travels down a loss chain: agent 1 (the initiator) causes a loss $\ell_1$ to agent 2, which can then no longer honour its agreement with agent 3 and causes it a loss $\ell_2$ , and so on until agent $n$ causes a loss $\ell_n$ to a final victim. Gudmundsson, Hougaard and Ko (Management Science 70(3), 2024) ask how the total loss should be shared among the agents of the chain, with the aim of encoding the answer in a smart contract that settles such disputes automatically.

Their central rules are the fixed-fraction rules $\varphi^\lambda$ , under which every agent pays a fixed fraction $\lambda$ of the loss it caused and the initiator also pays the remaining fraction $1-\lambda$ of the total. An axiomatic characterization (their Theorem 1) singles out the whole family $\lambda\in[0,1]$ but not a particular member. This mission formalizes the paper's game-theoretic argument for the midpoint $\lambda=1/2$ : the cooperative game that records which coalitions are liable for which losses has the midpoint allocation as both its Shapley value and its nucleolus, the two best-known single-valued solutions of cooperative game theory.

Setting

Agents are $N=\{1,\dots,n\}$ with $n\ge1$ . A loss problem is a vector $\ell=(\ell_1,\dots,\ell_n)\in\mathcal L^n=\mathbb R^n_{\ge0}$ , where $\ell_i$ is the loss agent $i$ causes to agent $i+1$ ; the total loss is $L=\ell_1+\dots+\ell_n$ . The fixed-fraction rule with parameter $\lambda$ assigns

\varphi^\lambda_1(\ell)=\lambda\ell_1+(1-\lambda)L,\qquad \varphi^\lambda_i(\ell)=\lambda\ell_i\ (i\ge2).

A transferable-utility game on $N$ is a function $v$ assigning a real number to every coalition $S\subseteq N$ . The liability game of $\ell$ is

v_\ell(S)=\begin{cases}\sum_{i\in S}\ell_i&\text{if }1\in S,\\0&\text{otherwise:}\end{cases}

a coalition is liable for the losses its members caused only if it contains the initiator, without whom no loss would have occurred. The contribution of player $i$ to the grand coalition is $M_i(v)=v(N)-v(N\setminus\{i\})$ .

Two solution concepts are compared with $\varphi^{1/2}(\ell)$ .

The Shapley value gives player $i$

\mathrm{Sh}_i(v)=\sum_{S\subseteq N\setminus\{i\}}\frac{|S|!\,(n-|S|-1)!}{n!}\bigl(v(S\cup\{i\})-v(S)\bigr).

An imputation is a vector $x$ with $\sum_{i\in N}x_i=v(N)$ and $x_i\ge v(\{i\})$ for all $i$ . The excess of $S$ at $x$ is $e(S,x)=v(S)-\sum_{i\in S}x_i$ , and the excess vector $\theta(x)$ lists the $2^n$ excesses in non-increasing order. The nucleolus $\nu(v)$ (Schmeidler 1969) is the imputation whose excess vector is lexicographically smallest.

A game is convex if $v(\emptyset)=0$ and $v(S)+v(T)\le v(S\cup T)+v(S\cap T)$ for all $S,T$ .

Formalization targets

Goal: Theorem 3 (p. 15)

For every $\ell\in\mathcal L^n$ ,

\mathrm{Sh}(v_\ell)=\varphi^{1/2}(\ell)\qquad\text{and}\qquad \nu(v_\ell)=\varphi^{1/2}(\ell),

where the second identity is stated as: $\varphi^{1/2}(\ell)$ is an imputation of $v_\ell$ with lexicographically minimal sorted excess vector, and every imputation with this property equals $\varphi^{1/2}(\ell)$ .

Milestones (Appendix A, proof of Theorem 3, p. 24)

$v_\ell$ is convex.
(§4.2, p. 15; referenced, already on the platform) the core of a convex game is non-empty and contains its Shapley value, which applies to $v_\ell$ by milestone 1.
$M_1(v_\ell)=L$ and $M_i(v_\ell)=\ell_i$ for $i\ne1$ .
Every nucleolus $\nu$ of $v_\ell$ satisfies $\nu_i=M_i(v_\ell)/2=\ell_i/2=\varphi^{1/2}_i(\ell)$ for $i\ne1$ , and $\nu_1=\varphi^{1/2}_1(\ell)$ .
The Shapley value of $v_\ell$ is a nucleolus of $v_\ell$ .

Significance

The result. Theorem 1 of the paper leaves a one-parameter family of rules, and a designer who writes the rule into a contract must still choose $\lambda$ . Theorem 3 gives a reason for $\lambda=1/2$ that does not depend on the axioms: if losses are treated as the worth of coalitions, two solution concepts that are built on unrelated principles (average marginal contributions on one side, lexicographic minimization of complaints on the other) both return $\varphi^{1/2}(\ell)$ . Together with the paper's Theorem 2 (equidistance from ideal allocations, a separate mission of this series) it makes the midpoint the paper's recommended rule.

Formalizing it. The paper proves Theorem 3 by citing two results of Muto, Nakayama, Potters and Tijs (1988) on big boss games, the nucleolus formula (their Theorem 4.2) and the coincidence of Shapley value and nucleolus for convex big boss games (their Theorem 4.5); the paper itself proves only convexity and the marginal contributions. A formal proof must therefore establish the nucleolus of $v_\ell$ directly from Schmeidler's lexicographic definition. No nucleolus definition exists on the platform (searched 2026-10-14), and this mission introduces one. The Shapley value and convex games are reused from the platform's published definitions, and the core-contains-the-Shapley-value theorem for convex games is already proved there.

Difficulty

Convexity and the marginal contributions are finite case analyses. The Shapley identity is a weighted sum over all coalitions not containing a player and needs the combinatorial identities behind the weights; for $v_\ell$ only the position of the initiator relative to $S$ matters.

The substantive part is the nucleolus. Its definition compares sorted vectors of $2^n$ excesses, so an argument cannot treat one coalition at a time: showing that a candidate is lexicographically minimal requires controlling how a change of the payoff vector permutes the order of all excesses. The obvious first idea, minimizing only the largest excess (the least core), generally leaves a set of imputations rather than a point, so the comparison must go down the whole sorted list. Coalitions whose excesses tie, which happens for instance when some $\ell_i=0$ , are where informal arguments about sorted excess vectors are usually loose, and a formal proof has to handle them.

Formalization scope

Agents are Fin n with the initiator at index 0; [NeZero n] encodes $n\ge1$ , which the paper assumes because $1\in N$ . Loss vectors are Fin n → ℝ with the hypothesis IsLoss ℓ ( $\ell_i\ge0$ ).
A game is Finset (Fin n) → ℝ. The Shapley value is the published Supermodularity.Cooperative.ShapleyValue and convexity the published Supermodularity.Cooperative.IsConvexGame (the supermodular form, equivalent to the increasing-marginal-contribution form used on p. 24).
The nucleolus is defined in the mission's Nucleolus file: imputations with efficiency and individual rationality, excess $v(S)-\sum_{i\in S}x_i$ over all $2^n$ coalitions including $\emptyset$ and $N$ , sorted by ≥ into a list, and compared with List.Lex (· < ·). Including $\emptyset$ and $N$ does not change the order, since their excess is $0$ at every imputation.
The goal states both identities. The nucleolus half is not replaced by membership in the core or by minimality of the largest excess, the minimization runs over imputations and not over all of $\mathbb R^n$ , and uniqueness is stated explicitly rather than taken from Schmeidler's theorem.
The paper never defines the nucleolus or the Shapley value; the standard definitions (Schmeidler 1969; Shapley 1953) are used.

Contributions welcome: proofs of the milestones; general lemmas on sorted excess vectors (e.g. Kohlberg-type criteria for the nucleolus), which are reusable for every later nucleolus result; Schmeidler's existence and uniqueness theorem; and the general big boss game results of Muto et al. (1988).

Selected references

J. Gudmundsson, J. L. Hougaard, C. Y. Ko, Sharing Sequentially Triggered Losses: Automated Conflict Resolution Through Smart Contracts, Management Science 70(3):1773–1786, 2024 (accepted manuscript, 2023). https://doi.org/10.1287/mnsc.2023.4772
S. Muto, M. Nakayama, J. Potters, S. Tijs, On big boss games, The Economic Studies Quarterly 39(4):303–321, 1988. https://doi.org/10.11398/economics1986.39.303
D. Schmeidler, The nucleolus of a characteristic function game, SIAM Journal on Applied Mathematics 17(6):1163–1170, 1969. https://doi.org/10.1137/0117107
L. S. Shapley, A value for n-person games, in Contributions to the Theory of Games II, Annals of Mathematics Studies 28, Princeton University Press, 1953, pp. 307–317. https://doi.org/10.1515/9781400881970-018
L. S. Shapley, Cores of convex games, International Journal of Game Theory 1:11–26, 1971. https://doi.org/10.1007/BF01753431

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Linear Optimization Operations Research Probability+1·Captain: mikedeng1

Selection and Ordering Policies for Hiring Pipelines via Linear Programming 3: In Simultaneous Offering With Values at Least τ, a Truncated-LP Value-Ordered Policy Earns ≥ α^τ_k Times the LP BoundResearch Paper

Motivation

A firm that fills a fixed number of positions from a pool of applicants faces uncertainty at every stage: applicants differ in value, and an offer may be declined. When offers must go out at once, as in a recruiting season with a common deadline, the firm decides a single offer set and lives with however many candidates accept. Sending too few offers leaves positions empty; sending too many hires beyond capacity at a cost. This trade-off is the same as overbooking in revenue management (airline seats, hotel rooms), with the difference that candidates are heterogeneous in both value and acceptance probability.

Epstein and Ma (arXiv:2210.04059v3, Operations Research 2024) study three hiring-pipeline models through linear programming relaxations. This mission formalizes their lower bound for the third model, Simultaneous Offering (§3.3 and §6 of the paper). The question it answers is practical: how much is lost if the firm restricts itself to value-ordered policies, which send offers to the $m$ most valuable candidates for some $m$ , instead of searching over all $2^n$ offer sets? Value-ordered policies are natural, because a firm rarely wants to withhold offers from its best candidates, and the optimal one is computable by a short dynamic program. The paper shows (Example 1, p. 25) that they can be arbitrarily bad in general, and that they are provably good when every candidate's value is bounded below.

Setting

An instance $I=(k,n,p,v)$ has $k$ positions and $n$ candidates, $1\le k\le n$ . Candidate $i$ accepts an offer independently with probability $p_i\in[0,1]$ and then contributes value $v_i\ge0$ . Candidates are labelled so that $v_1\ge v_2\ge\dots\ge v_n$ . The firm chooses an offer set $S\subseteq[n]$ ; with $A\subseteq S$ the random set of acceptances, the reward is

\sum_{i\in A}v_i-\big[\,|A|-k\,\big]^+,

the cost per candidate hired beyond capacity being normalised to $1$ . $\mathrm{OPT}_{\mathrm{sim}}(I)$ is the best expected reward over offer sets. For $\tau>0$ the instance is $\tau$ -bounded, $I\in\mathcal I^\tau_{\mathrm{sim}}$ , if $v_i\ge\tau$ for all $i$ ; the interesting range is $\tau\in(0,1)$ , since for $\tau\ge1$ offering to everyone is optimal.

The benchmark is the LP relaxation

\mathrm{LP}_{\mathrm{sim}}(I)=\max_{y,z}\ \sum_{i}v_ip_iy_i+z\quad\text{s.t.}\quad z\le k-\sum_ip_iy_i,\ \ z\le0,\ \ 0\le y_i\le1,

which charges the overage cost on the expected number of hires. Write $V_{\mathrm{high}}=\{i:v_i>1\}$ and $P_{\mathrm{high}}=\sum_{i\in V_{\mathrm{high}}}p_i$ .

The algorithm $\mathrm{ALG}^s_{\mathrm{sim}}$ , for $s\in[0,1]$ , takes an optimal LP solution $y$ , scales its total mass $\sum_iy_ip_i$ by $s$ to $\kappa$ , and fills mass $\kappa$ from the most valuable candidate down: $y'_i=1$ while the prefix mass $\sum_{l\le i}p_l$ stays at most $\kappa$ , one fractional entry, then zeros. Offers are then sent independently with probabilities $y'_i$ , so the algorithm randomises between two value-ordered policies. Its expected reward is $R_{\mathrm{ALG}^s_{\mathrm{sim}}}(I)$ .

The guarantee involves the truncated Poisson mean $\mathbb E[\min\{\mathrm{Pois}(\lambda),k\}]=\sum_{j\ge0}\min\{j,k\}e^{-\lambda}\lambda^j/j!$ and

\alpha^\tau_k=\sup_{0\le s\le1}\Big(s-\frac{s}{\tau}+\frac{\mathbb E[\min\{\mathrm{Pois}(sk),k\}]}{\tau k}\Big).

Formalization targets

Goal: Theorem 3 (p. 30)

For $\tau\in(0,1)$ , every $I\in\mathcal I^\tau_{\mathrm{sim}}$ , every optimal solution of $\mathrm{LP}_{\mathrm{sim}}(I)$ and a maximiser $s^*\in[0,1]$ of the objective defining $\alpha^\tau_k$ ,

R_{\mathrm{ALG}^{s^*}_{\mathrm{sim}}}(I)\ \ge\ \alpha^\tau_k\,\mathrm{LP}_{\mathrm{sim}}(I).

Milestones

Lemma 10 (p. 27): $\mathrm{LP}_{\mathrm{sim}}(I)\ge\mathrm{OPT}_{\mathrm{sim}}(I)$ .
Lemma 11 (p. 27): some optimal LP solution is a filled prefix $y=(1,\dots,1,y_j,0,\dots,0)$ with $y_j>0$ .
Lemma 12 (p. 28): the total mass of such a solution is $P_{\mathrm{high}}$ if $P_{\mathrm{high}}>k$ and $\min\{k,P_{\mathrm{total}}\}$ otherwise (posed under the disclosed hypothesis $v_i>0$ , $v_i\ne1$ , $p_i>0$ ).
Monotonicity (App. D.5, p. ec16): $\mathbb E[\min\{\mathrm{Pois}(x),k\}]/x$ is decreasing in $x$ .
Proposition EC.2 (p. ec3): $\mathbb E[\min\{\mathrm{Pois}(k),k\}]=k(1-e^{-k}k^k/k!)$ .
Lemma 13 (p. 29): if $P_{\mathrm{high}}\le k$ , then $R_{\mathrm{ALG}^s_{\mathrm{sim}}}(I)\ge\big(s-s/\tau+\mathbb E[\min\{\mathrm{Pois}(sk),k\}]/(\tau k)\big)\mathrm{LP}_{\mathrm{sim}}(I)$ for every $s\in[0,1]$ .
The $\mathrm{ALG}^1$ bound (p. 30): on every $\tau$ -bounded instance, $R_{\mathrm{ALG}^1_{\mathrm{sim}}}(I)\ge(1-e^{-k}k^k/(k!\tau))\,\mathrm{LP}_{\mathrm{sim}}(I)$ .
Lemma 14 (p. 29): if $P_{\mathrm{high}}>k$ , deflating the acceptance probabilities to an instance with $P_{\mathrm{high}}=k$ does not increase the ratio $R_{\mathrm{ALG}^{s^*}_{\mathrm{sim}}}/\mathrm{LP}_{\mathrm{sim}}$ (posed under the disclosed hypothesis $v_i\ne1$ ).

Significance

Theorem 3 makes value-ordered policies an $\alpha^\tau_k$ -approximation on $\tau$ -bounded instances, through Lemma 10. The factor tends to $1$ as $k\to\infty$ for fixed $\tau$ : at $s=1$ it is at least $1-e^{-k}k^k/(k!\tau)=1-O(1/\sqrt k)$ (Corollary 4). The paper's Theorem 4, the subject of the companion mission in this series, shows that no value-ordered policy beats the same expression with the supremum over $s\ge0$ , so the analysis is tight wherever the two suprema coincide. The interpretation of $\tau$ is how soft the capacity is: the bound degrades exactly as the overage cost dominates candidate values.

The result is proved in the paper; nothing in this mission is open mathematics. To our knowledge none of it has been machine-checked. A complete development would give the platform an overbooking-type stochastic selection model with an exact expected reward, an LP benchmark, and the truncated-Poisson constant that recurs in prophet-inequality and online-matching analyses (the $k$ -uniform correlation gap of Yan 2011). Proposition EC.2 and the monotonicity of $\mathbb E[\min\{\mathrm{Pois}(x),k\}]/x$ are reusable on their own.

Difficulty

The obvious argument compares the algorithm's random number of hires with the LP's expected number and loses everything to the overage term: the LP pays $[\mathbb E|A|-k]^+$ , the algorithm pays $\mathbb E[|A|-k]^+$ , and Jensen's inequality points the wrong way. The gap must be controlled by a sum of independent Bernoulli variables against a Poisson variable of the same mean, uniformly over how the mass is spread across candidates. A second difficulty is the regime $P_{\mathrm{high}}>k$ , where the LP optimum deliberately exceeds capacity; a direct bound there does not reduce to the first case, and the paper's reduction (Lemma 14) changes the instance rather than the algorithm. Tie cases $v_i=1$ make the LP optimum non-unique, which the formal statements must handle explicitly.

Formalization scope

Candidates are Fin n (0-based), all quantities are real. The sorted labelling $v_1\ge\dots\ge v_n$ is part of instance validity (Antitone I.v), as the paper assumes after relabelling; ties may be ordered arbitrarily. Every expectation is an explicit finite sum over the $2^n$ hire sets with independent probabilities $y'_ip_i$ ; there is no measure theory. The truncated Poisson mean is the series of the Poisson pmf. $\mathrm{LP}_{\mathrm{sim}}(I)$ is the supremum of the objective over the feasible set, which is the LP value; $\alpha^\tau_k$ is the supremum over $[0,1]$ . $s^*$ is any maximiser of the $\alpha^\tau_k$ objective over $[0,1]$ ; one exists by continuity. A candidate with $p_i=0$ receives $y'_i=0$ , which changes no reward.

The algorithm's reward is the exact expected reward of the run with independent offers, never the bound produced in the paper's proof, and the comparison is with the LP value at an optimal solution, not with the objective at an arbitrary feasible point; $\alpha^\tau_k$ is the supremum of the specific Poisson expression, not a free constant.

Two statements carry disclosed hypotheses, because the printed versions fail at ties: Lemma 12 assumes $v_i>0$ , $v_i\ne1$ , $p_i>0$ , and Lemma 14 assumes $v_i\ne1$ . Counterexamples are in the items' Formalization Notes. Theorem 3 and Lemma 13 need neither and hold for every optimal LP solution.

A complete development needs Poisson series manipulations (the closed form, concavity of $\lambda\mapsto\mathbb E[\min\{\mathrm{Pois}(\lambda),k\}]$ ), a Bernoulli-to-Poisson comparison for $\mathbb E\min\{\sum_iD_i,k\}$ , and fractional-knapsack structure for the LP. Contributions of alternative proofs of any milestone, and of the Poisson facts as standalone results, are welcome.

Selected references

B. Epstein, W. Ma, Selection and Ordering Policies for Hiring Pipelines via Linear Programming, arXiv:2210.04059v3 (2023); Operations Research (2024). https://arxiv.org/abs/2210.04059
Q. Yan, Mechanism Design via Correlation Gap, Proceedings of SODA 2011, 710–719. https://doi.org/10.1137/1.9781611973082.56
M. T. Goodrich, R. Tamassia, Algorithm Design: Foundations, Analysis, and Internet Examples, Wiley, 2001, Chapter 5 (fractional knapsack).

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Analysis Operations Research Optimization+1·Captain: mikedeng1

Sample-Driven Optimal Stopping: From the Secretary Problem to the i.i.d. Prophet Inequality III: The Explicit Lower Bound on the Competitive Ratio Is at Least α* Times pResearch Paper

Motivation

Optimal stopping with partial information sits between two classical problems. In the secretary problem a decision maker sees values in random order, knows nothing about how they are generated, and must stop on the best one; the optimal guarantee is $1/e$ . In the i.i.d. prophet inequality the values are independent draws from a known distribution, and the best guarantee, relative to the expected maximum, is the constant $\alpha^*\approx 0.745$ : Hill and Kertz (1982) and Kertz (1986) showed that no stopping rule does better on every distribution, and Correa, Foncea, Hoeksma, Oosterwijk and Vredeveld (2017, doi:10.1145/3033274.3085137) gave a rule that achieves it. Practical settings, such as pricing from historical data, lie in between: the distribution is unknown, but samples of it are available.

Correa, Cristi, Epstein and Soto (arXiv:2011.06516v2) interpolate between the two problems with the $p$ -sample-driven optimal stopping problem: each of $N$ adversarial values is independently revealed in advance (a sample) with probability $p$ , and the remaining values arrive online in random order. At $p=0$ this is the secretary problem; as $p\to1$ it approaches the i.i.d. prophet inequality. They show that the optimal competitive ratio $\alpha(p)$ of the limit problem is the value of an infinite-dimensional linear program, and they construct an explicit threshold rule whose guarantee $\tilde\alpha(p)$ is the root of a scalar equation.

