Operations Research

Motivation

A gambler watching a sequence unfold has to decide, at each moment and knowing only the past, whether to take what is on the table or wait for something better. That is the whole of optimal stopping, and it is one of the few problems in stochastic control with a clean and completely general answer: the value of the problem is the smallest superharmonic function dominating the immediate payoff. Snell (1952) proved the martingale form; the dynamic-programming form is due to Chow, Robbins and Siegmund. It is the structure behind the pricing of American options, the secretary problem, sequential hypothesis testing, and the bandit problems of Chapter 5.

Bäuerle and Rieder's Chapter 10 (Markov Decision Processes with Applications to Finance, Springer, 2011) derives this from their own Markov-decision machinery rather than from martingale theory, which makes the whole development elementary and self-contained: a stopping problem is a Markov Decision Problem whose action space is {continue, stop}, so Chapter 2's finite-horizon theory and Chapter 7's unbounded-horizon theory apply to it verbatim. The chapter then runs the resulting theory on three classical problems and solves each one in closed form.

Setting

The problem. A Markov process (X_n) on a Borel space E is observed. A stopping time is a random time τ with {τ ≤ n} ∈ F_n — "upon observing the process until time n we can decide whether or not τ has already occurred". Stopping at τ collects

$$R_\tau := \sum_{k=0}^{\tau-1} c_k(X_k) + g_\tau(X_\tau),$$

a running reward c_k while continuing and a stopping reward g_τ at the end, and the problem is to find V_N^*(x) := sup_{τ ≤ N} E_x[R_τ] (10.1). Assumption (B_N) — finiteness of the supremum of the positive parts — is what makes this well posed.

The reduction (Theorem 10.1.2). Take A = {0,1}, let a = 0 mean continue and a = 1 mean stop, and make the transition law uncontrollable on continuation and absorbing on stopping. A policy π = (f_0,…,f_{N-1}) induces the stopping time τ_π = inf{n | f_n(X_n) = 1} ∧ N, and conversely every stopping time is a history-dependent policy. The theorem says the two suprema agree: the extra history buys nothing.

The recursion (Theorems 10.1.3, 10.1.5). The Bellman operator becomes a two-branch maximum,

$$\mathcal{T}v(x) = \max\Big\{g(x),\ c(x) + \beta\int v(x')Q^X(dx'|x)\Big\},$$

with no action variable left in it. In the stationary case J_0 = g, J_n = \mathcal{T}J_{n-1}; the J_n increase, the sets S_n^* = {J_n = g} shrink — "the tendency to stop is non-decreasing as time goes by" — and the optimal rule is "stop on first entry into S_{N-n}^*".

The unbounded horizon (§10.2). Now the reward is discounted, R_τ = Σ β^k c(X_k) + β^τ g(X_τ) for τ < ∞, the value is V_∞^*(x) = sup_{τ<∞} E_x[R_τ], and there is no terminal condition to induct from. Three candidate values present themselves: V_∞^*; G = sup_π liminf_n J_{nπ}, a supremum over policies of limits of finite-horizon values; and J = lim_n J_n, which exists by monotonicity. Theorem 10.2.2, the goal, says all three coincide, that the common value solves J = \mathcal{T}J and satisfies 0-free bounds, and — the characterization — that it is the smallest c-superharmonic function majorizing g.

Turning the value into a rule (Theorems 10.2.3, 10.2.7, Corollaries 10.2.6, 10.2.8). Knowing the value is not knowing when to stop. Theorem 10.2.3 produces the stopping region as S^* = {J = g} = {d ≥ 0} where d = lim_n d_n, under two conditions that Corollary 10.2.6 then gives three checkable sufficient conditions for. Theorem 10.2.7 is the practical one, the One-Step-Look-Ahead Rule: if the set where stopping now beats stopping one step later is closed under the transition law, then the myopic rule is globally optimal. Corollary 10.2.8 adds monotonicity and gets a threshold.

Three applications (§10.3). The house seller who receives i.i.d. offers and pays maintenance on each rejection should accept the first offer above an explicit threshold, obtained as the maximiser of a one-dimensional function (Theorem 10.3.1). The secretary problem's value function is computed exactly (Proposition 10.3.2), giving the classical rule — reject the first k^*, then take the first leader — with success probability (k^*/N)h(k^*) and k^*(N)/N → 1/e (Theorem 10.3.3). And when the offers' distribution has an unknown parameter, MTP_2 of the likelihood propagates into monotonicity of the value in the information state (Theorem 10.3.4), with a fully explicit solution for the exponential/Inverse-Gamma conjugate pair (Theorem 10.3.6).

What is being asked

Formalize Theorem 10.2.2 in full: the three-way equality of V_∞^*, G and J, the fixed point equation, and — the part that carries the theorem — minimality among all functions that are both c-superharmonic and above g. Asserting only that J is such a function, or only one of the two conditions, is a strictly weaker and different claim.

The twelve milestones are the rest of the chapter, in attack order: the reduction and the two recursions, then the unbounded-horizon apparatus, then the three worked problems.

The stopping-time apparatus is built rather than assumed — the chain's law pinned by its finite-dimensional distributions, stopping times valued in ℕ ∪ {∞}, rewards vanishing at ∞ — because every theorem here is the identification of a supremum over stopping times with something computable, and carrying the value as an abstract function would make them vacuous. Every supremum is taken as a least upper bound against an explicit set of achievable values rather than by sSup, so that a set unbounded above is not silently given the value 0.

Motivation

Two problems in this mission, both about markets that do not behave the way the textbook Black–Scholes market does, and both solved by the same technique.

The first is portfolio choice in a pure jump market. Prices move by jumps at the epochs of a Poisson process, not by continuous Brownian fluctuation. This is not a technical variation: the market is incomplete, there is no replicating portfolio, and the machinery of stochastic analysis that makes the diffusion case tractable is unavailable. What is available instead is that the wealth process is piecewise deterministic — between jumps it follows an ODE, and all the randomness is in when the jumps happen and how big they are. Chapter 8's technique embeds such a process in its jump chain and turns the continuous-time control problem into a discrete-time Markov Decision Model with infinite horizon; Chapter 7's contracting theory then solves that.

The second is trade execution in an illiquid market. An agent must sell a large block of shares by a deadline. Placing the whole order at once moves the price against them, and in a traditional order book other participants can see the intention and trade against it — so the order goes to a dark pool, where there is no order book and matches arrive at random. The agent can only sell when a counterparty happens to appear, and whatever is unsold at the deadline must be dumped on the traditional market at once. The question is how much to offer at each opportunity.

The two problems have opposite curvature — the first is a concave maximisation of utility, the second a convex minimisation of cost — and the section is a good demonstration that the same embedding technique handles both, with each problem's structure entering only through which set of functions the value function is sought in.

Setting

The jump market (§9.3). The bond is S⁰_t = e^{ρt}; the risky assets follow dS^k_t = S^k_{t-}(μ_k dt + dC^k_t) where C_t = Σ_{n≤N_t} Y_n is a compound Poisson process of intensity λ whose jumps Y_n are supported in (-1,∞)^d, which keeps prices positive. Short-sellings are prohibited, so the admissible fractions of wealth form the compact set 𝒰 = {u ≥ 0, u·e ≤ 1}, and the wealth follows

$$dX_t = X_{t-}\big((\rho + \pi_t\cdot(\mu-\rho e))dt + \pi_t dC_t\big). \tag{9.10}$$

The investor maximises E^π_{tx}[U(X_T)] for a strictly increasing, strictly concave U.

The embedded model's state is (t,x) — a jump time and the wealth just after it — and its action is a whole control path α : [0,T] → 𝒰, followed until the next jump. Between jumps the wealth is φ^α_t(x) = x exp(∫₀^t (ρ + α_s·(μ-ρe))ds) (9.13), and the transition kernel is substochastic: with probability e^{-λ(T-t)} no further jump arrives before the horizon, and the reward r(t,x,α) = e^{-λ(T-t)}U(φ^α_{T-t}(x)) is collected instead.

