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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
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.
Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
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.
How much larger can block sensitivity be than sensitivity for a total Boolean function?
For a total Boolean function , a block of coordinates is sensitive at if flipping every bit in that block changes . Let count sensitive singletons and be the maximum number of pairwise disjoint sensitive blocks. Define and .
This campaign seeks larger exponents for which there exists an infinite family of total Boolean functions , indexed by , satisfying
Here denotes a quantity tending to zero as . Higher is better. The classical exponent is , and Huang's bound gives a ceiling of .
OpenAI's construction achieves an exponent strictly greater than .
One sufficient way to establish an exponent is to find a single total Boolean function with and . Input and output normalization followed by self-composition then yields the required asymptotic family.
Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.
Harvey and van der Hoeven established an algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.
For two -bit integers, the target is
A positive beats asymptotically; larger 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.
Classical algorithms solve 3SUM in time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic 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 -bit words, and pursues smaller exponents.
Classical algorithms solve all-pairs shortest paths in time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic algorithm. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.
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.
The sharp Hlawka inequality for Schatten -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 . We conjecture that the same formula holds for all .
What is the smallest cutoff for which this formula holds for every real ?
References:
Is every odd number a sum of primes? This campaign tracks formalized proofs of the smallest that suffices.
Schnirelmann (1930) showed some finite works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved unconditionally. Helfgott (2013) proved that every odd number greater than is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: is neither prime nor + prime.
Schoolbook matrix multiplication takes operations. The exponent is the infimum of all such that two matrices can be multiplied in arithmetic operations; trivially , and is conjectured but open.
Strassen gave the first nontrivial bound, , in 1969, and introduced the laser method in 1986 to reach . Coppersmith and Winograd's 1990 bound of stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached in 2025, and the current record is , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?
Many social services are rationed by waiting rather than by price: public health clinics, food banks, shelters, legal aid. Users who need the service most often have no outside option, while users with less need could be served elsewhere but join the queue anyway and lengthen everyone's wait. The provider usually cannot force anyone to leave, and charging money defeats the purpose of the service. What the provider does control is information: it observes the queue and decides what to tell arriving users about it.
Anunrojwong, Iyer and Manshadi (2023) model this as a Bayesian persuasion problem in a queue, building on the revenue-maximization model of Lingenbrink and Iyer (2019, Operations Research 67(5):1397–1416). Their first structural result, Theorem 1, says that whatever trade-off the provider makes between the two user groups, the efficient way to signal is a simple threshold rule, and that the threshold never exceeds the one under full information. This mission formalizes Theorem 1 and the steps of its proof.
A single server serves users first-come-first-served; service times are exponential with rate one. Users of two types arrive by independent Poisson processes: low-need users (type ) at rate and high-need users (type ) at rate , with total rate . A type- user who joins with users already present receives utility , where , for large , is strictly decreasing, and the increments are non-increasing (Assumption 1). High-need users always join; low-need users take an outside option worth unless joining is at least as good in expectation.
By the revelation principle and Lemma 1 of the paper, an obedient binary signaling mechanism is identified with its steady-state distribution of the queue length. The admission policies are the probability distributions on satisfying the detailed-balance constraints
The signaling mechanisms additionally satisfy the obedience constraints
The welfare of the two groups is and . A mechanism Pareto-dominates if for both , with at least one strict inequality; is Pareto-efficient within if no dominates it (Definition 1).
A distribution has a threshold structure (Definition 2) if for some , for and for : low-need users are told to join below , to leave above , and to join with probability at . The threshold is . The full-information threshold is the smallest with : when the queue is revealed, low-need users join exactly below it.
For every that is Pareto-efficient within , there is a finite such that
No weight is fixed: the statement covers the whole Pareto frontier.
Theorem 1 reduces an infinite-dimensional design problem (a distribution on subject to obedience) to a one-parameter family of threshold policies . Every later result of the paper is computed in that family: the bound of the optimal signaling mechanism against full information when all users are low-need (Proposition 1), the condition under which information design Pareto-dominates full information (Theorem 4 and Proposition 3), and the conditions under which signaling achieves the first-best admission policy (Theorems 5 and 6). The bound also has a direct reading: efficient persuasion only ever discourages low-need users relative to full information, never encourages them.
The theorem is proved in the paper (App. A). This mission produces a machine-checked version of the proof, on top of a reusable Lean model of obedient signaling in an -type queue with two user types. To our knowledge no part of this model or of its results has been formalized before.
The statement quantifies over the full Pareto frontier, so it cannot be obtained from first-order conditions of a single weighted objective. Both steps are perturbation arguments on infinite sequences: each must build an explicit competitor, keep it inside the detailed-balance constraints and both obedience constraints at once, and show weak improvement for one group with strict improvement for the other. The obedience constraint (LEAVE) cuts against (JOIN): moving mass toward shorter queues raises but can raise , and the bound on is where is needed. Since is allowed, the geometric tails are not automatically summable and the infinite sums have to be handled with care. The threshold must be excluded explicitly; it is not ruled out by the structure argument alone.
All definitions live in NeedPersuasion.Threshold (file Def_NeedPersuasion_Threshold_Model). Queue lengths are ℕ, everything else is ℝ; the two types are two fields of a Model structure. The standing assumptions are the predicate Model.Standing: , , , , for some (equivalent to for a strictly decreasing sequence, and allowing ), both utilities strictly decreasing, and the -increments non-increasing. Every theorem assumes it; no other hypothesis is added, and in particular is not assumed.
Committed conventions:
HasSum, and finite welfare: and summable. The paper treats the welfare sums as real numbers; without this, a policy with welfare (possible at ) would receive welfare from Lean's tsum.HasThresholdStructure; HasThresholdAtMost π y asks for a finite with .ParetoEfficientIn), never maximization of .Trivializing encodings are ruled out: the threshold-structure predicate is not satisfied by every admission policy, efficiency is taken over all of (including the competitors the proof builds), and no sum whose value is compared can be a divergent tsum. A sorry-free sanity file exhibits a model satisfying Standing whose full-information mechanism lies in , so the hypotheses are not vacuous.
A complete development needs manipulation of tsum over shifted and truncated sequences, geometric tails, and first-order stochastic dominance for sequences on ; these pieces are reusable for the other missions of this series (missions 2–4 restate the same model). Contributions of proofs of individual milestones, or of general lemmas on detailed-balance sequences, are welcome.
Many models in statistics, machine learning and engineering are nonconvex, nonsmooth and coupled by constraints: a sparsity penalty such as SCAD, MCP or capped- on one block of variables, a smooth loss coupling the blocks, and a nonlinear equality constraint tying them together. Augmented-Lagrangian methods and ADMM-type splittings are the workhorses for such problems, but most convergence theory for them assumes linear constraints or convexity. Zhu, Zhao and Zhang (arXiv:2007.12219v1, published in Mathematics of Operations Research, 2024) propose a first-order primal-dual method, NAPP-AL (nonconvex auxiliary problem principle with augmented Lagrangian), for a nonconvex problem with a nonlinear coupling constraint, and prove that it reaches approximate stationarity at a sublinear rate. This mission targets that rate result and the chain of lemmas behind it.
Fix dimensions . Problem (P) is
where and are differentiable with Lipschitz gradients, is lower semicontinuous and possibly nonconvex and nonsmooth, is closed and convex, is Lipschitz with constant , each component of has a Lipschitz gradient (constants , ), and is an matrix with . These are Assumption 1 (H1)–(H7); (H7) asks to be bounded below and coercive on the feasible set .
The augmented Lagrangian of (P) at is
Its stationarity set (p. 6) collects the triples with, for ,
where is the limiting subdifferential and the normal cone of . A stationary point is a with , and , with .
NAPP-AL takes a -strongly convex with Lipschitz gradient (Bregman distance ), a penalty (Assumption 2) and . With it sets to a global minimizer over of the linearized model
to the minimizer of a strongly convex quadratic, and . The step lies in , with given by the explicit formula (8), which depends on . The analysis runs through the potential
with explicit constants .
Under Assumptions 1 and 2, with set by (8),
The result gives NAPP-AL an iteration complexity of to reach an -stationary point (Remark 4.1), for problems with a nonlinear, nonconvex coupling constraint and a nonsmooth nonconvex regularizer, using only gradient evaluations and one proximal-type subproblem per iteration. It is also the base layer for the paper's later results: linear convergence under an error bound (Theorem 5.1) and the sufficient conditions for that error bound (Propositions 6.1–6.2) all start from Theorem 4.1 and Lemma 4.3.
The result is proved in the paper; nothing here is formalized elsewhere. A formal development adds a machine-checked Lyapunov analysis of an augmented-Lagrangian method with nonlinear constraints, with every constant explicit. It also records a correction: Proposition 3.1 as printed is false by a factor of at most , and the corrected bound is what the paper's proof establishes.
The difficulty is the dual variable. The multiplier update has no step size of its own, so the augmented Lagrangian itself need not decrease along the iterates, and the obvious descent argument for fails; the step of (8) depends on the current multiplier estimate , so even the step sizes are not uniformly bounded a priori. Boundedness of the iterates is not a hypothesis, and the iterates are infeasible, so coercivity of on the feasible set does not apply to them directly. The rate needs the residual bound of Proposition 3.1 with a constant uniform in , while depends on and . Passing to cluster points involves the limiting subdifferential of a nonconvex, nonsmooth , which is not closed under arbitrary convergence of its arguments.
EuclideanSpace ℝ (Fin k). The norm of a triple is the Euclidean norm of the stacked vector, defined explicitly; Mathlib's sup norm on products is never used. is a continuous linear map with its operator (spectral) norm and adjoint; is the Rayleigh minimum over the unit sphere.Metric.infDist (zero on ) is not used; with it, every residual bound would hold whenever is empty. The goal is stated as in , not as a big-O bound.NonconvexSplitting.Shared.LimitingSubdiff) and the normal cone of , which is empty off , so that stationarity forces .LocallyLipschitzOn); (full column rank, invoked in the proof of Lemma 4.2); (H7) is coercivity on the feasible set only.Contributions welcome: proofs of the milestones in any order, general facts about limiting subdifferentials and normal cones of convex sets (sum rule with a locally Lipschitz function, closedness under attentive convergence), and the "non-increasing summable sequence is " lemma. These are reusable beyond this mission. The source is the preprint arXiv:2007.12219v1, not the published version; all labels and constants are those of v1.
Retailers and restaurants increasingly deliver orders from their own stores within an hour or less. Orders arrive throughout the day, a fixed pool of drivers is based at the store, and in each dispatch wave the operator must decide how many drivers to send out now and along which routes. Sending many drivers now gives short delivery times for the current orders but leaves few drivers for orders that arrive while those drivers are still on the road. Liu and Luo (arXiv:2107.13058, Manufacturing & Service Operations Management, 2023) model this trade-off as a finite-horizon stochastic dynamic program and propose a rollout policy built on a tractable approximation of its cost-to-go. Their Theorem 1 justifies the approximation: with suitable parameters it lies below the true expected cost-to-go, with other parameters it lies between that cost and the cost of any feasible simple policy, and in the last period it is exact. This mission formalizes that theorem.
A store (the depot ) serves a finite set of customer locations over periods with strictly increasing decision epochs . A fleet of drivers carries at most items per trip; travel times , an on-site service time and a delivery target are given. At epoch the operator observes the demand of period : the set of locations with an order and the quantities , drawn from a finitely supported law , independently across periods.
The driver status records how many drivers are still en route in each period because of earlier dispatches, so drivers are available at . A decision is a set of depot-to-depot routes , one per dispatched driver. It is feasible if it uses at most drivers, serves every current order exactly once, loads at most items per route, and delivers every order within . The delivery times are and ; the cost of is . A driver whose route lasts longer than is counted as en route in period ; this is the transition (9) from to .
The cost-to-go is the minimum over feasible of the period's cost plus at the resulting status, with the period cost alone ((7)–(8)). The single-period cost is the minimum cost of serving the demand with at most drivers.
The approximation (10) replaces the future by single-period problems coupled only through driver counts:
where and the nonnegative parameters , , estimate how many of the drivers dispatched in period are still en route in period . A static myopic policy dispatches a state-independent number of drivers in each period and routes them to minimize the single-period cost; it is feasible from if it never asks for more drivers than are available and its single-period problems are always feasible.
For and a status with :
and for every .
The rollout policy AJRP of the paper solves, at each epoch, the current routing problem together with as a stand-in for the future. Theorem 1 is what connects that stand-in to the dynamic program: it shows that the en route parameters move the approximation between a valid lower bound and the cost of an implementable policy, and its exactness part yields the paper's Corollary 1 (AJRP is optimal for ). Lemma 1 shows the approximation keeps the monotonicity of the true value in the driver status.
The paper's proofs are in its Online Appendix (Appendix D). They are short but informal, and the formal statement here differs from the printed one in three pinned-down respects: the single-period cost is read with at most drivers, as the proofs use it; the upper bound is capped by the policy's cost, since otherwise it is vacuous; and the parameter choice printed for the upper bound has to be corrected in the last period. No part of the result has a machine-checked proof. A formal proof makes these readings definitive, and it produces a reusable Lean model of multi-period dispatching with routes, en route drivers and a Bellman recursion over a random demand.
Every part is a backward induction on the remaining periods, but the inductions run over a state that grows: the status depends on the durations of the routes chosen in period , and the infimum in (7) is over finite lists of routes, not over a finite action set. The lower bound needs that a larger status can only raise the cost; the upper bound needs that a policy which is feasible on its reachable states stays feasible after every realized demand. Exchanging the expectation with the minimum over the driver count in the last period is not a general fact and holds only because the single-period cost is nonincreasing in the number of available drivers. The obvious choice of parameters for the upper bound, the one printed in the proof, does not bound the last period's count and fails.