Prior work on learning a distribution from samples obtained guarantees $\alpha^*-O(\varepsilon)$ with $O(n^2/\varepsilon)$ samples (Correa, Dütting, Fischer, Schewior, 2019) and with $O(n/\varepsilon^6)$ samples (Rubinstein, Wang, Weinberg, 2020). Theorem 9 of the paper, the goal of this mission, gives such a guarantee with $O(n/\varepsilon)$ samples.

Setting

Fix $p\in(0,1)$ . The guarantee of the explicit threshold rule is the number $\tilde\alpha(p)\in(0,1)$ solving

\ln p+\frac{\tilde\alpha(1-p)^2}{p}=\sum_{i=1}^{\infty}\frac{\ln\bigl(1-\tilde\alpha+\tilde\alpha[(i+1)p^i-ip^{i+1}]\bigr)}{i(i+1)}.\tag{19}

Lemma 7 of the paper shows that (19) has exactly one root in $(0,1)$ . The i.i.d. prophet constant $\alpha^*\in(0,1)$ is the solution of

\int_0^1\frac{dy}{y(1-\ln y)+1/\alpha^*-1}=1 .

The proof of Theorem 9 (Appendix B.9) works with three auxiliary functions of $p$ , each depending on $\alpha^*$ as a parameter:

a(p)=\sum_{i\ge1}\frac{\ln\bigl(\tfrac1p-\alpha^*(1-p^i(1+i(1-p)))\bigr)}{i(i+1)}+\alpha^*(2p-p^2),\qquad b(p)=\frac1{p^2}\sum_{i\ge1}\frac{1}{i(i+1)\bigl(\tfrac1p-\alpha^*(1-p^i(1+i(1-p)))\bigr)},

c(p)=\sum_{i\ge1}\frac{p^{i-1}-p^i}{\tfrac1{p\alpha^*}-1+p^i(1+i(1-p))}.

Appendix B.7 also names the two sides of (19): $f(p,\alpha)=\ln p+\alpha(1-p)^2/p$ and $g(p,\alpha)=\sum_{i\ge1}\ln(1+\alpha(p^i(1+i(1-p))-1))/(i(i+1))$ . In Lean, IsAlphaTilde p a and IsAlphaStar s are shared predicates in SampleDrivenOS.Limit; aB9 s p, bB9 s p, cB9 s p, fB7 p α, and gB7 p α are in SampleDrivenOS.Linear.

Formalization targets

Goal: Theorem 9 (p. 17)

\text{for every }p\in(0,1):\qquad \tilde\alpha(p)\ \ge\ \alpha^*p .

It is stated for every root $\tilde\alpha$ of (19) and every solution $\alpha^*$ of the integral equation; neither existence nor uniqueness is assumed.

Milestones (Appendix B.7 and B.9, in the order the proof uses them)

B.7: $f(p,\cdot)$ is strictly increasing on $[0,1]$ and $g(p,\cdot)$ non-increasing on $[0,1)$ .
B.9: if (19) holds with " $\le$ " at $\alpha^*p$ in place of $\tilde\alpha$ (after the substitution $\tilde\alpha=f(p)p$ ), then $\tilde\alpha(p)\ge\alpha^*p$ .
B.9: that inequality is equivalent to $\alpha^*\le a(p)$ .
B.9: $a$ is differentiable on $(0,1)$ with $a'(p)=2\alpha^*(1-p)-b(p)+c(p)$ , also in term-by-term form.
B.9: $b(p)\ge 1/p$ .
B.9: $c(p)\le\frac1p\int_0^1\frac{dy}{1/(p\alpha^*)-1+y(1-p\ln y)}-\alpha^*(1-p)$ .
B.9: using the defining equation of $\alpha^*$ ,

\frac1p\int_0^1\frac{dy}{\tfrac1{p\alpha^*}-1+y(1-p\ln y)}\le\frac1p-\frac{1-p}{p}\Bigl(\frac1{p\alpha^*}-\frac1e\Bigr)\ln\frac{\tfrac1{p\alpha^*}+\tfrac pe}{\tfrac1{p\alpha^*}+\tfrac pe-1}.

B.9: for all $s\in[0.74,0.75]$ and $p\in(0,1)$ , $\bigl(\frac1{ps}-\frac1e\bigr)\ln\frac{1/(ps)+p/e}{1/(ps)+p/e-1}\ge0.8$ .
B.9: if $\alpha^*\in[0.74,0.75]$ , then $c(p)\le\frac1p-\alpha^*(1-p)(1+\frac1p)$ .
B.9: under the same hypothesis, $\frac{d}{dp}a(p)\le0$ on $(0,1)$ .

Significance

The explicit rule's guarantee $\tilde\alpha(p)$ is a lower bound on the optimal ratio $\alpha(p)$ , and Theorem 8 (a separate mission of this series) shows $\tilde\alpha(p)\to\alpha^*$ as $p\to1$ . Theorem 9 makes that convergence quantitative and uniform: the guarantee is at least linear in $p$ , so with $p=1-\varepsilon$ , an online set of size $n=\varepsilon N$ and $O(n/\varepsilon)$ samples the rule earns $\alpha^*-O(\varepsilon)$ of the expected maximum. This improves the sample complexity of the earlier learning-based results by a polynomial factor, in a model more general than theirs.

The theorem is proved in the paper; it has no machine-checked proof. The proof is elementary real analysis on explicit series and integrals, and one of its steps is a numerical check that the paper does not carry out in print. A formalization turns that numerical claim into a certified inequality and checks a chain of estimates in which one inequality sign is misprinted (see Formalization scope).

Difficulty

The statement is about two implicitly defined numbers. $\tilde\alpha(p)$ has no closed form, and $\alpha^*$ is defined by an integral equation. Evaluating both numerically at finitely many $p$ proves nothing for the remaining $p$ , and the inequality is tight at $p=1$ ( $\tilde\alpha(1)=\alpha^*$ ), so no uniform slack is available near $p=1$ .

Concavity of $\tilde\alpha$ would give the result immediately, and the paper reports it numerically, but the authors were unable to prove it; the argument must avoid it. The remaining work handles infinite series whose terms depend on $p$ through both $p^i$ and $i(1-p)$ : differentiating such a series term by term requires uniform control of the derivative series, comparing it with an integral needs a monotonicity argument, and one step needs a bound valid on an entire two-parameter region rather than at a point.

Formalization scope

Numbers, not processes. No probability, program or algorithm appears in Lean. Everything is a real-valued function on $\mathbb R$ .
Predicates, not choice. $\tilde\alpha(p)$ and $\alpha^*$ enter as IsAlphaTilde p a (with $0<a<1$ ) and IsAlphaStar s (with $0<s<1$ ). Statements quantify over every solution. Both exist, so the goal is not vacuous: existence of $\tilde\alpha$ is Lemma 7 (posed in the second mission of this series), and existence and uniqueness of $\alpha^*=1/\beta^*$ are posed as CorreaThreshold.Adaptive.beta_star_existsUnique (512ff915-7ec1-4d24-87b3-bc8adfee6bad).
Series. Every series is a Lean tsum over i : ℕ with the paper's index $i\ge1$ shifted to i + 1. For parameters in $(0,1)$ all series are summable, so tsum is the true sum; summability is part of what solvers prove, never a hypothesis. Differentiability of $a$ is likewise a conclusion (HasDerivAt), never assumed.
Total functions. Real.log 0 = 0 makes the integrands' values at $y=0$ equal their limits, so the interval integrals are the true integrals. Every division ( $1/p$ , $1/(ps)$ ) is at a positive argument under the stated ranges, and every logarithm's argument is positive.
$\alpha^*\approx0.745$ . The paper uses this approximation through the interval $\alpha^*\in[0.74,0.75]$ of its numerical check. Milestones 9 and 10 carry that interval as a hypothesis; the goal does not. The lower bound $\alpha^*>0.745$ is posed as CorreaThreshold.Adaptive.inv_beta_star_gt (ce70487c-b155-4e47-9ce8-8348dc1b1645). The upper bound $\alpha^*\le0.75$ (numerically $\alpha^*\approx0.74544$ ) is posed nowhere and is welcome as a lemma.
A printed slip. The paper prints the conclusion of milestone 9 as " $c(p)\ge\dots$ ". The surrounding argument requires and announces " $\le$ ", and the formal statement uses $\le$ .
Ruled-out trivializations. Fixing $p$ , replacing $\alpha^*$ by the numeral $0.745$ (a weaker statement, since $0.745<\alpha^*$ ), stating $a$ 's monotonicity through deriv (which is $0$ at non-differentiable points), or replacing $\tilde\alpha$ by a chosen value all change the theorem. None is used.
Reusable infrastructure. Welcome contributions include term-by-term differentiation of tsums under a dominating series, Riemann-sum comparisons for monotone integrands, a Chebyshev-type integral inequality for two decreasing functions, and certified interval arithmetic for a two-variable inequality such as milestone 8.

Selected references

J. Correa, A. Cristi, B. Epstein, J. A. Soto, Sample-Driven Optimal Stopping: From the Secretary Problem to the i.i.d. Prophet Inequality, arXiv:2011.06516v2, 2021 (journal version: Mathematics of Operations Research, 2024). https://arxiv.org/abs/2011.06516v2
J. Correa, P. Foncea, R. Hoeksma, T. Oosterwijk, T. Vredeveld, Posted Price Mechanisms for a Random Stream of Customers, ACM EC 2017; Mathematics of Operations Research 46(4), 2021. https://doi.org/10.1145/3033274.3085137
T. P. Hill, R. P. Kertz, Comparisons of stop rule and supremum expectations of i.i.d. random variables, Annals of Probability 10(2), 1982 (reference [26] of the paper). https://doi.org/10.1214/aop/1176993767
R. P. Kertz, Stop rule and supremum expectations of i.i.d. random variables: a complete comparison by conjugate duality, Journal of Multivariate Analysis 19, 1986 (reference [28] of the paper).
J. Correa, P. Dütting, F. Fischer, K. Schewior, Prophet Inequalities for I.I.D. Random Variables from an Unknown Distribution, ACM EC 2019. https://doi.org/10.1145/3328526.3329627
A. Rubinstein, J. Z. Wang, S. M. Weinberg, Optimal Single-Choice Prophet Inequalities from Samples, ITCS 2020. https://doi.org/10.4230/LIPIcs.ITCS.2020.60

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Algorithmic Game Theory Economics Operations Research+1·Captain: mikedeng1

Climate Impact Investing 1: With an Average-Emissions Interaction Effect the Unique Equilibrium Emission Schedule Is ψ* − ε/(2n+ε)·(ψ* + 2n²ψ̄*/(2n+(n+1)ε)), Below the Schedule Without InteractionResearch Paper

Motivation

Green investors hold a growing share of financial wealth, and the stated purpose of much of this capital is to change how companies behave, not only to avoid exposure to climate risk. Whether exclusionary or impact-oriented portfolios actually move corporate emissions is an equilibrium question: investors' preferences shift the cost of capital, and companies choose emissions knowing this. De Angelis, Tankov and Zerbib, Climate Impact Investing (SSRN 3562534; Management Science 2022, doi:10.1287/mnsc.2022.4472), build a continuous-time model in which regular and green investors trade the shares of $n$ companies, green investors penalize climate externalities, and each company chooses an emission schedule to maximize its expected market value.

This mission formalizes the companies' side of that model: the reduction of each company's problem to a date-by-date payoff (Proposition 2), the explicit emission rate it yields when externalities depend on a company's own emissions (Corollary 3), and the equilibrium among companies when externalities also depend on the economy's average emissions (Proposition 4, §3.4 "Interaction effects"). The last result is the one the paper uses to argue that internalizing aggregate emissions pushes every company to emit less.

Setting

There are $n\ge1$ companies, a final date $T>0$ and a rate of time preference $\rho>0$ . A schedule $\psi$ assigns to each date $s\in[0,T]$ the vector $\psi_s\in\mathbb R^n_+$ of emission rates. Company $i$ has a marginal abatement cost $c^i_s\ge0$ . Green investors hold a wealth share $\alpha\in(0,1)$ . The climate externalities $\theta^i_s(\psi_s)$ (as internalized by green investors) and $\theta^{c,i}_s(\psi_s)$ (as internalized by the companies) are functions of the whole emission profile.

Company $i$ 's profit function is its discounted expected equilibrium price,

\mathcal J^i(\psi)=\int_0^T e^{-\rho t}\Big(d^i+\int_0^T c^i_s(\psi^i_s-\psi^i_b)\,ds+\int_0^t\theta^{c,i}_s(\psi_s)\,ds+\alpha\int_t^T\theta^i_s(\psi_s)\,ds-\gamma^*\int_t^T[\Sigma_s\mathbf 1]_i\,ds\Big)dt,

with $d$ the expected terminal dividends, $\psi_b$ the baseline emissions, $\gamma^*$ an aggregate risk-aversion constant and $\Sigma_s$ the dividend covariance. An optimal emission schedule (9) is a Nash equilibrium: no company raises $\mathcal J^i$ by changing its own schedule.

With the time factors

\beta^c_t=\frac{e^{-\rho t}-e^{-\rho T}}{1-e^{-\rho T}},\qquad\beta_t=\frac{1-e^{-\rho t}}{1-e^{-\rho T}},

the date- $t$ payoff (12) of company $i$ is $F_i(x)=\beta^c_t\theta^{c,i}_t(x)+\alpha\beta_t\theta^i_t(x)+c^i_tx^i$ for a profile $x\in\mathbb R^n$ .

Without interaction, $\theta^i_t(x)=\kappa_{0,t}-\frac{\kappa_t}{2}(x^i)^2$ and similarly for $\theta^{c,i}$ with $\kappa^c$ , where $\kappa_t,\kappa^c_t>0$ are sensitivities; the resulting rate is

\psi^{*,i}_t=\frac{c^i_t}{\beta^c_t\kappa^c_t+\alpha\beta_t\kappa_t}.\tag{13}

With an interaction effect of elasticity $\varepsilon\ge0$ , the externalities become

\theta^i_t(\psi)=\kappa_{0,t}-\frac{\kappa_t}{2}\big[(\psi^i_t)^2+\varepsilon\psi^i_t\bar\psi_t\big],\qquad\bar\psi_t=\frac1n\sum_{j=1}^n\psi^j_t,

and similarly for $\theta^{c,i}$ . The stage game at date $t$ has players $i=1,\dots,n$ , strategies $x^i$ and payoffs $F_i$ .

Formalization targets

Goal: Proposition 4(i)

For every $t\in[0,T]$ , the stage game with interaction effect has exactly one Nash equilibrium, namely

\psi^{*,\varepsilon,i}_t=\psi^{*,i}_t-\frac{\varepsilon}{2n+\varepsilon}\Big(\psi^{*,i}_t+\frac{2n^2}{2n+(n+1)\varepsilon}\bar\psi^*_t\Big),\tag{15}

and it lies weakly below the schedule without interaction: $\psi^{*,\varepsilon,i}_t\le\psi^{*,i}_t$ for all $i$ .

Milestones

(39): an integration by parts that rewrites the reduced objective as $\int_0^T$ of a weighted date- $t$ payoff, and the equivalence between maximizing $\mathcal J^i$ and the reduced objective.
Proposition 2: a schedule that maximizes (12) at every date is optimal for $\mathcal J^i$ against every admissible unilateral deviation.
Corollary 3: (13) is the unique maximizer of the date- $t$ payoff with own-emission externalities over $x\ge0$ .
(40): a profile is a stage equilibrium if and only if $x^i=\psi^{*,i}_t-\frac{\varepsilon}{2n}\big(\sum_jx^j+x^i\big)$ for all $i$ .
Sum identity: summing (40) gives $\sum_ix^i=\frac{2n}{2n+(n+1)\varepsilon}\sum_i\psi^{*,i}_t$ .

Companions

Proposition 4(ii): as $n\to\infty$ with $\psi^{*,i}_t,\bar\psi^*_t$ fixed, (15) tends to $\psi^{*,i}_t-\frac{\varepsilon}{\varepsilon+2}\bar\psi^*_t$ (16).
Corollary 3 at schedule level: (13) is optimal for $\mathcal J^i$ .
Proposition 4 at schedule level: a nonnegative, admissible (15) satisfies the Nash condition (9) for the profit functions.

Significance

The result quantifies how a shared concern for aggregate emissions changes individual behaviour. Each company's emission falls by an amount proportional to its own no-interaction rate and to the economy's average, and in a large market the reduction is $\frac{\varepsilon}{\varepsilon+2}$ times the average emission. With $\varepsilon=1$ , equal weight on own and average emissions, a company cuts its emissions by one third of the average (p. 30). The comparison with (13) is the paper's argument that interaction effects reinforce the impact of green investing.

Formalizing it makes the modelling layer precise: what "optimal schedule" means once the stochastic layer is evaluated, under which integrability conditions the integration by parts holds, and on which domain the equilibrium lives. The results are proved on the page with short arguments; none of them is machine-checked anywhere. The time-factor module is shared with the companion mission on climate uncertainty.

Difficulty

The algebra of (15) is short; the care lies elsewhere. First, the average $\bar\psi_t$ contains company $i$ 's own emission, so the derivative of $\theta^i_t$ in $\psi^i_t$ includes the term $\varepsilon\psi^i_t/n$ . Treating $\bar\psi_t$ as fixed, the mean-field reading, gives a different equilibrium, $\psi^{*,i}_t-\frac{\varepsilon}{2}\bar\psi_t$ , which is not (15). Second, the page's proof imposes first-order conditions, which characterize equilibria of the game on $\mathbb R^n$ , while the model's schedules are nonnegative and (15) can be negative (for instance when $c^i_t=0$ and $\varepsilon>0$ ). Third, Proposition 2 is an exchange of the order of integration on $\{0\le s\le t\le T\}$ that must be justified from integrability alone, without continuity of the schedule.

Formalization scope

The paper's model is stochastic: Brownian cash-flow news, subjective measures obtained by Girsanov's theorem, and CARA investors. None of this is formalized, and Propositions 1 and 5 are out of scope as equilibrium statements. The formal statements start from the companies' objective in the evaluated form of p. 64, the expected equilibrium price under the companies' measure.

Conventions committed to in Lean:

Companies are indexed by Fin n with $n>0$ ; profiles are Fin n → ℝ; $\Sigma_s\mathbf 1$ is Matrix.mulVec (Sig s) 1, and only its $i$ -th coordinate is used.
Time factors, the quadratic externality and (13) live in the shared module TimeFactors. $\rho>0$ and $T>0$ are standing hypotheses; $\kappa_t,\kappa^c_t>0$ on $[0,T]$ (Corollary 3's standing assumption, inherited by Proposition 4 through (13)) and $c^i_t\ge0$ (implicit on the page) are disclosed. Positivity of $\kappa_{0,t},\kappa^c_{0,t}$ is not used and is dropped.
A schedule is admissible for company $i$ if it is Borel measurable and nonnegative on $[0,T]$ and $c^i\psi^i$ , $\theta^i(\psi)$ , $\theta^{c,i}(\psi)$ are integrable there; $c^i$ and $[\Sigma\mathbf 1]_i$ are assumed integrable. Measurability is required by footnote 7; Lean encodes it through the schedule's zero extension outside $[0,T]$ . Integrability is not on the page; without it the integrals would silently default to $0$ .
The page writes $d$ in $\mathcal J^i$ for the coordinate $d^i$ .
The goal's stage game is played on $\mathbb R^n$ ; this is a disclosed correction of the page's domain $\mathbb R^n_+$ , which its own first-order proof does not respect. The schedule-level companion restores $\mathbb R_+$ under the proviso that (15) is nonnegative.
"Lower" is the weak inequality: at $\varepsilon=0$ or $c\equiv0$ the two schedules coincide.
Proposition 4(ii)'s " $\simeq$ for $n\gg\varepsilon$ " is pinned down as a limit $n\to\infty$ with the no-interaction quantities held fixed.

A statement that assumes (40) or defines equilibrium as the solution of the first-order system would be trivial; the goal quantifies over all profiles and uses the Nash predicate, and no milestone assumes the conclusion of another.

The development needs interval integrals and Fubini on a triangle (both available in Mathlib's measure theory), and elementary algebra over Fin n sums. The profit-function layer and the stage-game predicates are reusable for other emission games built on the same model. Proofs of any item are welcome, as are proofs of the necessity direction of Proposition 2.

Selected references

T. De Angelis, P. Tankov, O. D. Zerbib, Climate Impact Investing, SSRN 3562534, version of 25 June 2021. https://ssrn.com/abstract=3562534
T. De Angelis, P. Tankov, O. D. Zerbib, Climate Impact Investing, Management Science, 2022. https://doi.org/10.1287/mnsc.2022.4472
W. Nordhaus, Estimates of the Social Cost of Carbon: Concepts and Results from the DICE-2013R Model and Alternative Approaches, Journal of the Association of Environmental and Resource Economists 1(1/2), 2014. https://doi.org/10.1086/676035

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Combinatorics Operations Research Probability·Captain: mikedeng1

Secretary Problems with Biased Evaluations Using Partial Ordinal Information I: ProxyLabel with a Width Estimate ω′ ≥ ω/c Is e^c(ω′+1)/(1−1/k)^{k−1}-CompetitiveResearch Paper

Motivation

In the classical secretary problem, candidates arrive one at a time in uniformly random order, and a decision maker must accept or reject each one irrevocably. The decision maker sees exact scores, or at least a total ranking of the candidates seen so far. In hiring, admissions and procurement the observed scores are often biased: evaluations of candidates from different groups, or produced by different evaluators, cannot be compared reliably, even when comparisons inside a group can. Salem and Gupta (SSRN 3444283; Management Science, 2023, doi:10.1287/mnsc.2023.4926) model this by giving the algorithm only the comparisons it can trust. These form a partial order on the candidates, and the true scores remain hidden.