The trade execution model (§9.4). A Poisson process of intensity λ delivers the trading epochs; selling a shares costs C(a) with C strictly increasing and strictly convex (the discrete form (9.19)), C(0) = 0; the inventory X_t = x₀ - ∫₀^t π_s dN_s is what remains, and C(X_T) is the terminal dump. Here the flow is uncontrolled — the inventory does not move between epochs — which makes the embedded model simpler.

What is being asked

The goal is Theorem 9.3.4, the main result for the terminal wealth problem, in all six of its parts: the value function is the limit of the value iteration and lies in IM_cv; it is the unique fixed point of the dynamic programming operator there; value iteration converges at the explicit geometric rate α_b^n/(1-α_b); there exists an optimal Markov portfolio strategy given by a single decision rule; policy iteration holds; and Howard's policy improvement algorithm holds. Parts a)–c) describe the value; parts d)–f) produce the strategy, and a formalization of the first three alone would omit the entire control half of the theorem.

The seven milestones are the rest of §9.3–9.4: the reduction from continuous to discrete time, the bounding function and its explicit contraction modulus, the invocation of Chapter 7's Structure Theorem, the iff-characterization of when holding only the bond is optimal, the stability of the value and of the optimal policies under perturbation of the utility, and then the trade execution problem's own bounding function and its monotone, unit-Lipschitz optimal execution rate.

Two formalization conventions run through everything here. The operator of §9.3 is a supremum over a space of control paths, and since Mathlib's sSup of a set unbounded above is 0 — with an unbounded reward that is a live risk, not a formality — it is carried as a relation defined by least upper bounds against explicit sets of achievable values, with its iterates a chain of such relations. And the continuous-time side is built, not assumed: Theorem 9.3.1 is the identification of the continuous-time value with the discrete-time one, so the law of the embedded jump chain is pinned by the one-step conditional law the book displays, and the terminal wealth is read off that chain.

Motivation

A correlated equilibrium (Aumann, 1974) is a joint distribution over the players' strategy profiles from which no player gains by deviating from a recommended strategy. A central question of learning in games is which equilibria repeated play of simple, myopic rules can reach. Foster and Vohra (1997) answered it for calibrated forecasting: if each player forecasts the opponent with a calibrated rule and best-responds to the forecast, the empirical distribution of play approaches the set of correlated equilibria (their Theorem 1). That result has become a basic reference point for no-regret and calibration-based learning in games (Hart and Mas-Colell, 2000).

This mission formalizes the paper's converse. Theorem 1 says calibrated learning ends up in the correlated equilibria; the converse says it can end up at any of them, for almost every game. Together the two results characterize exactly which long-run outcomes calibrated learning with best responses can produce.

Setting

Two players choose strategies from finite sets $S(1)$ with $m$ elements and $S(2)$ with $n$ elements; player $i$ receives payoff $u_i(x,y)\in\mathbb{R}$ and maximizes it. A game $G=(u_1,u_2)$ is a pair of real $m\times n$ matrices, that is, a point of $\mathbb{R}^{2mn}$; a set of games has measure zero if it is Lebesgue-null in $\mathbb{R}^{2mn}$.

A joint distribution $D$ on $S(1)\times S(2)$ is a correlated equilibrium if for every map $\Phi:S(1)\to S(1)$, $\sum_{x,y}D(x,y)u_1(\Phi(x),y)\le\sum_{x,y}D(x,y)u_1(x,y)$, and symmetrically for player 2. $\pi(G)$ is the set of correlated equilibria.

Play is repeated in rounds $t=0,1,2,\dots$. Before each round, player 1 issues a forecast $f_1(t)$, a probability vector over $S(2)$, produced by a deterministic forecasting rule from the history of play so far; player 2 likewise forecasts player 1. For a forecast sequence $f$ and the opponent's plays $z$, let $N(p,t)$ be the number of the first $t$ rounds with forecast $p$, and $\rho(p,j,t)$ the fraction of those rounds in which the opponent played $j$ (zero if $N(p,t)=0$). The forecasts are calibrated if for every $j$

$$\sum_p |\rho(p,j,t)-p_j|\,\frac{N(p,t)}{t}\ \longrightarrow\ 0 \qquad (t\to\infty).$$

Each player then plays $R_i$ of its forecast, where the best-reply function $R_i$ picks a best response to every forecast and does not depend on the round. $D_t(x,y)$ is the fraction of the first $t$ rounds in which $(x,y)$ was played.

$\lambda(G)$, the set of limit points of calibrated forecasts, consists of the $D$ for which some best-reply functions $R_1,R_2$ and some calibrated forecasting rules make $D_t(x,y)\to D(x,y)$ for all $(x,y)$.

Formalization targets

Goal: Theorem 2 (p. 47)

$$\text{for Lebesgue-almost every } G\in\mathbb{R}^{2mn}:\qquad \lambda(G)=\pi(G).$$

Milestones

1. $\lambda(G)\subseteq\pi(G)$ for every game (Theorem 1 restated, p. 46).
2. Every joint distribution $D$ is the limiting empirical distribution of a deterministic play sequence supported on $\{D>0\}$ (p. 47).
3. Along such a sequence the conditional forecasts $p_{1,t}=D(x_t,\cdot)/\sum_yD(x_t,y)$ and $p_{2,t}=D(\cdot,y_t)/\sum_xD(x,y_t)$ are calibrated (p. 47).
4. In a correlated equilibrium each recommended strategy is a best response to its conditional forecast (p. 47).
5. For almost every payoff matrix, each set $M_b(x)$ of forecasts to which $x$ is a best response is either empty or contains a forecast in the open simplex at which $x$ is the unique best response (p. 47).
6. The perturbed forecasts $p_i=(1-1/i)p^*+(1/i)q$ converge to $p^*$ and keep a unique best reply (p. 48).
7. For the $3\times3$ game of p. 48, a correlated equilibrium lies outside $\lambda(G)$, and $\lambda(G)$ is the single point mass on $(C,2)$.

Significance

Theorem 1 by itself leaves open whether calibration selects among correlated equilibria, for instance toward Nash equilibria or toward particular payoffs. Theorem 2 closes that question negatively for generic games: every correlated equilibrium is the genuine limit, not merely an accumulation point, of calibrated play with stationary best replies. As the paper notes, adding the assumption that the limit exists therefore does not refine the equilibrium reached, in contrast with Fudenberg and Kreps's result for asymptotically myopic Bayesian play. The $3\times3$ example shows that the genericity hypothesis cannot be dropped.

The results are proved in the 1997 paper; none is formalized. A complete development produces a reusable layer for repeated two-player games (calibration, empirical distributions, best-reply maps, correlated equilibria), a genericity lemma for best-response regions that is useful beyond this paper, and a machine-checked version of an argument that the paper gives only in outline.

Difficulty

The natural construction takes a correlated equilibrium $D$, a play sequence realizing $D$, and forecasts equal to the conditional distributions of $D$. The forecasts are then calibrated and each played strategy is a best response. What fails is the best-reply function: two strategies $x'\ne x''$ can have the same conditional forecast $p^*$, while a stationary $R_1$ maps $p^*$ to only one strategy. Separating them needs forecasts near $p^*$ at which each is the unique best response, and that exists only when the best-response regions have nonempty relative interior, a property that fails on a null set of games (the $3\times3$ example) and whose genericity must be proved. The perturbed forecasts are no longer exactly equal to the conditional frequencies, so calibration has to be re-established with errors that vanish along the sequence.