Locations are a type with the depot as none; periods are 1-based natural numbers and every statement carries . Costs and values lie in ; every minimum is an infimum, equal to on an empty feasible set; expectations are sums against the probability mass function of . A decision is a list of nonempty routes covering the current orders exactly once. The parameters are nonnegative reals by their type, with the diagonal written into (10). The available drivers in the dynamic program are in ; in APT the right-hand side is the real .
Standing assumptions, shared by every statement: ; strictly increasing epochs; finitely supported, period-independent demand laws; off . Added hypothesis: for (the en route drivers belong to the fleet). Disclosed readings: travel times do not depend on the period; delivery times follow the recursion of Appendix C with equality, starting from the depot leg; a policy's expected cost is defined recursively along its decisions.
Ruled out: defining or through the APT objective (exactness would become definitional); decision sets that may leave orders unserved (every cost would vanish); an upper bound without the cap by the policy's cost; negative parameters .
The development needs the model and its recursion (provided as one definition file), monotonicity of in , and induction principles for the well-founded recursions. Proofs of the milestones in any order are welcome; Lemma 1 is not used by the paper's proof of Theorem 1 but is a natural first step.
Repair access changes more than the cost of fixing a broken product. It also changes how many used products reach a resale market, what those products sell for, and whether a manufacturer prefers to sell another new unit or earn a repair margin. Jin, Yang, and Zhu study these connected pricing decisions in a two-period durable-goods model in Right to Repair: Pricing, Welfare, and Environmental Implications. Their Proposition 1 identifies the manufacturer's prices and the resulting market outcome when independent and manufacturer repair restore the same product quality. This mission formalizes that proposition, including all seven rows of its Table 1.
The question is relevant to work on durable-goods markets because a repaired product can compete with a new one in the next period. A lower independent-repair cost can affect the manufacturer's behavior even if consumers do not choose independent repair in the final equilibrium. The paper's setting makes that effect precise through the used-product price, new-product demand, and the manufacturer's repair decision.
A new product provides one period of use and then fails with probability . If it remains functional, or if a failed unit is repaired, it can be used for one more period as a used product. After two periods of use it has no value. Consumers have types uniformly distributed on . A new product gives per-period gross utility ; a functional used product gives , with . This mission takes the paper's identical-repair-quality case and zero resale transaction cost .
The manufacturer chooses a new-product price and a manufacturer repair price . Producing a unit costs and repairing it costs . A consumer can instead pay for independent repair. A manufacturer repair price of means that service is unavailable. The used-product price emerges from the secondary market. Consumers who face a failure repair with probability : they repair whenever the resale gain exceeds , do not repair when it falls short, and may mix at equality.
Consumers can be inactive, buy a used unit every period, or buy a new unit every period. Their corresponding per-period utilities are , , and
The thresholds partition these choices. In a market with a positive used price, used demand equals the supply generated by new buyers: . At , used supply may exceed demand. The manufacturer's per-period profit combines new-unit margin with its repair margin on units it actually repairs. These primitives and Assumption 1 appear in §4.1 of the paper.
Lemma 1 states that consumer utility has the three-segment form above. The appendix then compares reduced pricing problems for no repair, full repair, and partial repair. One milestone records the full-repair optimum:
at a feasible full-repair maximizer. Other milestones record the profit-gap identities, the ordering of two production-cost cutoffs, and the appendix results for the two higher ranges of independent-repair cost. Their statements follow Online Appendix G.
The goal is the complete equilibrium characterization of Proposition 1 and Table 1. When , the optimal outcome changes at and from no repair to partial repair to full repair. When , three cutoffs give four rows: two no-repair regimes, partial repair, and full repair. Each row specifies , , , and ; the partial-repair rows require . The target fixes the paper's explicit formulas rather than only the number or order of the regimes.
The theorem gives a simultaneous answer for manufacturer pricing and the secondary-market outcome. It identifies when manufacturer repair is unused, offered to some consumers, or offered to all consumers facing failure. In the low- case it also distinguishes two no-repair regimes with different used prices. Those distinctions matter for subsequent welfare and environmental comparisons in the paper: a price change can alter new production and resale quantities even when the repair count does not change.
The paper proves Proposition 1 analytically; this mission asks for a machine-checked proof of its exact statement. A complete development would also establish reusable interfaces for a market-clearing equilibrium with a zero-price excess-supply corner, price optimization across several feasible regimes, and comparisons between algebraic cutoff formulas. The draft theorem statements are open Lean goals; the local sanity check proves that one specified market equilibrium is inhabited, not the proposition itself.
The manufacturer's choice affects both the direct repair margin and the resale price that determines demand for new and used goods. Optimizing the new-product price in isolation therefore does not give the global result. The appendix's candidate solutions come from distinct repair and market-clearing regimes, and their feasible sets meet at cutoffs where different prices can tie. A global claim has to compare each candidate against all admissible price pairs and their equilibria. The radical and rational cutoff formulas add boundary cases: a formula may be used only in the regime where its denominator and square-root argument have the intended meaning.
Lean uses real prices, costs, probabilities, and consumer thresholds. The standing assumptions are , , , and the paper's Assumption 1, and . It also pins the three unit costs to , consistent with the paper's nonnegative-cost appendix domain. An unavailable manufacturer repair service is represented by none, not by an arbitrary large real number. A real repair price must be nonnegative.
The equilibrium predicate requires actual utility-maximizing consumer segments, the repair rule, and market clearing. It allows excess used-product supply only at . It also admits the appendix's zero-demand outcome, in which neither product yields positive utility, no units sell, and profit is zero. An optimal outcome contains an equilibrium and beats every equilibrium at every admissible competing price pair. Thus an empty equilibrium relation cannot make the goal true. At cutoff ties the theorem asserts that the displayed prices are an optimum, without claiming uniqueness. The two partial-repair rows assert strict bounds on , excluding a vacuous value.
The appendix's Problems 1, 4, and 5 are separate reduced objectives and feasible sets. The rational partial-repair objective is accompanied by its polynomial continuation for the profit-gap identity; that continuation is used only where the printed rational form could produce . The source's second identity in (G.41) omits a factor in its denominator. The Lean statement uses the corrected algebraic identity, while the milestone preserves the printed line. Contributions that prove the consumer, reduced-problem, cutoff, and global-optimality milestones are all within scope. The remaining propositions on welfare, quality differences, and environmental effects are outside this mission.
b869b33a21376a298a2355fb4de697d59b1362d5895f166f3255348812bc1763.On-demand delivery platforms for meals and groceries promise delivery within minutes, and the main operational lever they control is how a courier is dispatched. A courier can carry each order as soon as it is ready (dedicated delivery), or wait until two orders are available and deliver them in one trip (batch delivery). Batching saves travel per order but makes the first customer wait for a second order to arrive. Which mode gives the shorter wait, and hence the higher revenue, is a question practitioners face daily; it is the subject of M. Chen and M. Hu, Courier Dispatch in On-Demand Delivery (SSRN 3675063; published in Management Science, DOI 10.1287/mnsc.2023.4858).
The paper answers the question in closed form for a stylized disk model. This mission formalizes its first headline result, for the case in which the demand rate is fixed exogenously (for example by a growth target or by kitchen capacity), as stated in Propositions 1 and 2 of the authors' version dated 2022-09-21.
A vendor sits at the centre of a disk of radius and employs one courier with unit travel speed, so travel time equals travel distance. Customer orders arrive as a Poisson process of rate , the demand rate, and each customer's location is independent and uniformly distributed on the disk.
Under dedicated delivery each trip is a round trip to one customer, of length with uniform on the disk. Its moments are and , so the courier is an M/G/1 server with
where is the service rate and the summed coefficients of variation. The paper defines the expected wait under dedicated delivery as the M/G/1 in-line delay
Under batch delivery the courier leaves only with two orders and visits both, so a trip has length for two independent uniform points, with . This gives
and a variability coefficient that the paper obtains by numerical integration. The paper defines the batch wait as the time for a second order to arrive, plus Kingman's approximation of the E/G/1 in-line delay, plus the expected delay of being second in a batch:
Customers have valuations with c.d.f. on and a wait cost per unit time; at price the potential arrival rate thins to . The vendor's revenue rate at demand rate and expected wait is
Dedicated delivery is stable exactly when ; batch delivery when .
For every there is a constant such that for all , with ,
and the same two equivalences hold for the revenues and , for every , every with , and every . The paper reports ; the goal asserts only the existence and the location in .
The result says that, with demand fixed, the choice between dedicated and batch delivery depends on the market only through the product : dedicated delivery is better when demand is low or the area small, batching when demand is high or the area large. The threshold is a single universal number, independent of the valuation distribution and of the wait-cost rate, because at a fixed demand rate the strategy with the shorter wait always earns more. This is the efficiency-driven benchmark against which the paper's second main result (endogenous demand in a crowded market, where the order reverses) is contrasted.
To our knowledge none of this has been machine-checked. The mission produces a checked version of the paper's closed-form comparison and, in milestones 1–2, checked computations of classical geometric-probability moments of the disk (the mean distance between two uniform points). The paper's own argument for the single crossing is numerical ("solved using standard methods"); a formal proof must replace it by an exact argument with rigorous bounds on .
The reduction to one variable is algebra. The difficulty is the sign analysis of on : it is a difference of two rational functions with poles at and , involving , and clearing denominators gives a cubic whose roots the paper only reports numerically. The natural first idea, showing is monotone on the whole interval, does not apply directly, since starts at , first decreases, and only later rises to ; a proof must establish that it crosses zero exactly once. A further point is that is not a closed-form number, so the crossing must be shown uniformly over an interval of values.
The moment milestones need integration in polar coordinates on the plane; the mean distance between two uniform points of a disk is a classical but nontrivial four-dimensional integral.
All objects live in the namespace CourierDispatch.Exogenous. Real numbers are ℝ, is lam, is Lam, and is an arbitrary function Q : ℝ → ℝ (no property of it is needed, since at equal the price terms cancel). The plane is EuclideanSpace ℝ (Fin 2) and the uniform law on the disk is Lebesgue measure conditioned on the closed ball.
Committed conventions:
Q stands for ; its properties cancel when the two strategies are compared at the same demand rate.The threshold (and , ) is existential in every statement; it is never a definition and never an infimum of a sign set, and , are never defined through , so the goal cannot be made true by construction.
Needed infrastructure: polar-coordinate integration on the plane (Mathlib's integral_comp_polarCoord_symm), and rigorous rational bounds on (Real.pi_gt_d6, Real.pi_lt_d6). The goal does not depend on the moment milestones 1–2, which ground the constants , , in the geometry of the disk and are welcome as separate contributions.
Blood platelets, fresh produce, dairy and many pharmaceuticals have a fixed shelf life: a unit that has not been used within periods of its arrival must be discarded. Managing such stock is a classical problem of inventory theory, going back to Nahmias (1975), and it remains of practical interest for hospital blood banks and grocery retail. The difficulty is structural. The state of the system is the whole age profile of the stock, a vector of dimension , and the optimal policy depends on all of it. Dynamic programming over this state is intractable even for moderate lifetimes, so practice relies on simple heuristics, above all base-stock policies, which raise the total stock to a fixed level every period.
For decades the performance of base-stock policies in this system was understood only numerically. Chazan and Gal (1977) and Cooper (2001) bounded the long-run outdating of a base-stock policy; Nandakumar and Morton (1993) built a base-stock policy from bounds on the optimal order-up-to level and showed numerically that it performs close to optimal. Bu, Gong and Chao (SSRN 3724966, accepted by Management Science, 2022) give the first theoretical guarantees, comparing a simple base-stock policy with the optimal policy in several asymptotic regimes. This mission formalizes the first of them: the optimality gap decays exponentially in the lifetime .
Time is discrete, . Demands are i.i.d. copies of a nonnegative random variable with and distribution function ; .
At the start of period the state is , where is the stock whose remaining lifetime is at most periods. The firm orders , which arrives immediately and raises the total level to (with ). Demand is served oldest-first (FIFO issuance) and unmet demand is lost. Stock reaching the end of its life is outdated:
The system starts empty. Each period costs
with holding cost , lost-sales penalty and additional outdating cost . A policy is admissible if each order is a nonnegative measurable function of the current state; is the set of admissible policies. The long-run average cost is , and .
The base-stock policy orders ; its cost is . The heuristic level is a minimizer over of
and is the newsvendor level.
For every and ,
The bound holds for every minimizer and every demand law with positive finite mean. Since , the gap decays exponentially in .
Theorem 1 says that a single explicit base-stock level, computable from the demand law by a one-dimensional convex search, is nearly optimal once the lifetime is moderately long, with an explicit, distribution-dependent rate. It explains why the non-perishable newsvendor intuition carries over to long-lived products, and it is the first of seven guarantees in the paper. Its ingredients are reused throughout: Lemmas 1 and 2, Proposition 2 and the gap bound (21) are posed in this mission and used, without restatement, by missions 2 to 4 of this series (demand population size, penalty cost, outdating cost).
The result is proved on paper; nothing about perishable inventory has been machine-checked. A formal development would provide a reusable model of FIFO perishable dynamics with long-run average costs, sample-path inequalities for outdating, and the first verified optimality-gap bound for an inventory heuristic in a high-dimensional state space.
Most of the argument is sample-path bookkeeping plus Jensen's inequality, but two steps are not routine. Lemma 1 asserts that the long-run average outdating of exists and does not depend on the initial state. The natural route through a stationary distribution fails: when the chain can be periodic and has no steady state, so the identity must hold for Cesàro averages directly; in the aperiodic case the paper uses a minorization condition (Meyn and Tweedie, Theorem 16.0.2). Proposition 2 compares every admissible policy with the heuristic cost, which needs the independence of the order-up-to level from current and future demands, and a conditional Jensen step for nonlinear expectations of a policy-dependent quantity.