The question is how much is lost when only partial ordinal information is available. The paper's answer is governed by the width of the partial order, the largest number of mutually incomparable candidates. Width 1 is the classical totally ordered problem. Width $N$ means that no comparison is available at all. This mission formalizes the paper's first positive result with explicit constants: an algorithm, ProxyLabel, whose competitive ratio grows linearly in a known estimate of the width.

Setting

A finite set $C$ of $N$ elements carries a partial order $\preceq$ . Each element $a$ has a hidden true score $w(a) \ge 0$ . The scores are pairwise distinct and consistent with the order: $a \prec b$ implies $w(a) < w(b)$ . The width $\omega$ of $(C, \preceq)$ is the largest size of an antichain, a set of pairwise incomparable elements.

The elements arrive in a uniformly random order. The algorithm knows $N$ and a selection capacity $k \ge 1$ . At each time it sees the comparisons of $\preceq$ among the elements that have arrived. It may select at most $k$ elements, and every decision is irrevocable. The offline optimum $\mathrm{OPT}$ is the largest total score of at most $k$ elements. An algorithm is $\rho$ -competitive if, on every instance,

\mathbb E\Big[\frac{w(\mathrm{Alg})}{\mathrm{OPT}}\Big] \ge \frac1\rho,

where $w(\mathrm{Alg})$ is the total score of the selected elements and the expectation is over the arrival order and the algorithm's randomness.

ProxyLabel (Algorithm 1 of the paper) takes a width estimate $\omega' \in \mathbb N$ . It works as follows.

Each element receives a label $\ell(a) \in \{1, \dots, k\}$ , uniformly and independently.
It draws $m \sim \mathrm{Bin}(N, \omega'/(\omega'+1))$ and lets the sample $S$ be the first $m$ arrivals. Sampled elements are never selected.
A later element $a$ is selected if it is maximal among all previously seen elements with its label, sampled or not, and no element with its label has been selected yet.

Each label class therefore contributes at most one element: the first non-sampled one that beats everything of its label seen before.

Formalization targets

Goal: Proposition 5.2

If $\omega' \ge \omega / c$ for some $c \ge 1$ , then ProxyLabel has competitive ratio at most

\rho = \frac{e^{c}(\omega'+1)}{\big(1 - \frac1k\big)^{k-1}} \ \le\ e^{c+1}(\omega'+1),

with the convention $0^0 = 1$ . That is, $\mathbb E[w(\mathrm{Alg})] \ge \mathrm{OPT}/\rho$ .

Milestones, from the proof (pp. 16–17)

Under uniform labels, the $j$ -th best element is the maximum of its label class with probability $(1 - 1/k)^{j-1} \ge (1 - 1/k)^{k-1} \ge 1/e$ .
Display (3): conditional on that event, it is selected with probability at least $\frac{1}{\omega'+1}\big(1 - \frac{1}{\omega'+1}\big)^{\omega}$ .
If $\omega/c \le \omega'$ with $c \ge 1$ , this quantity is at least $1/(e^{c}(\omega'+1))$ .
Hence each of the top $k$ elements is selected with probability at least $(1 - 1/k)^{k-1}/(e^{c}(\omega'+1))$ .
$\mathbb E[w(\mathrm{Alg})] \ge \sum_{i=1}^k \mathbb P(a_i \in \mathrm{Alg})\, w(a_i)$ over the top $k$ elements $a_1, \dots, a_k$ .

Companion: Corollary 5.1

With the exact width $\omega' = \omega$ , the ratio is at most $e(\omega+1)/(1 - 1/k)^{k-1} \le e^2(\omega+1)$ , and it is $e(\omega+1)$ for $k = 1$ .

Significance

Proposition 5.2 shows that partial ordinal information costs at most a factor linear in the width, provided a width estimate within a constant factor is available. Proposition 5.1 of the same paper shows that no algorithm beats ratio $\omega$ , so the bound is order-optimal. ProxyLabel is also the building block of the paper's width-free algorithm PosetLabel (§5.4), which estimates the width from a first sample and then runs ProxyLabel's selection phase.

The result is proved in the paper and has not been machine-checked. A complete development would give a formal model of online selection on a poset under random arrival, with an explicit selection rule and exact finite averages for probabilities. It would also contain a formal use of Dilworth's theorem inside a probabilistic argument, and the elementary inequalities $(1 - 1/k)^{k-1} \ge 1/e$ and $(1 - 1/(\omega'+1))^{\omega} \ge e^{-c}$ in the form the proof needs.

Difficulty

The events of the algorithm are tangled. The sample is a prefix of a random order with a random length. Whether an element is selected depends on everything that arrived before it, on the labels, and on whether an earlier element of the same label was already selected. The bound must be proved for every finite poset and every consistent scoring, with exact constants, on the finite probability space of arrival orders, labellings and sample sizes. The width enters only through the structure of a single label class, a subposet whose width is at most $\omega$ , so the argument has to pass from the global width to an arbitrary random subposet.

A tempting simplification lets every passing unsampled element be selected. That algorithm can select several elements of one label, so its expected score is larger and the bound becomes easier. It is not ProxyLabel.

Formalization scope

Elements are a finite type α with a PartialOrder, so $N$ is Fintype.card α. The width $\omega$ is the published definition AppliedComb.Posets.width α, the maximum size of an antichain. Dilworth's theorem is the published, proved theorem AppliedComb.Posets.dilworth.
A consistent scoring is $w : \alpha \to \mathbb R$ with $w \ge 0$ , $w$ injective, and $a < b \Rightarrow w(a) < w(b)$ . These three conditions are the paper's standing assumptions (§3.1, §3.3), not additions.
Arrival orders are bijections α ≃ Fin N, and labellings are functions α → Fin k. Labels are independent of the order, which is the same distribution as assigning labels on arrival.
Probabilities and expectations are explicit finite averages with weights $1/N!$ , $1/k^N$ and the binomial weights $\binom{N}{m}p^m(1-p)^{N-m}$ with $p = \omega'/(\omega'+1)$ .
The selection rule is encoded exactly. An element is selected iff it is unsampled, no earlier element of its label (sampled or not) is above it, and no earlier unsampled element of its label passed the same test.
OPT is the maximum of $\sum_{a \in s} w(a)$ over sets with $|s| \le k$ , so it is the total score when $k \ge N$ .
The competitive ratio is stated as $\mathrm{OPT}/\rho \le \mathbb E[w(\mathrm{Alg})]$ . This agrees with Definition 3.4 when $\mathrm{OPT} > 0$ ; the page's ratio is undefined when $\mathrm{OPT} = 0$ .
$(1 - 1/k)^{k-1}$ has a natural exponent, so $k = 1$ gives $0^0 = 1$ . The width estimate $\omega'$ is a natural number.
The proposition's "interval order" is a slip of the page: the algorithm and the proof concern arbitrary posets, and the goal is stated for every finite poset.
The order-optimality clause of Corollary 5.1 refers to Proposition 5.1, the subject of mission II of this series, and is not stated here.
The goal is a statement about the full expectation of the selected score over arrival order, labels and sample size. It is not the conditional bound (3). A formalization in which several elements per label can be selected, or in which "previously seen" excludes the sample, proves a statement about a different algorithm.

Contributions are welcome on every milestone. The prefix-of-a-random-order lemma (a binomially sized prefix is an i.i.d. Bernoulli sample) is reusable well beyond this mission.

Selected references

J. Salem and S. Gupta, Secretary Problems with Biased Evaluations Using Partial Ordinal Information, SSRN 3444283 (version of 2022-04-17); Management Science, 2023. https://ssrn.com/abstract=3444283, https://doi.org/10.1287/mnsc.2023.4926
R. P. Dilworth, A Decomposition Theorem for Partially Ordered Sets, Annals of Mathematics 51(1), 1950, 161–166. https://doi.org/10.2307/1969503
R. Kleinberg, A Multiple-Choice Secretary Algorithm with Applications to Online Auctions, SODA 2005. https://dl.acm.org/doi/10.5555/1070432.1070519

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Number Theory Operations Research Optimization·Captain: mikedeng1

Integer Factorization: Why Two-Item Joint Replenishment Is Hard 1: For Odd Composite M, Every Optimal Plan of an Explicit Two-Item Aperiodic JRP Instance Has q₂ = M and 1 < gcd(q₁, M) < MResearch Paper

Motivation

The joint replenishment problem (JRP) is one of the basic models of multi-item inventory control. Each item follows the economic order quantity (EOQ) model of Harris (1913): ordering costs a fixed amount, holding stock costs a rate proportional to the stock, and the classical trade-off between the two determines how often to reorder. In a JRP, items replenished together share part of the fixed cost, so the reorder cycles of the items should be coordinated. The problem has been studied for decades, with a large literature of heuristics and approximation algorithms (see the survey of Khouja and Goyal 2008; Porras and Dekker 2008 compare the two ways of charging joint costs), yet its exact complexity remained unclear even for very few items.

Schulz and Telha (arXiv:2007.09045, 2020; Operations Research, 2024) give a striking answer for the smallest non-trivial case, two items. They show that solving the two-item problem exactly is at least as hard as integer factorization (their Theorem 1), and, assuming the Riemann hypothesis, that the two-item periodic JRP is NP-hard under randomized reductions (Theorem 2). This mission formalizes the mathematical core of Theorem 1(a), for the aperiodic JRP: the paper's Lemma 1.

Timeline. The complexity of both JRP variants was open for a long time. Schulz and Telha announced the reduction from integer factorization to the two-item aperiodic JRP (Theorem 1(a)) in 2011. Cohen-Hillel and Yedidsion (2018) then proved that the aperiodic JRP with an arbitrary number of items is strongly NP-hard (they call it the periodic JRP). The 2020 preprint adds the reduction for the two-item periodic JRP (Theorem 1(b)) and, under the Riemann hypothesis, a randomized reduction from Partition to it (Theorem 2).

Setting

For an item with fixed ordering cost $K$ and holding cost rate $H$ , replenished every $q$ time units, the long-run cost per unit time is

\mathrm{EOQ}_{K,H}(q) = \frac{K}{q} + Hq .

Reorder cycles are positive integers $q \in \mathbb Z^+$ . In the aperiodic two-item JRP (objective (1) of the paper) items 1 and 2 have data $(K_1,H_1)$ and $(K_2,H_2)$ , and every joint replenishment earns a discount $K_0 \le K_1, K_2$ . With cycles $q_1, q_2$ , joint replenishments happen every $\mathrm{lcm}(q_1,q_2)$ time units, so the discount per unit time is $\mathrm{DC}(q_1,q_2;K_0) = K_0/\mathrm{lcm}(q_1,q_2)$ and the problem is

\min\left\{\mathrm{JRP}(q_1,q_2) = \mathrm{EOQ}_{K_1,H_1}(q_1) + \mathrm{EOQ}_{K_2,H_2}(q_2) - \frac{K_0}{\mathrm{lcm}(q_1,q_2)} \ :\ q_1,q_2 \in \mathbb Z^+\right\}.

For an integer $M$ , the instance of Lemma 1 has

K_1 = K_0 = M(M-1),\qquad H_1 = 4,\qquad K_2 = M^3(M^2-1),\qquad H_2 = M(M^2-1).

Item 2's own EOQ optimum is $q_2 = M$ . Fixing $q_2 = M$ leaves the one-variable problem (2) with objective

\mathrm{JRP}^{\mathrm{red}}(q_1) = M(M-1)\left(\frac{1}{q_1} - \frac{1}{\mathrm{lcm}(q_1,M)}\right) + 4q_1 .

In Lean these are eoq, aperiodicCost, lemma1Cost, eoq2 (item 2's cost) and jrpRed, and an optimal solution (IsOptimal) is a pair of positive integers whose value is no larger than that of any other pair of positive integers.

Formalization targets

Goal: Lemma 1

For every odd composite number $M$ , the instance above satisfies $K_0 \le K_1$ , $K_0 \le K_2$ , has all coefficients at most $M^5$ , has an optimal solution, and

\text{every optimal } (q_1,q_2) \text{ satisfies } \quad q_2 = M \quad\text{and}\quad 1 < \gcd(q_1,M) < M .

So $\gcd(q_1, M)$ is a non-trivial divisor of $M$ .

Milestones (the steps of the paper's proof, pp. 9–10)

$\mathrm{EOQ}_{K_2,H_2}(M\pm1) = \mathrm{EOQ}_{K_2,H_2}(M) + H_2/(M\pm1) \ge \mathrm{EOQ}_{K_2,H_2}(M) + K_0$ .
For $q_2 \ne M$ : $\mathrm{JRP}(q_1,q_2) \ge \mathrm{EOQ}_{K_2,H_2}(M) + K_0 + 4 > \mathrm{JRP}(1,M)$ .
$\mathrm{JRP}(q_1, M) = \mathrm{EOQ}_{K_2,H_2}(M) + \mathrm{JRP}^{\mathrm{red}}(q_1)$ .
For $q_1$ coprime with $M$ : $\mathrm{JRP}^{\mathrm{red}}(q_1) \ge 4(M-1)$ .
For a divisor $p$ of $M$ with $3 \le p < M$ and a multiple $q$ of $p$ : $\mathrm{JRP}^{\mathrm{red}}(q) \le (M-1)(M-p)/q + 4q$ .
Some multiple $q$ of $p$ in $\left[\frac{M-1}{2} - \frac{p-1}{2}, \frac{M-1}{2} + \frac{p-1}{2}\right]$ has $\mathrm{JRP}^{\mathrm{red}}(q) < 4(M-1)$ .
Multiples $q_1$ of $M$ have $\mathrm{JRP}^{\mathrm{red}}(q_1) = 4q_1 \ge 4M$ , while $\mathrm{JRP}^{\mathrm{red}}((M-1)/2) \le 4(M-1)$ .

Significance

The result. Lemma 1 turns any exact algorithm for the two-item aperiodic JRP into a factoring algorithm: solve the instance, read off $\gcd(q_1, M)$ , and recurse on the two factors. Since the instance's coefficients have size $O(\log M)$ , a polynomial-time JRP algorithm would factor integers in polynomial time (Theorem 1(a)). This explains, in a precise sense, why the many exact and approximation methods for the JRP never produced a provably polynomial exact algorithm even for two items, and why the problem's difficulty is number-theoretic rather than combinatorial.

Formalizing it. The lemma has a complete, elementary proof in the paper; it has not been machine-checked. The formalization pins down every quantifier the prose leaves loose: that optimal solutions exist, that the claim is about every optimum, which coefficients are used, and that periods range over positive integers only. The proof steps (the item-2 pin, the coprime bound, the interval argument) are stated as separate milestones so they can be checked individually.

Difficulty

Each step is elementary, but the argument mixes discrete and continuous reasoning. The objective is not convex in $q_1$ : the term $-K_0/\mathrm{lcm}(q_1,M)$ jumps down at every period sharing a factor with $M$ . The obvious approach, comparing against the continuous EOQ minimizer, gives the right threshold $4(M-1)$ for coprime periods but says nothing about where a cheaper non-coprime period sits. The interval step needs the exact size of a window of integers around $(M-1)/2$ , that $p$ consecutive integers contain a multiple of $p$ , and a strict inequality between a quadratic's roots and that window. The existence of an optimum is not stated in the paper and needs a coercivity argument over $\mathbb Z^+ \times \mathbb Z^+$ .

Formalization scope

This mission formalizes the correctness of the reduction behind Theorem 1(a); polynomial running time is not formalized, and neither are Theorem 1 as a statement about algorithms, the recursion to a full factorization, nor primality testing.

Conventions: periods are natural numbers, and every quantification over a period requires it to be positive, because Lean evaluates $K/0 = 0$ , $\mathrm{lcm}(0,M) = 0$ and $\gcd(0,M) = M$ . Costs are real numbers. Coefficients are natural numbers cast to $\mathbb R$ . "Odd composite" is Odd M ∧ 1 < M ∧ ¬ M.Prime. "Size at most $5\log M$ " is stated as "every coefficient is at most $M^5$ ", equivalent in any logarithm base.

Two additions relative to the printed statement are disclosed in the goal's Formalization Note. The goal states the explicit instance of the proof (p. 9) rather than "there is an instance", since an existential instance could be chosen to make the claim easy and loses the polynomial-size construction the reduction needs. The goal also asserts that an optimal solution exists, so that "every optimal solution satisfies …" is not vacuously true. Neither periods equal to zero nor junk values of division can satisfy the optimality predicate.

All objects are defined in one file, JRPHard.Aperiodic.Setting; nothing beyond Mathlib's Nat.gcd, Nat.lcm, Nat.Prime and real arithmetic is needed. Proofs of any milestone, or alternative proofs of the goal, are welcome.

Selected references

A. S. Schulz and C. Telha, Integer factorization and Riemann's hypothesis: Why two-item joint replenishment is hard, arXiv:2007.09045v1, 2020; published as Integer Factorization: Why Two-Item Joint Replenishment Is Hard, Operations Research, 2024. https://arxiv.org/abs/2007.09045
F. W. Harris, How many parts to make at once, Factory, The Magazine of Management 10(2), 1913; reprinted in Operations Research 38(6), 1990. https://doi.org/10.1287/opre.38.6.947
E. Porras and R. Dekker, A solution method for the joint replenishment problem with correction factor, International Journal of Production Economics 113(2), 834–851, 2008. https://doi.org/10.1016/j.ijpe.2007.11.008
M. Khouja and S. Goyal, A review of the joint replenishment problem literature: 1989–2005, European Journal of Operational Research 186(1), 2008. https://doi.org/10.1016/j.ejor.2007.03.007
M. Cohen-Hillel and L. Yedidsion, The periodic joint replenishment problem is strongly NP-hard, Mathematics of Operations Research 43(4), 2018. https://doi.org/10.1287/moor.2017.0904

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Economics Operations Research Optimization+1·Captain: mikedeng1

Accelerability vs. Scalability: R&D Investment Under Financial Constraints and Competition I: Comparative Statics of an Unconstrained Monopoly's Optimal R&D IntensityResearch Paper

Motivation

Firms deciding how much to spend on research and development face two different kinds of return to spending. More money can make the eventual innovation more valuable, or it can make the innovation arrive sooner. Lin (Management Science, 2022, doi:10.1287/mnsc.2022.4503) calls these two properties scalability and accelerability, and shows that they predict how R&D intensity responds to financing frictions and to competition. The benchmark of the whole paper is a single financially unconstrained firm with no rival. Its comparative statics, Proposition 1 of the paper, are what the constrained and competitive cases are compared against, and they already contain one of the paper's headline observations: a higher interest rate raises the optimal R&D intensity exactly when the project is accelerable.

Setting

A technology is a tuple $T=(A,\beta,\eta,\gamma)$ . At investment intensity $R$ (spending per unit of time), the expected payoff of the innovation and the arrival rate of discovery are

u(R)=A\,R^{\beta},\qquad \lambda(R)=\eta\,R^{\gamma},

with the standing assumptions $A>0$ , $\eta>0$ , $\beta,\gamma\in[0,1)$ and $\beta+\gamma<1$ (decreasing returns to scale). The project is scalable if $u'>0$ , here $\beta>0$ , and accelerable if $\lambda'>0$ , here $\gamma>0$ . Intensities are bounded below by $\underline R=1$ .

The firm discounts at rate $r>0$ . The discovery time $\tau_d$ is exponentially distributed with rate $\lambda(R)$ ; the firm pays $R$ per unit of time until $\tau_d$ and then receives $u(R)$ . Its R&D value at intensity $R$ is

\Pi(R)=\mathbb E\Big[e^{-r\tau_d}\,u(R)-R\int_0^{\tau_d}e^{-rt}\,dt\Big].

Lemma 1 of the paper identifies this with the perpetuity value

V(R)=\frac{u(R)\lambda(R)-R}{\lambda(R)+r}. \tag{5}

The value is positive only below $\bar R=(A\eta)^{1/(1-(\beta+\gamma))}$ , so the firm chooses $R$ from the range $[\underline R,\bar R]=[1,\bar R]$ of (7). The optimal intensity set is

R^*_{uc}=\{R\in[1,\bar R]:\ \Pi(R')\le\Pi(R)\ \text{for all } R'\in[1,\bar R]\}.

Its comparative statics are studied through $\ln V(R)=\ln(u\lambda-R)-\ln(\lambda+r)$ .