Formalization scope

• Strategies are Fin m and Fin n; payoffs are real matrices; players maximize. A game is a point of (Fin m → Fin n → ℝ) × (Fin m → Fin n → ℝ) with Mathlib's volume (Lebesgue measure on $\mathbb{R}^{2mn}$), and "almost every" is ∀ᵐ.
• A correlated equilibrium is the joint-distribution form of p. 44 (a joint distribution with the two deviation inequalities).
• Forecasting rules map finite histories to probability vectors; the play is generated recursively from the rules and the best-reply functions. Best-reply functions are arbitrary stationary selections of a best response at every probability vector; they may not depend on the round, since round-dependent tie-breaking enlarges $\lambda(G)$ (the matching pennies example of p. 46).
• Rounds are $0,\dots,t-1$; $D_0=0$. The calibration sum runs over the forecasts issued so far, which is the paper's sum over all $p$ since the other terms vanish.
• $\lambda(G)$ requires convergence of $D_t$, not a subsequence. Replacing the limit by a limit point, or stating only $\pi(G)\subseteq\lambda(G)$, does not formalize the theorem.
• The page's sentence "Almost every game has the property that all the sets $M_b(x)$ have non-empty interior" is false for dominated strategies ($M_b(x)=\emptyset$ on an open set of games). Milestone 5 states the dichotomy the page's argument proves: empty or with relative interior.
• The page prints the denominator of $p_{2,t}$ as $\sum_{x}D(x_t,y)$; the mission uses $\sum_xD(x,y_t)$.

The theorems are stated from the mission's own definitions of the game, calibration and $\lambda(G)$; no statement is vacuous: at $m=0$ or $n=0$ both sides of the goal are empty, and the $3\times3$ example exercises every definition. Contributions welcome: proofs of the milestones in any order, the genericity lemma (Lebesgue-null sets of linear degeneracies), and the calibration estimates for the perturbed forecasts.

Selected references

• D. P. Foster and R. V. Vohra, Calibrated learning and correlated equilibrium, Games and Economic Behavior 21 (1997), 40–55. https://doi.org/10.1006/game.1997.0595
• R. J. Aumann, Subjectivity and correlation in randomized strategies, Journal of Mathematical Economics 1 (1974), 67–96. https://doi.org/10.1016/0304-4068(74)90037-8
• A. P. Dawid, The well-calibrated Bayesian, Journal of the American Statistical Association 77 (1982), 605–610. https://doi.org/10.1080/01621459.1982.10477856
• D. Fudenberg and D. M. Kreps, Learning mixed equilibria, Games and Economic Behavior 5 (1993), 320–367. https://doi.org/10.1006/game.1993.1021
• S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica 68 (2000), 1127–1150. https://doi.org/10.1111/1468-0262.00153

Motivation

Convexity on the integer lattice cannot be defined by the classical secant-line inequality alone: a function can be midpoint-convex along every line and still admit no useful global optimality theory, because integer points off a chosen line are invisible to it. M-convex functions, introduced by Murota, resolve this by replacing the secant condition with an exchange axiom directly generalizing the basis-exchange property of matroids and the convex-hull structure of network flows: a function on the integer lattice is M-convex if, whenever two points can be improved by moving one coordinate up and a compensating coordinate down, at least one such move weakly improves the sum of the two function values. This single axiom turns out to be equivalent to several strikingly different-looking properties — invariance under a wide family of domain operations, supermodularity in the M♮ (translation-invariant) case, and, most importantly, a local-to-global optimality principle: a point is a global minimizer of an M-convex function if and only if no single coordinate exchange improves it. This mission develops the algebraic core of that theory — the exchange axiom's basic consequences, its equivalent local and dynamic reformulations, and the operations that preserve it — building toward the theorem that recasts M-convexity itself as an algorithmically meaningful local-search guarantee.

Companion mission 06-mconvex-functions-i (Discrete Convex Analysis V) covers this chapter's own primary line of development: the equivalence of M-convexity and M♮-convexity with their respective exchange axioms (Theorem 6.2), the M-optimality criterion (Theorem 6.26), a minimizer-cut lemma (Theorem 6.28), and the M-proximity theorem (Theorem 6.37, its goal). This mission builds the vocabulary those results also need (redeclared here, since sibling drafts cannot yet import one another) and proves the results that chapter leaves for a second pass: the domain structure of M- and M♮-convex functions, worked examples (quadratic forms, quasi-separable functions), the operations that preserve M-convexity, supermodularity of the M♮-convex case, the descent-direction property, and — this mission's goal — the equivalence of the exchange axiom with a dynamic sequential-improvement property.

Setting

Fix a finite ground set $V$. A function $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain $\operatorname{dom} f$ is M-convex if it satisfies the exchange axiom (M-EXC[Z]): for $x, y \in \operatorname{dom} f$ and $u \in \operatorname{supp}^+(x-y)$ (coordinates where $x$ exceeds $y$), there is $v \in \operatorname{supp}^-(x-y)$ with

$$f(x) + f(y) \ge f(x - \chi_u + \chi_v) + f(y + \chi_u - \chi_v).$$

Writing $\tilde f(x_0, x) = f(x)$ when $x_0 = -x(V)$ and $+\infty$ otherwise (a lift to one extra coordinate), $f$ is M$^\natural$-convex if $\tilde f$ is M-convex; M♮-convexity is a genuine generalization of M-convexity (every M-convex function is M♮-convex, but not conversely) and coincides with it exactly when $\operatorname{dom} f$ lies on a single hyperplane. The linear-weighted function $f[p](x) = f(x) - \langle p, x \rangle$ (for $p \in \mathbb R^V$) is the standard device for testing local optimality under an arbitrary reweighting.

Formalization targets

Goal: the exchange axiom as sequential improvement

$$f \text{ is M-convex} \iff \forall p \in \mathbb R^V,\ \forall x, y \in \operatorname{dom} f,\ f[p](x) > f[p](y) \implies f[p](x) > \min_{u \in \operatorname{supp}^+(x-y)}\ \min_{v \in \operatorname{supp}^-(x-y)} f[p](x - \chi_u + \chi_v),$$

with the analogous statement for M♮-convexity (Theorem 6.24). This is the weakest stable form: it makes no reference to a specific algorithm, only to the existence of an improving single exchange whenever the current point is suboptimal under any linear reweighting — a property a faster algorithm could exploit without invalidating the characterization itself.

Supporting structural targets

Eleven further results build the vocabulary and toolkit this goal draws on: the domain structure of M-convex and M♮-convex functions (Propositions 6.1, 6.7), the equivalence of the exchange axiom with a local, bounded-distance version (Theorem 6.4), worked examples establishing M-convexity for quadratic forms, univariate, conservation-law, and quasi-separable functions (Propositions 6.8-6.9), the domain and range operations preserving M-convexity (Theorem 6.13, Proposition 6.14), supermodularity of the M♮-convex case (Theorem 6.19), the descent-direction property (Proposition 6.23) that Theorem 6.24 generalizes, and a discrete subgradient inequality (Proposition 6.25).

Significance

Theorem 6.24 is the bridge between the static exchange axiom (a property of function values at pairs of points) and the dynamic behavior of local-search algorithms: it says a greedy single-coordinate-exchange step, applied to any linearly reweighted version of an M-convex function, always finds a strict improvement when one exists. This is exactly the guarantee that makes steepest-descent-type algorithms for M-convex function minimization correct, and it is the theorem chapter 10's algorithmic analysis (Schrijver-type methods) relies on implicitly whenever it argues that local exchange steps make global progress. The descent-direction property (Proposition 6.23) is the special case $p=0$, isolating the core combinatorial fact before the reweighting machinery is added. The operations catalog (Theorem 6.13) is the practical toolkit that lets later chapters build complex M-convex functions (network flow costs, matroid rank functions composed with linear maps) from simple pieces without re-verifying the exchange axiom from scratch each time.