The Lean development fixes the following conventions.
Fin (m - 1) → ℝ holding the cumulative levels, with , so (empty state, all leftover stock outdates) is included; , , , are fields of the parameter record.ℝ≥0).CappedBaseStock_Model and LostSalesOrderUpTo.Asymptotic.Backorder.Trivializing readings are excluded: is the simulated cost of , not a formula; ranges over all admissible policies, not only base-stock ones; is the argmin predicate, not a closed form; , and the demand law are general; and no gap is stated with truncated subtraction. Contributions of any of the milestones, of general facts about Cesàro limits of Markov chains, or of the convexity of , are welcome.
Decentralized exchanges on public blockchains do not match buyers with sellers through an order book. Most of them trade against a liquidity pool: a smart contract that holds reserves of two tokens and quotes a price by a fixed formula. The most widely used formula is the constant product rule of Uniswap (Adams, Zinsmeister and Robinson, 2020), under which a trade must leave the product of the two reserves unchanged. Pools of this kind settle a large share of on-chain token trading, and their pricing rule is fixed in code, visible to every participant, and executed in the order in which transactions are included in a block.
That last feature creates a market-structure question. Pending transactions are public before they execute, so another trader can see an order and place her own trade ahead of it. The empirical literature documents this behaviour at scale (Daian et al., 2020; Zhou et al., 2021). Park's paper asks which properties of the pricing rule itself make such strategies profitable, independently of any information the front-runner has. The paper compares the constant product rule with the cost function of a traditional market maker whose marginal price is linear in quantity, in the tradition of inventory models such as Biais (1993), and then derives four properties of constant product pricing in its Proposition 3.
Timeline:
A pool holds units of a token and units of a token . A trader who takes units of out of the pool pays units of , where is fixed by the constant product invariant
A positive is a buy of ; a negative is a sale of units of into the pool, and then is negative: the trader receives . Solving the invariant gives the paper's cost function, equation (3):
After a buy of the pool holds units of and units of . Selling units of into a pool therefore earns units of . The paper writes for on the current pool.
For the companion comparison the mission also defines the linear market maker's cost , where is the fundamental value, the number of market makers, their excess inventory and a liquidity factor.
For , and :
Part (1) says that splitting an order in time does not change its cost; part (3) that splitting liquidity into identical pools, with the order split pro rata, does not change it either. Part (2) is the front-runner's profit: she buys , the victim buys , and she sells into the pool left by both buys. Part (4) is the profit of selling into a pool just moved by a buy of and buying on an identical untouched pool, for every .
A companion item poses Proposition 1: with , , and , constant product pricing has a larger signed cost than linear pricing for , and a smaller one for , .
Parts (1) and (3) are the consistency properties one asks of any market maker: an order's cost does not depend on how it is sliced in time, or on how liquidity is fragmented across identical venues. Parts (2) and (4) show that the constant product rule, despite these properties, rewards anyone who can trade ahead of an observed order or across two identical pools. These are properties of the pricing curve alone, with no informational advantage, which is what makes them relevant to the design of on-chain exchanges, transaction ordering rules and fee schedules.
All statements are proved in the paper by direct algebra, and they are not open. What this mission adds is a machine-checked record of the exact statements, with the paper's printed sign slips resolved explicitly, and a reusable definition of the constant product cost function in Lean on which statements about fees, multi-hop routing or other invariant curves can build. No machine-checked version of Proposition 3 is known.
The mathematics is rational-function algebra; the difficulty is in stating it correctly. Three points need care. First, the sale convention: the paper's cost function returns a negative number for a sale, while the printed statements of parts (2) and (4) add that number as if it were the positive proceeds. Read literally, both printed left-hand sides are negative. Second, the pool after a trade must be tracked exactly: part (4) prints the first pool's reserve as , whereas after a buy of it is . Third, every quantity is a ratio with a denominator such as , or , and the statements hold only on the domain where those are positive. A solver who simplifies before checking signs will prove the wrong inequality.
Everything lives in the namespace CPAMMFlaws.Properties over the real numbers. The definition pcmm x X Y := x * Y / (X - x) is equation (3) verbatim, argument order ; ptmm x V ℓ N I is the printed closed form of . Conventions committed to:
Trivializing formalizations are ruled out: Lemma 2 is the invariant characterization over all , not the defining equation of pcmm; parts (2), (4) and equation (4) are strict inequalities on the signed proceeds, not and not absolute values; and no statement drops .
Out of scope: the paper's CARA-normal market-maker equilibrium (Lemma 1), its results on the linear market maker (Proposition 2), the fee analysis of §IV and the calibration of §V. The linear cost enters only through the companion Proposition 1. Contributions welcome beyond the milestones: proofs of the goal's parts directly, and further invariant-curve properties built on pcmm.
Best subset selection fits a linear predictor using at most of available columns. Its explicit limit on the number of selected variables makes the model easy to interpret, but the search can still fit noise: among many subsets of the same size, one can align unusually well with a noisy response. Mazumder, Radchenko and Dedieu study this effect in Subset Selection with Shrinkage. They add a penalty on the size of the selected coefficients and show how this changes prediction error when the signal-to-noise ratio is low.
An penalty is familiar from the Lasso. The estimator here also keeps the subset-size constraint, so its comparison class and its tuning parameter have a different role from those in an unconstrained Lasso. The paper's Theorem 3 gives three deterministic error bounds for this estimator, each valid on a specified event of the Gaussian noise. These bounds are the target of this mission. The separate probability estimates for the events are Theorem 1 of the paper and belong to another mission in this series.
Let be a fixed design matrix whose columns have Euclidean norm one. The observation is , where is a fixed mean and the coordinates of are independent random variables, with . The result below fixes one realized noise vector and assumes it lies in an event stated below. No assumption that is itself a linear combination of columns of is required.
For , write , where counts nonzero coordinates. An -regularized best subset estimator is any minimizer
The objective has no factor . The assertion applies to every minimizer, so tied solutions are covered. The sparse minimum singular value is . When and its domain is nonempty. Write for the decreasing rearrangement of .
The paper defines three noise events used here. bounds the largest coordinate of by . On , every satisfies
On , every satisfies . Here and all logarithms are natural. These definitions, including the constant in , are the printed deterministic events of p. 13.
At , Theorem 3(A) states
For , a positive universal constant gives
For every , a positive universal constant gives
The multiplying distinguishes this rate from the -penalized estimator's bound. The three claims together, with their separate tuning conditions, are Theorem 3 on p. 16. The milestones follow the displays of its proof in Supplement B.6, pp. 42–43.
Each inequality compares the prediction of any selected minimizer with that of every sparse competitor. The first bound has an exact leading constant and no additive noise term on . The second keeps a linear penalty term when meets a fixed noise-dependent threshold. The third replaces the linear term with a quadratic term controlled by the sparse minimum singular value. This distinguishes regimes in which coefficient shrinkage changes the error rate. The paper also combines deterministic bounds with event probabilities in subsequent corollaries; those probability statements are separate from this goal.
The paper proves Theorem 3. The formalization task is to verify its model definitions, event interpretation and all three inequalities in Lean, with the same constants and quantifiers. The milestone statements isolate the basic inequality, the bound, the rearranged-coordinate bound, the case split and the sparse step. A complete proof would also provide reusable finite-dimensional norm and matrix estimates. This mission does not include the paper's numerical experiments, data-driven model-size selection, the -penalized estimator, or the event-probability theorem.
The constraint makes the estimator a minimizer over a nonconvex set. A standard unconstrained Lasso argument does not directly supply the paper's bound for every feasible sparse comparator. The slow rates rely on two different noise controls: a coordinatewise bound on and a maximum of a sorted-coordinate term and a prediction-norm term on . The fast rate also needs the sparse minimum singular value to relate coefficient error to prediction error. The constants in these steps matter: replacing the exact first tuning equality or dropping the factor in the fast rate would state a different theorem.
Lean uses Fin n → ℝ and Fin p → ℝ for vectors and Matrix (Fin n) (Fin p) ℝ for . Norms are explicit finite sums, rather than the function type's default supremum norm. The paper's one-based sorted index is j.val + 1; the sum using is encoded as a maximum over coordinate permutations, which agrees with the decreasing rearrangement. The events are predicates on realized noise vectors. The Gaussian law explains the model but is not needed in these deterministic statements. The estimator is an argmin predicate, never a chosen minimizer.
The universal constants precede all dimensions and data in the Lean goal. The infima in the three displayed bounds are expressed as inequalities for every , avoiding the default value of an empty or unbounded real infimum. The standing conditions are , unit-norm columns, and . The first slow-rate branch retains the paper's equality for . The fast branch additionally requires or : this makes the term meaningful under Lean's total division, which returns zero at a zero denominator. The displayed events use the paper's original constant; this mission makes only a deterministic claim on that event.
The sparse domain for is nonempty under these conditions. All square-root radicands and logarithm arguments in the target are in their intended ranges. The formalization must rule out a universal constant that depends on , , , or , and must apply to every minimizer. Contributions to the shared norm inequalities, rearrangement identity and matrix calculations are useful beyond this theorem.
A retailer can pay a supplier when it places an order or after the supplier delivers it. When production yield is uncertain, the choice changes who bears the cost of missing units. It can also change the supplier's production decision and, through the amount available for sale, the surplus received by consumers. Geng, Guo, Xiao and Yang study these effects in a decentralized supply chain and show that payment postponement can help all three parties, hurt all three, or benefit some while hurting others, depending on production cost and the retailer's pricing rule. Their Proposition 4 gives the precise cost intervals for these outcomes in the SSRN author version, pp. 19–20.
The result matters to contract design in supply chains where realized output can be below planned production. Paying for units ordered and paying for units delivered allocate that shortage differently. A comparison based only on the retailer's expected profit would miss the effects on the upstream supplier and final consumers. Proposition 4 keeps all three outcomes visible while holding the retailer's pricing policy fixed.
There is one supplier and one retailer. The supplier chooses a wholesale price and plans production at cost . The retailer orders and sells to a market with linear demand , where and is the retail price. The yield factor equals with probability and with probability , with . Thus . The retailer receives units; the production cost applies to planned , including units lost to yield risk §3.1–3.2.
Under pay-by-order, the retailer transfers to the supplier and the supplier produces . Under pay-by-delivery, the transfer is and the supplier can choose a larger production quantity. Under ex ante pricing, the retailer fixes before yield is known. Under responsive pricing, the retailer sets after observing delivery. Combining payment and pricing gives OA (order, ex ante), OR (order, responsive), DA (delivery, ex ante), and DR (delivery, responsive). The supplier first sets , the retailer next orders and possibly fixes a price, the supplier chooses production, yield is realized, and responsive pricing occurs last Table 1, p. 9.
Both firms maximize expected profit at every stage. A subgame-perfect equilibrium is a complete plan of choices after every history in which each choice is optimal at that history. For scenario , write and for equilibrium retailer and supplier profit. The consumer surplus is expected surplus from sales: for delivered stock and market price , the model integrates inverse demand less from zero to . This is the paper's priority allocation convention for consumers with the highest willingness to pay p. 10.
The main target is Proposition 4, which compares OA with DA and OR with DR. Let mean the delivery-payment outcome minus its order-payment counterpart. A plus sign below means a strict increase, a minus sign a strict decrease, and zero equality. The admissible cost range is .
For ex ante pricing, with no restriction , the paper gives a threshold and the following pattern:
For responsive pricing, assuming , the pattern is on , on , on , and on . The thresholds are those specified in the paper: is equation (6), is equation (8), and the scenario cost thresholds come from Appendix B. The first ex ante interval can be empty for some parameters; its printed endpoints are retained. The milestones establish the supplier's production and responsive pricing rules, the four scenario outcomes, and the threshold ordering used in the comparison Appendices B–C.
The theorem separates the effect of changing the payment contract from the effect of changing the pricing rule. For advance pricing it identifies an interval where payment postponement helps both firms and consumers, then an interval where consumers lose despite both firms gaining. For responsive pricing the direction of every effect can reverse as cost passes successive thresholds. At high cost the change in payment timing leaves all three equilibrium outcomes unchanged Proposition 4, pp. 19–20.
The paper derives the equilibrium formulas in Appendix B and proves the comparisons in Appendix C. The result is therefore a known mathematical claim, while this mission asks for a machine-checked development from the sequential game through the cost-dependent equilibrium outcomes and final comparisons. The definitions of a two-point yield game and its complete contingent strategies can support other formal studies of payment or pricing postponement. There is no claim that Proposition 4 already has a machine-checked proof.
The supplier's production payoff has kinks at and . The retailer's best order changes when those production choices change, and responsive pricing adds another kink when delivered stock crosses . As wholesale price varies, the supplier must compare outcomes on both sides of these changes. Consequently a single first-order condition for a smooth objective does not characterize the equilibrium. The cost thresholds also depend on square roots and comparisons among several formula pieces, including the larger root chosen for Appendix B, pp. 33–37.
Lean represents expectation by the two weighted yield outcomes. Production cost is a separate argument, so thresholds depend only on . Parameters satisfy , , and . Wholesale and retail prices obey the strict admissibility bounds of equation (3). Orders are nonnegative; pay-by-delivery production is nonnegative; pay-by-order production equals the order. Consumer surplus is a real interval integral. Every comparison quantifies over actual subgame-perfect equilibria, and the goal separately asserts that the compared equilibria exist at every admissible cost. Equilibrium profits are computed from strategies and transfers, rather than defined to equal the Appendix B formulas.