Formalization targets

Goal: Proposition 1 (p. 5)

For a set $S'$ to lie above a set $S$ means: every $x\in S$ and $y\in S'$ satisfy $x\le y$ , and $x<y$ whenever $x>\underline R$ . The goal states that $R^*_{uc}$ is nonempty and

if $\beta=\gamma=0$ then $R^*_{uc}=\{\underline R\}$ ;
if $\beta>0$ or $\gamma>0$ , then $R^*_{uc}$ at $A'>A$ lies above $R^*_{uc}$ at $A$ ;
if $\beta>0$ , then $R^*_{uc}$ at $\beta'>\beta$ (with $\beta'+\gamma<1$ ) lies above $R^*_{uc}$ at $\beta$ ;
if $\gamma=0<\beta$ , then $R^*_{uc}$ at $\eta'>\eta$ lies above $R^*_{uc}$ at $\eta$ ;
if $\gamma=0$ , $R^*_{uc}$ does not depend on $r$ ; if $\gamma>0$ , $R^*_{uc}$ at $r'>r$ lies above $R^*_{uc}$ at $r$ .

Milestones (p. 5 and Appendix A.2, p. 18)

Lemma 1 (5): $\Pi(R)=V(R)$ for every $R>0$ .
Without accelerability and with $A\beta\eta\ge1$ : $R^*_{uc}=\{(1/A\beta\eta)^{1/(\beta-1)}\}$ , a formula free of $r$ .
The cross partials of $\ln V$ on $1\le R$ , $u\lambda>R$ :

\frac{\partial^2\ln V}{\partial R\,\partial A}=\frac{\lambda}{(\lambda u-R)^2}\frac{\partial u}{\partial A}(1-\beta-\gamma)>0,\qquad \frac{\partial^2\ln V}{\partial R\,\partial r}=\frac{\lambda'}{(\lambda+r)^2},

together with (A.3) in $\beta$ and (A.4) in $\eta$ , with the signs the paper states.

Significance

Proposition 1 is the reference point for the rest of the paper: Proposition 2 (financially constrained monopoly), Corollary 1 (constrained firms under-invest without accelerability) and Propositions 3–4 (R&D races) are all read against it. Two of its clauses are not what a reader would guess. The speed factor $\eta$ has an unambiguous effect only when the project is not accelerable, because for accelerable projects a higher $\eta$ also raises the marginal cost of expanding $R$ through the heavier effective discount rate $\lambda+r$ (p. 5). And the interest rate raises intensity for accelerable projects: in the paper's words (p. 5), "a higher $r$ reduces the marginal cost of expanding $R_{uc}$ through the discount rate."

The paper proves Proposition 1 in Appendix A.2 with an informal appeal to Topkis's monotonicity theorem; the result has no machine-checked proof. Formalizing it makes three things explicit that the printed proof leaves implicit: the expectation-to-perpetuity identity (5) for an exponential discovery time, the domain on which $\ln V$ is defined and has increasing differences, and the fact that the range $[1,\bar R]$ itself moves with $A$ , $\beta$ and $\eta$ . The derivative claims of the paper are also made precise for optima that need not be unique or interior.

Difficulty

The statement is about an optimal set, and the parameter moves both the objective and the feasible range. The first obstacle is that the paper's derivative form $\partial R^*_{uc}/\partial s>0$ is false as printed when the optimum sits at the corner $\underline R$ : with $\gamma=0$ , $\beta=0.1$ , $\eta=1$ , $r=1$ and $A\in\{2,2.2\}$ the optimum is $1$ in both cases. Any proof therefore has to separate corner optima from interior ones. The second is that $\ln V$ is only defined where $u\lambda>R$ ; the increasing-differences inequalities hold there, but the comparison of two optima at two parameter values can pass through points where the value at one of the parameters is not positive. The third is that $V$ is not concave in general, so first-order conditions alone do not identify the optimal set.

Formalization scope

All declarations live in the namespace AccelScal.UCMonopoly. Lean conventions:

A Tech structure with fields A β η γ : ℝ; Tech.Standing bundles (2) and the added condition $A\eta>1$ , which the paper expresses as "conditional on investment taking place" (p. 5) and which makes the value at $\underline R$ positive and the range nondegenerate. The discount rate is a separate hypothesis $r>0$ .
$u$ , $\lambda$ and $\bar R$ use real powers (Real.rpow); every statement keeps $R\ge1$ (or $R>0$ for Lemma 1).
The R&D value ucPayoff is the expectation (4) under ProbabilityTheory.expMeasure (λ R), with the random payoff $\tilde u$ replaced by its mean $u(R)$ and the cash-flow term of (4), which does not depend on $R$ , dropped.
RstarUC T r is the set of maximizers of ucPayoff T r over Set.Icc 1 (Rbar T), the range of the same parameters.
" $\partial R^*_{uc}/\partial s>0$ " is OrderedAbove: weak order between all optima, strict when the smaller optimum exceeds $1$ . " $\partial R^*_{uc}/\partial r=0$ " is equality of the sets.
$u'=0\Leftrightarrow\beta=0$ and $\lambda'=0\Leftrightarrow\gamma=0$ on $R\ge1$ , so the hypotheses are stated on $\beta$ , $\gamma$ .
The cross partials $\partial^2\ln V/\partial R\,\partial s$ are the $s$ -derivative of the $R$ -derivative, on $1\le R$ , $R<u(R)\lambda(R)$ .

Trivializing formalizations are ruled out: the value is defined as an expectation, not by the closed form (5); the optimal set is a set of maximizers, never a selected optimum or a closed-form formula; nonemptiness is part of the goal, so no clause is vacuous; the range moves with the parameter; and every ordering keeps its strict part at interior optima.

Not posed: Lemma 1 (6) (the financially constrained value, which needs the first passage of a geometric Brownian motion), (A.5) (Proposition 1 makes no claim about $\gamma$ ), and a sign of (A.4) for $\gamma>0$ (the paper says it cannot be signed). A complete development needs the Laplace transform of the exponential law, derivatives of real powers and logarithms, and a monotone comparative statics argument on intervals; a reusable lemma of the form "strictly increasing differences on a moving interval domain orders the argmax sets" would serve other missions as well. Contributions of any of the milestones, or of such general lemmas, are welcome.

Selected references

D. Lin, Accelerability vs. Scalability: R&D Investment Under Financial Constraints and Competition, Management Science, Articles in Advance, 2022. https://doi.org/10.1287/mnsc.2022.4503
D. M. Topkis, Minimizing a submodular function on a lattice, Operations Research 26(2), 1978. https://doi.org/10.1287/opre.26.2.305
X. Vives, Oligopoly Pricing: Old Ideas and New Tools, MIT Press, 1999. https://mitpress.mit.edu/9780262720403/
S. Athey, Monotone comparative statics under uncertainty, Quarterly Journal of Economics 117(1), 2002. https://doi.org/10.1162/003355302753399481

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Operations Research Probability·Captain: mikedeng1

Lightning Network Economics: Channels I: The Expected Discounted Cost of a Bidirectional Payment Channel in Closed FormResearch Paper

Payment channels and the cost of locked capital

Blockchains such as Bitcoin settle every transaction on a public ledger. Each on-chain transaction pays a fee and waits for confirmation, so frequent small payments are expensive. The Lightning Network moves such payments off-chain: two parties lock funds in a shared payment channel and pay each other by updating the channel's balance, touching the blockchain only to open, close or re-fund it. A channel is not free either. The locked funds cannot be used elsewhere, and once one side's balance is exhausted the channel has to be reset on-chain.

Guasoni, Huberman and Shikhelman (Lightning Network Economics: Channels, Management Science, 2023) quantify this trade-off in a stochastic model and compare three ways of settling two opposite streams of payments: on-chain transactions, unidirectional channels, and one bidirectional channel. Their comparisons all rest on a single computation, the expected discounted cost of a channel in closed form (Theorem 1). This mission formalizes that computation.

The model

Two nodes exchange unit payments. Node 1 pays node 2 one unit at each jump of a Poisson process $N^{\lambda_1}_t$ with rate $\lambda_1$ ; node 2 pays node 1 one unit at each jump of an independent Poisson process $N^{\lambda_2}_t$ with rate $\lambda_2$ . The paper normalizes $\lambda_1\le\lambda_2$ . The net balance of node 1 is

X_t=N^{\lambda_2}_t-N^{\lambda_1}_t,\qquad X_0=0 .

In a bidirectional channel node 1 commits $l_1$ units and node 2 commits $l_2$ units. The money is the channel's capacity: the channel works as long as $-l_1<X_t<l_2$ . Locked capital has an opportunity cost of $r(l_1+l_2)$ per unit of time, where $r>0$ is the continuously compounded interest rate, and all costs are discounted at rate $r$ . When the balance reaches $-l_1$ (node 1 is exhausted) or $+l_2$ (node 2 is exhausted), the channel is reset at cost $B$ and the balance restarts at $0$ with the same commitments.

Let $J(n)$ be the expected total future cost when the balance equals $n$ . Conditioning on the first payment, which arrives after an exponential time with rate $\lambda_1+\lambda_2$ , gives the first-step equation (15)

J(n)=(l_1+l_2)\frac{r}{r+\lambda_2+\lambda_1}+J(n+1)\frac{\lambda_2}{r+\lambda_2+\lambda_1}+J(n-1)\frac{\lambda_1}{r+\lambda_2+\lambda_1},\qquad -l_1<n<l_2,

with the reset conditions $J(-l_1)=J(0)+B$ and $J(+l_2)=J(0)+B$ . The cost of the channel is $c^{l_1,l_2}(\lambda_1,\lambda_2)=J(0)$ .

In a unidirectional channel only node 2 pays (rate $\lambda_2$ ) and only node 2 commits capital ( $l_2$ units). The cost satisfies $J(n)=l_2\frac{r}{r+\lambda_2}+J(n+1)\frac{\lambda_2}{r+\lambda_2}$ for $0\le n<l_2$ with $J(l_2)=J(0)+B$ , and $c^{0,l_2}(0,\lambda_2)=J(0)$ .

The roots of the characteristic equation $\lambda_2\alpha^2-(\lambda_1+\lambda_2+r)\alpha+\lambda_1=0$ are

\alpha_\pm=\frac{\lambda_1+\lambda_2+r\pm\sqrt{(\lambda_2-\lambda_1)^2+r^2+2r(\lambda_2+\lambda_1)}}{2\lambda_2}. \tag{4}

Formalization targets

Goal: Theorem 1 (ii)

For $0<\lambda_1\le\lambda_2$ , $r>0$ , any $B$ and natural commitments $l_1,l_2\ge1$ , the bidirectional cost system has a solution, and every solution satisfies

c^{l_1,l_2}(\lambda_1,\lambda_2)=l_1+l_2-B\,\frac{\alpha_-^{l_1}(1-\alpha_+^{l_1+l_2})-\alpha_+^{l_1}(1-\alpha_-^{l_1+l_2})}{\alpha_-^{l_1}(1-\alpha_+^{l_1+l_2})-\alpha_+^{l_1}(1-\alpha_-^{l_1+l_2})+\alpha_+^{l_1+l_2}-\alpha_-^{l_1+l_2}}. \tag{3}

Milestones

Competing exponential clocks (proof of Theorem 1, p. 10): the minimum $\tau$ of independent exponential times with rates $\lambda_2,\lambda_1$ is exponential with rate $\lambda_2+\lambda_1$ , and $\tau^{\lambda_2}$ rings first with probability $\lambda_2/(\lambda_2+\lambda_1)$ . This is a published, proved platform theorem, reused as is.
The first term (p. 10): $E[e^{-r\tau}]=(\lambda_2+\lambda_1)/(\lambda_2+\lambda_1+r)$ and $E[\int_0^\tau e^{-rs}r(l_1+l_2)\,ds]=(l_1+l_2)\,r/(r+\lambda_2+\lambda_1)$ .
The last term (p. 10): $E[e^{-r\tau}J(X_{t+\tau})\mid X_t=n]=J(n+1)\frac{\lambda_2}{r+\lambda_2+\lambda_1}+J(n-1)\frac{\lambda_1}{r+\lambda_2+\lambda_1}$ .
General solution of (15) (§5.1): on the lattice $-l_1\le n\le l_2$ , (15) holds at the interior points if and only if $J(n)=l_1+l_2+k_1\alpha_-^n+k_2\alpha_+^n$ for constants $k_1,k_2$ .
The cost function (§5.1): the closed form of $J(n)$ at every balance $n$ , which gives (3) at $n=0$ .
Theorem 1 (i), (1): $c^{0,l_2}(0,\lambda_2)=l_2+B\big(((r+\lambda_2)/\lambda_2)^{l_2}-1\big)^{-1}$ .
Theorem 1 (i), (2): with $k=B\log(1+r/\lambda_2)$ and $B>0$ , the minimum of (1) over real sizes is $\frac{B}{2k}\big(-k+\sqrt{k(k+4)}+2\log(\tfrac12(k+\sqrt{k(k+4)}+2))\big)$ , attained at $l_2=\frac Bk\log(\tfrac12(k+\sqrt{k(k+4)}+2))$ .

What the closed form gives

Formula (3) is what the rest of the paper works with. Lemma 1 prices on-chain settlement of a stream of rate $\lambda$ at $C\lambda/r$ . Comparing that price with (1) and (3) yields the paper's thresholds on the on-chain fee below which no channel pays off (Theorem 2), the small-interest-rate asymptotics of the optimal symmetric channel, whose minimal cost scales like $3(2B\lambda/r)^{1/3}$ (Theorem 3), and the long-run number of channel payments per on-chain transaction (Proposition 2). Without an exact expression for the cost these comparisons would rest on simulation.

The paper proves Theorem 1 by a first-step argument and the solution of a linear difference equation. No part of it has been machine-checked. A formal proof would supply what the published argument passes over: that the first-step equation (15) and the reset conditions together have a unique solution on the lattice, that the denominator of (3) does not vanish, and the measure-theoretic computation of the coefficients of (15) from the exponential clocks. The later missions of this series (rate bounds, small-rate asymptotics, the balance walk) build on the same model.

Where the difficulty lies

The algebra of solving a second-order linear recurrence is routine. Two points are not. First, the solution has to be shown to exist and to be unique for every admissible set of parameters. This requires $\alpha_-\ne\alpha_+$ and a non-singular $2\times 2$ system for $k_1,k_2$ , whose determinant is the denominator of (3), and the paper does not discuss when that determinant vanishes. Second, the coefficients of (15) come from expectations over exponential times: the joint expectation of the discount factor $e^{-r\tau}$ and the indicator of which clock rang first. The paper factors it as $E[e^{-r\tau}]\,P(\tau=\tau^{\lambda_2})$ , which uses that the minimum is independent of the identity of the first clock, a fact it does not state.

Formalization scope

The rates are written lam₁, lam₂; the sizes $l_1,l_2$ are natural numbers with $l_1,l_2\ge1$ , because payments are of one unit and the balance lives on the lattice $\{-l_1,\dots,l_2\}$ . The cost function is a map $J:\mathbb Z\to\mathbb R$ whose values off the lattice are unconstrained.
The cost is defined by its equations, (15) with the reset conditions (IsBiCostFn) and the §5.2 system (IsUniCostFn), never by (3) or (1). Every statement about "the cost" asserts both that a solution exists and that every solution takes the stated value, so it is neither vacuous nor dependent on a choice of solution.
The unidirectional system is a separate definition: the bidirectional one at $l_1=0$ would force $B=0$ . For the same reason the Remark after Theorem 1 ("(1) follows from (3) by substituting $\lambda_1=l_1=0$ ") is not posed: with $\alpha_-=0$ and $l_1=0$ the denominator of (3) vanishes.
Powers are natural powers in (1) and (3), integer powers (zpow) in the general solution and in $J(n)$ , and real powers (Real.rpow) in the minimization (2), which is over real sizes $m>0$ as in the paper's proof.
The paper prints the exponent in the numerator of the $J(n)$ display as $l+n$ ; the statement uses $l_1+n$ , which is what the derivation gives and what reduces to (3) at $n=0$ .
The probabilistic milestones use Mathlib's expMeasure for the laws (through push-forward measures of measurable random variables) and IndepFun. Integrability is part of what is to be proved.
The hypothesis $\lambda_1\le\lambda_2$ is kept in the goal as printed. No sign of $B$ is assumed except $B>0$ in (2), without which the infimum is not attained.
Ruled out as trivializing: defining the cost as its closed form (which turns the goal into a definitional equality), fixing numerical parameters, restricting to $\lambda_1=\lambda_2$ , and treating the sizes as real numbers in the cost system.

Contributions welcome: proofs of the milestones in any order, and reusable lemmas on linear difference equations with constant coefficients and on Laplace transforms of exponential laws, neither of which Mathlib currently has in this form.

Selected references

P. Guasoni, G. Huberman, C. Shikhelman, Lightning Network Economics: Channels, Management Science, Articles in Advance, 2023. https://doi.org/10.1287/mnsc.2022.01664
J. Poon, T. Dryja, The Bitcoin Lightning Network: Scalable Off-Chain Instant Payments, technical report, 2015 (as cited by Guasoni et al.). https://lightning.network/lightning-network-paper.pdf

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Algorithmic Game Theory Economics Operations Research+1·Captain: mikedeng1

Proof-of-Work Cryptocurrencies: Does Mining Technology Undermine Decentralization? 1: The Mining Game Has a Unique Equilibrium Whose Number of Active Miners Is Fixed by an Explicit Cost ConditionResearch Paper

Motivation

Proof-of-work blockchains such as Bitcoin keep their ledger secure by a contest: miners spend computing power ("hash rate") to solve a cryptographic puzzle, and the miner who solves it first earns the block reward. The probability of winning is proportional to a miner's share of the total hash rate, so mining is a rent-seeking contest in the sense of Tullock (1967), with heterogeneous costs. Whether such a contest stays decentralized, that is, whether many miners remain active or a few efficient ones take over, is a question about the structure of its equilibrium.

Capponi, Ólafsson and Alsabah (arXiv:2106.09783v1, 2021) model a two-stage game: miners first invest in hardware, then compete in hash rate. This mission formalizes the second stage, the mining game, for a fixed cost vector. Its results (Propositions 4.1–4.4) are the foundation of the paper's comparative statics (Propositions 4.6, 4.8) and of its investment stage (Proposition 5.1), which are posed in the later missions of this series.

Setting

There are $N\ge 2$ miners, indexed $i=1,\dots,N$ . Miner $i$ has a cost-per-hash $c_i>0$ , and the miners are sorted so that $c_1\le c_2\le\dots\le c_N$ . All miners compete for a total reward $R>0$ and share a capacity parameter $\gamma\ge 0$ : a larger $\gamma$ means a tighter hashing capacity.

Each miner chooses a hash rate $h_i\ge 0$ . The aggregate hash rate is $H=\sum_{j=1}^N h_j$ , and miner $i$ 's objective is

\pi_i(h)=\begin{cases}\dfrac{h_i}{H}R-c_ih_i-\dfrac{\gamma}{2}h_i^2, & H>0,\\[4pt] 0, & H=0.\end{cases}

An equilibrium hash rate profile $h^*$ is a nonnegative profile in which no miner can increase its objective by changing its own hash rate to any other nonnegative value. A miner is active if $h^*_i>0$ ; $n$ denotes the number of active miners. The cumulative cost of the first $k$ miners is $c^{(k)}=c_1+\dots+c_k$ , and the marginal cost of miner $i$ at the equilibrium is $MC^*_i=c_i+\gamma h^*_i$ .

In Lean these objects are Params N (fields c, R, γ), Params.Standing, aggH, payoff, IsMiningEq, activeCount, cumCost, aggForm, partCond, margCost and profit, in the namespace PoWMining.Equilibrium.

Formalization targets

Goal: the number of active miners (Proposition 4.4)

An equilibrium exists, and in every equilibrium the number of active miners is the largest $2\le n\le N$ with

c_n<\frac{c^{(n)}+R\gamma/c_n}{n-1}.

This number is weakly increasing in $R$ and in $\gamma$ (with the costs held fixed). The goal identifies the degree of decentralization of the mining stage purely from the model parameters.

Milestones, in the order of the paper's proof

Every equilibrium has at least two active miners (p. 38).
If miner $k+1$ is active in an equilibrium, so is miner $k$ : the active miners are the $n$ cheapest (p. 38).
The first-order condition of an active miner (4.2):

\frac{R}{H^*}\Bigl(1-\frac{h^*_i}{H^*}\Bigr)=c_i+\gamma h^*_i.

Summing (4.2) over the active miners (B.1): $(n-1)R-c^{(n)}H^*-\gamma (H^*)^2=0$ .
Proposition 4.1: the equilibrium exists and is unique, $h^*_i=H^*(R-c_iH^*)/(R+\gamma (H^*)^2)$ for $i\le n$ and $0$ otherwise, and $H^*$ is the positive root

H^*=\frac{\sqrt{(c^{(n)})^2+4(n-1)R\gamma}-c^{(n)}}{2\gamma}\ (\gamma>0),\qquad H^*=\frac{(n-1)R}{c^{(n)}}\ (\gamma=0).\tag{4.1}

Proposition 4.2: $MC^*_i$ increases in $i$ , $MC^*_i<R/H^*\iff c_i<R/H^*\iff i\le n$ , and $c_i/MC^*_i$ increases while $\gamma h^*_i/MC^*_i$ decreases.