None of these results are open — they are Murota's own systematic development of the exchange- axiom theory, with worked examples drawn from classical quadratic and separable function theory. What this mission contributes is a faithful, machine-checked formal statement of each, sharing the Lean vocabulary (MExchangeAxiom, MNaturalConvex, LinearWeight) the rest of the Discrete Convex Analysis series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The forward direction of Theorem 6.24 (M-convex $\implies$ sequential improvement) follows in one step from Proposition 6.23 applied to $f[p]$, itself M-convex by Theorem 6.13(3) — routine once those two pieces are in hand. The converse is the substantial direction: it must derive the full static exchange axiom from a property that only ever exhibits some improving exchange at some linear weighting, for every pair of suboptimal points — the proof constructs an explicit adversarial weighting $p$ designed so that failure of the local exchange step at that specific $p$ forces the domain itself to be M-convex (via Theorem 4.3) and then forces the local exchange axiom (M-EXCloc[Z]) via a bipartite-matching argument on the coordinates that differ, finally invoking Theorem 6.4 to lift locality to the full exchange axiom. No shortcut bypasses this two-stage reduction (domain structure, then local exchange) — attempting to verify (M-EXC[Z]) directly from (M-SI[Z]) without first pinning down that $\operatorname{dom} f$ is M-convex fails because the exchange axiom's own statement presupposes a well-structured domain.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; functions are (V → ℤ) → WithTop ℝ. SuppPos/SuppNeg are Finset V (not Set V), matching how the (M-SI[Z])/(M♮-SI[Z]) axioms and the descent-direction property use Finset.inf, whose value on an empty index set is ⊤ — exactly the book's own stated convention for an empty minimum. No Module ℝ or ConvexOn machinery is used for WithTop ℝ-valued arithmetic; scalar actions by positive reals (PosScalarMul, Theorem 6.13(1)) and by naturals (FCheck's flow coefficients, Proposition 6.25) are built directly from the native order and AddMonoid structure. No numeric constants are hard-coded anywhere in this mission (rule 7 is vacuous); Proposition 6.8's quadratic-form conditions are stated with the book's own literal coefficients ($0$, and the min/≥ structure of Eq. (6.25)-(6.28)), not a special case. Theorem 6.13's parts (7) (aggregation) and (8) (integer infimal convolution) are not restated here since the book itself proves them only later via Chapter 9's network-transformation machinery — see the Difficulty note and MODERATION_NOTES.md; this is not a trivializing omission, since the six operations that are included already exercise every domain- and range-transformation technique this mission's goal needs. This mission's definitions (MExchangeAxiom, MNaturalConvex, CharVec, DomZ, SuppPos, SuppNeg, CharVecOpt) are redeclared from chunk 06-mconvex-functions-i rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the twelve sorrys are welcome; the goal's converse direction and Theorem 6.13's operations are the two with the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• K. Murota, "Discrete convex analysis," Mathematical Programming, 83 (1998), pp. 313-371 (the exchange axiom and its equivalent local/dynamic reformulations).

Motivation

Financial regulators and risk managers summarise the loss distribution of a position by a single number, the capital that must be held against it. The two standards in practice are Value-at-Risk (VaR), a quantile of the loss, and expected shortfall (ES), an average of the upper quantiles; ES is the standard of the Basel framework. Both depend on the position only through its probability law, a property called law invariance, which is what allows them to be estimated from data.

The axiomatic theory of risk measures, starting with Artzner, Delbaen, Eber and Heath (1999) and Föllmer and Schied (2002), centres on convex risk measures. VaR is not convex, however, and neither are its robust variants used in practice: the maximum or the median of VaR over a set of scenario models, or the benchmark-loss VaR of Bignozzi, Burzoni and Munari (2020). Castagnoli, Cattelan, Maccheroni, Tebaldi and Wang (Operations Research 70(5), 2022) propose star-shapedness as the common property of all of these measures: doubling the exposure to a risky position at least doubles the required capital. Their Section 7 identifies exactly which law-invariant risk measures are star-shaped.

Timeline:

• 1999: Artzner et al. introduce coherent risk measures and show that ES-type measures dominate VaR (doi:10.1111/1467-9965.00068).
• 2002: Föllmer and Schied introduce convex risk measures.
• 2020: Mao and Wang characterise the risk measures consistent with second-order stochastic dominance as infima of ES-based functionals.
• 2022: Castagnoli et al. prove Theorem 5, the corresponding characterisation of star-shaped law-invariant risk measures through VaR.

Setting

Let $(\Omega,\mathcal F,P)$ be a probability space with $P$ atomless: every event of positive probability contains an event of strictly smaller positive probability. A position is a bounded measurable function $X:\Omega\to\mathbb R$, read as a loss: $X(\omega)>0$ is money lost in state $\omega$. The positions form a linear space $\mathcal X$ that contains the constants and carries the pointwise order $X\geqq Y$.

A risk measure is a function $\rho:\mathcal X\to\mathbb R$ that is monotone ($X\geqq Y\Rightarrow\rho(X)\ge\rho(Y)$), translation invariant ($\rho(X-m)=\rho(X)-m$ for real $m$) and normalized ($\rho(0)=0$). It is star-shaped if $\rho(\lambda X)\ge\lambda\rho(X)$ for all $X$ and all $\lambda>1$, and law-invariant if $X$ and $Y$ with the same law under $P$ satisfy $\rho(X)=\rho(Y)$. Its acceptance set is $\mathcal A_\rho=\{X\mid\rho(X)\le 0\}$. A set $S$ in a vector space is star-shaped if $\lambda s\in S$ whenever $s\in S$ and $\lambda\in[0,1]$.

For $\alpha\in(0,1)$ the Value-at-Risk of $X$ is

$$\mathrm{VaR}_\alpha(X)=\inf\{x\in\mathbb R : P(X>x)\le 1-\alpha\}.$$

Write $F_X(x)=P(X\le x)$. A loss $X$ first-order stochastically dominates $Y$, written $X\succsim_{\mathrm{FSD}}Y$, if $F_X\ge F_Y$ pointwise, so $X$ is the smaller loss.

Formalization targets

Goal: Theorem 5, (i) ⇔ (ii)

For a function $\rho:\mathcal X\to\mathbb R$ the following are equivalent:

1. $\rho$ is a star-shaped and law-invariant risk measure;
2. there is a star-shaped set $\mathcal G$ of increasing functions $g:(0,1)\to\mathbb R$ with $g(0+)\le 0$ such that

$$\rho(X)=\inf_{g\in\mathcal G}\ \sup_{\alpha\in(0,1)}\ \{\mathrm{VaR}_\alpha(X)-g(\alpha)\}\qquad X\in\mathcal X.\tag{25}$$

The goal fixes neither $\mathcal G$ nor any constant. The paper's "Moreover" clause, closure of the class under the operations of Theorem 1, is excluded: its proof is a citation of Liu et al. (2020, Theorem 2) plus "the rest is straightforward".

Milestones

• Eq. (7): $\rho(X)=\min\{m\in\mathbb R\mid X-m\in\mathcal A_\rho\}$ for every risk measure.
• Proposition 2: for a risk measure $\rho$, the following are equivalent: $\rho$ is star-shaped; $\mathcal A_\rho$ is star-shaped; $\rho=\rho_{\mathcal A}$ for a star-shaped acceptance set $\mathcal A$.
• Eq. (A.1): $F_X\ge F_Y$ if and only if $\mathrm{VaR}_\alpha(X)\le\mathrm{VaR}_\alpha(Y)$ for all $\alpha\in(0,1)$.
• FSD consistency (proof of Theorem 5): if $P$ is atomless and $\rho$ is monotone and law-invariant, then $X\succsim_{\mathrm{FSD}}Y\Rightarrow\rho(X)\le\rho(Y)$.