The paper's statements require care at ties. The model selects when that is a profit-maximizing pay-by-delivery production choice, including the indifferent DR regime. The printed middle DR consumer-surplus expression combines outcomes from different production choices; the equilibrium-consistent expression is used here, and Proposition 4(R)'s comparison remains valid under either expression. Equation (2) swaps the scenario labels of committed and responsive prices, so consumer surplus uses the actual ex ante price in OA and DA and the price after delivery in OR and DR. The printed in Appendix B.1 is read as , as in Lemma 1. Outcome claims use open cost intervals because uniqueness at exact thresholds fails; equilibrium existence remains part of the targets there. The threshold is the larger root of the relevant quadratic factor, and and use the paper's explicit formulas. These conventions are recorded in the local moderation notes.
The development calls for real inequality reasoning, optimization of piecewise objectives, and careful handling of interval integrals. Contributions that prove the production and price best responses, establish the four equilibrium lemmas, or verify the threshold order each advance the final comparison.
A retailer that buys from a supplier with unreliable production has two contractual levers against supply risk, and both postpone a decision until after the yield is known. Payment postponement means paying only for the units actually delivered (pay-by-delivery) instead of the units ordered (pay-by-order). This moves the yield risk upstream. Pricing postponement means setting the retail price after the delivery is observed (responsive pricing) instead of committing to it in advance (ex ante pricing). This moves the risk downstream to consumers. Both practices are common in procurement, and the natural intuition is that two ways of shedding the same risk should be substitutes: once one lever is in use, the other should be worth less.
Geng, Guo, Xiao and Yang (MSOM 25(3), 2023; author version SSRN 3546872) compute the subgame-perfect equilibria of all four combinations of the two levers in a stylized supplier–retailer game. Their Proposition 3 shows that the intuition fails at intermediate production cost. When full yield is likely and yield loss is mild, the two postponements are complements there: postponing payment is worth more when pricing is also postponed. This mission formalizes that result together with the four equilibrium characterizations it rests on.
A supplier sells one product to a retailer, who sells it in a market with linear demand , . The supplier plans production at unit cost and realizes the fraction of it, where
and is the mean yield. The retailer orders and receives .
The four scenarios are OA, OR, DA and DR. The first letter is the payment rule: by order (O, the transfer is ) or by delivery (D, the transfer is ). The second letter is the pricing rule: ex ante (A, the price is fixed before the yield) or responsive (R, the price is set after is observed). Each scenario is a sequential game. The supplier posts a wholesale price . The retailer orders , and also commits to a price in the ex ante scenarios. The supplier then chooses ; under pay-by-order it produces exactly . The yield realizes, and under responsive pricing the retailer then sets . The retailer earns minus the transfer, and the supplier earns the transfer minus . Both are risk neutral. Admissible actions follow the paper's standing ranges: , , and between (or under pay-by-delivery) and .
Write for the retailer's expected profit in a subgame-perfect equilibrium of scenario . The paper's Definition 1 measures the interaction of the two postponements by
the value of payment postponement under responsive pricing, minus its value under ex ante pricing. The postponements are substitutes if and complements if .
The parameter region of the result is together with , where is the largest real root of the cubic . The cost thresholds (scenario DR) and (scenario DA) are explicit algebraic functions of given in the paper's Appendix B.
For and , every scenario has an equilibrium at every cost , and there is a cost such that, at every choice of the four equilibria,
is existential, as in the paper, and depends only on the parameters.
The result separates the first-order question (which lever the retailer prefers, the subject of the paper's Proposition 2) from the second-order one (whether one lever raises the value of the other). It shows that two risk-shifting instruments can reinforce each other. On , adopting one postponement raises the value of the other. So whenever the retailer gains from adopting either lever alone, it gains even more from adopting both. Below and above the reverse holds. Above , payment postponement is irrelevant, because the supplier never inflates production there.
The paper proves Lemmas 1–4 by backward induction in Appendix B and the proposition by comparing closed forms in Appendix C. None of it is machine-checked. A formal proof requires the four backward inductions with every kink and tie handled, and the polynomial inequalities between the thresholds over the parameter region. The formalization also corrects the record: the paper's explicit formula (7) for does not locate the sign change of (at it exceeds ), while the existential statement of Proposition 3 holds at every sampled point.
Once Lemmas 1–4 are available, is a difference of explicit piecewise functions of . The difficulty lies in those lemmas and in the thresholds. Each lemma is a three-stage optimization. The supplier's profit is piecewise linear in , with kinks at and . The retailer's profit is piecewise in , with kinks where it becomes worth inflating the order. The supplier's value function in is piecewise quadratic, and its maximum can sit at a kink. In scenario DR it sits exactly at the price where the supplier is indifferent to inflating production. A naive reading of the paper's tie rule removes the equilibrium there altogether. On the middle DR interval the sign of changes inside the interval, at a root of a polynomial with nested square roots. On the other pieces, the signs of and the ordering are polynomial inequalities in , with defined implicitly as a cubic root.
The equilibrium profits are outcomes of the equilibrium predicate. They are never defined by their closed forms, and the goal carries the existence of an equilibrium in each scenario. Without that, the comparison would be vacuous whenever the predicate is unsatisfiable.
Every object is elementary: real arithmetic, square roots, min/max, one interval integral and one sSup. The equilibrium predicate and the four lemmas are shared with the three companion missions of this paper (Propositions 2, 4 and 5). Proofs of the lemmas, of the threshold inequalities and of the goal are all welcome.
Machine-learning systems are increasingly placed beside human experts — physicians reading scans, loan officers screening applications, judges assessing risk — rather than replacing them. The machine processes the structured data it can see; the human also has access to information the algorithm cannot observe, but her attention is limited. Whether such a partnership improves the decision is a question about how the human reallocates scarce cognitive effort once part of the information is handed to her for free.
Boyacı, Canyakmaz and de Véricourt (ESMT Working Paper 20-02, 2020; published in Management Science, 2024) model the human as a rationally inattentive decision-maker in the sense of Sims (2003): she chooses how much information to process, and pays a cost proportional to the mutual information between the state and her decision. In the binary-choice setting, Matějka and McKay (2015) showed that such problems reduce to choosing conditional choice probabilities, with logit-form solutions. The paper uses this reduction to compare a decision-maker who decides alone with one who first sees a machine's assessment. This mission formalizes its first comparison, Proposition 1: the machine never lowers the decision-maker's accuracy or her expected value.
A decision-maker (DM) must announce whether the state of the world is good () or bad (). A correct choice pays , an incorrect one . Her prior that the state is good is , and is the information cost.
A strategy is a pair of conditional probabilities of choosing in state and in state . It induces the choice probability , the accuracy , the cognitive cost
(the mutual information between state and action, in nats) and the expected value . The DM maximizes . The optimal accuracy and optimal value are the accuracy and value at a maximizer. Two thresholds organize the solution:
The machine. Information is aggregated into two binary variables: , which the machine and the DM can both evaluate, and , which only the DM can assess. Realizations are and the state is good iff . The DM's prior on is , with for all four realizations, so . The machine reveals at no cost; the DM updates by Bayes' rule to and then solves her problem with prior . The machine-assisted accuracy and value are the ex-ante expectations
For every and every prior with positive entries,
The result. Proposition 1 is the baseline against which the paper's other findings are measured: although accuracy and value always improve, the machine can lower the probability of choosing the good state, raise false positives, and increase the DM's cognitive effort (Theorems 2–4 of the paper). Without Proposition 1 those effects would not be separable from a simple loss of decision quality. The closed forms (13) and (15) also show that, where the DM processes information, her accuracy depends on alone and not on her prior.
Formalizing it. The proposition and its milestones are proved in the paper, in Appendix A, and the closed forms rely on the logit characterization of the optimum. None of this is machine-checked. A complete development produces a verified closed-form solution of the binary rational-inattention problem with entropy cost, with accuracy and value as explicit functions of the prior, and a verified value-of-information inequality for that problem. Both are reusable by the other four missions of this series and by any binary rational-inattention model.
The obvious argument — more information cannot hurt — is not available off the shelf: the DM is not a Bayesian with free information but optimizes an entropy-penalized program, and the comparison is between optimal values of two such programs at different priors. and are values at an optimum of a non-linear program over . Their closed forms require identifying exactly when the corner solutions and are optimal and handling the degenerate priors , where the maximizer is not unique. Both functions are piecewise in , so their shape on does not follow from the shape of each piece: the behaviour at the kinks and decides it. Accuracy is a component of the objective, not the objective itself, so statements about do not follow from optimality alone.
Real.binEntropy (natural logarithm); is lam.Classical.epsilon); every maximizer yields the same values. They are defined for every real and used on .Bool with true ; the prior is π : Bool → Bool → ℝ with positive entries summing to . Posteriors are defined by Bayes' rule; , are expectations over .Real.binEntropy (derivative, concavity) is in Mathlib. Needed: existence of a maximizer on the square, the first-order characterization of the interior optimum, and convexity of piecewise functions glued with matching slopes. Contributions of general lemmas on these points are welcome.When a supplier's production is subject to random yield, only part of a planned production run turns into usable output. A retailer sourcing from such a supplier bears the resulting shortage risk unless the contract or the selling strategy moves it elsewhere. Random yield is a classical topic in production planning (Yano and Lee, 1995), and postponing a decision until uncertainty resolves is a classical risk-mitigation tool (Van Mieghem and Dada, 1999).
Geng, Guo, Xiao and Yang study two postponements available to a retailer facing an unreliable supplier. Payment postponement replaces payment for the ordered quantity by payment for the delivered quantity, which shifts the yield risk to the supplier. Pricing postponement (responsive pricing) sets the retail price after the delivery is known, which shifts the risk to consumers. Their Proposition 2 answers which of the four combinations the retailer prefers, as a function of the production cost, when the yield loss is not too severe. This mission formalizes that proposition and the equilibrium results it rests on.
A supplier sells to a retailer, who sells to a market with linear demand , . The supplier's yield factor takes the value with probability and with probability , with , and has mean . The supplier plans production , pays per planned unit, and delivers against an order . Throughout, .
There are four scenarios, written :
Each scenario is a Stackelberg game. The supplier sets the wholesale price , the retailer orders (and, ex ante, prices), the supplier produces, the yield realizes, and under responsive pricing the retailer prices. Both firms are risk neutral. The solution concept is subgame-perfect equilibrium, and denotes the retailer's expected equilibrium profit in scenario . Admissible prices are those of the paper's standing assumption (3): , and under pay-by-order or under pay-by-delivery.
The cost thresholds of the comparison are
For and every , equilibria of OR, DA and DR exist, and every choice of equilibria satisfies
The proposition describes a preference cycle. Payment postponement (DA) is best when production is cheap, because the supplier then absorbs yield risk by inflating production. Pricing postponement (OR) is best at intermediate cost. Using both (DR) is strictly best on , where the two postponements are worth more together than either alone. Above the supplier stops inflating, DR and OR coincide, and pricing postponement suffices. These are the paper's managerial conclusions on contract and pricing choice under supply risk, and the other three missions of the series (strategic complementarity, and the effect of each postponement on the whole chain) rest on the same equilibria.
The paper derives the four equilibria in its Appendix B and proves the propositions in Appendix C as direct comparisons of closed forms. None of this has been machine-checked. The mission produces a formal model of the four games and a verified link from the game to the closed forms. Lemmas 2–4 are reusable by every statement about this model.
Once the closed forms are known, the comparison is polynomial algebra in with square roots. The work lies upstream. Each equilibrium requires a backward induction over piecewise-linear objectives in and and piecewise-quadratic objectives in and . Optima occur at kinks such as the inflation threshold , and the supplier's value function is discontinuous there, so the maximum can sit at a boundary rather than at the center of a piece. In scenario DR the equilibrium wholesale price on sits exactly at , where the supplier is indifferent among all productions in , so existence depends on how that tie is resolved. Uniqueness of outcomes, which the comparison needs, holds only on open cost intervals, because at each threshold two wholesale prices tie with different outcomes. The threshold inequalities for are not visible from the formulas and need careful estimates of the radicals.
Parameters are a structure with validity predicate , ; the cost is a separate real argument. The expectation over is the two-term sum , with no measure theory. A strategy profile consists of a wholesale price and rules for production , the order , the ex ante price and the ex post price . Equilibrium requires optimality at every admissible history over the admissible sets of (3). Consumer surplus is the integral , evaluated at the ex post price under responsive pricing and at the committed price under ex ante pricing.
Conventions and disclosed departures from the page:
is the retailer's profit along an equilibrium of the game, never its closed form; a formalization that defines the profits by the Appendix B formulas, or an equilibrium predicate that no profile satisfies, would make the goal trivial. The goal therefore also asserts existence of equilibria of the three scenarios.
Contributions welcome: proofs of the milestones; general lemmas on maximizing piecewise-linear and piecewise-quadratic functions with kinks; the threshold inequalities. Proposition 1 (OA versus the others) is not part of this mission.
A life annuity pays instalments conditional on an annuitant surviving to specified instants. As discussed in Dickson, Hardy and Waters (2020), actuarial value depends on the exact timing of the payments. For a pension payable in arrears, the payment event is at the end of the period. A contractual guarantee can preserve payments for the estate after death, but only once the entitlement has commenced. Those conditions must be explicitly modelled before calculating actuarial factors.
Take a finite distribution over last survived anniversaries with nonnegative probabilities summing to one. The final state is right-censored: it does not assert death at the terminal age. Define . A schedule has its first index , up to instalments, an amount for each instalment, and an annual discount factor with . Advance instalments fall at and arrears instalments at .
The first scheduled instalments are guaranteed only when survival to commencement activates the annuity. They continue to the estate after death during that period. Later instalments depend on survival at their actual payment instant. No spouse benefit is assumed.
Define the realised random variable by summing actual payment indicators across the finite schedule. Derive its expectation independently from the lifetime-state mass function. Define a second valuation using survival to each payment instant, and prove the equivalence by interchanging finite sums.
The instalment sum is truncated at the finite horizon. The timing offset equals zero for advance and one for arrears.