A companion statement, Proposition 4.3 (equilibrium profits are decreasing in $i$ , positive exactly for the active miners, and profit-per-hash of the active miners is decreasing), is posed as a theorem of the mission but is not on the goal's proof path.

Significance

The result. Proposition 4.4 says that participation in proof-of-work mining is decided by a single explicit inequality comparing the marginal miner's cost to the average cost of the cheaper miners, corrected by the capacity term $R\gamma/c_n$ . With $\gamma=0$ the condition reads $c_n<\frac{n}{n-1}\cdot\frac{c^{(n)}}{n}$ : the marginal miner's cost may exceed the average active cost only slightly, so cost heterogeneity drives inefficient miners out. The capacity constraint is what keeps them in. The paper's discussion of centralization, its comparative statics of hash rate shares (Propositions 4.6 and 4.8) and its analysis of hardware investment and entry (Proposition 5.1) all take the equilibrium of Proposition 4.1 and the active count of Proposition 4.4 as given.

Formalizing it. The results are proved in the paper (Appendix B, pp. 38–39); none of them is machine-checked. A formal development produces a complete, checked equilibrium analysis of a heterogeneous-cost Tullock contest with quadratic capacity costs: existence and uniqueness of a pure equilibrium, the explicit hash rates, and the participation threshold. The platform currently has no contest, rent-seeking or mining game.

Difficulty

The closed forms are straightforward to verify once the active set is known; the difficulty is the active set itself. The objective is concave in the own hash rate only when the other miners' aggregate is positive, and it jumps at $H=0$ , so the standard concave-game existence theorem does not apply directly on the whole strategy space, and the equilibrium has to be located among profiles with at least two active miners. Uniqueness requires excluding every equilibrium with a different active set, not only the one the formulas produce. For Proposition 4.4, the obvious route, substituting the closed form of $H^*$ into " $c_n<R/H^*$ ", produces a condition in terms of the equilibrium's own $n$ ; showing that the set of $k$ satisfying the printed inequality is an initial segment $\{2,\dots,n\}$ , so that "largest" picks out exactly the equilibrium count, is the step that needs care, and the monotonicity in $R$ and $\gamma$ is a comparison of two different equilibria.

Formalization scope

Miners are Fin N, with Lean index $i$ standing for the paper's miner $i+1$ . The standing assumptions Params.Standing are $N\ge 2$ , $R>0$ , $\gamma\ge 0$ , $c_i>0$ and Monotone c (the paper's sorting, p. 12).
The objective omits the fixed entry cost $K$ of (3.4): with the investment profile fixed it does not depend on the miner's own hash rate. The case $H=0$ is an explicit branch, not Lean's $x/0=0$ .
The equilibrium (Definition 3.2(i)) is the best-response property "for every $x\ge 0$ , deviating to $x$ does not raise the objective", which is equivalent to the paper's " $=\sup_{h_i\ge 0}$ " and avoids junk values of sSup.
"Increasing/decreasing" is weak (Monotone/Antitone), as in the paper's footnote 12. "Largest $n$ " is IsGreatest, not sSup on $\mathbb N$ . Both branches of (4.1) are kept as printed; $\gamma=0$ is allowed throughout.
Ruling out trivializations: the equilibrium is never defined by its closed form and the active count is never defined by the participation condition; the goal includes existence, so its statements about every equilibrium are not vacuous; $N$ and $n$ are general.
Needed infrastructure: elementary real analysis (concavity and one-variable first-order conditions), finite sums over Fin N, quadratic roots. Lemmas about the objective's concavity and best responses, and the initial-segment property of the participation condition, are reusable in the later missions of this series. Contributions of any milestone, or of further intermediate lemmas, are welcome.

Selected references

A. Capponi, S. Ólafsson, H. Alsabah, Proof-of-Work Cryptocurrencies: Does Mining Technology Undermine Decentralization?, arXiv preprint, 2021. https://arxiv.org/abs/2106.09783v1
G. Tullock, The Welfare Costs of Tariffs, Monopolies, and Theft, Western Economic Journal 5(3), 1967. https://doi.org/10.1111/j.1465-7295.1967.tb01923.x

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Combinatorics Machine Learning Operations Research·Captain: mikedeng1

Offline Multi-Action Policy Learning: Generalization and Optimization 2: Depth-L Decision Trees Satisfy log N_H(ε,Π_L) ≤ (2^L−1)log p + 2^L log d + (2^L−1)log(L/ε+1)Research Paper

Motivation

Offline policy learning chooses a treatment rule from logged data: given $n$ observations of features $X_i$ , the action $A_i$ taken and the reward obtained, pick a policy $\pi:\mathcal X\to\{1,\dots,d\}$ from a class $\Pi$ whose expected reward is close to the best in the class. Zhou, Athey and Wager (arXiv:1810.04778v2) bound the regret of their cross-fitted augmented inverse-propensity-weighted learner by a quantity of order $\kappa(\Pi)\sqrt{V_*/n}$ , where $\kappa(\Pi)$ is an entropy integral built from Hamming covering numbers of $\Pi$ . A bound of this kind is only useful for a concrete policy class if $\kappa(\Pi)$ is finite and explicit.

Decision trees are the class the paper uses in practice: they are interpretable, they are the standard object in personalized-treatment applications, and the paper's computational sections (a mixed-integer program and a tree-search algorithm) optimize over them. Lemma 4 of the paper is the bridge between the general regret theory and trees: it bounds the Hamming covering number of the class $\Pi_L$ of depth- $L$ trees, and hence $\kappa(\Pi_L)$ . Covering and complexity bounds for trees in the 0–1 (two-action) setting go back to VC-dimension arguments; the multi-action Hamming-cover version here is the one this paper's regret bound consumes.

Setting

Features are points $x\in\mathbb R^p$ with components $x(1),\dots,x(p)$ ; there are $d$ actions, written $1,\dots,d$ . A deterministic policy is a map $\pi:\mathbb R^p\to\{1,\dots,d\}$ .

Hamming distance and covers. Fix points $x_1,\dots,x_n$ (repetitions allowed). For two policies,

H(\pi_1,\pi_2)=\frac1n\sum_{j=1}^n\mathbf 1\{\pi_1(x_j)\ne\pi_2(x_j)\}.

The $\varepsilon$ -Hamming covering number $N_H(\varepsilon,\Pi,\{x_1,\dots,x_n\})$ is the least $K$ such that some $\pi_1,\dots,\pi_K\in\Pi$ have every $\pi\in\Pi$ within Hamming distance $\varepsilon$ of some $\pi_i$ . The covering number of the class is the worst case over point sets,

N_H(\varepsilon,\Pi)=\sup\{N_H(\varepsilon,\Pi,\{x_1,\dots,x_m\}) : m\ge1\},

and the entropy integral is $\kappa(\Pi)=\int_0^1\sqrt{\log N_H(\varepsilon^2,\Pi)}\,d\varepsilon$ .

Decision trees. A depth- $L$ tree has $L$ layers: $L-1$ layers of branch nodes and a final layer of leaves, so $2^{L-1}-1$ branch nodes and $2^{L-1}$ leaves; a depth-1 tree is a single leaf. A branch node carries a split variable $i\in\{1,\dots,p\}$ and a threshold $b\in\mathbb R$ , and sends $x$ to its left child if $x(i)<b$ and to its right child otherwise. Each leaf carries an action. $\Pi_L$ is the set of policies computed by depth- $L$ trees. In the Lean development these are DTree p d L, DTree.eval and treePolicies p d L; the Hamming layer is hammingDist, hammingCoverNumOn, hammingCoverNum and entropyIntegral.

Formalization targets

Goal: Lemma 4, part 1 (p. 27)

For $p,d,L\ge1$ and $\varepsilon>0$ , $N_H(\varepsilon,\Pi_L)$ is finite and

\log N_H(\varepsilon,\Pi_L)\le(2^L-1)\log p+2^L\log d+(2^L-1)\log\Big(\frac L\varepsilon+1\Big).

Milestones (App. F)

Display (66), p. 67: for any reals $r_1,\dots,r_m$ there are $\lceil L/\varepsilon\rceil$ thresholds $b_i$ such that every indicator $\mathbf 1_b(r)=\mathbf 1\{r<b\}$ is within Hamming distance $\varepsilon/L$ of some $\mathbf 1_{b_i}$ with respect to $r_1,\dots,r_m$ .
App. F, Step 2, p. 69: for any $n\ge1$ points, $p^{2^L-1}\lceil L/\varepsilon\rceil^{2^L-1}d^{2^L}$ depth- $L$ trees $\varepsilon$ -cover $\Pi_L$ , i.e. $N_H(\varepsilon,\Pi_L,\{x_1,\dots,x_n\})\le p^{2^L-1}\lceil L/\varepsilon\rceil^{2^L-1}d^{2^L}$ .
App. F, Step 2, p. 69: $N_H(\varepsilon,\Pi_L)\le p^{2^L-1}\lceil L/\varepsilon\rceil^{2^L-1}d^{2^L}$ .

Companion: Lemma 4, part 2 (p. 27)

\kappa(\Pi_L)\le\sqrt{(2^L-1)\log p+2^L\log d}+\tfrac43L^{1/4}\sqrt{2^L-1}.

It is posed as printed, as a second theorem of the mission rather than a milestone of the goal.

Significance

The result. Lemma 4 makes the abstract regret theory of the paper concrete: with part 2 it gives Theorem 4, the regret bound $O_p\big((\sqrt{(2^L-1)\log p+2^L\log d}+\tfrac43L^{1/4}\sqrt{2^L-1})\sqrt{V_*/n}\big)$ for learning depth- $L$ trees, which for fixed $L,p,d$ is $O_p(\sqrt{V_*/n})$ . The covering bound grows only logarithmically in $1/\varepsilon$ , which is the property the paper's Assumption 3 asks of a policy class. The same counting gives covering bounds for any class of piecewise-constant rules on axis-aligned partitions of bounded depth.

Formalizing it. The result is proved in the paper (App. F) but, as far as known, has no machine-checked proof. The paper omits the induction that combines per-node threshold grids into a cover of whole trees ("a straightforward but lengthy induction (omitted)"), and its appendix counts $2^L-1$ branch nodes where §6.1's trees have $2^{L-1}-1$ . A formal proof fixes both: it supplies the omitted induction and confirms that the printed bound holds for the class actually defined in §6.1. The milestones are reusable statements about threshold indicators and tree covers. The companion part 2 is posed with the printed constant $4/3$ , although the printed derivation evaluates $\int_0^1(L/\varepsilon^2)^{1/4}d\varepsilon$ as $\frac43L^{1/4}$ where the integral equals $2L^{1/4}$ ; whether the printed bound holds is part of what the mission settles.

Difficulty

The one-dimensional step (66) is elementary once the Hamming distance between two threshold indicators is identified with a count of points between the thresholds. The difficulty is in passing from single thresholds to whole trees. Replacing each branch node's threshold independently by its nearest grid threshold, with grids built on all $n$ points, does not give an $\varepsilon$ cover: the errors at different nodes of the same layer add up, and the naive bound is $(2^{L-1}-1)\varepsilon/L$ rather than $\varepsilon$ . The points that reach a node depend on the thresholds above it, so the approximation has to respect the tree structure, while the number of candidate trees must still be counted as a product over nodes. The passage from finite point sets to $N_H(\varepsilon,\Pi_L)$ , which is a supremum over all point sets in $\mathbb N\cup\{\infty\}$ , and the arithmetic with $\lceil L/\varepsilon\rceil$ and real logarithms are routine but must be done in the stated types.

Formalization scope

Features range over all of $\mathbb R^p$ (Fin p → ℝ); the paper's $\mathcal X$ is a bounded subset of $\mathbb R^p$ , and a bound over $\mathbb R^p$ implies the bound over every $\mathcal X\subseteq\mathbb R^p$ . Actions are Fin d, split variables Fin p (0-based); policies are deterministic.
Point sets are sequences Fin m → (Fin p → ℝ) with repetitions allowed, $m\ge1$ ; covers are proper (centres in $\Pi_L$ ). $N_H(\varepsilon,\Pi)$ is in ℕ∞, $\kappa(\Pi)$ a lower Lebesgue integral in ℝ≥0∞ over $(0,1]$ .
Depth follows §6.1: DTree p d 1 are leaves, DTree p d 0 is empty. The bounds are the printed ones, which over-count for this convention; the Lean was not switched to the appendix's convention, which would change $\Pi_L$ .
Standing hypotheses: $p\ge1$ , $d\ge1$ , $L\ge1$ ("Let $L$ be any fixed positive integer"), $\varepsilon>0$ (needed for $\lceil L/\varepsilon\rceil$ ).
The goal asserts finiteness of $N_H(\varepsilon,\Pi_L)$ explicitly. Without it the inequality would be trivially true for an infinite covering number, because Lean's toNat ⊤ = 0 and Real.log 0 = 0. The covering number is an infimum that is $0$ only for an empty class, and $\Pi_L$ is nonempty under the hypotheses, so the cover is not trivialized either.
Not posed: the $O(\log 1/\varepsilon)$ remark of Lemma 4.1 (a remark on the right-hand side); Theorem 4 (p. 28, the combination with the regret theorem, which belongs to the companion mission on regret); the mixed-integer program and tree search of §6.2–6.3 and App. G (computational).

Contributions welcome: proofs of the milestones, the omitted per-node induction as a stand-alone lemma, and the companion part 2.

Selected references

Z. Zhou, S. Athey, S. Wager, Offline Multi-Action Policy Learning: Generalization and Optimization, arXiv preprint, 2018 (v2). https://arxiv.org/abs/1810.04778
L. Breiman, J. Friedman, R. Olshen, C. Stone, Classification and Regression Trees, Wadsworth, 1984. https://doi.org/10.1201/9781315139470
D. Bertsimas, J. Dunn, Optimal classification trees, Machine Learning 106, 1039–1082, 2017. https://doi.org/10.1007/s10994-017-5633-9
S. Athey, S. Wager, Policy Learning With Observational Data, Econometrica 89(1), 133–161, 2021. https://doi.org/10.3982/ECTA15732

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Operations Research Optimization Probability+1·Captain: mikedeng1

Pricing in On-Demand and One-Way Vehicle-Sharing Networks 1: Static Prices Earn at Least K/(N+K−1+Λ/μ) of the Fluid Optimum, via Increasingly Tight Availability BoundsResearch Paper

Why static prices in vehicle sharing

In one-way car sharing, bike sharing and ride hailing, a vehicle that serves a trip from location $i$ to location $j$ ends up at $j$ . Prices therefore do two jobs: they collect revenue and they steer the fleet. The best dynamic (state-dependent) pricing policy reacts to where every vehicle is, but computing it is a Markov decision problem over a state space that grows combinatorially with the fleet. A static policy, one fixed price per origin–destination pair, is easy to compute and to post, and the question is how much it gives up.

Waserhole and Jost (2016) bounded the loss when travel times are negligible. Banerjee, Freund and Lykouris (arXiv:1803.07841, Operations Research 2022) gave guarantees for large fleets. Benjaafar and Shen (SSRN 3998297; Operations Research Technical Note, 2023, doi:10.1287/opre.2023.2446) prove a bound valid for every fleet size $K$ that accounts for travel times, and then a sequence of progressively tighter bounds. This mission formalizes that chain.

Setting

There is a finite set $\mathcal N$ of $N$ locations and $K\ge 1$ vehicles. Trip requests from $i$ to $j$ arrive at rate $\Lambda_{ij}\ge 0$ ; the pairs with positive demand form $\mathcal A=\{(i,j):\Lambda_{ij}>0\}$ , and the network $(\mathcal N,\mathcal A)$ is strongly connected. A trip from $i$ to $j$ lasts on average $1/\mu_{ij}$ . A quantile policy $q=(q_{ij})$ , $q_{ij}\in[0,1]$ , is the fraction of requests that accept the posted price; the effective rates are $\lambda_{ij}=q_{ij}\Lambda_{ij}$ and $\lambda_i=\sum_j\lambda_{ij}$ . Each served trip earns $I_{ij}(q_{ij})$ , and $q\mapsto qI_{ij}(q)$ is concave.

The fluid problem (2) is

\max_q\ R_s(q)=\sum_{(i,j)\in\mathcal A}\Lambda_{ij}q_{ij}I_{ij}(q_{ij})\quad\text{s.t.}\quad\sum_j\Lambda_{ij}q_{ij}=\sum_j\Lambda_{ji}q_{ji}\ \forall i,\quad \sum_{(i,j)\in\mathcal A}\frac{\Lambda_{ij}q_{ij}}{\mu_{ij}}\le\Delta,\quad q_{ij}\in[0,1],

with $\Delta\ge K$ . Its optimal solution $q^s$ is the static policy; it is assumed to induce a strongly connected network. The value $R_s(q^s)$ is the fluid optimum.

Under a static policy the fleet moves through a closed queueing network: a single-server pick-up queue at each location, served at rate $\lambda_i$ , and an infinite-server transit queue for each pair, with mean time $1/\mu_{ij}$ . The paper invokes the BCMP theorem for this network; the mission writes its stationary law as the product form. With $r$ a positive solution of the traffic equations (9), $r_i=\sum_j r_j\lambda_{ji}/\lambda_j$ , a state $x$ (vehicle counts at locations and in transit, total $K$ ) has weight

w(x)=\prod_i\Bigl(\frac{r_i}{\lambda_i}\Bigr)^{x_i}\prod_{(i,j)}\frac{1}{x_{ij}!}\Bigl(\frac{r_i\lambda_{ij}}{\lambda_i\mu_{ij}}\Bigr)^{x_{ij}}.

The availability $\alpha_i(K)$ is the weight share of the states with $x_i\ge 1$ : the probability that a customer at $i$ finds a vehicle. The reward rate of the static policy is (3):

R(q^s)=\sum_{(i,j)\in\mathcal A}\alpha_i\,\Lambda_{ij}q^s_{ij}I_{ij}(q^s_{ij}).

Finally $\Lambda=\sum_{\mathcal A}\Lambda_{ij}$ , $1/\mu=\sum_{\mathcal A}\Lambda_{ij}/(\Lambda\mu_{ij})$ , and $\lambda/\mu=\sum_{\mathcal A}\Lambda_{ij}q^s_{ij}/\mu_{ij}$ .

Formalization targets

Goal: Theorem 2, static half (p. 9)

With $B^j_0(q^s)=B^j_0=0$ and, for $1\le i\le K$ ,

B^K_i(q^s)=\frac{K}{N+K-1+\frac{\lambda}{\mu}\bigl(1-B^{K-1}_{i-1}(q^s)\bigr)},\qquad B^K_i=\frac{K}{N+K-1+\frac{\Lambda}{\mu}\bigl(1-B^{K-1}_{i-1}\bigr)},

the goal is: for every $i=1,\dots,K$ ,

\frac{R(q^s)}{R_s(q^s)}\ \ge\ B^K_i(q^s)\ \ge\ B^K_i\ \ge\ \frac{K}{N+K-1+\Lambda/\mu},

and $B^K_{i+1}>B^K_i$ for $i=1,\dots,K-1$ .

The paper's Theorem 2 begins with one more inequality, $R(q^s)/R(q^*)\ge R(q^s)/R_s(q^s)$ , where $q^*$ is the best state-dependent policy. That inequality is Proposition 1 (p. 6). Its proof needs the stationary behaviour of the network under an arbitrary state-dependent policy with matrix-exponential trip times, together with Jensen's inequality and Little's law; that model is not formalized here. Proposition 1 is not part of this mission. The goal is the remaining chain, and it implies the paper's theorem once Proposition 1 is available.

Milestones

Equal availability under balanced flows (p. 7, Appendix p. 23): $\alpha_i(K)=\alpha_j(K)$ .
The ratio equals the availability (pp. 7–8): $R(q^s)/R_s(q^s)=\alpha$ .
Equation (5) (p. 8): $\alpha(K)=K/\bigl(K+N-1+\frac{\lambda}{\mu}(1-\alpha(K-1))\bigr)$ .
Theorem 1, static half (p. 7): $R(q^s)/R_s(q^s)\ge K/(K+N-1+\Lambda/\mu)$ .
Equation (6) and $0\le\alpha(K-i)\le(K-i)/(N+K-i-1)$ (p. 9).
$B^K_K(q^s)=\alpha(K)$ (p. 10).
$B^K_{i+1}(q^s)>B^K_i(q^s)$ (p. 10).

Significance

The bound $K/(N+K-1+\Lambda/\mu)$ holds for every fleet size, tends to $1$ at rate $O(1/K)$ , and reduces to the Waserhole–Jost bound when travel times vanish. The sequence $B^K_i(q^s)$ gives a computable certificate between that bound and the exact ratio $B^K_K(q^s)=\alpha(K)$ , which an operator can evaluate from $N$ , $K$ and the fluid solution alone. Equation (5), a one-step recursion in the fleet size for a network mixing single-server and infinite-server stations, is the reusable core: the two sibling missions (the demand-free bound of Theorem 3 and the split-network bound of Theorem 4) rest on the same recursion.

The results are proved in the paper; none is formalized elsewhere as far as a search of the platform shows. The platform has the multiclass BCMP framework and single-server closed Jackson networks, but not availability in networks with infinite-server stations, nor equation (5). This mission adds a product-form availability layer, the equal-availability lemma, the recursion (5), and the bound chain.