Significance

Theorem 5 says that the star-shaped law-invariant risk measures are exactly the robustifications of VaR: each benchmark $g$ in $\mathcal G$ gives a capital requirement $\sup_\alpha\{\mathrm{VaR}_\alpha(X)-g(\alpha)\}$, and $\rho$ takes the most favourable benchmark. The class contains VaR, ES, their scenario-based maxima and the benchmark-loss VaR. The theorem parallels Theorem 4 of the same paper, derived from Mao and Wang (2020), in which ES replaces VaR and SSD-consistency replaces law invariance. Together the two results separate the two classes: VaR is star-shaped and law-invariant but not SSD-consistent. Section 7 also records that star-shaped law-invariant measures are in general not minima of law-invariant convex risk measures, which is why a representation specific to VaR is needed.

On the formal side, the result is proved in the paper; no machine-checked proof is known. The mission yields a Lean library of law-invariant risk measures on bounded measurable positions, a quantile-based Value-at-Risk with its basic order properties, and the FSD-consistency of monotone law-invariant functionals on atomless spaces. That last statement is the probabilistic core that other law-invariant representation theorems reuse.

Difficulty

The direction (ii) ⇒ (i) is a direct computation. The direction (i) ⇒ (ii) rests on FSD consistency. The obvious argument writes $X\succsim_{\mathrm{FSD}}Y$ as $X\le Y$ and applies monotonicity, but first-order dominance compares only laws, and two positions ordered in law need not be ordered state by state. One must construct positions with the laws of $X$ and $Y$ that are ordered pointwise, and this uses atomlessness in an essential way: on a space with atoms, the construction can fail. The remaining steps are bookkeeping of extended-real infima and suprema, including levels $\alpha$ near $0$ where $g$ may diverge.

Formalization scope

Positions are the bounded measurable functions $\Omega\to\mathbb R$ (a Submodule ℝ (Ω → ℝ), definition Positions) with the pointwise order, rather than equivalence classes in $L^\infty(\Omega,\mathcal F,P)$. For a law-invariant $\rho$ the two readings agree: almost surely equal positions have the same law, and a position that dominates another almost surely has a pointwise modification with the same law that dominates it everywhere. Atomlessness is defined locally (IsAtomless), because Mathlib's NoAtoms only states that singletons are null, which is weaker. Law invariance compares push-forward measures P.map X; measurability is part of Positions, so these are never degenerate.

VaR is an sInf of reals and is applied only to bounded positions at levels in $(0,1)$, where the infimum is over a nonempty set that is bounded below. In (25) each function $g$ has domain exactly $(0,1)$, "increasing" means weakly increasing, and $g(0+)\le 0$ is stated as $\inf_{\alpha\in(0,1)}g(\alpha)\le 0$ in the extended reals. Both sides of (25) are compared in the extended reals, since the inner supremum can be $+\infty$. The minimum in Eq. (7) is an attained minimum (IsLeast), and the supremum condition on acceptance sets is a least upper bound (IsLUB).

These choices rule out the trivialising formalizations of (25). A real-valued supremum would read an unbounded supremum as $0$. Dropping monotonicity of $g$ or the condition $g(0+)\le 0$ describes a larger class that includes non-normalized functionals. Allowing $\mathcal G=\emptyset$ is excluded because the left side of (25) is a real number and the right side would be $+\infty$.

The following are not part of the mission: the "Moreover" clause of Theorem 5, Theorem 4 (its main direction is Mao and Wang 2020, Theorem 3.1), and Proposition 7 (ES as the smallest SSD-consistent risk measure dominating $\mathrm{VaR}_\alpha$), which would need second-order dominance and ES as additional definitions.

Reusable infrastructure: VaR on bounded measurable functions and its homogeneity and translation properties; the equivalence (A.1); existence of a uniform random variable on an atomless probability space and the quantile coupling it provides. Proofs of these as separate lemmas are welcome.

Selected references

• E. Castagnoli, G. Cattelan, F. Maccheroni, C. Tebaldi, R. Wang, Star-Shaped Risk Measures, Operations Research 70(5):2637–2654, 2022. https://doi.org/10.1287/opre.2022.2303
• P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent Measures of Risk, Mathematical Finance 9(3):203–228, 1999. https://doi.org/10.1111/1467-9965.00068
• H. Föllmer, A. Schied, Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter, 2016. https://doi.org/10.1515/9783110463453
• T. Mao, R. Wang, Risk Aversion in Regulatory Capital Principles, SIAM Journal on Financial Mathematics 11(1):169–200, 2020 (cited as Mao and Wang 2020 in the source paper).
• V. Bignozzi, M. Burzoni, C. Munari, Risk Measures Based on Benchmark Loss Distributions, Journal of Risk and Insurance, 2020 (cited as Bignozzi et al. 2020 in the source paper).

Motivation

A risk measure turns the random loss of a financial position into a single number: the amount of capital a regulator or a risk manager requires to hold the position. Two families of risk measures dominate practice and theory. Value-at-Risk (VaR), a quantile of the loss distribution, is used in banking and insurance regulation; it is positively homogeneous but not convex, so it can penalize diversification. Convex risk measures (Föllmer and Schied 2002; Frittelli and Rosazza Gianini 2002), and their positively homogeneous subclass of coherent risk measures (Artzner, Delbaen, Eber and Heath 1999), reward diversification and come with a duality theory, but they exclude VaR and many of its robust variants.

Castagnoli, Cattelan, Maccheroni, Tebaldi and Wang (Operations Research 70(5), 2022) study the class that contains both: star-shaped risk measures, those for which increasing the exposure to a position never decreases the risk per unit of exposure. The class is closed under the aggregation operations used in practice (averages across models, worst cases across scenarios, medians, risk sharing), which convexity is not. It contains VaR, Expected Shortfall, their scenario-based robustifications such as $\mathrm{MaxVaR}$, the benchmark-loss VaR of Bignozzi et al. (2020), and utility-based shortfall risk for utilities with the Landsberger–Meilijson property.

Timeline. Artzner et al. (1999) axiomatize coherent risk measures. Föllmer and Schied (2002) and Frittelli and Rosazza Gianini (2002) introduce convex ones. Föllmer and Schied (2016, Proposition 4.47) show that VaR is the minimum of the convex risk measures dominating it. Castagnoli et al. (2015) state the representation below without proof. The 2022 paper proves it for every star-shaped risk measure, and shows that the property characterizes the class.

Setting

Fix a set $\Omega$ of states. The space of positions $\mathcal X$ is a linear space of bounded functions $X:\Omega\to\mathbb R$ containing every constant function; the constant $m$ is identified with the position that pays $m$ in every state. A value $X(\omega)>0$ is a loss. No probability measure is fixed. $\mathcal X$ is ordered pointwise: $X\geqq Y$ means $X(\omega)\ge Y(\omega)$ for every $\omega$.

A risk measure is a function $\rho:\mathcal X\to\mathbb R$ that is monotone ($X\geqq Y\Rightarrow\rho(X)\ge\rho(Y)$), translation invariant ($\rho(X-m)=\rho(X)-m$ for all real $m$) and normalized ($\rho(0)=0$). It is

• star-shaped if $\rho(\lambda X)\ge\lambda\rho(X)$ for all $X$ and all $\lambda>1$;
• convex if $\rho(\lambda X+(1-\lambda)Y)\le\lambda\rho(X)+(1-\lambda)\rho(Y)$ for all $X,Y$ and all $\lambda\in(0,1)$;
• positively homogeneous if $\rho(\lambda X)=\lambda\rho(X)$ for all $\lambda>0$;
• coherent if it is positively homogeneous and subadditive, $\rho(X+Y)\le\rho(X)+\rho(Y)$.