Partition a non-guaranteed whole-horizon annuity into temporary and deferred periods. For a guaranteed annuity, split payments into the activated period-certain amount and the strictly later survival-contingent amount. If , its entire value is survival to commencement multiplied by the corresponding annuity-certain present value. The benefit kernel supports level, arithmetic increasing or decreasing, geometric and piecewise step-up instalments.
The exact finite level-annuities relationship includes the terminal payment:
Expected squares of realised present values yield second moments and variance. The uniform distribution of deaths (UDD) provides a separate linear interpolation of observed annual survival probabilities at fractional instants.
At a fixed payment frequency, split each annual death-state mass equally across its subperiods under UDD. The same resulting micro-lifetime distribution can be used for fractional instalments and continuous benefit-rate integrals, so their actual cashflow representations share exact event boundaries. The construction agrees with annual survival at integer anniversaries. Placing the unsplit annual masses directly at integer death times would give a different, stepwise survival model within a year.
The zero-interest level annuity has an exact finite correction to the integral of linearly interpolated UDD survival. If is that integral, annual absolute survival and , the due value equals , and the arrears value equals . Both have an explicit error and converge to . Under the fixed micro-atom lifetime law, continuously paid GBP 1 per year instead equals the same-frequency arrears value at zero interest, because the real-time survival indicator is constant between atom times. The limit of these finite-atom laws recovers the UDD linear-survival valuation.
A separate real-time lifetime-state model represents a continuous benefit rate in money per year rather than discrete money per instalment. Its actual realised annuity present value is integrated across the entitlement interval, with a guarantee activated by survival at commencement and payments then continuing to the estate before expiry. A finite probability expectation of the realised integral is reconciled with independently integrated survival weights. The continuous certain value is a comparison under the same deterministic discount force.
The discrete and real-time laws are connected at annual and fractional instalment boundaries through exact survivor-event identities. A bounded continuous payment-rate function admits an appropriate Riemann-sum limit under the specific regularity assumptions in the formal declaration. A twice differentiable integrand with uniformly bounded second derivative admits the two-term Woolhouse-type endpoint correction with a finite-horizon quadrature error bound. These limiting and approximation statements remain open formal theorem obligations, with no universal error claim for arbitrary discontinuous mortality laws.
The unified capstone reconciles actual annual advance and arrears payments and UDD subperiod payments with independent survival-weighted sums. It also reconciles the same finite lifetime-state law with real-time survival at anniversaries and continuous integrated benefit valuations, including an activated guarantee. Separate milestones handle second moments, terminal-censored assurance-annuity identity, Riemann convergence and the explicit quadrature error bound. The earlier finite due-minus-arrears relationship remains a supporting target. These obligations follow from the separately defined probability masses and cashflow events, rather than assuming any valuation equality.
The classical distinction between annuity-certain and life-annuity present values is explained by Promislow (2015) through an explicit interest-and-survivorship discount mechanism. The present model makes this accounting precise at the level of individual payment events. It is consequently possible to compare a certain payment stream with a contingent one using exactly the same instalment schedule. It is also possible to alter the instalment amount without silently altering the survival law or guarantee entitlement.
The resulting mathematical foundation is suitable for retirement income and defined benefit pension cashflows where timing and benefit continuation clauses materially affect value. It separates annual unit amount from each instalment, and survival at commencement from survival at later payment events.
A death before commencement cannot activate a deferred guarantee. A death during the guarantee permits payments to the estate. An arrears instalment requires survival at the subsequent instant and the final one cannot silently disappear. Proving these relationships from true payment indicators is harder than defining annuity factors so the desired equations hold automatically. Finite lifetime censoring and the last surviving anniversary must remain visible in all derivations.
The proposed Lean representation is an indexed real-valued finite probability mass function, a survival indicator and Finset-based cashflow sums. The canonical scope is a single namespace and targeted imports of Mathlib's real-number and finite-big-operator files. The environment revision is pinned in the manifest, but remains subject to live remote verification. Periods are integer anniversaries, amounts are per actual instalment, and the valuation origin is zero. Only finite sums are asserted without convergence hypotheses. The existing classical annuity formulae appear in the cited books. Their underlying mathematics is known, while the original finite cashflow encoding and its Lean theorem obligations are open and unproved.
These primitives are intended to supply later work on DB pension guarantee periods, future life-insurance premium annuities and pension escalation. A future separate mission could reuse the actual realised-cashflow event semantics to value survival-contingent spouse benefits with a two-life entitlement law, while keeping spouse entitlement fixed at the policyholder's death under the scheme rules. The current work limits itself to the single-life estate-continuation benefit and therefore does not risk silently importing joint-life assumptions.
The extended finite discrete formalisation constructs an actual UDD subperiod death-state law. Its fractional annuity valuations follow from realised subperiod payment events, with each instalment equal to the nominal annual pension divided by frequency. Separate theorems reconcile second moments with pairwise survival-event intersections, derive the incremental value of guarantees and reconcile finite annual assurance with the life-annuity due while retaining the terminal censored-state residual. The separate continuous law and a quantitative two-term Woolhouse-type error inequality are now included as remotely elaborated open theorem declarations, still requiring genuine proofs. General commutation-column identities, noninteger-year deferrals under arbitrary interpolation, unbounded whole-life convergence and classical higher-order Woolhouse estimates need further formal models. An unconfirmed Foundation A owner Draft is not an importable dependency. Every theorem declaration is an open proof obligation and no accepted proofs are asserted.
Best subset selection fits a linear model using at most of available predictors, choosing the that minimize the residual sum of squares. Mixed-integer optimization made it computable at moderate scale (Bertsimas, King & Mazumder, 2016), which reopened the statistical question of when it should be used. Empirical comparisons (Hastie, Tibshirani & Tibshirani, 2020) found that best subsets predicts poorly when the signal-to-noise ratio (SNR) is low: by fitting the noise exactly within its chosen support it overfits, and shrinkage methods such as the Lasso do better there.
Mazumder, Radchenko & Dedieu propose to add a shrinkage penalty to the cardinality-constrained problem. Their Theorem 2 quantifies the effect of an (Euclidean-norm) penalty: it proves prediction-error bounds that are linear in the penalty level and therefore remain small in the low-SNR regime where best subsets fails. This mission formalizes that theorem.
The data follow the model
where , is an unknown deterministic mean and is a noise vector (in the paper, with independent entries, ). The design is deterministic and its columns have unit Euclidean norm. Write for the Euclidean norm, for the number of nonzero coordinates of , and .
For a model size and a penalty level , the -regularized best-subsets estimator is any solution
The penalty is the norm , not its square, and the loss carries no factor . The quantity of interest is the prediction error .
The minimal -sparse eigenvalue of is . For , is the non-increasing rearrangement of . Given , three noise events are sets of noise vectors:
The paper's Theorem 1 shows these events have high probability; Theorem 2 is a deterministic statement on each of them. The symbol means "up to a positive universal multiplicative constant", one that does not depend on or .
Let .
(A) Slow rate. If , then on
and on
(B) Fast rate. For every , on
The bounds hold for every minimizer and for misspecified models: need not lie in the column span of .
The steps of the printed proof (Supplement B.2, pp. 39–40), in the order the proof uses them: the basic inequality (29); the bound (30) on and its consequence (31); the dichotomy on , where (30) implies (31) and (32) implies (33); the elementary inequality (34); the basic inequality on (p. 40); and the bound of through (p. 40).
The result. The slow-rate bound of part (A) is linear in . When the signal is small compared with , the term is smaller than the rate that best subsets attains, and the paper's Theorem 4 shows that in this regime the prediction error of best subsets is at least of order . Together the two results explain the empirical finding that shrinkage helps best subsets at low SNR. Part (B) shows nothing is lost at high SNR: the fast rate is the minimax rate over -sparse vectors plus the shrinkage term . Combined with Theorem 1, the bounds give the high-probability statements of the paper's Corollaries 1 and 3.
Formalizing it. The theorem is proved in the paper; no machine-checked version is known to exist. The mission produces a checked deterministic core: once the events are granted, every constant (, the factor , the threshold ) is verified exactly. Together with the companion mission on Theorem 1 it would yield fully checked high-probability oracle inequalities for a cardinality-constrained estimator, which no current formal library contains.
Every step is elementary, but the bookkeeping is exact. The obvious reading "insert the event's inequality into (29)" does not by itself work on : the event's right-hand side is a maximum of a sorted weighted sum and a prediction-norm term, and the first branch has to be compared with for the -sparse difference , including when exceeds . That comparison is a combinatorial estimate on sorted sums, not a substitution. On , the penalty difference is not controlled by the event at all and has to be converted into prediction errors through . In both parts the final constant must not depend on , or .
All objects are explicit Lean definitions in the namespace SubsetShrink.L2:
Fin n and Fin p with 0-based indices, and is a real Matrix (Fin n) (Fin p);Fin m → ℝ is the sup norm;The model consists of with unit columns, , , and . The estimator is a predicate that every minimizer satisfies; no minimizer is chosen and existence is not assumed. "On the event" is a hypothesis that the noise vector satisfies the event's printed inequality, so no probability measure appears. is the event as printed on p. 13; the correction of its probability bound belongs to the mission on Theorem 1 and does not affect these deterministic statements.
Committed conventions:
Part (B) assumes or . When the paper's bound is vacuous (), but Lean's would turn it into a false claim.
A formalization that lets the constant depend on , or , fixes a particular minimizer, weakens an event (for example drops "for all "), or computes the infimum as a real ⨅ over an unbounded set would trivialize the theorem and is ruled out.
Not formalized here: the probability bounds of Theorem 1, the -penalized Theorem 3, the lower bound of Theorem 4, the data-driven choice of (Theorem 5), and the corollaries and numerical experiments. These are separate missions or follow-ups. Contributions of reusable lemmas (the rearrangement bound for the sorted sum, properties of , the elementary absorption inequality (34)) are welcome.
Best subset selection fits a linear model using at most of the available features: it minimizes the residual sum of squares subject to the sparsity constraint , where counts the nonzero coordinates of . Mazumder, Radchenko and Dedieu (arXiv:1708.03288v4; Operations Research, 2023) show that best subsets overfits when the signal-to-noise ratio is low, and propose adding a shrinkage penalty with to the same constrained problem. Their statistical results (oracle inequalities, a lower bound for best subsets, a BIC-type choice of ) are the subject of the companion missions II–VI of this series.
The estimator is a mixed-integer program. To compute it in practice, and to warm-start the mixed-integer solver that certifies optimality, the paper uses a discrete first-order (DFO) method: a proximal-gradient iteration in which the proximal step is a projection-like operation onto the set of -sparse vectors with a norm penalty. The method descends from iterative hard thresholding (Blumensath & Davies, 2009), from proximal gradient methods for composite problems (Nesterov, 2013), and from the discrete first-order algorithms of Bertsimas, King and Mazumder (Annals of Statistics, 2016). This mission formalizes the paper's convergence guarantee for it, Proposition 2.
Vectors live in with the Euclidean inner product and norm . Write for the number of nonzero coordinates and . For the penalty is the norm itself, not its -th power.
Let be a differentiable convex function whose gradient is -Lipschitz:
The motivating case is with . The composite problem (7) is
For the quadratic model of around is
and the thresholding operator is the set of minimizers
which can contain several points when the have ties. A DFO sequence with constant starts at a -sparse and, for , picks any
Assume in addition and . For every DFO sequence:
(a) if , then is non-increasing in and converges to some ;
(b) if , then for every
The statement holds for any of the stated class and any choice in the set-valued operator; nothing about least squares is assumed.
Proposition 2 is what makes the DFO method a sound component of the paper's computational pipeline. The objective values never increase, so every iterate is a feasible point at least as good as the warm start. Part (b) says that the step length is small after iterations. A point whose step vanishes is a fixed point of the update, which the paper calls a first-order stationary point (Definition 1). The stopping rule of the algorithm () is therefore reached in at most steps.
The result is proved in the paper, in half a page that defers its central inequality (22) to Proposition 6 of Bertsimas, King & Mazumder (2016). It has not been formalized. A machine-checked version produces three reusable pieces: a descent lemma for functions with Lipschitz gradient on , the variational characterization of the sparse proximal step, and a sufficient-decrease-to-rate argument for any descent method. These are the same pieces that iterative hard thresholding, proximal gradient, and block-coordinate methods for problems need.
The difficulty is moderate, and it lies in the infrastructure rather than in any single step. The descent lemma (9) needs the fundamental theorem of calculus along a segment for a function on a Euclidean space, combined with the Lipschitz bound on the gradient. The feasible set is not convex, so no convex-analysis argument (subgradient optimality, firm nonexpansiveness of a proximal map) applies to the thresholding step. Inequality (22) must instead be obtained by comparing the minimizer with the feasible competitor itself, which is why the hypothesis appears. Part (a) also needs a lower bound on for its limit to exist and be nonnegative, which the general class of does not provide on its own.
is EuclideanSpace ℝ (Fin p), the gradient is Mathlib's gradient f, and coordinates are -based. Membership in is a predicate: feasibility plus minimality against every -sparse competitor. No selection from the set is fixed, and existence of minimizers is not assumed. A DFO sequence is any β : ℕ → ℝᵖ satisfying the update for every , indexed from as printed. The stopping rule and the support-restricted polishing Step 2 are not part of it. The penalty is for and the Euclidean norm for , and every theorem assumes and . The minimum over is a finite infimum over a nonempty index set.
Three hypotheses are added to Proposition 2 as printed, each disclosed in the item. The proof's claim "" is false for a general convex with Lipschitz gradient ( sends to ), so and are assumed. The update divides by , and allows , so is assumed. "Decreasing" is read as non-increasing. Convexity of is kept as in the paper's setting (7)–(8), though only the gradient bound is used. Proposition 1 is stated with an existential ordering: for a fixed sorting permutation the printed claim fails under ties.