Difficulty

The paper derives (5) from the random observer (arrival) theorem and Little's law, and neither is available as a formal tool, so the recursion has to be obtained from the product form itself, as an identity between sums over the state spaces with $K$ and $K-1$ vehicles. Equal availability needs uniqueness, up to scale, of the positive solution of the traffic equations on a strongly connected network (a Perron–Frobenius-type fact for an irreducible stochastic matrix). In the bound chain, $B^K_i(q^s)\ge B^K_i$ needs the recursion map to be monotone in both the workload and the previous value, which holds only once $0\le B\le 1$ is established; strict monotonicity in $i$ is a double induction.

The obvious shortcut, defining $\alpha$ by the recursion (5), makes Theorem 1 a one-line consequence of $\alpha(K-1)\ge 0$ and deletes the content of the first three milestones; the mission rules it out by defining $\alpha$ as the product form.

Formalization scope

Locations form a Fintype with Nonempty; $N$ is its cardinality. Demand, travel rates, rewards and policies are functions on all ordered pairs, self-pairs included; sums over $\mathcal A$ are sums over all pairs, which agree because each summand carries $\Lambda_{ij}$ . Strong connectivity, of $(\mathcal N,\mathcal A)$ and of the network induced by $q^s$ , uses nonempty paths (Relation.TransGen), so every location has positive accepted demand and $\mathcal A\ne\emptyset$ . Every ordered pair has a transit queue; a pair with $\lambda_{ij}=0$ has load $0$ and stays empty ( $0^0=1$ ). All arithmetic on $K$ , $N$ and $i$ is in $\mathbb R$ ; $B(K,0)=0$ for every $K\ge 0$ , including the $B^0_0$ that the paper's definition omits. $R_s(q^s)>0$ is a hypothesis of the ratio statements, since otherwise the ratio is $0/0$ . The statements keep the paper's hypothesis that $q^s$ is optimal for (2) with $\Delta\ge K$ ; the proofs use only feasibility and balanced flows, and the queueing milestones are stated for every balanced quantile policy inducing a strongly connected network.

The product form is a definition: the mission does not prove that it is the stationary law of the network's Markov chain (the BCMP theorem). The general closed product-form network (single-server and infinite-server stations on an arbitrary finite station set) is a separate definition and is reusable beyond this mission. Contributions welcome: proofs of (5) via normalizing constants, the traffic-equation uniqueness lemma, and the elementary bound-sequence inequalities.

Selected references

S. Benjaafar, X. Shen, Pricing in On-Demand (and One-Way) Vehicle Sharing Networks, SSRN abstract 3998297, version of 2023-01-10. https://ssrn.com/abstract=3998297 ; Technical Note, Operations Research, 2023. https://doi.org/10.1287/opre.2023.2446
F. Baskett, K. M. Chandy, R. R. Muntz, F. G. Palacios, Open, Closed, and Mixed Networks of Queues with Different Classes of Customers, J. ACM 22(2), 1975. https://doi.org/10.1145/321879.321887
S. Banerjee, D. Freund, T. Lykouris, Pricing and Optimization in Shared Vehicle Systems: An Approximation Framework, Operations Research 70(3), 2022. https://arxiv.org/abs/1803.07841
A. Waserhole, V. Jost, Pricing in Vehicle Sharing Systems: Optimization in Queuing Networks with Product Forms, EURO J. Transportation and Logistics 5, 2016. https://doi.org/10.1007/s13676-014-0054-4
D. K. George, C. H. Xia, Fleet-Sizing and Service Availability for a Vehicle Rental System via Closed Queueing Networks, European J. Operational Research 211(1), 2011. https://doi.org/10.1016/j.ejor.2010.12.015
M. Reiser, S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, J. ACM 27(2), 1980. https://doi.org/10.1145/322186.322195

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Linear algebra Quantum Information·Captain: mikedeng1

Improvements in Quantum SDP-Solving with Applications 1: A Unitary Preparing a Purification of a Subnormalized Density Operator ϱ Gives an Exact (1, w+a+1, 0)-Block-Encoding of ϱResearch Paper

Motivation

Quantum algorithms for semidefinite programming need a way to use an input matrix inside a quantum circuit. In the quantum state input model, an oracle prepares a purification of a density operator. In the quantum operator input model, a unitary instead exposes that operator as a block of a larger unitary. The two kinds of access serve different subroutines. Lemma 12 of van Apeldoorn and Gilyén’s 2018 preprint connects them for a subnormalized density operator. The same paper uses this connection when comparing its input models for quantum SDP solving.

The result concerns a precise finite-dimensional construction. It matters even without the paper’s query-complexity estimates: it specifies which registers must be present, what happens to probability mass outside the accepted flag state, and which block of the resulting unitary equals the desired operator. These details are easy to lose when a statement is shortened to “purification gives a block-encoding.”

Setting

A w-qubit register has basis labels in $\{0,1\}^{w}$ . A subnormalized density operator $\varrho$ on that register is a positive semidefinite complex matrix with $\operatorname{Tr}\varrho\leq1$ . Its trace may be strictly below one. A purification is a unit vector $|\varphi\rangle$ on three registers: the w-qubit system, one flag qubit, and a c-qubit junk register. The operator $\varrho$ is the flag-zero block after tracing out the junk. In entries,

\varrho_{ij}=\sum_{t\in\{0,1\}^{c}}\varphi_{i,0,t}\,\overline{\varphi_{j,0,t}}.

Thus probability in the flag-one branch does not contribute to $\varrho$ . Let $a=c+1$ , so a unitary $G$ preparing $|\varphi\rangle$ acts on $w+a$ qubits and $|\varphi\rangle=G|0\rangle^{\otimes(w+a)}$ . The result adds an initial w-qubit system and one extra qubit. Its register swap $S$ exchanges the initial $(\text{system},\text{extra})$ register with the $(\text{prepared system},\text{flag})$ register and leaves the c junk qubits fixed. The construction is

V=(I_{2^{w+1}}\otimes G^{\dagger})S(I_{2^{w+1}}\otimes G).

A $(\alpha,k,\varepsilon)$ block-encoding of an operator $A$ is a unitary on the system and k ancilla qubits whose all-zero ancilla block differs from $A/\alpha$ in operator norm by at most $\varepsilon/\alpha$ . Here the scale is one and the error is zero. The block of $V$ is taken at the all-zero state of the extra qubit and the entire register of $G$ .

Formalization targets

The goal is Lemma 12 with the ancilla count given by its displayed construction:

\left\|\varrho-(I_{2^w}\otimes\langle0|^{\otimes(w+a+1)})V (I_{2^w}\otimes|0\rangle^{\otimes(w+a+1)})\right\|=0.

The target includes that $V$ is unitary, making it an exact $(1,w+a+1,0)$ block-encoding of $\varrho$ . The theorem identifies $V$ explicitly, so its use of one $G$ and one $G^{\dagger}$ is part of the construction. The source’s Lemma 12 prints $(1,w+a,0)$ , while its proof says $V$ has $2w+a+1$ qubits and takes the block at $1+w+a$ ancillas. This mission uses the proof’s consistent count and labels the correction. The four milestones follow the proof’s stated steps: unitarity of $V$ , identification of its zero block, elimination of the flag-one component, and equality of the remaining SWAP sandwich with $\varrho$ .

Significance

The exact identity shows that a state-preparation unitary can be used where the paper’s quantum operator model expects a block-encoding. It also covers genuinely subnormalized operators, for which a normalized-state argument is insufficient: a preparation may have nonzero amplitude in the flag-one branch even though only the flag-zero branch defines $\varrho$ . The explicit ancilla count prevents such lost probability mass from being silently included in the output block.

Formalizing this known result gives a reusable interface between a purification predicate and a block-encoding predicate. The mission statement is open on Prove2Me; the paper’s mathematical argument is known. A related open block-encoding statement in Nannicini’s treatment concerns a normalized density matrix and a circuit-level construction with a different ancilla arrangement. It does not state this paper’s subnormalized, flag-qubit identity. The finite-register definitions and SWAP identities can be reused by later matrix-level quantum algorithms, whether or not the paper’s larger complexity bounds are formalized.

Difficulty

The first apparent shortcut is to use the ordinary purification identity without the flag qubit. That recovers the trace over both flag outcomes, which can have trace one even when $\operatorname{Tr}\varrho<1$ . The required block must select flag zero. A second source of error is register order: the paper displays the system first, while the existing Lean block-encoding predicate places the ancilla register first. Equality of matrix blocks depends on the explicit reindexing. Complex conjugation also matters: swapping the vector and covector amplitudes would produce a transpose of $\varrho$ instead of $\varrho$ for complex states.

Formalization scope

The Lean system register is QAlgorithms.Qubits w; the flag and extra register are each Bool; and the junk is QAlgorithms.Qubits c. The identity $a=c+1$ makes the flag register present even at $c=0$ . The all-zero state is the all-false basis label. Matrices are over $\mathbb C$ , tensor products are matrix Kronecker products, and $S$ is a permutation matrix fixing the junk. The paper’s printed identity factor beside the swap has size $2^a$ ; the consistent factor is $2^{a-1}=2^c$ , since the swap already acts on two $w+1$ -qubit registers.

The referenced QAlgorithms.IsBlockEncoding definition includes unitarity and measures error with the operator norm. Lean reindexes $V$ to put its $(\text{extra},\text{prepared system},\text{flag},\text{junk})$ ancilla first. The theorem requires $G$ unitary, $\varrho$ subnormalized, and the exact purification relation between $G|0\rangle$ and $\varrho$ . The purification relation prevents a vacuous claim about an unrelated matrix. The optional quantum gate-count clause of the printed lemma is outside this matrix-level formalization; the register swap is explicitly the swap of $w+1$ qubit pairs. A circuit-level gate-count theorem would require an additional gate-list representation.

The needed reusable components are finite complex matrices, unitary permutation matrices, Kronecker products, partial traces, and the operator norm. Contributions can address the four milestone identities, the main theorem, or a circuit-level refinement of the gate count. None of the statements introduces the proof’s Schmidt decomposition as an assumption.

Selected references

J. van Apeldoorn and A. Gilyén, Improvements in Quantum SDP-Solving with Applications, arXiv preprint, 2018, arXiv:1804.05058v1, Definition 1 (p. 3), Definition 2 (p. 4), Lemma 12 and proof (pp. 15–16).
G. Nannicini, Quantum Algorithms for Optimizers, arXiv preprint, 2024, arXiv:2408.07086, Definition 7.11 and the normalized density-matrix construction.

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Dynamical Systems Economics Operations Research+1·Captain: mikedeng1

Minimum Earnings Regulation and the Stability of Marketplaces I: The Marketplace Is Stable Exactly When M ≤ w* Under Tight Labor Supply (p* ≤ w) and M ≤ H(R(p)) Under Loose SupplyResearch Paper

Motivation

Minimum hourly earnings rules for ride-hailing drivers are meant to protect income during periods when drivers spend part of their time idle. A platform pays a wage per hour carrying riders, yet drivers care about earnings per hour logged onto the platform. When many drivers enter after a wage increase, utilization can fall and offset the increase. A regulation that uses the preceding period's utilization to set the next wage can therefore create feedback in the marketplace. Asadpour, Lobel, and van Ryzin model this feedback and identify the exact minimum earnings targets compatible with bounded wages and continuing service (Asadpour, Lobel, and van Ryzin, 2022, §§3–4).

The question matters to marketplace design because a target that cannot be sustained within the model cannot be reached through a bounded wage policy, even if the platform changes prices over time. The theorem separates two economic regimes. When labor supply is tight, the decisive number is the price where demand equals supply. When supply is loose, the decisive number is the greatest revenue the platform can obtain from riders, converted into affordable driver earnings. This mission concerns the paper's threshold theorem and the four lemmas that state its main analytic ingredients.

Setting

In each discrete period $t$ , a platform chooses a rider price $p_t\ge0$ and a wage per ride hour $w_t\ge0$ . At price $p$ , riders demand $D(p)$ hours of trips. At expected hourly earnings $z$ , drivers supply $S(z)$ hours. The quantity of completed rides and driver utilization are

Q(p,z)=\min\{D(p),S(z)\},\qquad u(p,z)=\frac{Q(p,z)}{S(z)}.

Drivers earn wages only while carrying riders. Thus an earnings equilibrium is a positive sequence $z_t$ satisfying $z_t=w_tu(p_t,z_t)$ in every period. The platform's period profit is $\pi(p,w,z)=(p-w)Q(p,z)$ . A minimum earnings target $M>0$ requires

w_{t+1}\ge\frac{M}{u(p_t,z_t)}.

The rule uses utilization in one period to set a wage floor in the next. A policy is stable if its prices and wages are feasible, it satisfies equilibrium and the wage rule, wages have a finite upper bound across all periods, and profit is nonnegative in all but finitely many periods. This definition permits a finite transition with losses but excludes a wage sequence that grows without bound (Asadpour, Lobel, and van Ryzin, 2022, p. 8, Definitions 1–2 and (5)).

The revenue function is $R(p)=pD(p)$ , and the labor cost function is $C(z)=zS(z)$ . Write $H$ for the inverse of $C$ on positive expenditure. The paper assumes that $D$ is positive, continuous, and strictly decreasing; $S$ is strictly increasing, differentiable, and positive except possibly at zero; $R$ has a unique peak at $p^*$ ; and demand and supply cross at $w^*$ , so $D(w^*)=S(w^*)$ (Asadpour, Lobel, and van Ryzin, 2022, p. 8, Regularity assumptions).

Each theorem in this mission retains these standing assumptions, including when a particular result uses only some of them.

Formalization targets

Exact stability threshold

The goal is Theorem 1 of the paper. For every $M>0$ , stability is characterized by

\operatorname{Stable}(D,S,M)\quad\Longleftrightarrow\quad M\le z^*,\qquad z^*=\begin{cases}w^*,&p^*\le w^*,\\H(R(p^*)),&p^*>w^*.\end{cases}

The equality case $p^*=w^*$ belongs to the first branch. The theorem gives an if and only if statement about all feasible sequences of prices and wages, rather than a condition for one chosen stationary policy (Asadpour, Lobel, and van Ryzin, 2022, p. 11, Theorem 1 and (7)).

Supporting results

The milestones reproduce Lemmas 1–4 and two claims in the Online Appendix. Lemma 1 establishes the inverse labor cost's existence, regularity, and monotonicity. Lemma 2 identifies continuous, unique equilibrium earnings for a given price and positive wage. Lemma 3 states that stability admits a policy with price equal to wage. Lemma 4 rewrites the wage rule in that regime. The appendix claims compare the loose-supply threshold with $w^*$ and show that the tight-supply wage-growth factor exceeds one above $w^*$ (Asadpour, Lobel, and van Ryzin, 2022, pp. 9–10 and Online Appendix p. 4).

Significance

The theorem identifies the full stable range of targets in this model. In the tight-supply case, raising the target above the demand–supply crossing cannot be stabilized by choosing a different sequence of prices. In the loose-supply case, the highest stable target is tied to the cost of supplying drivers at the platform's peak potential revenue. The distinction explains why the same regulation can have different limits across markets with different demand and supply curves (Asadpour, Lobel, and van Ryzin, 2022, p. 11 and Figure 2).

The paper proves this result mathematically. The remaining formalization task is to represent its sequence model and analytic assumptions in Lean and prove the threshold statement and its supporting lemmas there. Its inverse-cost, equilibrium, and policy-stability interfaces can also be reused for the companion results on utilization shocks and static stability in the same paper. Those companion theorems are separate missions in this series.

Difficulty

The short threshold formula hides two distinct obligations: a policy must satisfy the earnings equilibrium and lagged wage rule at every period, while the impossibility direction must exclude every stable feasible policy, including policies that vary prices over time. The equilibrium earnings formula changes regime when supply at the posted wage crosses demand. At the boundary the two formulas must agree, and the argument must still handle wages that approach or cross the demand–supply point. Boundedness and profit conditions also interact with a finite initial period in which profit may be negative.

The paper's printed explanation for one strict inequality in the tight-supply converse is insufficient: two decreasing factors do not determine the direction of their product. The formal target keeps the inequality the paper uses, while the appendix quotation remains as printed (Asadpour, Lobel, and van Ryzin, 2022, Online Appendix p. 4).

Formalization scope

Prices, wages, earnings, demand, supply, cost, and revenue are real valued. The curves' regularity is imposed on $[0,\infty)$ ; differentiability of supply is imposed on $(0,\infty)$ together with continuity at zero. Revenue is strictly increasing up to $p^*$ and strictly decreasing afterward, encoding the unique peak. The crossing $w^*$ and inverse $H$ are parameters with their characterizing properties, so the statements hold for their uniquely determined values. The inverse is only used at positive costs.

Lean indexes the first period by $0$ , so its rule applies to $w_{t+1}$ for every $t\ge0$ and places no floor on $w_0$ . The paper starts at period $1$ and likewise leaves the first wage free. “Bounded wages” means a single real upper bound for every period; “all but finitely many periods” is represented by an eventually true condition on $\mathbb N$ .

An equilibrium requires $z_t>0$ . This explicit domain condition prevents a zero-supply input from making $u=0/0$ and then $M/u=0$ under Lean's total real division, which would make the regulation vacuous. Lemma 2 correspondingly uses $w>0$ . Its branch condition $S(w)<D(p)$ agrees with the paper's $w<S^{-1}(D(p))$ wherever that inverse exists and remains meaningful when demand is outside the range of supply. These domain choices are stated with the items they affect. A complete development needs real-analysis facts about continuous strictly monotone functions and inverses, eventual properties of sequences, and elementary bounds on the utilization ratio. Contributions establishing these interfaces or the stated milestones are within scope.

Selected references

Arash Asadpour, Ilan Lobel, and Garrett van Ryzin, Minimum Earnings Regulation and the Stability of Marketplaces, SSRN preprint 3502607, version of August 24, 2022; published in Manufacturing & Service Operations Management, 2022. SSRN preprint. The local formalization uses the dated preprint's page and theorem numbering.

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Theoretical Computer Science·Captain: Chutong

CLRS — Chapter 20, §20.5: Strongly Connected Components (Kosaraju) — Correctness and Running TimeTextbook

The mutual-reachability partition

Two vertices belong to the same strongly connected component when each can reach the other by a directed path. The partition separates the mutually reachable parts of a directed graph without requiring the whole graph to be connected. This mission formalizes the two-pass DFS algorithm of CLRS, fourth edition, Chapter 20, §20.5, often called Kosaraju's algorithm. The fourth-edition contents place this material after DFS in §20.3.

Graph and returned component labels

The input graph has $n$ vertices numbered from zero and $m$ adjacency occurrences. Its representation is an array of integer adjacency arrays plus a degree sequence. Parallel arcs and self-loops are allowed; duplicate occurrences contribute separately to $m$ . Empty and disconnected graphs are included. Symmetric entries can represent an undirected graph, but no symmetry is assumed.

The source scc calls the shared dfs entry, constructs the transpose by reversing every adjacency occurrence, and calls the same dfs_ordered entry on that transpose. The first pass produces increasing finish order. A linear pass reverses the order prefix for the second traversal; no comparison sort is performed. The returned object is the second traversal's DFSResult.

Its component array contains integer labels, and components contains their count $k$ . Labels are contiguous identifiers $0,\ldots,k-1$ , each used by at least one vertex. They are not canonical vertex representatives and need not be stable under adjacency reordering. Only the component fields are specified by the SCC postcondition; the remaining result fields describe the second DFS pass on the transpose.

Formalization targets

The headline CLRS.SCC.scc_correctness_and_running_time requires a single terminating scc.Call whose result labels exactly the initial graph's mutual-reachability classes. For every valid pair $u,v$ ,

\operatorname{component}[u]=\operatorname{component}[v] \iff (u\leadsto v)\land(v\leadsto u).

The component count must satisfy

0\le k\le n,\qquad k=0\iff n=0,

and every label below $k$ must occur. The component array and result object are fresh, and the complete call preserves all previously allocated caller storage. Its named-event cost obeys one uniform bound:

\exists C\in\mathbb N,\ C>0:\quad T\le C(1+n+m).

A supporting SCC milestone requests the explicit bound

T\le20+21n+6m

for the same call and complete postcondition. This includes both DFS traversals, input-order construction, transpose construction, finish-order reversal, and fixed setup work.

The other supporting milestone, transpose_correctness_and_running_time, requests a fresh graph with exactly reversed adjacency occurrences, unchanged vertex and occurrence counts, caller-storage preservation, and a uniform linear event bound. Reversal retains multiplicity, not just the existence of edges. These are open goals; no mathematical or execution proof is supplied.

Composing a verified traversal

The desired result gives later callers an exact test for component membership by comparing labels. Its count and coverage clauses make the labels usable for a compact representation of the partition. An algorithm using the partition need not repeat the analysis of recursive exploration or inspect compiler transitions.