The acceptance set of $\rho$ is $\mathcal A_\rho=\{X\in\mathcal X\mid\rho(X)\le0\}$. More generally, an acceptance set is a subset $\mathcal A\subseteq\mathcal X$ with $\sup\{m\in\mathbb R\mid m\in\mathcal A\}=0$ that is closed downwards ($X\in\mathcal A$, $Y\leqq X$ imply $Y\in\mathcal A$). It is convex if convex and coherent if a convex cone, and it generates $\rho_{\mathcal A}(X)=\inf\{m\mid X-m\in\mathcal A\}$. A set $S$ is star-shaped if $\lambda s\in S$ for all $s\in S$ and $\lambda\in[0,1]$.

In the Lean development the space of positions is PositionSpace Ω, positions are elements of 𝒳.carrier, and the predicates are IsRiskMeasure, IsStarShaped, IsConvexRiskMeasure, IsCoherentRiskMeasure, acceptanceSet, IsAcceptanceSet, IsConvexAcceptanceSet.

Formalization targets

Goal: Theorem 2 (p. 2643)

For a risk measure $\rho$, the following are equivalent:

1. $\rho$ is star-shaped;
2. there is a set $\Gamma$ of convex risk measures with

$$\rho(X)=\min_{\gamma\in\Gamma}\gamma(X)\qquad\text{for all }X\in\mathcal X;$$

3. there is a family $\{\mathcal A_\beta\}_{\beta\in B}$ of convex acceptance sets with

$$\rho(X)=\min\{m\in\mathbb R\mid X-m\in\mathcal A_\beta\text{ for some }\beta\in B\}\qquad\text{for all }X\in\mathcal X.$$

Moreover, for star-shaped $\rho$, $\Gamma$ may be taken to be the set of all convex risk measures $\gamma\geqq\rho$, and the family to be their acceptance sets. The minima are attained. The goal fixes no particular $\Omega$, $\mathcal X$ or $\Gamma$.

Milestones

• Proposition 1 (p. 2642): star-shapedness is equivalent to $\rho(\alpha X)\le\alpha\rho(X)$ for $\alpha\in(0,1)$, and to the risk-to-exposure ratio $\beta\mapsto\rho(\beta X)/\beta$ being increasing on $(0,\infty)$.
• Eq. (7) (p. 2642): $\rho(X)=\min\{m\mid X-m\in\mathcal A_\rho\}$.
• Proposition 2 (p. 2642): $\rho$ is star-shaped iff $\mathcal A_\rho$ is star-shaped iff $\rho=\rho_{\mathcal A}$ for a star-shaped acceptance set $\mathcal A$.
• Theorem 1 (p. 2643), in four parts: the infimum, supremum, $\mu$-average and inf-convolution of star-shaped risk measures are star-shaped risk measures.
• Proposition 3 (p. 2642): for subadditive risk measures, star-shaped, positively homogeneous and convex coincide.
• Theorem 2, positively homogeneous case: the same equivalence with "positively homogeneous", "coherent risk measures" and "coherent acceptance sets".
• Corollary 1 (p. 2645): $\inf_{X\in\mathcal Y}\rho(X)=\inf_{\gamma\in\Gamma}\inf_{X\in\mathcal Y}\gamma(X)$ for any $\mathcal Y\subseteq\mathcal X$.

Significance

Theorem 2 identifies star-shaped risk measures as exactly the lower envelopes of convex risk measures. Consequences drawn in the paper: minimizing a star-shaped risk measure over a set of positions reduces to a family of convex risk-minimization problems (Corollary 1, Proposition 6), each convex risk measure in the envelope carries its dual representation (Proposition 5), and VaR-type measures inherit a tractable structure without being convex. The positively homogeneous case gives the analogous statement for VaR-like measures in terms of coherent ones, generalizing Föllmer–Schied Proposition 4.47.

The paper proves these results; the mission adds a machine-checked proof on a general space of bounded positions, with the attainment of every minimum made explicit. No formalization of star-shaped risk measures is known to exist. The platform mission Coherent Measures of Risk formalizes the Artzner et al. axioms on finitely many states with a gain convention; its definitions are not reused here. The definitions of this mission (risk measures on a space of bounded functions, acceptance sets, the aggregation operations) are reusable by any later development of monetary risk measures without a reference probability.

Difficulty

The equivalence (2)⇔(3) and the direction (2)⇒(1) reduce to closure properties of the class; the content is (1)⇒(2) together with the "Moreover" clause. The obvious first idea, taking the convex hull or convex envelope of $\rho$ or of $\mathcal A_\rho$, fails: the convex hull of $\mathcal A_\rho$ generates a convex risk measure below $\rho$, not above it, and a single convex risk measure cannot equal a non-convex $\rho$. What is needed is, for each position, a convex risk measure that dominates $\rho$ everywhere and touches it at that position, and it must be monotone, translation invariant and normalized, not merely a convex functional. Attainment of the minimum, rather than an infimum, is part of the claim.

Formalization scope

• $\mathcal X$ is any Submodule ℝ (Ω → ℝ) containing the constants whose elements are bounded, bundled as PositionSpace Ω. It is not specialised to all bounded functions, and no measurability or probability is imposed; positive values are losses.
• "min" is always IsLeast (attained). The acceptance-set axiom $\sup\{m\mid m\in\mathcal A\}=0$ is IsLUB, not sSup … = 0. $\rho_{\mathcal A}$ uses the real sInf, which is genuine for acceptance sets on bounded positions.
• Every $\gamma\in\Gamma$ is a full risk measure: monotone, translation invariant, normalized and convex. A representation of $\rho$ as a minimum of arbitrary, non-normalized convex functionals holds for every monotone translation-invariant map and is not this theorem; such a formalization is ruled out.
• The family in (3) is a set of subsets of $\mathcal X$, with no finiteness or nonemptiness assumption.
• A coherent acceptance set is convex and closed under multiplication by every $t>0$.
• Theorem 1 adds the hypotheses the page leaves implicit: a nonempty index set for supremum and infimum; the power-set σ-algebra and a countably additive probability measure for the average (the paper's proof also covers capacities with Choquet integrals, which are not stated here); $n\ge1$ and the normality condition (10) for the inf-convolution.
• Corollary 1 computes infima in the extended reals, so the set $\mathcal Y$ may be empty and the infima may be $-\infty$.

Contributions welcome: proofs of any milestone, and in particular general lemmas on risk measures on a space of bounded functions (the bounds $\inf X\le\rho(X)\le\sup X$, properties of $\rho_{\mathcal A}$, convexity of $\rho_{\mathcal A}$ for convex $\mathcal A$), which are reusable across risk-measure missions.

Selected references

• E. Castagnoli, G. Cattelan, F. Maccheroni, C. Tebaldi, R. Wang, Star-Shaped Risk Measures, Operations Research 70(5):2637–2654, 2022. https://doi.org/10.1287/opre.2022.2303
• P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent Measures of Risk, Mathematical Finance 9(3):203–228, 1999. https://doi.org/10.1111/1467-9965.00068
• H. Föllmer, A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics 6(4):429–447, 2002. https://doi.org/10.1007/s007800200072
• M. Frittelli, E. Rosazza Gianin, Putting order in risk measures, Journal of Banking & Finance 26(7):1473–1486, 2002. https://doi.org/10.1016/S0378-4266(02)00270-4
• H. Föllmer, A. Schied, Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter, 2016. https://doi.org/10.1515/9783110463453

Motivation

Convexity is normally defined additively — a function's value at a mixture is bounded by the mixture of its values — but many of the properties that make convexity useful in optimization (a local minimum is global, level sets are well-behaved) survive under a much weaker, purely ordinal notion: quasi-convexity, which compares function values rather than adding them. A nondecreasing rescaling of a convex function is generally not convex, but it is always quasi-convex — so a theory built only on ordinal comparisons automatically covers every such rescaling for free, at the cost of a more delicate proof architecture (since the algebraic cancellations available to additive convexity are no longer available).