The statement cannot be trivialized by restricting or the run. The goal quantifies over every of the class and every sequence obeying the update, the limit in (b) is the actual limit of the objective values, and the rate holds for every . Not formalized here: the mixed-integer formulations, the continuation and local-search heuristics of §2.4, Definition 1, and the paper's statistical results (missions II–VI). Contributions of reusable infrastructure, such as a general descent lemma or a lemma turning summed decrease into a rate, are welcome as separate theorems.
Most online advertising is sold by auction, one impression at a time, to advertisers who manage a budget over many auctions rather than a value for a single one. Since 2019 the major ad exchanges have moved from second price to first price auctions, in which the winner pays its own bid. A budget-constrained advertiser in a first price auction must therefore shade its bids twice: once because of the budget, and once because of the payment rule. Whether these two effects admit a coherent equilibrium, and of what form, is the question behind Balseiro, Kroer and Kumar, Contextual Standard Auctions with Budgets: Revenue Equivalence and Efficiency Guarantees (Management Science, 2023, doi:10.1287/mnsc.2023.4719).
Earlier work on pacing equilibria treated second price auctions, where truthful bidding is dominant without budgets (Balseiro, Besbes and Weintraub 2015; Conitzer et al. 2022). In first price auctions with a single deterministic item, an equilibrium may fail to exist (the paper's Online Appendix A). The paper restores existence by letting items carry a random context and values depend on it linearly. This mission formalizes its first main result, Theorem 1: an equilibrium of a simple, interpretable form exists.
An item has a context , a feature vector drawn from a law with a density. There are buyers. Each buyer has a type , a weight vector and a budget , drawn independently from a law with a density on
Buyer values item at , and . The seller reveals , then runs a first price auction with a public reserve price : a bid wins if , where is the strategy of the competitors, and the winner pays its bid. A budget only needs to hold in expectation over and the competitors' types.
A strategy is a symmetric first price equilibrium (SFPE, Definition 1) if for -almost every type the bid function solves
A pacing function scales values down to paced values . Let be the distribution function of the highest of i.i.d. paced values, and for let
with if and if . The value pacing–based strategy (Definition 2) is
For a buyer facing competitors who use , the dual function is for .
The paper states that Proposition 2 combined with Proposition 1 implies Theorem 1.
Theorem 1 says that budget-constrained buyers in contextual first price auctions have an equilibrium that is described by one scalar per type: shade the value by the factor , then bid as an unconstrained buyer would in an i.i.d. first price auction against paced competitors. This is the object on which the rest of the paper is built: revenue equivalence across standard auctions (Theorem 2) and a liquid-welfare price of anarchy of at least (Theorem 3) are both statements about pacing-based equilibria. Practically, it supports the use of pacing for budget management in first price auctions, with pacing multipliers computed as for second price auctions.
The result is proved in the paper; every proof is in its online appendix, which is not part of the source this mission cites. Nothing in the paper is formalized. A formalization would check a strong-duality argument for a non-convex problem and an infinite-dimensional fixed-point argument whose topology the authors chose with care, and it would produce reusable pieces: multivariate total variation, compactness of bounded-variation sets in , and the first price equilibrium bid against an arbitrary (possibly atomic) value distribution with a reserve price.
The buyer's problem is not convex in the bid function, so standard convex duality does not give strong duality; Proposition 1 needs a direct argument combining Lagrangian optimality (Lemma 1), differentiability of in , and complementary slackness. Lemma 1 itself cannot quote the textbook first price equilibrium, because the paced value distribution may have atoms and the reserve price creates discontinuities.
The fixed point lives in an infinite-dimensional space of pacing functions. The obvious topologies fail: the set of bounded pacing functions is not compact in any , and in the weak-* topology of upper hemicontinuity of the best-response correspondence is hard to obtain; restricting to Lipschitz functions loses the property that best responses stay in the set. The paper restricts to functions of bounded total variation, which requires the monotonicity of Lemma 2 and the variation bound of Lemma 3.
Contexts are Fin d → ℝ, types are pairs (w, B) : (Fin d → ℝ) × ℝ, and is the explicit sum . A single Model structure holds , and Model.Standing collects the assumptions of §2.2 together with four that the paper uses without stating them: , , nonempty and bounded (so ), and measurability of and . Pacing functions are measurable functions on that are nonnegative on ; values off are irrelevant since .
The conventions fixed in Lean:
IsMinOn, over in Propositions 1–2 and over in Lemma 5 and , as printed.EReal, with Euclidean and test fields supported in ; is with respect to Lebesgue measure. Lemma 4 is stated on the image of in Lp ℝ 1 (volume.restrict Θ), in its norm topology.A formalization in which is defined by its closed form, in which non-integrable payments count as zero, or in which the strategy need not be measurable would make the milestones easier without proving the paper's theorem; these readings are excluded by the definitions above.
A complete development needs: first price equilibrium bids for general atomic distributions with reserves; measurability of parametric integrals; a duality argument for one-dimensional Lagrangian relaxations; multivariate bounded variation and its compact embedding in of a box; and the Kakutani–Fan–Glicksberg theorem in a locally convex space. The bounded-variation and fixed-point parts are reusable beyond this mission. Proofs of any milestone, and alternative arguments, are welcome.
A person using a machine's advice may decide more accurately, but attention is costly. Before asking what a machine changes, the baseline decision has to be understood: how often would the person choose the action associated with a good state without machine input? Boyacı, Canyakmaz, and de Véricourt study this question in a model of a decision maker who can acquire information about an unknown binary state and pays for the expected reduction in uncertainty. Their working paper derives a closed form for the decision maker's choice probability. That baseline is used later in the paper to compare decisions with and without machine input.
The information cost follows the rational inattention approach: a decision maker chooses how much information to process while considering its effect on the final action. In the discrete choice setting, Matějka and McKay (2015) characterize optimal conditional choice probabilities. The paper adopts their equivalent choice probability representation in §4. This mission isolates the first consequence of that representation, Theorem 1, before the paper introduces the machine.
The unknown state is good, , with prior probability , and bad, , with probability . The decision maker selects either (declare good) or (declare bad). A correct action earns one unit, and an incorrect action earns zero. Write for the probability of choosing when the state is , and for the probability of choosing when the state is . A strategy is any pair ; the pair may lie on the no-information diagonal or on the boundary of the square.
The unconditional choice probability is . The accuracy is . Binary entropy, measured with the natural logarithm, is , with . Given an information-cost parameter , the cognitive cost in the paper's choice probability representation is
The decision maker maximizes net value over the whole square. An optimal pair is one whose value is at least that of every feasible pair. The source originally introduces a general distribution of information signals and a Bayes consistency condition in problem (2). Section 4 and footnote 8 adopt the equivalent conditional choice probability representation used here. The equivalence itself is an input from Matějka and McKay (2015), rather than a claim of this mission.
For and , define the thresholds
The goal is the paper's Theorem 1: every optimal pair has the same unconditional choice probability,
The same theorem states , that is non-decreasing in the prior, and that the lower threshold increases while the upper threshold decreases as increases. This mission includes all these clauses in its single goal. The intermediate milestones formalize the paper's conditional choice equations, the algebraic interior candidate, its threshold crossings, and its slope and threshold derivatives.
The formula partitions prior beliefs into three regimes. With a sufficiently low prior, the decision maker always chooses ; with a sufficiently high prior, the decision maker always chooses . Between the thresholds, the probability of choosing changes continuously with the prior. The threshold movements describe how a higher information cost changes the region in which information processing affects the choice. In the paper, these probabilities feed into accuracy, error rate, effort, and machine input comparisons.
The paper proves the result mathematically. The statements here are open Lean targets, so compilation of their declarations does not amount to machine-checked proofs. A complete development would verify the optimality characterization and the closed form, then provide reusable binary entropy and constrained optimization lemmas. It would also expose a precise baseline that subsequent missions can import after publication.
Equations (4)–(6) alone do not select the optimum: both and satisfy them for every prior. The printed uniqueness phrase in Lemma 1 therefore cannot be carried into a theorem about all solutions of those equations. The formal target says instead that every maximizing strategy satisfies the conditional choice equations, and Theorem 1 identifies the probability selected by maximization. The endpoint strategies, where entropy is evaluated at zero or one, must remain in the feasible set. One must also distinguish the algebraic interior root from the optimizer when that root leaves .
Lean represents probabilities and costs as real numbers. The strategy space is the full closed square , and IsOptimal compares net value against every point in that square. The prior in the goal and Lemma 1 lies strictly between zero and one, as in the paper's §4 setting; is strictly positive. Boolean state and action types are unnecessary in this first mission because the paper's §4 representation is already the pair of conditional probabilities. The general-signal distribution of problem (2) is not encoded here.
The entropy function is Mathlib's Real.binEntropy, with natural logarithms and the endpoint values used by the paper. The threshold definitions reproduce Theorem 1 exactly. pBar is the interior algebraic expression from Appendix A, and pClosed is only the right-hand side of equation (11). Neither defines the optimizer. In particular, defining the starred optimal probability by the closed form would remove the theorem's main claim. The goal quantifies over every optimal pair and asserts that its computed choice probability equals that expression.
Contributions to the proof will need results about binary entropy at the boundary, derivatives of the objective in the interior, existence and characterization of maximizers on a compact square, and elementary inequalities for the exponential thresholds. The entropy and compact optimization infrastructure can support later formalizations beyond this paper. The four listed milestones follow the authors' Appendix A argument; alternative proofs of the same faithful statements are welcome.
Certify the separation exponent using OpenAI's construction. Sensitivity counts individual bit flips that change the output; block sensitivity counts disjoint sensitive blocks. Both are maximized over inputs. The notation fixes one input, and denotes the all-zero input.
The goal requires arbitrarily large sensitivity and for every . OpenAI's depth-nine seed, with , supplies a slightly larger exponent; the displayed decimal must be certified exactly.
This mission is the campaign's foundation. Its central reusable milestone turns a finite certificate , , , and into the asymptotic claim at exponent . Normalization and sensitivity growth are proof obligations. Later missions import this theorem and the shared definitions.
OpenAI's quantitative separation is already proved on Prove2Me. This mission certifies the numerical exponent and the reusable bridge; its goal remains Open until all required proofs are accepted.
The price of anarchy (PoA) of a nonatomic congestion game is the ratio of the total cost of a Wardrop equilibrium to the minimum total cost. Worst-case bounds on it over whole classes of cost functions are classical (Roughgarden and Tardos, 2002; Correa, Schulz and Stier-Moses, 2004). In practice the demands and cost functions of a traffic model are estimated, never known exactly, so a second question matters as much as the worst case: how much does the PoA of a given game move when its data move?
Wu and Möhring (arXiv:2007.13979v3, published in Mathematics of Operations Research, 2023) answer this with a metric on games that compares demands and cost functions simultaneously. They show that the PoA is continuous, that it is Hölder continuous with exponent near every game whose costs are Lipschitz, and that it is not uniformly Hölder continuous. This mission formalizes their Theorem 12 (§4.3): at two special classes of games, games with constant costs and games with strictly increasing continuously differentiable costs, the exponent improves to , so the PoA is locally Lipschitz. The authors motivate the result by the differentiability results of Cominetti, Dose and Scarsini (2020), which suggest Lipschitz behaviour at games whose costs have strictly positive derivatives.
Fix a finite set of arcs, a finite nonempty set of origin–destination (O/D) pairs, and for each a set of at least two paths, each a nonempty set of arcs; every arc lies on some path. A game consists of arc cost functions , continuous, non-decreasing and non-negative on , and demands with total demand ; moreover for . These games form the game space .
A flow routes each demand, . The arc flow is , the path cost , and the total cost . A flow is a Wardrop equilibrium (WE) if every used path of a pair is a cheapest path of that pair. The SO cost is the minimum total cost, the WE cost is the total cost of a WE flow (all WE flows have the same cost), and the PoA is
The distance of two games and is
which compares cost functions only on the range both games use, plus their values at the respective total demands.
Let .
then there are and such that
The goal leaves the constants unspecified; only their existence, depending on alone, is asserted.
At a general game with Lipschitz costs the PoA is only Hölder continuous with exponent in this metric, and the paper shows that no exponent works uniformly over the game space. Theorem 12 identifies two natural classes of games at which the sensitivity is linear: constant costs (the PoA equals and grows at most linearly under perturbation) and strictly increasing costs, which include affine costs with positive slopes. For these games an estimation error of size in demands and costs changes the PoA by .
The result is proved in the paper; it is not formalized anywhere. A complete development would also provide machine-checked equilibrium-comparison estimates between two games of one demand slice ((4.39)–(4.40)), which are reusable for any stability analysis of Wardrop equilibria. The paper's explicit constants in (4.41)–(4.46) are off by a factor (the last step of (4.40) uses ); the existential Theorem 12 is unaffected, and a formalization with correct explicit constants is a welcome addition.
The PoA is a ratio of two optimal values, and the perturbed game may change the demand vector and the costs at once, including the costs on , which 's hypotheses do not control. A direct comparison of and therefore fails: the WE flows live in different feasible sets, and the hypotheses on say nothing beyond . The general Hölder- argument loses a square root exactly where it bounds the distance between the arc flows of the two equilibria through an approximate-equilibrium estimate; obtaining exponent requires an estimate in which the flow difference enters linearly, and this is where strict monotonicity of the costs is needed. Constant costs are degenerate in the opposite direction: the WE is far from unique, and only costs, not flows, can be compared.
The model follows §2–§3.2 of the paper and reuses the published definition KellyStochasticNetworks_Wardrop for arc flows, feasibility and the Wardrop condition. Paths are nonempty incidence columns (a generalization of loop-free paths that the proofs do not use); is nonempty and Condition 1 is standing (, every arc on a path). Costs are functions constrained on only. is an infimum over all flows, a supremum over WE flows, ; the distance is (3.3) with the sup norm on demand and endpoint vectors and a supremum over the compact set . The derivative in Theorem 12b) is derivWithin on , one-sided at the endpoints, and is ContDiffOn ℝ 1 on that interval.