The separate DFS mission supplies CLRS.DFS.ordered_correctness_and_explicit_running_time as an open dependency, with the full forest and storage contract and the numerical budget $4+6n+2m$ for any valid supplied root permutation. Its identity-order companion, CLRS.DFS.correctness_and_explicit_running_time, requests $7+8n+2m$ for the first pass. These explicit budgets can be combined with transpose construction and finish-order reversal to establish this mission's complete event bound. This mission imports the same definitions and requests the new component theorem; it does not duplicate the DFS program. The main statement contains no hypothesis asserting that DFS or SCC is already correct. The mathematical two-pass algorithm and its linear bound are textbook results; formal verification of these particular generated calls is the contributor task. See CLRS, fourth edition, §§20.3 and 20.5.

What must be justified

A DFS tree in an arbitrary directed graph is not necessarily a strongly connected component. The second traversal's particular order and reversed edges are essential to the requested equivalence. Both directions matter: vertices assigned the same label must be mutually reachable, and mutually reachable vertices must not receive different labels. The cost obligation cannot omit the transpose or order-processing loops, and repeated arcs must be counted rather than treated as free duplicates.

Formalization scope

Reachability is defined by finite paths in the original graph, including zero-edge paths at valid vertices. Transposition preserves each edge occurrence. Every transpose row, including an empty row, is separately allocated before any edges are appended, so the program does not accidentally share mutable row storage. Caller input rows may alias because they remain read-only.

The shared prerequisites are the published Algo2Lean runtime, module-call support, and unweighted graph representation and specifications. All three missions reuse these same definitions by reference, without introducing duplicate prerequisite bundles. The existing Dijkstra graph definition uses weighted, object-linked adjacency lists; the shared unweighted module supplies the arrays-of-arrays storage contract used here.

The definitions use Algo2Lean v0.1.4, source commit 0f79d5e4242e6477531c9ee4410409f5b1c92f76, in project mode with standalone=false. The source root contains the explicitly named modules below; the entry file is identified for each mission. The reproduction manifest records raw generation and platform import-path adaptations separately. Stable generated definitions supply the public Call relations.

The clock graph_work counts the marked source blocks, function entries, and evaluated loop guards, including final false guards. Each marked block contains a fixed amount of source work. Costs are differences between the caller's actual initial and final counters; other clocks are preserved. This model uses mathematical integer sequences and unbounded integers. It does not assert machine wall-clock time, integer bit complexity, stack-space bounds, or Lean proof-checking time. Construction of the caller's input graph is outside the traversal entry point.

All theorem bodies are open contributor tasks. Supplied definitions and well-formed goal statements are not correctness, termination, or running-time proofs. The requested results concern the displayed program's actual executions, not an assumed-correct replacement algorithm or a hypothesis containing its conclusion.

The complete entry is scc.algo; its dependencies are dfs.algo and graph_input.algo. The DFS source is shown unchanged to make the imports and counted work explicit. Supporting theorem references reuse the DFS mission's definitions and open contract.

`graph_input.algo`

# Unweighted directed graph: vertices are 0, ..., size-1.
# Row adj[u][0:degree[u]] lists outgoing targets, with one slot per arc.
# Traversals require nonnegative degrees and in-range targets. Duplicate arcs
# and self-loops are allowed. The constructor stores the supplied references.

class UnweightedGraph:
    size: Int
    adj: Seq[Seq]
    degree: Seq

    def __init__(self, size: Int, adj: Seq[Seq], degree: Seq) -> None:
        self.size = size
        self.adj = adj
        self.degree = degree

`dfs.algo`

# CLRS fourth edition, Section 20.3: depth-first search.
# dfs_ordered takes a permutation of the vertices as its outer-loop order.
# Result.order lists vertices in increasing finish time; reverse its valid
# prefix before using it as the second-pass order in the SCC algorithm.
# graph_work counts the explicitly marked blocks and loop-guard evaluations.

from graph_input import UnweightedGraph


class DFSResult:
    color: Seq
    parent: Seq
    discover: Seq
    finish: Seq
    order: Seq
    component: Seq
    clock: Int
    finished: Int
    components: Int

    def __init__(self) -> None:
        tick("graph_work")
        self.color = const(0)
        self.parent = const(-1)
        self.discover = const(0)
        self.finish = const(0)
        self.order = const(-1)
        self.component = const(-1)
        self.clock = 0
        self.finished = 0
        self.components = 0


def visit(graph: UnweightedGraph, result: DFSResult, u: Int, label: Int) -> None:
    tick("graph_work")
    result.clock += 1
    result.discover[u] = result.clock
    result.color[u] = 1
    result.component[u] = label
    neighbors: Seq = graph.adj[u]
    j: Int = 0
    tick("graph_work")
    # @scan_neighbors
    while j < graph.degree[u]:
        tick("graph_work")
        v: Int = neighbors[j]
        if result.color[v] == 0:
            tick("graph_work")
            result.parent[v] = u
            visit(graph, result, v, label)
        j += 1
        tick("graph_work")
    tick("graph_work")
    result.color[u] = 2
    result.clock += 1
    result.finish[u] = result.clock
    result.order[result.finished] = u
    result.finished += 1


def dfs_ordered(graph: UnweightedGraph, order: Seq) -> DFSResult:
    tick("graph_work")
    result: DFSResult = DFSResult()
    i: Int = 0
    tick("graph_work")
    # @scan_roots
    while i < graph.size:
        tick("graph_work")
        u: Int = order[i]
        if result.color[u] == 0:
            tick("graph_work")
            visit(graph, result, u, result.components)
            result.components += 1
        i += 1
        tick("graph_work")
    tick("graph_work")
    return result


def dfs(graph: UnweightedGraph) -> DFSResult:
    tick("graph_work")
    order: Seq = const(0)
    i: Int = 0
    tick("graph_work")
    # @identity_order
    while i < graph.size:
        tick("graph_work")
        order[i] = i
        i += 1
        tick("graph_work")
    tick("graph_work")
    return dfs_ordered(graph, order)

`scc.algo`

# CLRS fourth edition, Section 20.5: strongly connected components.
# The first DFS records increasing finish order. The second DFS visits the
# transpose in decreasing first-pass finish order. Its forest labels are SCCs.
# Adjacency rows in the transpose are separately allocated, including empty rows.
# graph_work counts the explicitly marked blocks and loop-guard evaluations.

from graph_input import UnweightedGraph
from dfs import DFSResult, dfs, dfs_ordered


def transpose(graph: UnweightedGraph) -> UnweightedGraph:
    tick("graph_work")
    adj: Seq[Seq] = const(const(0))
    degree: Seq = const(0)
    u: Int = 0
    tick("graph_work")
    # @allocate_rows
    while u < graph.size:
        tick("graph_work")
        adj[u] = const(0)
        u += 1
        tick("graph_work")

    u = 0
    tick("graph_work")
    # @scan_sources
    while u < graph.size:
        tick("graph_work")
        neighbors: Seq = graph.adj[u]
        j: Int = 0
        tick("graph_work")
        # @reverse_edges
        while j < graph.degree[u]:
            tick("graph_work")
            v: Int = neighbors[j]
            adj[v][degree[v]] = u
            degree[v] = degree[v] + 1
            j += 1
            tick("graph_work")
        u += 1
        tick("graph_work")

    tick("graph_work")
    return UnweightedGraph(graph.size, adj, degree)


def scc(graph: UnweightedGraph) -> DFSResult:
    tick("graph_work")
    first: DFSResult = dfs(graph)
    tick("graph_work")
    reversed_graph: UnweightedGraph = transpose(graph)
    tick("graph_work")
    order: Seq = const(-1)
    i: Int = 0
    tick("graph_work")
    # @decreasing_finish_order
    while i < graph.size:
        tick("graph_work")
        order[i] = first.order[graph.size - 1 - i]
        i += 1
        tick("graph_work")
    tick("graph_work")
    return dfs_ordered(reversed_graph, order)

Selected references

Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, Introduction to Algorithms, fourth edition, MIT Press, 2022. Publisher; fourth-edition table of contents.
Lunjia Hu and Chutong Yang, Algo2Lean, software release v0.1.4. Lunjia Hu is the project creator. Hosted compiler; fixed release source.

13 thms3 active usersReviewed

Completed

Theoretical Computer Science·Captain: Chutong

CLRS — Chapter 20, §20.3: Depth-First Search — Correctness and Running TimeTextbook

A traversal forest that later algorithms can reuse

Depth-first search records a directed graph's exploration as a predecessor forest with discovery and finish times. Those outputs are used by algorithms that depend on the ordering of exploration, including strongly connected components. This mission targets CLRS, fourth edition, Chapter 20, §20.3; its companion component mission targets §20.5. Both references are fixed by the fourth-edition contents.

Ordered exploration and event times

The input is an unweighted directed graph on $n$ vertices, represented by integer adjacency arrays of arrays and a degree sequence. The number $m$ counts adjacency occurrences, including parallel arcs. Self-loops, disconnected graphs, and the empty graph are included. Prefix entries identify vertices; entries outside those prefixes have no specified meaning.

The reusable entry dfs_ordered receives an allocated root-order sequence whose first $n$ entries form a permutation of the vertices. The ordinary entry dfs constructs identity order and invokes it. Both use the same recursive visit routine. The result includes parent, discovery, finish, color, finish-order, and forest-component arrays, together with event, finished-vertex, and component counts.

Let $d[v]$ and $f[v]$ be discovery and finish times. An ancestor is defined by zero or more returned parent-child links. The output order lists vertices in increasing finish time. The companion SCC source explicitly reverses this prefix; the DFS result itself is not advertised as decreasing order.

Formalization targets

The headline CLRS.DFS.correctness_and_running_time concerns the complete identity-order call. It requests the complete forest and storage contract and one uniform runtime bound:

\exists C\in\mathbb N,\ C>0:\quad T\le C(1+n+m).

The constant is independent of the graph, heap, and initial clocks. Two supporting milestones make the counted work available for composition. CLRS.DFS.ordered_correctness_and_explicit_running_time covers every valid supplied root permutation and requests

T_{\mathrm{ordered}}\le 4+6n+2m.

CLRS.DFS.correctness_and_explicit_running_time covers the complete identity-order entry and requests

T_{\mathrm{dfs}}\le 7+8n+2m.

The identity-order wrapper contributes $3+2n$ marked events outside its ordered call, so the numerical targets compose directly. The explicit ordered budget also supports the second pass of SCC, where the caller supplies decreasing first-pass finish order. Each milestone includes the full postcondition and bounds that same terminating call. All three statements are open contributor tasks; no DFS correctness or runtime proof is supplied.

For every vertex, the returned event times must satisfy

1\le d[v]<f[v]\le2n,

with all $2n$ discovery and finish events distinct. Every non-root parent is an actual input predecessor. Ancestors correspond exactly to nested event intervals:

u\text{ is an ancestor of }v\iff d[u]\le d[v]\ \land\ f[v]\le f[u].

Unrelated intervals are disjoint. The white-path property additionally equates descendants of $u$ with vertices reachable from $u$ using only vertices whose discovery time is at least $d[u]$ . Root-order priority requires every earlier input-order vertex to finish before a later tree root is discovered. These are required structural properties of the ordered traversal; the generated Call relation fixes the concrete adjacency-scanning order.

The finish-order prefix is a vertex permutation with strictly increasing finish times. All output colors are black, the event clock is $2n$ , and the finished count is $n$ . Component labels are contiguous nonempty forest-tree identifiers; equal labels mean the same predecessor tree, not necessarily the same strongly connected component. Every target is an open theorem for contributors.

Reuse beyond a single traversal

A caller can retain the fresh result and continue with its heap, using the unchanged graph and all other pre-existing storage. The caller-facing predicate CLRS.DFS.Post before after result g inputOrder includes both the mathematical result and caller-storage preservation, matching the argument order and preservation placement used by BFS. It requires a fresh result object and six distinct fresh arrays. The goals combine that postcondition with the same terminating Call and its cost. Once proved, the explicit ordered contract supports a later caller's own root permutation without copying the DFS implementation or repeating its analysis.

This scope matters for the textbook SCC construction: its second traversal needs an order supplied by a first traversal. Merely returning timestamps would omit the structural properties that make those timestamps useful. Formalizing the complete contract makes the existing mathematical DFS analysis available through the generated public Call interface; no DFS proof is supplied here.

What must be justified

Valid parent edges and distinct timestamps alone do not identify a DFS forest. The open contract also relates ancestry, interval nesting, white paths, and the outer loop's priority. Directed reachability must not be confused with membership in one DFS tree: an edge may point into a tree that finished earlier. The event-count argument must cover recursive calls and the full outer scan, including already visited vertices, while the empty graph still returns allocated outputs.

Formalization scope

The graph and root-order sequence are read-only and may share input storage where their representation conditions permit it. Recursive calls share a fresh result object. Its component count equals the number of nonempty forest trees; labels follow tree-discovery order. The identity-order entry includes constructing that order in its own cost. The ordered entry's input permutation is supplied by its caller and is preserved.

The definitions use Algo2Lean v0.1.4, source commit 0f79d5e4242e6477531c9ee4410409f5b1c92f76, in project mode with standalone=false. The source root contains the explicitly named modules below; the entry file is identified for each mission. Raw generated definitions and platform import-path adaptations are recorded separately in the reproduction manifest. Stable generated definitions supply the public Call relations; optional named-line interfaces are not part of the goal statements.

The complete entry is dfs.algo, with graph_input.algo as its shared dependency. CLRS.DFS.Forest and Post expose the reusable mathematical and storage contracts. Contributions should establish these open obligations against the displayed source.

`graph_input.algo`

# Unweighted directed graph: vertices are 0, ..., size-1.
# Row adj[u][0:degree[u]] lists outgoing targets, with one slot per arc.
# Traversals require nonnegative degrees and in-range targets. Duplicate arcs
# and self-loops are allowed. The constructor stores the supplied references.

class UnweightedGraph:
    size: Int
    adj: Seq[Seq]
    degree: Seq

    def __init__(self, size: Int, adj: Seq[Seq], degree: Seq) -> None:
        self.size = size
        self.adj = adj
        self.degree = degree

`dfs.algo`

# CLRS fourth edition, Section 20.3: depth-first search.
# dfs_ordered takes a permutation of the vertices as its outer-loop order.
# Result.order lists vertices in increasing finish time; reverse its valid
# prefix before using it as the second-pass order in the SCC algorithm.
# graph_work counts the explicitly marked blocks and loop-guard evaluations.

from graph_input import UnweightedGraph


class DFSResult:
    color: Seq
    parent: Seq
    discover: Seq
    finish: Seq
    order: Seq
    component: Seq
    clock: Int
    finished: Int
    components: Int

    def __init__(self) -> None:
        tick("graph_work")
        self.color = const(0)
        self.parent = const(-1)
        self.discover = const(0)
        self.finish = const(0)
        self.order = const(-1)
        self.component = const(-1)
        self.clock = 0
        self.finished = 0
        self.components = 0


def visit(graph: UnweightedGraph, result: DFSResult, u: Int, label: Int) -> None:
    tick("graph_work")
    result.clock += 1
    result.discover[u] = result.clock
    result.color[u] = 1
    result.component[u] = label
    neighbors: Seq = graph.adj[u]
    j: Int = 0
    tick("graph_work")
    # @scan_neighbors
    while j < graph.degree[u]:
        tick("graph_work")
        v: Int = neighbors[j]
        if result.color[v] == 0:
            tick("graph_work")
            result.parent[v] = u
            visit(graph, result, v, label)
        j += 1
        tick("graph_work")
    tick("graph_work")
    result.color[u] = 2
    result.clock += 1
    result.finish[u] = result.clock
    result.order[result.finished] = u
    result.finished += 1


def dfs_ordered(graph: UnweightedGraph, order: Seq) -> DFSResult:
    tick("graph_work")
    result: DFSResult = DFSResult()
    i: Int = 0
    tick("graph_work")
    # @scan_roots
    while i < graph.size:
        tick("graph_work")
        u: Int = order[i]
        if result.color[u] == 0:
            tick("graph_work")
            visit(graph, result, u, result.components)
            result.components += 1
        i += 1
        tick("graph_work")
    tick("graph_work")
    return result


def dfs(graph: UnweightedGraph) -> DFSResult:
    tick("graph_work")
    order: Seq = const(0)
    i: Int = 0
    tick("graph_work")
    # @identity_order
    while i < graph.size:
        tick("graph_work")
        order[i] = i
        i += 1
        tick("graph_work")
    tick("graph_work")
    return dfs_ordered(graph, order)

Selected references

Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, Introduction to Algorithms, fourth edition, MIT Press, 2022. Publisher; fourth-edition table of contents.
Lunjia Hu and Chutong Yang, Algo2Lean, software release v0.1.4. Lunjia Hu is the project creator. Hosted compiler; fixed release source.

10 thms3 active usersReviewed

Completed

Theoretical Computer Science·Captain: Chutong

CLRS — Chapter 20, §20.2: Breadth-First Search — Correctness and Running TimeTextbook

Shortest paths in an unweighted graph

Breadth-first search computes which vertices can be reached from a source and the fewest edges needed to reach each one. Its predecessor information also records a shortest route. This mission belongs to the CLRS fourth-edition formalization series and targets Chapter 20, §20.2, including the algorithm's linear running-time guarantee. The section placement is given in the publisher's table of contents.

Graph, source, and outputs

A directed graph consists of vertices numbered $0,\ldots,n-1$ and outgoing adjacency rows. The field adj[u] is an integer sequence, and degree[u] determines its meaningful prefix. Every entry in that prefix must name a valid vertex. Let $m$ be the sum of these row lengths: parallel arcs count separately, and self-loops are permitted. An undirected graph is represented by symmetric directed entries, with both directions counted in $m$ .

The caller supplies a valid source $s$ , with $0\le s<n$ . Consequently, the source contract excludes the empty graph. The distance output uses $-1$ for an unreachable vertex; otherwise it records a nonnegative number of edges. The parent output uses $-1$ at the source and unreachable vertices. Other reached vertices have a predecessor. No particular choice between equally short routes is required.

Formalization targets

The headline CLRS.BFS.correctness_and_running_time asks for one actual terminating bfs.Call whose result satisfies correctness, preservation, allocation, and running time together. Write $D$ and $P$ for its output arrays and $s\leadsto v$ for a finite directed walk in the initial graph. For every vertex $v$ ,

D[v]=-1\iff\neg(s\leadsto v).

When $D[v]\ge0$ , the returned distance is attained and minimal:

\exists\text{ a walk }s\leadsto v\text{ of length }D[v],\qquad D[v]\le\text{length}(p)\text{ for every walk }p:s\leadsto v.

Moreover, $D[s]=0$ , $P[s]=-1$ , and each reached $v\ne s$ satisfies

P[v]\to v,\qquad D[P[v]]+1=D[v].

The running-time target is uniform over inputs, heaps, adjacency order, and initial counters:

\exists C\in\mathbb N,\ C>0:\quad T\le C(1+n+m).

Here $T$ is the increment of graph_work for that same call. The supporting milestone correctness_and_explicit_running_time retains the stronger numerical target

T\le 6n+2m+5

with the identical correctness and storage contract. Both statements are open; the numerical bound is a requested obligation, not an established result.

A reusable shortest-path contract

The result allows a later caller to read shortest distances and predecessor edges while continuing with the returned heap. The caller-facing predicate CLRS.BFS.Post before after result g source includes both the mathematical result and caller-storage preservation. It requires a fresh result object and two distinct fresh arrays, and preserves every previously allocated sequence, object, and collection. The goals combine that postcondition with the same terminating Call and its cost. Input graph storage therefore remains available for further traversals.

The textbook supplies the mathematical algorithm and analysis; the work here is their formal verification against a mutable source program. The finite-prefix specification gives callers useful facts without prescribing irrelevant sequence entries outside the vertex range. The source's discovery state is encoded by distance $-1$ , and its queue is a fresh sequence with head and tail indices. These are implementation choices within the stated BFS contract. The familiar correctness and complexity claims are discussed in CLRS, fourth edition, §20.2.

What must be justified

Returning a reachable predecessor alone does not establish shortestness. The contract also requires minimum distance and a correct unreachable sentinel. Cycles, repeated adjacency entries, and edges to vertices already discovered must not produce repeated queue insertion or invalidate the cost bound. An execution theorem must establish termination and the result together; reasoning only about executions that happen to return leaves a missing obligation.

Formalization scope

CLRS.GraphTraversal.GraphRep connects the input heap to a finite mathematical graph. Adjacency rows may share references, because traversal only reads them. Validity includes allocated rows and nonnegative degrees. The explicit initialization loop is counted even though the result constructor already gives the same default values. Outputs are fresh, so their mutation cannot change aliased input storage.

The definitions use Algo2Lean v0.1.4, source commit 0f79d5e4242e6477531c9ee4410409f5b1c92f76, in project mode with standalone=false. The source root contains the explicitly named modules below; the entry file is identified for each mission. Raw generated definitions and platform import-path adaptations are recorded separately in the reproduction manifest. Stable generated definitions supply the public Call relations; optional named-line interfaces are not part of the goal statements.

The complete source entry is bfs.algo; its shared dependency is graph_input.algo. Contributors may add proof lemmas or named observations while retaining the main source-facing contract.