Chapter 6's second half asks exactly how far this idea extends in the discrete setting: does the M-convexity exchange axiom have an ordinal, quasi-convex relaxation that still supports the same strong minimization theory — an optimality criterion, a minimizer-cut lemma, and, most significantly, a proximity theorem with the same explicit distance bound? This mission formalizes the chapter's answer: yes, and the relevant relaxed class, functions satisfying condition (SSQM$_{\ne}$), is large enough to include every strictly increasing rescaling of an M-convex function, a class the M-convex theory of chunk 06 alone says nothing about.

Setting

Let $V$ be a finite ground set and $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain. Building on chunk 06's M-convex exchange axiom (M-EXC[Z]), this chapter introduces several ordinal relaxations. $f$ is weakly quasi M-convex, satisfying (QMw), if for every pair of distinct points $x, y \in \operatorname{dom} f$ there exist $u$ in the positive support and $v$ in the negative support of $x - y$ with $f(x - \chi_u + \chi_v) \le f(x)$ or $f(y + \chi_u - \chi_v) \le f(y)$ — an "or" where (M-EXC[Z]) demands an additive inequality. Two further conditions restrict attention to points of different function value and sharpen the conclusion to a three-way trichotomy (strictly better on one side, or exactly tied on both): (SSQM$_{\ne}$) quantifies universally over $u$ (as in (M-EXC[Z])), while (SSQM$_{\ne,w}$) quantifies existentially over both $u$ and $v$ (as in (QMw)). The linear perturbation of $f$ by $p : V \to \mathbb R$ is $f[p](x) = f(x) - \langle p, x \rangle$.

Formalization targets

Goal: Theorem 6.78 (the quasi M-proximity theorem)

Let $f$ satisfy (SSQM$_{\ne}$), $n = |V|$, $\alpha$ a positive integer. If $x_\alpha \in \operatorname{dom} f$ satisfies $f(x_\alpha) \le f(x_\alpha + \alpha(\chi_v - \chi_u))$ for all $u, v \in V$, then $\arg\min f \ne \emptyset$ and there is $x^* \in \arg\min f$ with $\|x_\alpha - x^*\|_\infty \le (n-1)(\alpha - 1)$ — verbatim the same conclusion, and the same exact bound, as chunk 06's Theorem 6.37(1), now established for the strictly larger class satisfying (SSQM$_{\ne}$) rather than the M-convex exchange axiom itself.

Milestones: Theorems 6.68(2), 6.76, 6.77

Theorem 6.68(2): $f$ satisfies (M-EXC[Z]) if and only if every linear perturbation $f[p]$ satisfies (QMw) — quantifying exactly how much weaker (QMw) is pointwise, and how the gap closes once quantified over every perturbation. Theorem 6.76 (the quasi M-optimality criterion): the direct analogue of chunk 06's Theorem 6.26 for the quasi-convexity classes — a purely pairwise local check still characterizes global (or, in the (QMw) case, strict unique) optimality. Theorem 6.77 (the quasi M-minimizer cut): chunk 06's Theorem 6.28 continues to hold verbatim when its M-convexity hypothesis is replaced by (SSQM$_{\ne}$) — the structural fact the proximity theorem's proof is built from survives the relaxation intact.

Significance

The result itself. The proximity theorem is the result algorithms actually use: a scaling algorithm for minimizing quasi-convex functions of this kind inherits exactly the same correctness guarantee, with exactly the same distance bound, as the M-convex case — this is a genuine broadening of chapter 10's algorithmic reach, not a restatement dressed in weaker hypotheses. Every strictly increasing scalar transformation of an M-convex objective (a common modeling device — re-expressing a cost in utility units, or applying a monotone risk measure) now falls under a proximity theorem, whereas prior to this chapter's relaxation such a transformation would generally destroy M-convexity itself and leave optimization theory silent on the transformed problem.

Formalizing it. No matching item exists on the platform for quasi M-convexity in any of its forms. Formalizing Theorem 6.78 requires first pinning down (SSQM$_{\ne}$) exactly (there are six closely related axiom variants in this section of the book, only three of which — (QMw), (SSQM$_{\ne}$), (SSQM$_{\ne,w}$) — are needed for this mission's chosen results), and this mission also captures, via Theorem 6.68(2), the precise sense in which these relaxed conditions are strictly weaker than plain M-convexity while remaining tightly connected to it.

Difficulty

The natural first instinct, given how close the quasi-convexity axioms look to (M-EXC[Z]), is to try to prove Theorem 6.78 by directly imitating chunk 06's proof of Theorem 6.37 line by line. This mostly works — the proof structure (fix a target coordinate, build a chain of strictly decreasing values via repeated exchange steps, bound the chain's length using the scaled hypothesis) survives verbatim — but every step that chunk 06's proof took by adding two instances of the exchange inequality together must be replaced by an ordinal argument, since (SSQM$_{\ne}$) only ever asserts a disjunction of value comparisons, never an additive inequality relating four function values simultaneously the way (M-EXC[Z])'s $f(x)+f(y) \ge f(x-\chi_u+\chi_v)+f(y+\chi_u-\chi_v)$ does. The book's proof handles this by working with strict inequalities and the trichotomy structure of (SSQM$_{\ne}$) directly rather than algebraic cancellation — the same overall architecture, but every arithmetic step rebuilt as a case analysis on which disjunct of (SSQM$_{\ne}$) fires.

Formalization scope

This mission builds directly on chunk 06's published items (CharVec, SuppPos, SuppNeg, DomZ, MExchangeAxiom, ArgMin), per the platform's textbook convention that a later chapter of the same book imports an earlier one's definitions rather than redrafting them; its own namespace DiscreteConvex.MConvexFunctions.Quasi nests under chunk 06's DiscreteConvex.MConvexFunctions accordingly. Δf(z;v,u) (Eq. (6.2)) is never reified as a separate object; every occurrence is unfolded directly into an f-value comparison, avoiding WithTop ℝ subtraction throughout, consistent with chunk 06's own convention.

A trivializing formalization of the goal would silently strengthen (SSQM$_{\ne}$) back to plain M-convexity (making this mission redundant with chunk 06's Theorem 6.37) or loosen the exact bound $(n-1)(\alpha-1)$ to an unspecified function of $n, \alpha$; neither is done. Six axiom variants appear in this section of the book ((QM), (SSQM), (QMw), (SSQM$_w$), (SSQM$_{\ne}$), (SSQM$_{\ne,w}$)); only the three actually needed by this mission's four items are drafted, and Theorem 6.68's first part (an implication chain among the other three) is left out — see MODERATION_NOTES.md. Contributions building the polyhedral M-convex-function bridge (§6.11–6.12, Theorems 6.59–6.64), the level-set characterizations (Theorems 6.72, 6.74), or the scaled quasi M-minimizer cut (Theorem 6.79, the direct generalization of Theorem 6.77 drafted here) are welcome.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• M. Avriel, W. E. Diewert, S. Schaible, I. Zang, Generalized Concavity, Plenum Press, 1988.

Motivation

Chapter 7's abstract machinery — contracting Markov Decision Models, the Structure Theorem, value iteration with an explicit convergence rate — earns its keep by solving concrete problems. Section 7.6 works through four kinds of application: a return to the classical cash-balance inventory problem, now over an infinite horizon; the "red-and-black" gambling problem, where a player tries to reach a target fortune before going bankrupt; and, most substantially, the infinite-horizon two-armed bandit, where the general theory reveals something genuinely surprising — the qualitatively optimal policy can be computed one arm at a time.

Setting

Every application here specializes the general infinite-horizon, contracting-model machinery of chunks 07a/07b to a concrete transition structure. The cash-balance model orders inventory up to a level $a$ at linear cost, incurs a holding/shortage cost, then absorbs a random demand. The red-and-black model bets a fraction of a bounded fortune on a biased coin, absorbing at bankruptcy or at the target. The bandit model reconsiders the Beta-Bernoulli two-armed bandit of chunk 05b, now over an infinite horizon with a genuine discount $\beta<1$: the key new tool is the K-stopping problem, a fictitious single-arm decision problem where, at every stage, the decision maker may either pull the arm or retire with a fixed payment $K$. The Gittins index $I(m,n)$ is the smallest such payment at which retiring immediately is already as good as continuing.