Every statement quantifies over games in the game space only: off it, would be a junk ratio such as . The paper prints the conclusions of Theorem 12 and (4.33) with a strict inequality, which fails at ; the non-strict form is posed, and no hypothesis is added (distinct games can have distance ). Milestones (4.39) and (4.40) are posed for two arbitrary games of one demand slice, the general form of the paper's step for its auxiliary game.
A complete development needs existence of SO and WE flows on the game space, the variational characterization of Wardrop equilibria, the mean value theorem on a closed interval with one-sided derivatives, and continuity of in the metric; contributions of any of these as separate lemmas are welcome. The source is the arXiv preprint v3; the published Mathematics of Operations Research version is not the cited artifact.
Traffic users choose routes for themselves, while a planner may choose routes that minimize everyone's combined travel cost. The price of anarchy measures the cost of the users' equilibrium relative to the planner's optimum. Travel demand and arc latency functions are estimated from data and change over time, so a ratio that changes abruptly under a small input change would be difficult to interpret or use. Wu and Möhring study this sensitivity for nonatomic congestion games, where each individual user has negligible influence on arc flows. Their preprint proves that the optimum cost, equilibrium cost, and their ratio vary continuously over the game space they define. This mission isolates that foundational continuity result. It supplies the qualitative starting point for the paper's later Hölder and Lipschitz estimates.
The cited artifact is arXiv:2007.13979v3, dated 2021. The article subsequently appeared in Mathematics of Operations Research in 2023, but theorem numbers and pages here refer to the preprint.
Fix finite sets of arcs, of routes, and of origin–destination pairs. Each route serves one pair and contains at least one arc. Every arc belongs to some route, and every pair has at least two routes. A game assigns each arc a nonnegative, nondecreasing, continuous latency function and each pair a nonnegative demand . The total demand is . These are the paper's basic model and standing combinatorial assumptions (pp. 10–11 and 14).
A feasible flow puts nonnegative traffic on routes so that the traffic for each pair sums to its demand. The induced arc flow is . Its total cost is
A social optimum (SO) is a feasible flow minimizing total cost. A Wardrop equilibrium (WE) is a feasible flow in which every route carrying positive traffic has minimum cost among the routes serving the same pair. Write for the SO cost and for the WE cost. The paper shows that all SO flows have the same cost and all WE flows have the same cost (pp. 11–12 and 18).
A generalized game has the stated nonnegative demands and regular latency functions. The game space further requires and for every arc and every , the paper's Condition 2. On this space, the optimum cost is positive and the price of anarchy is defined by
For and , the paper's distance is the maximum of three discrepancies: the sup norm of , the largest for and , and the sup norm of . The last term compares costs at the two games' respective demand endpoints. Games with equal demands and equal latency functions up to that demand are identified as equivalent (pp. 15–17, (3.1)–(3.3)).
The supporting milestones establish a positive lower bound for and an upper bound for (Lemma 3), the distance properties of generalized games (Lemma 4), and behavior of feasible, SO, and WE flows under convergent game sequences (Lemma 5). The goal is the paper's Theorem 6, p. 19: for any sequence with and ,
The theorem establishes that sufficiently small changes in both demand and latency data produce small changes in the two total costs and their ratio. It allows the PoA to be treated as a stable observable of a game, including where the identity of an optimum or equilibrium flow changes. The result is qualitative: it does not supply a uniform rate of change. The same paper investigates sharper rates under additional assumptions (§4).
The mathematical result is proved in the paper. In this mission its statements and definitions are compiled Lean drafts, while proofs of Theorem 6 and its milestone lemmas remain open. A completed development would make the game-space metric, limit-of-flow statements, and cost continuity available for later sensitivity results on nonatomic routing.
Pointwise continuity of each does not by itself control an optimum or equilibrium when demands change: the feasible-flow sets change with , and a flow for one game need not be feasible for another. A converging sequence of games can also have different optimal or equilibrium flows at every index. The distance compares functions only on demand-relevant intervals, so limits of costs must respect those varying intervals. Finally, continuity of the quotient requires a positive optimum cost throughout the game space; the paper's Condition 2 is what makes that denominator meaningful (pp. 15–19).
Lean uses finite types for arcs, routes, and pairs; a route is a nonempty – incidence column and is assigned to one pair. This admits arbitrary route collections rather than insisting on loop-free directed graph paths. The standing network predicate records the nonempty pair set, nonempty routes, every arc's membership in a route, and at least two routes per pair. Costs are total functions on the reals, with continuity, monotonicity, and nonnegativity required only on . Flow feasibility, arc flow, the Wardrop condition, and the potential reuse the published KellyStochasticNetworks_Wardrop definition. The total cost uses the arc formula; is an infimum over feasible flows and a supremum over WE flows. On valid games these recover the paper's common SO and WE costs.
The distance uses the sup norm on finite vectors and a real supremum over the bounded shared demand interval. Convergence is stated sequentially, following the paper, without imposing a metric-space instance on raw games that may be equivalent. Every theorem involving quantifies over the game space, so a zero denominator or an empty equilibrium set cannot supply a spurious value. The route-count and arc-count factors in Lemma 3 are the actual finite cardinalities. Contributions that establish existence and compactness of relevant flow sets, equivalence of the arc and path cost formulas, and the stated limit properties are useful beyond this theorem.
Dividing indivisible goods fairly among agents is a basic problem of fair division. Most of the literature assumes that the agents' valuations are known to the algorithm. In practice the algorithm only sees what the agents report, and an agent may misreport to get a better bundle. For allocation without money, truthful mechanisms are very restricted: for two agents with additive valuations, truthfulness and fairness are essentially incompatible (Amanatidis, Birmpas, Christodoulou and Markakis, EC 2017, reference [3] of arXiv:2109.08644v2).
Amanatidis, Birmpas, Fusco, Lazos, Leonardi and Reiffenhäuser (arXiv:2109.08644v2; Mathematics of Operations Research, 2024) drop truthfulness and ask a different question: do the pure Nash equilibria of a mechanism lead to allocations that are fair with respect to the agents' true valuations? They show that Round-Robin's equilibria are EF1 and that a modified cut-and-choose mechanism for two agents has equilibria that are MMS and EFX. This mission formalizes the last result of §4.2: a statement about every mechanism whose equilibria always exist and are always EFX.
There are two agents, and , and a finite set of indivisible goods. Agent has a non-negative value for each good and values a bundle additively, . Zero values are allowed. An allocation is a partition of : every good goes to exactly one agent.
A mechanism takes a bid profile , with the values agent claims, and returns an allocation of all goods. A bid profile is a pure Nash equilibrium (PNE) for the valuations (Definition 2.7) if no agent can obtain a bundle of strictly larger true value by switching to any other non-negative bid vector .
The class studied here (p. 18) consists of the always-EFX mechanisms: for every valuation profile on , has at least one PNE, and every PNE leads to an allocation that is EFX with respect to . The class is non-empty: the paper's Mod-Cut&Choose belongs to it (Theorem 4.3).
For every always-EFX mechanism , every valuation profile and every PNE ,
The number may depend on the instance and the equilibrium. The theorem asserts the strict separation from , with no explicit constant.
Theorem 2.6 is tight: for every there are EFX allocations that are not -MMS. Theorem 4.7 says that such allocations never arise as equilibria of a mechanism in the class, so strategic behaviour in an always-EFX mechanism yields strictly more than the generic EFX guarantee. For four goods the paper proves that equilibria of the class are fully MMS (Theorem 4.5); whether that holds for more goods, or whether a uniform exists, is left open in the paper's discussion (p. 22). A full MMS result would imply that computing best responses in any polynomial-time mechanism of the class is hard.
The result is proved on paper. As far as the platform's records show, none of these statements has a machine-checked proof. Formalizing it produces a reusable layer for strategic fair division: bids, best responses and PNE over all non-negative bids, EFX in the positive-goods form, and the two-agent EFX ⇒ -MMS bound.
The goal is a statement about an arbitrary mechanism, known only through two properties that hold on all valuation profiles. The natural first attempt, arguing from the allocation alone, cannot work: an EFX allocation can be exactly -MMS, so the equilibrium property must be used. The proof has to couple the equilibrium of the given instance with an equilibrium of an auxiliary instance on the same goods, about which nothing is known except that it exists and is EFX. Controlling what the mechanism outputs on mixed bid profiles, built from the bids of the two equilibria, is where the work lies. Zero-valued goods are essential to the argument, because EFX ignores them.
Fin 2 (agent 1 is 0), goods are Fin m for an arbitrary m; good (or
) is j - 1. Valuations are real per-good values with non-negativity as a hypothesis;
bundle values are sums, so additivity is built in. Bids are real vectors with non-negative
entries.LittleCharity.MMS.mms with two labelled, possibly empty,
bundles.Trivializing readings are ruled out: the conclusion requires strictly and the same for both agents; measuring fairness with bids instead of true values, restricting deviations, or asking for a uniform over the class would each change the theorem.
Contributions welcome: proofs of Theorem 2.6 and the two counting steps (pure finite combinatorics over additive valuations), the case analysis for , and the final coupling argument. The maximin-share facts used (such as ) are also useful elsewhere.
Dividing indivisible goods fairly is a central problem of computational social choice. With additive valuations, the fairness notions of interest are relaxations of envy-freeness, since exact envy-freeness is often impossible with indivisible items. The strongest relaxation in common use is envy-freeness up to any good (EFX) (Caragiannis et al., 2019); a different, share-based benchmark is the maximin share (MMS) (Budish, 2011).
Most algorithms that produce such allocations take the agents' valuations as input. When agents report these valuations themselves, they may misreport, and truthful mechanisms without payments are known to be very limited (Amanatidis, Birmpas and Markakis, 2016). Amanatidis, Birmpas, Fusco, Lazos, Leonardi and Reiffenhäuser (arXiv:2109.08644, Mathematics of Operations Research 2024) instead ask what fairness survives at the pure Nash equilibria of a non-truthful mechanism, measured with the agents' true values. This mission formalizes one of their answers: for two agents and four goods, every equilibrium of a mechanism whose equilibria always exist and are always EFX is in fact an MMS allocation.
There are two agents, and , and a set of indivisible goods. Agent has a non-negative value for each good , and values a bundle additively, . An allocation is a partition of : every good goes to exactly one agent.
A mechanism receives a bid profile , where is the value agent claims for each good, and outputs an allocation of all goods. A profile is a pure Nash equilibrium (PNE) if no agent can obtain a bundle of strictly larger true value by changing her bid to any other .
The class of always-EFX mechanisms (§4.2) consists of the mechanisms that, for every valuation profile on , have at least one PNE, and whose every PNE yields an EFX allocation with respect to . The class is non-empty: the paper's Mod-Cut&Choose mechanism belongs to it (Theorem 4.3).
For and every always-EFX mechanism , every valuation profile and every PNE of for that profile,
Without incentives, an EFX allocation for two agents guarantees only of the maximin share, and this is tight already for four goods (Amanatidis, Birmpas and Markakis, 2018; Theorem 2.6 of the paper). Theorem 4.5 shows that the equilibrium structure of any always-EFX mechanism closes this gap completely on four goods: strategic behaviour, rather than degrading fairness, forces the outcome from EFX up to MMS. It is the first step of the paper's separation between equilibrium EFX allocations and generic ones; Theorem 4.7, a companion mission of this series, extends it to any number of goods with an approximation factor strictly above . The authors suspect the exact statement fails for five or more goods, so the four-good case is where the full MMS guarantee is known.
The theorem is proved in the paper. To our knowledge, neither it nor its lemmas have a machine-checked proof. The mission adds the statement over an abstract class of mechanisms, quantified over all mechanisms and all non-negative bids, which is a reusable template for "every equilibrium of every mechanism in a class" results in mechanism design without money.
The theorem is about an arbitrary mechanism of which nothing is known except the two equilibrium properties of the class. The direct approach, analysing an EFX allocation of the given instance, cannot succeed on its own: Theorem 2.6 exhibits EFX allocations on four goods that miss the maximin share by a factor , and the equilibrium property for the given valuation alone does not exclude them. The argument has to use the class hypothesis on a second, auxiliary valuation profile on the same goods, and couple the equilibria of the two profiles through deviations of a single agent. The remaining difficulty is the finite case analysis on the bundles of four goods and the maximin share, which is a maximum over all two-bundle partitions.
Lean represents agents as Fin 2 and goods as Fin m (here Fin 4), both 0-based. Valuations are per-good real values v i g with the hypothesis ; bundle values are finite sums, so additivity holds by definition. A mechanism is any function from bid profiles Fin 2 → Fin 4 → ℝ to bundle assignments Fin 2 → Finset (Fin 4). The maximin share reuses the published definition LittleCharity.MMS.mms with two labelled, possibly empty bundles. The goods of the lemmas are a bijection of Fin 4 sorting agent 1's values.
The class hypothesis is the paper's in full: the mechanism allocates all goods on every non-negative bid profile, has a PNE for every non-negative valuation profile on the same four goods, and every such PNE is EFX. EFX is Definition 2.1, removing only goods the envious agent values positively; best responses range over all non-negative bid vectors; EFX and MMS are measured with the true valuations, not the bids. A formalization that restricted deviations to a subset of bids, measured EFX or MMS on bids, or used the stronger EFX that removes every good (in the hypothesis, this would weaken the theorem) would be a different and weaker statement and is excluded.
Lemma 4.6 on the page writes in its conclusion where is meant; the Lean states it for agent 1. No hypothesis beyond the page's is added.