`graph_input.algo`

# Unweighted directed graph: vertices are 0, ..., size-1.
# Row adj[u][0:degree[u]] lists outgoing targets, with one slot per arc.
# Traversals require nonnegative degrees and in-range targets. Duplicate arcs
# and self-loops are allowed. The constructor stores the supplied references.

class UnweightedGraph:
    size: Int
    adj: Seq[Seq]
    degree: Seq

    def __init__(self, size: Int, adj: Seq[Seq], degree: Seq) -> None:
        self.size = size
        self.adj = adj
        self.degree = degree

`bfs.algo`

# CLRS, fourth edition, Chapter 20, Section 20.2: breadth-first search.
# Adjacency rows contain vertex identifiers; degree gives each finite row length.
# The graph is read-only. A valid source lies between 0 and graph.size - 1.

from graph_input import UnweightedGraph


class BFSResult:
    distance: Seq
    parent: Seq

    def __init__(self) -> None:
        tick("graph_work")
        self.distance = const(-1)
        self.parent = const(-1)


def bfs(graph: UnweightedGraph, source: Int) -> BFSResult:
    tick("graph_work")
    result = BFSResult()
    vertex: Int = 0

    tick("graph_work")
    while vertex < graph.size:
        tick("graph_work")
        result.distance[vertex] = -1
        result.parent[vertex] = -1
        vertex += 1
        tick("graph_work")

    tick("graph_work")
    queue: Seq = const(-1)
    head: Int = 0
    tail: Int = 1
    queue[0] = source
    result.distance[source] = 0

    tick("graph_work")
    while head < tail:
        tick("graph_work")
        vertex = queue[head]
        head += 1
        index: Int = 0

        tick("graph_work")
        while index < graph.degree[vertex]:
            tick("graph_work")
            neighbor: Int = graph.adj[vertex][index]
            if result.distance[neighbor] == -1:
                tick("graph_work")
                result.distance[neighbor] = result.distance[vertex] + 1
                result.parent[neighbor] = vertex
                queue[tail] = neighbor
                tail += 1
            index += 1
            tick("graph_work")

        tick("graph_work")

    tick("graph_work")
    return result

Selected references

Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, Introduction to Algorithms, fourth edition, MIT Press, 2022. Publisher; fourth-edition table of contents.
Lunjia Hu and Chutong Yang, Algo2Lean, software release v0.1.4. Lunjia Hu is the project creator. Hosted compiler; fixed release source.

6 thms3 active usersReviewed

Completed

Algorithmic Game Theory Economics Operations Research·Captain: mikedeng1

Fair and Consistent Prize Allocation in Competitions 1: A Prize Allocation Rule Satisfies Order Preservation, Endowment Monotonicity and Consistency iff It Is an Interval RuleResearch Paper

Motivation

Sports leagues, tournaments, esports events and research competitions all end with a ranking of the participants and a sum of prize money to divide among them. How the money is split according to the ranking is a design choice with visible consequences: Winner-Takes-All schemes reward only the top, equal division ignores the ranking, and most real prize schedules sit in between. The economics of tournaments has studied prize structures as incentive devices, but it has not asked which schedules follow from principles of fairness and coherence.

Dietzenbacher and Kondratev (arXiv:2008.08985, Management Science 2023) take the axiomatic approach of fair allocation theory to this question. The central axiom is consistency, the variable-population principle that has characterized solutions in bankruptcy, taxation, bargaining and cooperative games (Aumann and Maschler 1985; Young 1987; Lensberg 1988; Thomson 1988; Hart and Mas-Colell 1989): if any subgroup of competitors pools its prizes and re-divides them among themselves by the same rule, nobody's prize changes. This mission formalizes the paper's first main result, which identifies exactly the rules that are consistent, weakly respect the ranking, and never take money away when the endowment grows.

Setting

Let $U$ be a countable set of at least three potential competitors. A competition is a triple $(N, \mathcal R, E)$ : a nonempty finite set $N \subseteq U$ , a ranking $\mathcal R : N \to \{1, \dots, |N|\}$ (a bijection; $i$ is ranked above $j$ if $\mathcal R(i) < \mathcal R(j)$ ), and a prize endowment $E \ge 0$ . For a nonempty $S \subseteq N$ , the subranking $\mathcal R_S$ lists $S$ in the order of $\mathcal R$ .

A prize allocation rule $\varphi$ assigns to every competition an allocation $\varphi(N, \mathcal R, E) \in \mathbb R^N_+$ with $\sum_{i \in N} \varphi_i(N, \mathcal R, E) = E$ . Two basic rules are Equal Division, $\mathrm{ED}_i = E/|N|$ , and Winner-Takes-All, $\mathrm{WTA}_i = E$ if $\mathcal R(i) = 1$ and $0$ otherwise.

The axioms of the main theorem are, for every competition and $i, j \in N$ :

order preservation: $\mathcal R(i) < \mathcal R(j) \Rightarrow \varphi_i(N, \mathcal R, E) \ge \varphi_j(N, \mathcal R, E)$ ;
endowment monotonicity: $E < E' \Rightarrow \varphi_i(N, \mathcal R, E) \le \varphi_i(N, \mathcal R, E')$ ;
consistency: for every nonempty $S \subseteq N$ and $i \in S$ ,

\varphi_i(N, \mathcal R, E) = \varphi_i\Big(S, \mathcal R_S, \sum_{j \in S} \varphi_j(N, \mathcal R, E)\Big).

Anonymity (prizes depend only on $|N|$ , the position and $E$ ) and endowment continuity are auxiliary properties.

An interval rule is given by pairwise disjoint open intervals $(a_k, b_k)$ , $k = 1, 2, \dots$ , with $a_k \in \mathbb R_+$ and $b_k \in \mathbb R_+ \cup \{+\infty\}$ . With $n = |N|$ and $r = \mathcal R(i)$ ,

\varphi_i(N, \mathcal R, E) = \begin{cases} a_k & \text{if } n a_k \le E \le (n - r + 1) a_k + (r - 1) b_k; \\ E - (n - r) a_k - (r - 1) b_k & \text{if } (n - r + 1) a_k + (r - 1) b_k \le E \le (n - r) a_k + r b_k; \\ b_k & \text{if } (n - r) a_k + r b_k \le E \le n b_k; \\ E / n & \text{otherwise.} \end{cases}

At endowment $n a_k$ everyone receives $a_k$ ; as $E$ grows, prizes are raised to $b_k$ one competitor at a time from the top; when $E/n$ lies in no $[a_k, b_k]$ the endowment is divided equally. ED has $a_k = b_k = 0$ ; WTA has $a_1 = 0$ , $b_1 = +\infty$ .

Formalization targets

Goal: Theorem 1 (p. 13)

\varphi \text{ satisfies order preservation, endowment monotonicity and consistency} \iff \varphi \text{ is an interval rule.}

Both directions are part of the goal.

Milestones, in the order the proof uses them

Lemma 1 (p. 10): endowment monotonicity implies endowment continuity.
Lemma 2 (p. 11): order preservation, endowment monotonicity and consistency imply anonymity.
Lemma 3 (p. 11): two endowment monotonic, consistent rules that agree on all two-competitor competitions agree on all competitions.
The "if" direction (p. 28): every interval rule satisfies the three axioms.
Step 1 (p. 28): for two competitors, if $\varphi(N, \mathcal R, x_1 + x_2) = (x_1, x_2)$ then $\varphi(N, \mathcal R, 2x_1) = (x_1, x_1)$ or $\varphi(N, \mathcal R, 2x_2) = (x_2, x_2)$ .
Step 2 (p. 29): the two-competitor restriction has the interval form (9),

\varphi(N, \mathcal R, E) = (E - a_k, a_k) \text{ on } [2a_k, a_k + b_k], \quad (b_k, E - b_k) \text{ on } [a_k + b_k, 2b_k], \quad (E/2, E/2) \text{ otherwise.}

Further statements of §2.3

Lemma 4 (endowment additivity ⇔ scale invariance) and Corollaries 1–4, which single out WTA (winner-loser strict order preservation), ED (strict endowment monotonicity), the Winner-Takes-Surplus rules (winner strict endowment monotonicity) and the pair {ED, WTA} (additivity) inside the family of Theorem 1, are included as supporting theorems.

Significance

Theorem 1 replaces three qualitative requirements by an explicit, countably parametrized family. The family is broad (Equal Division, Winner-Takes-All, the "dollar at a time" rule of Example 1, and their concatenations along the endowment axis), so the theorem is the starting point for every sharper characterization in the paper: strengthening one axiom at a time selects WTA, ED or a Winner-Takes-Surplus rule (Corollaries 1–4), and the second and third missions of this series replace consistency by its local and top variants to obtain the single-parametric and parametric rules. Lemma 3 also isolates a general principle: a monotone consistent rule is determined by its behaviour on pairs.

On the formal side, Mathlib and the platform have no model of variable-population allocation with rankings. This mission provides one (competitions over a universe, subrankings, prize allocation rules and the consistency family of axioms) together with the first machine-checkable statement of a characterization theorem in this area. The paper's proofs are complete; nothing here is open mathematics, and none of it has been formalized before.

Difficulty

Consistency only constrains the rule across competitions; on its own it says nothing about how two competitors share a given endowment. The obvious plan, to read the intervals off the two-competitor allocation path and extend by Lemma 3, fails at the first step: an order preserving, monotone two-competitor path need not be a staircase of horizontal and vertical segments, and nothing about two competitors alone forces it to be. The staircase shape (Step 1) is a genuinely multi-competitor phenomenon that also relies on anonymity, which is not assumed and must be derived (Lemma 2, a six-case analysis over how two competitions share competitors). The countable family of intervals then has to be extracted from the path (Step 2), including endpoints at $+\infty$ and the gaps where the rule is equal division.

Formalization scope

A ranked set $(N, \mathcal R)$ is a duplicate-free nonempty List U, best first; $|N|$ is its length and $\mathcal R(i)$ = pos l i, the 1-based index. The subranking is the sublist of members of $S$ . A rule is a function φ : List U → ℝ → U → ℝ; its values on non-rankings, negative endowments and non-members are irrelevant, and every axiom and the interval-rule definition quantify only over rankings, $E \ge 0$ and $i \in N$ . The standing assumption on $U$ is carried as [Countable U] plus three distinct elements on every statement whose proof needs a third competitor (Lemma 2, Steps 1–2, Theorem 1, Corollaries 1–4); it is unnecessary for Lemmas 1, 3, 4 and for the "if" direction. Interval endpoints $b_k$ are extended reals (EReal), the intervals are indexed by $\mathbb N$ (finitely many intervals are padded with $a_k = b_k = 0$ ), $a_k \le b_k$ , and disjointness is of the open intervals. The four cases are implications; "otherwise" means $E \notin [n a_k, n b_k]$ for every $k$ .

The following formalizations would trivialize or change the theorem and are ruled out: a fixed competitor set or a finite $U$ of size two (consistency then has no reduced competitions to act on, and Corollary 1(i) fails); consistency restricted to two-element subsets or to some subsets only; a one-directional goal; interval rules without $+\infty$ endpoints (which excludes WTA and makes the "only if" direction false); and anonymity as an extra hypothesis.

The definitions are reusable by the two companion missions of this series (local and top consistency), which use the same model. Contributions welcome: proofs of the milestones, a general lemma that consistent monotone rules are determined by pairs, and proofs that ED, WTA and the Example 1 rule are interval rules.

Selected references

B. J. Dietzenbacher and A. Y. Kondratev, Fair and Consistent Prize Allocation in Competitions, arXiv:2008.08985v2, 2021; Management Science 69 (2023). https://arxiv.org/abs/2008.08985, https://doi.org/10.1287/mnsc.2022.4520
R. J. Aumann and M. Maschler, Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36 (1985), 195–213. https://doi.org/10.1016/0022-0531(85)90102-4
W. Thomson, A study of choice correspondences in economies with a variable number of agents, Journal of Economic Theory 46 (1988), 237–254. https://doi.org/10.1016/0022-0531(88)90131-7
T. Lensberg, Stability and the Nash solution, Journal of Economic Theory 45 (1988), 330–341. https://doi.org/10.1016/0022-0531(88)90274-8
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Economics Operations Research Optimization·Captain: mikedeng1

Digital Privacy I: The Profit-Maximizing Data Collection and Protection Have a Closed Form, and Both Weakly Increase with the Data-Driven Revenue Share PResearch Paper

Why digital businesses choose how much data to keep and how well to guard it

Platforms that earn money from their users' activity make two linked decisions about the data that activity generates: how much of it to record and store, and how much to spend protecting what they store. Stored data lets a business tailor its services and sell data-driven products, but it also attracts adversaries (hackers, data brokers, unauthorized third parties) whose access harms users, who then use the service less. Fainmesser, Galeotti and Momot, Digital Privacy (Management Science 69(6), 2023, doi:10.1287/mnsc.2022.4513; author manuscript SSRN 3459274), build a tractable model of this trade-off and solve it in closed form. Their first main result, Theorem 1, gives the profit-maximizing data strategy explicitly and shows that a business whose revenue depends more on data both collects more and protects more. The result is the basis for the paper's policy analysis (data protection standards, Pigouvian taxes on stored data) and for its comparison with the social optimum, which is the subject of a companion mission.

Setting

A unit mass of identical users each choose an activity level $a \ge 0$ . The business stores a fraction $\xi \in [0,1]$ of every user's activity and protects it at level $C \ge 0$ , at cost $\psi C$ . With average activity $\bar a$ and a mass $\omega$ of active adversaries, a user's utility is (display (1))

U(a, \bar a) = a - \tfrac12 a^2 + \beta a\bar a + (\rho - \omega) a\xi ,

where $\beta \in [0,1)$ measures network effects and $\rho \in [0,1)$ the users' benefit from tailored services built on stored data. Adversaries have access costs $\gamma C$ with $\gamma$ uniform on $[0, M]$ and gain $\bar a\xi$ ; letting $M \to \infty$ , the mass of active adversaries is $\omega = \xi\bar a/C$ . Proposition 1 shows that for every data strategy $(\xi, C)$ with $C > 0$ this users–adversaries subgame has a unique equilibrium,

\bar a^*(\xi, C) = \frac{C(1+\rho\xi)}{C(1-\beta)+\xi^2}, \qquad \omega^*(\xi, C) = \frac{\xi\bar a^*(\xi, C)}{C},

with consumer surplus $\mathrm{CS}(\xi, C) = \tfrac12\bar a^*(\xi, C)^2$ ; when nothing is stored, $\bar a^*(0, C) = 1/(1-\beta)$ . The business's profit is (display (2))

\Pi(\xi, C) = (1-P)\,\bar a^*(\xi, C) + P\,\xi\bar a^*(\xi, C) - \psi C ,

where the revenue model $P \in [0,1]$ is the share of revenue that is data-driven: $P = 0$ is a purely usage-driven business, $P = 1$ a purely data-driven one. A business optimum is a strategy $(\xi, C) \in [0,1] \times [0,\infty)$ that maximizes $\Pi$ over $[0,1] \times [0,\infty)$ .

The Lean development uses the same names: abar, omegaStar, CS, profit, Feasible, IsBusinessOpt, the best-response quantities Ch ( $C_h$ ), kappa ( $\kappa$ ), Cl ( $C_l$ ), xiBR, the threshold psiL ( $\psi_L$ ), the interior root xiI ( $\xi_i$ ), and the closed forms xiStar ( $\xi^\star$ ) and CStar ( $C^\star$ ).

Formalization targets

Goal: Theorem 1

For fixed $0 \le \beta < 1$ , $0 < \rho < 1$ , $\psi > 0$ and every $P \in (0,1]$ , let

\xi^\star = \max\Big\{0, \min\Big\{1, \frac{1}{2P\rho}\Big(-(1-P)\rho - P + |P-(1-P)\rho|\sqrt{\frac{\psi}{\max\{0,\psi-P\rho\}}}\Big)\Big\}\Big\},

C^\star = \frac{\xi^\star}{1-\beta}\Big(-\xi^\star + \sqrt{\tfrac1\psi(1+\rho\xi^\star)(1-P(1-\xi^\star))}\Big).

Then $(\xi^\star, C^\star)$ is a business optimum, and both $P \mapsto \xi^\star$ and $P \mapsto C^\star$ are weakly increasing on $(0,1]$ .

Milestones

Proposition 1 (p. 10): the subgame equilibrium (3) is unique, and consumer surplus is (4).
Proposition 1 (a)–(b) with footnote 15: $\bar a^*$ and $\xi\bar a^*$ increase in $C$ , and are single-peaked in $\xi$ .
Lemma 1 (p. 29): for fixed $\xi$ , the unique optimal protection is $\max\{C_h(\xi), 0\}$ .
Lemma 2 (p. 29): for fixed $C > 0$ , the unique optimal collection is $-\kappa(C) + \sqrt{\kappa(C)^2 + C(1-\beta)}$ if $C < C_l$ and $1$ if $C > C_l$ .
App. A.2 summary (p. 31): with $P\rho \le \psi_L \le \psi_H$ , the business optimum is $(1, C_h(1))$ for $\psi \le \psi_L$ , $(\xi_i, C_h(\xi_i))$ with $\xi_i \in (0,1)$ for $\psi_L < \psi < \psi_H$ , and $(0,0)$ for $\psi \ge \psi_H$ .

Significance

Theorem 1 is the statement that a business's revenue model alone orders its data practices: holding users and costs fixed, moving toward data-driven revenue never lowers collection and never lowers protection. The paper uses it to explain why consumer surplus is maximized at an intermediate revenue model $P = \rho/(1+\rho)$ (Proposition 2), and as the benchmark against which it evaluates regulation in §6. The closed form also makes the regime structure explicit: no collection when protection is expensive, full collection when it is cheap, and an interior solution in between.

The result is proved in the paper (App. A.2–A.3, with the proofs of Lemmas 1–2 in Online App. B), but parts of that proof are informal: several steps are "easy to check" or "can be derived similarly", one case is treated and the other left to the reader, and the second-order conditions are referred to a general lemma in Online App. C.5. No machine-checked version exists. A formal proof certifies the closed form, including its boundary cases, which the printed formula handles through an infinite square root.

Difficulty

The profit is not jointly concave, so first-order conditions do not settle the problem. The argument intersects the two best responses (Lemmas 1–2), and two candidate strategies survive in the intermediate regime: $(0,0)$ and the interior point. Choosing between them requires comparing profits across all $\psi$ and showing the comparison flips exactly at $\psi_H$ . Monotonicity in $P$ is harder still: as $P$ moves, so do both thresholds $\psi_L(P)$ and $\psi_H(P)$ , so a fixed $\psi$ can cross from one regime to another, and $\xi^\star$ and $C^\star$ must be shown monotone within each regime and across the switches. Within the interior regime $C^\star = C_h(\xi^\star)$ depends on $P$ both directly and through $\xi^\star$ , so differentiating the closed form does not by itself give the sign.

Formalization scope

All parameters are explicit real numbers; every optimum is taken over $\xi \in [0,1]$ , $C \ge 0$ , never over $\mathbb R^2$ . The page's ranges are $0 \le \beta < 1$ , $0 \le \rho < 1$ , $\psi \ge 0$ , $0 \le P \le 1$ . The following are added and disclosed in each item: $\psi > 0$ everywhere (otherwise $C_h$ and (6) divide by zero and no maximizer exists); $P > 0$ and $\rho > 0$ in Theorem 1 and the App. A.2 summary (formula (5) divides by $2P\rho$ ); $C > 0$ in Proposition 1 and Lemma 2 (the adversary mass $\xi\bar a/C$ ); $\rho > 0$ in Lemma 2.

Three conventions are fixed. First, $\bar a^*(0, C) = 1/(1-\beta)$ for every $C$ , the paper's value at $(0,0)$ where (3) is $0/0$ . Second, $\xi^\star = 1$ whenever $\psi \le P\rho$ , the page's own reading of the infinite square root in (5). Third, $\psi \ge \psi_H$ is written as $(P+(1-P)\rho)^2 \le 4(1-P)\psi$ , so that $\psi_H = +\infty$ at $P = 1$ . Using Lean's $0/0 = 0$ at either of the first two places makes Theorem 1 false, not trivial, and is ruled out by these definitions. Theorem 1 asserts that $(\xi^\star, C^\star)$ is a maximizer: uniqueness fails at $P = \rho/(1+\rho)$ , $\psi = \rho^2/(1+\rho)$ . Monotonicity is MonotoneOn on $(0,1]$ with $\beta, \rho, \psi$ fixed; a Monotone statement with junk values outside $(0,1]$ would not be the paper's claim.

A complete development needs only real analysis from Mathlib: one-variable calculus for the best responses, square-root and absolute-value manipulations for the closed form, and a profit comparison across regimes. The best-response lemmas are reusable for the companion mission on the social optimum. Contributions of proofs of any milestone, and of auxiliary lemmas such as $C^\star = C_h(\xi^\star)$ , are welcome.

Selected references

I. P. Fainmesser, A. Galeotti, R. Momot, Digital Privacy, Management Science 69(6):3157–3173, 2023. doi:10.1287/mnsc.2022.4513; author manuscript (This Draft: April 2022), SSRN 3459274.
E. L. Glaeser, J. A. Scheinkman, Non-Market Interactions, NBER Working Paper 8053, 2000 (existence and uniqueness of the users' response, used in App. A.1). doi:10.3386/w8053

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