A complete development needs finite case analysis over the partitions of four goods and the definitions above. The fairness definitions and the class of mechanisms are reusable for the companion mission on Theorem 4.7. Proofs of any milestone are welcome independently.
Fair division of indivisible goods asks how to split a set of items among agents with different values so that nobody has strong grounds to complain. Algorithms with fairness guarantees, such as Round-Robin, which always produces an allocation that is envy-free up to one good (EF1) when agents report their true additive valuations (Caragiannis et al., 2019), assume that agents report truthfully. Real agents are strategic, and truthful reporting is generally not an equilibrium of sequential allocation mechanisms (Kohler and Chandrasekaran, 1971). Amanatidis, Birmpas, Fusco, Lazos, Leonardi and Reiffenhäuser (arXiv:2109.08644, Mathematics of Operations Research 2024) ask instead whether the outcomes of a mechanism's pure Nash equilibria are fair. For that question to be meaningful, equilibria must exist.
Timeline.
An instance is a triple : a set of agents, a set of goods, and for every agent a non-negative value for every good . Valuations are additive: . Zero values and ties are allowed. All goods must be allocated.
Agents do not report . Each agent submits a bid , and a mechanism maps the bid profile to an allocation. In Round-Robin (Mechanism 1) the agents take turns in the order . At each turn the active agent receives the still-available good with the largest bid. Ties are broken lexicographically: beats for agent if , or if and has the smaller index.
A bid is a best response to if for every . A profile is a pure Nash equilibrium (PNE) if every agent's bid is a best response (Definition 2.7). The mechanism reads only bids. The bundles are judged with the true values.
For every instance with non-negative additive valuations,
Theorem A.3 is half of the paper's headline result for Round-Robin. Theorem 3.3 says that every PNE is EF1 with respect to the true valuations, and Theorem A.3 says that a PNE always exists. Together they show that a designer using Round-Robin gets EF1 outcomes at equilibrium on every additive instance, not only on strict ones. Without existence, the fairness of equilibria would be vacuous on the instances with ties or zero values, which arise naturally (indifferent agents, worthless goods).
The result is proved in the paper, modulo the cited Theorem A.2. As far as is known, none of it is machine-checked. A complete formalization would check the bluff-profile equilibrium of Aziz et al. for bid-based Round-Robin with lexicographic tie-breaking and deviations over all real bid vectors. It would also repair a gap in the perturbation argument. The page measures the perturbation against the smallest difference between two goods, which does not bound the difference between two bundles (values : the good gap is , but and differ by ). The milestones use the bundle-level gap, which makes the argument go through.
The obstacle is Theorem A.2. Truthful bidding is not an equilibrium: on page 7 of the paper, with two agents and three goods, agent 1 gains by bidding for a good that agent 2 would otherwise take first. So an equilibrium must be constructed, and verifying it requires bounding the value of every deviation of every agent. A deviation changes the order in which all later goods are taken, so its effect is not local. The equilibrium property quantifies over a continuum of real bid vectors. Reducing to strict positive valuations also requires care: a perturbation that is small for each good can still reverse a comparison between two bundles.
Agents are Fin n with [NeZero n], goods are Fin m, both 0-based. Values are v : Fin n → Fin m → ℝ with 0 ≤ v i g, and val v i S is the sum over S. Bids are non-negative real vectors. IsPNE mech v b requires every bid to be non-negative and every agent's bid to be a best response against all non-negative deviations, with bundle values computed from the true v.
Round-Robin is computed step by step. rank c sorts goods by non-increasing bid with a stable sort, which implements the lexicographic tie-breaking. rrFrom s G runs Round-Robin from start agent s on goods G. The goal uses only roundRobin = rrFrom 0 univ (Mechanism 1). goodGap is the page's , and bundleGap is the repaired one used in milestones 2–3.
The goal has no positivity or injectivity hypothesis. Several statements would trivialize it and are excluded: an equilibrium over rankings or over a finite menu of bids instead of all of ; bundle values measured on bids instead of true values; and Round-Robin with arbitrary rather than lexicographic tie-breaking, which would change the mechanism.
Needed infrastructure: the run of Round-Robin as a function of rankings, invariance of the allocation under bid changes that preserve the ranking, and the bluff-profile construction. These are reusable for any sequential allocation mechanism. The equilibrium-transfer milestone holds for any mechanism and is elementary. Lemma A.1 (a strict perturbation preserving best responses, p. 23) is formalized in the companion mission on EF1 equilibria. Contributions of the infrastructure above, of a proof of Theorem A.2, and of alternative existence proofs are welcome.
A firm with open positions often cannot interview and offer sequentially: hiring seasons are short, and offers go out in one batch. Some recipients decline, so the firm overbooks, sending more offers than it has positions, and pays for every hire beyond capacity. The same trade-off appears in airline and hotel overbooking and in university admissions. Epstein and Ma (arXiv:2210.04059v3, Operations Research 2024) study this Simultaneous Offering problem as one of three hiring-pipeline models, and analyse the natural class of value-ordered policies, which offer to the best candidates first. Their Theorem 3 shows that a value-ordered policy built from a linear-programming relaxation guarantees a fraction of the optimum when every value is at least . This mission formalizes the converse, Theorem 4: no value-ordered policy can guarantee more than .
An instance has positions and candidates. Candidate accepts an offer with probability , independently of the others, and then adds value . The firm chooses a set of candidates and sends all offers at once. If is the random set of acceptances, the reward is
so each hire beyond capacity costs (the paper rescales values so that this cost is ). Write for the expected reward of offering to , and for the best expected reward; randomizing over offer sets cannot do better.
Candidates are labelled so that . A value-ordered policy with parameter offers to . An instance is -bounded, , if for all . Without such a bound value-ordered policies can be arbitrarily poor (§6.1); measures how soft the capacity is, from nearly hard ( close to ) to nearly compensated by any hire ( close to ), and both theorems take .
For let be the mean of a Poisson variable truncated at , and
In Lean these are SimInstance, reward, OPTsim, valueOrdered, TauBounded, expMinPois, ratioObj, alpha and beta in the namespace HiringLP.SimUpper.
For every , and there is a valid, sorted instance with positions such that
The instance has type-1 candidates with , , , and type-2 candidates with , . With and :
Proposition EC.5: optimal , exist, , and . Proposition EC.6: if and only if , which holds whenever .
Theorem 4 bounds what any value-ordered policy can guarantee on -bounded instances, against the true optimum rather than an LP relaxation. With Theorem 3 it brackets the best guarantee of the class between and , and by Proposition EC.6 the two coincide when , in particular for all when . In that region the paper's truncated-LP policy is the best value-ordered policy in the worst case. The instance also shows why the restriction to value-ordered policies costs something: the optimum offers to the reliable type-2 candidates, while a value-ordered policy must first exhaust the slightly more valuable unreliable ones.
The results are proved in the paper. None of them has a machine-checked proof; this mission produces a formal version of Theorem 4 and the finite- estimates on which its proof rests, together with a reusable finite-sum encoding of independent acceptances, truncated Poisson and Binomial expectations, and the constants , .
The milestones are exact identities and inequalities at a fixed ; the hard part is the passage to the limit. The page's proof lets with and replaces by . The goal, however, asks for one finite instance on which the bound holds for every simultaneously, including of order and the values that also offer to type-2 candidates. A pointwise limit for each fixed does not give this; the Binomial–Poisson error has to be controlled uniformly in , and the error must fit inside the slack , part of which the term already uses. Note also that the comparison runs the wrong way for the obvious argument: is concave and the Binomial is less spread than the Poisson, so is at least its Poisson counterpart and a one-sided comparison cannot replace an approximation bound.
All randomness is finite. The expected reward is an explicit sum over the accepting sets with product probabilities; the Binomial pmf, its truncated mean and its tail are finite sums written by hand. The truncated Poisson mean is the series with the Poisson pmf. All quantities are real. Candidates are Fin n, so the paper's candidate is index ; sortedness is the hypothesis Antitone v in Valid, which also contains , and . is the maximum over all offer sets, never over value-ordered ones. is the supremum over and over . The page leaves implicit in ; the goal quantifies over every and requires the instance to have exactly positions. The quantifiers are "for all , there is an instance, for all ".
The goal does not fix the hard instance: any instance proves it. A formalization in which the comparison is with the best value-ordered policy, in which is a free parameter, or in which the reward is defined by the proof's Binomial formula would be a different statement and is ruled out by the definitions above. The definitions are not specialized to the hard instance, which appears only as a separate definition used by the milestones.
A complete development needs: finite sums over subsets with product weights and the reduction to a Binomial count for identical candidates; summability and continuity of the truncated Poisson mean; a Binomial–Poisson approximation bound uniform in the number of trials; concavity of in . The Poisson and Binomial infrastructure is reusable well beyond this mission. Proofs of the milestones, of the companions, and of auxiliary approximation lemmas are all welcome.
Payment channels let two parties make repeated transfers while using an on-chain transaction only when a channel needs resetting. They require funds to be committed in advance. The economic choice is therefore between the transaction fees saved by using a channel and the opportunity cost of funds tied up in it. Guasoni, Huberman, and Shikhelman give a stochastic cost model in which this tradeoff can be compared for one-way and two-way payment streams (Guasoni et al., 2023, §§1 and 3). Their small-discount-rate result makes the difference between the two arrangements quantitative: the optimal one-way cost grows like , while the optimal symmetric two-way cost grows like .
The paper was published online in Management Science on November 6, 2023. This mission concerns Theorem 3(i)–(ii) in that published version, especially the cost expansion (8). The paper proves these asymptotic claims; the task here is to state them precisely in Lean and develop machine-checked proofs of the stated cost laws.
Two nodes exchange unit payments. Payments from node 1 to node 2 arrive at rate , and payments in the reverse direction arrive at rate . Each payment stream is a Poisson process, and the streams are independent. The continuously compounded discount rate measures the opportunity cost of committed capital. A channel reset costs , while an on-chain payment costs . Node 1 commits and node 2 commits . The channel balance moves by one unit at each payment; at a boundary, a reset replenishes it (Guasoni et al., 2023, §§1.1, 3.1 and 5).
The paper's closed-form bidirectional cost is given in equations (3)–(4). It uses two characteristic roots and , determined by the payment rates and . This mission uses the formula at equal rates . A separate unidirectional cost from equation (1) describes a stream in only one direction. Its optimizing size is denoted below. The exact channel model has integer balance states; the paper optimizes the closed forms over positive real channel sizes in its asymptotic analysis.
Write for the infimum of the unidirectional cost over , and for the infimum of the symmetric bidirectional cost over . These are functions of and as well as of ; the compact paper notation suppresses those arguments. The term has its usual uniform meaning as with and fixed: its absolute value is at most a constant times for all sufficiently small positive .
Theorem 3(i), equation (7), states
The same part states that an optimal size is . The mission records that size claim using the explicit optimizer displayed after equation (2) (Guasoni et al., 2023, pp. 6–7).
The mission's goal is the cost clause of Theorem 3(ii), equation (8):
This includes the constant correction and a two-sided error estimate of order . The theorem also states that optimal commitments are equal and each has size . That separate size clause is outside this proposal; the goal does not assert the existence or location of a minimizing pair (Guasoni et al., 2023, p. 7).
The different exponents distinguish the two designs even when both can make frequent off-chain payments. At small , the model's optimal symmetric bidirectional cost grows more slowly than its optimal unidirectional cost. The formulas also retain the next constant term, allowing a comparison beyond the leading growth rates. The paper relates this difference to the amount of capital that must be committed by the participants (Guasoni et al., 2023, p. 7).
A formal proof would connect the exact channel-cost formulas, a two-variable infimum, the diagonal specialization, bounds on the hyperbolic cosecant, and real asymptotics in one reusable development. The supporting statements are useful separately: the root expansion concerns a quadratic characteristic equation, and the diagonal reduction concerns a symmetric optimization problem. The paper supplies proofs of the mathematical results; the remaining work is their faithful machine-checked formalization.
The cost depends on and on sizes that grow without bound as approaches zero. An expansion valid for each fixed size does not by itself justify substituting a size that varies with . The target asks for a uniform remainder at the infimum, including its constant term. The cost must be controlled from both sides over every admissible positive size; evaluating one promising size supplies only one side of that comparison. The paper's proof of Theorem 3 uses formal small- expansions of the cost and first-order conditions, and the stated error rate calls for care when these are applied to moving optimizers (Guasoni et al., 2023, pp. 11–12).
The Lean definitions reproduce equations (1), (3), and (4). The one-way and two-way costs are separate functions, matching the paper's separate treatment of the unidirectional boundary case. Real powers are used for the real-size optimization; the balance lattice in the exact stochastic model uses integer sizes. Both optimal costs are infima over strictly positive real sizes. The symmetric infimum is not defined by choosing a particular size or by assuming that the minimum is attained. All rates in the asymptotic theorems satisfy , the reset cost satisfies , and the filter sends to zero through positive values. Consequently a Big-O constant may depend on and but cannot depend on .
The on-chain costs for one stream and for two equal streams supply the paper's economic comparison, although does not occur in this mission's cost goal. The diagonal milestone omits that on-chain term because it is constant in the channel sizes; it states a midpoint cost inequality so no minimizer is assumed. The proof's root expansion is recorded for both and . The paper's assertion that the symmetric minimizing sizes equal is explicitly omitted. The development does not fix a numerical value of , permit an -dependent Big-O constant, drop the term, or replace the infimum by cost at a chosen size.
The mission needs real-power identities, properties of and , nonempty bounded-below cost sets, infimum estimates, and Mathlib's Asymptotics.IsBigO on the positive-side neighborhood of zero. Results about real asymptotics and symmetric minimization can be reused beyond this channel model. Contributions proving the cost milestones, the root expansion, the diagonal inequality, and the goal are all within scope.