Formalization targets

The goal, Theorem 7.6.10, is the Gittins index theorem for this book's two-armed bandit: always pulling the arm with the higher index is optimal for the full infinite-horizon problem. The milestones build the machinery it needs — the index's definition (Definition 7.6.5) and its equivalent representation as a supremum over stopping times (Theorem 7.6.6), the K-stopping value function's monotonicity/convexity/differentiability properties (Proposition 7.6.7), the index's optimal-stopping-set and indifference characterizations (Corollary 7.6.8), the two-arm joint stopping value's parallel structure (Proposition 7.6.9), and a fixed-point recasting useful for computation (Proposition 7.6.11) — plus, independently, the cash-balance and casino-game applications (Theorems 7.6.1-7.6.4), which use the general theory but not the bandit-specific machinery.

Significance

The Gittins index theorem's real content, emphasized by the book's own remark, is not merely that an optimal policy exists but how little computation it needs: instead of solving one optimization problem over the bandit's full four-dimensional joint state space $\mathbb N_0^2 \times \mathbb N_0^2$, the decision maker solves two independent two-dimensional single-arm problems and compares two numbers. This mission's formalization of the goal is built specifically to keep that separation visible — each arm's index is computed from a single, shared KStoppingValue structure applied to that arm's own state alone, never from a function that happens to take the whole joint state as an argument. The proof route here (via the K-stopping problem's explicit fixed-point characterization, Definition 7.6.5 and Proposition 7.6.11) is a genuinely different construction from the platform's existing Gittins-index theorems (BanditAlgorithm.gittins_index_theorem and related), which are built via Whittle's retirement/charge-accounting argument — checked directly and found to define the index differently enough that this mission drafts its own theorems rather than treat that construction as prior art.

Difficulty

The K-stopping value function $J(m,n;K)$ and the two-arm joint value $\tilde J(x;K)$ are both genuine fixed points of an infinite-horizon Bellman equation with no finite backward recursion to fall back on (the "stopping" option, rather than a terminal condition, is what makes the horizon infinite); this mission bundles them as data satisfying their own defining fixed-point equations, the same convention this series uses throughout for such objects. A second difficulty is Theorem 7.6.6's supremum over stopping times: without a canonical path measure for the underlying Markov chain (not built anywhere in this series), the two expectations the theorem compares are represented as data satisfying the positivity a genuine expectation must have, over an explicit, elementary notion of stopping time (a function of the whole observed path, adapted in the sense that whether it has fired by time $n$ depends only on the path up to $n$) — a faithful, if representational, rendering of the theorem's genuinely path-dependent content.

Formalization scope

The cash-balance model (Theorem 7.6.1) explicitly cites chunk 02d's finite-horizon critical-level sequences as a hypothesis rather than re-deriving them, since this mission's own content is the infinite-horizon extension, not a second proof of the finite-horizon theory those sequences come from. The casino-game theorems (7.6.2-7.6.4) state optimality for the specific, named timid and bold strategies, not for an unnamed "some optimal policy" — the theorems' entire content is that these particular policies, not merely some optimal one, are best in their regime. The bandit model's posterior mean and Bayes-update operator are kept identical in substance to chunk 05b's finite-horizon Beta-Bernoulli model (restated, since chunks cannot import each other's Lean), so a reader can see this section is solving the same underlying statistical model, now over an infinite horizon.

Selected references

• N. Bäuerle and U. Rieder, Markov Decision Processes with Applications to Finance, Universitext, Springer, 2011. DOI: 10.1007/978-3-642-18324-9.
• J. C. Gittins, "Bandit processes and dynamic allocation indices," Journal of the Royal Statistical Society, Series B, 1979 (the original index construction this section's K-stopping-problem approach reformulates).
• P. Whittle, "Multi-armed bandits and the Gittins index," Journal of the Royal Statistical Society, Series B, 1980 (the retirement-option construction the platform's existing Gittins theorems use, a different proof route from this chunk's own).
• L. E. Dubins and L. J. Savage, How to Gamble If You Must: Inequalities for Stochastic Processes, McGraw-Hill, 1965 (the classical red-and-black problem, Theorems 7.6.2-7.6.4).

Motivation

Chunk 07a built the general theory of infinite-horizon Markov Decision Processes and its sharpest special case, contracting models, where Banach's fixed point theorem delivers existence, uniqueness, and an explicit convergence rate all at once. That theory answers "does an optimal policy exist, and can I compute it by iterating a fixed point equation?" This mission answers the two questions a practitioner asks next: what happens when the reward's negative part, rather than its positive part, is the one that needs controlling (positive models, §7.4), and — more strikingly — can finding an optimal policy be reduced to solving a genuine linear program, the single most heavily-optimized computational primitive in all of operations research (§7.5)?

Setting

A positive Markov Decision Model is the mirror image of chunk 07a's general setup: instead of bounding the reward's positive part with an upper bounding function, the negative part is bounded by an integrability quantity $\varepsilon$, and the roles of "largest subharmonic" and "smallest superharmonic" swap accordingly. The computational sections build on chunk 07a's contracting theory directly: Howard's policy improvement algorithm iteratively replaces a decision rule with a strict pointwise improvement; the linear-programming approach recasts the entire optimization problem — the value function and the optimal policy — as a primal/dual pair of linear programs, not over finite vectors but over an infinite-dimensional space of measurable functions ($v \in IM$) and finitely-additive-in-spirit measures ($\mu \in M_b$); and state-space discretization approximates an infinite (Borel) state space by a finite grid, with an explicit, computable bound on the resulting numerical error.

Formalization targets

The goal, Theorem 7.5.8 (Strong Duality), is the section's deepest result: under chunk 07a's contracting Structure Theorem's own hypotheses, the primal linear program $(P)$ is solved exactly by the true optimal value function $J_\infty$, the dual program $(D)$ is solved by the occupation measure of any optimal stationary policy, and the two optimal values coincide. The milestones build up to it in three groups: the positive-model mirror theory (Lemmas 7.4.1-7.4.2, Theorems 7.4.3 and 7.4.5); Howard's policy improvement and its termination guarantee (Theorem 7.5.1, Corollary 7.5.3); and the linear-programming machinery itself (weak duality, complementary slackness, and the finite-state specialization that recovers an ordinary finite linear program, Theorems 7.5.6, 7.5.7, 7.5.9) together with the discretization error bounds that make the whole theory numerically usable (Proposition 7.5.11, Theorem 7.5.12).

Significance

The strong duality theorem is genuinely new content relative to what is already on the platform: the existing finite-dimensional LP duality missions (SmaleNinth.lp_strong_duality, LinearOptimization.lp_general_weak_duality, and others in the linear-optimization field) all operate over $\mathbb R^n$-valued vectors, while this theorem's primal and dual variables are a measurable function on a general Borel space and a measure on a general Borel space respectively — an infinite-dimensional linear program in the fullest sense. Theorem 7.5.9, the finite-state specialization, is the one point of genuine hypothesis-for-hypothesis contact with that prior art (checked directly; see STATUS.md for why it was drafted fresh rather than cited as a reference item), and it is exactly there that the reduction to an ordinary finite LP — with the platform's familiar vertex/extreme-point vocabulary — becomes visible.

Difficulty

Constructing the occupation measure $\mu^{f^\infty}_p$ without a canonical infinite-horizon path measure is the central technical challenge: it must be a genuine Measure (E × A), not merely a real-valued functional, since the dual program optimizes over a space of such measures. This mission builds it from iterated Measure.bind (pushing the initial law $p$ forward through the model's kernel under a fixed stationary decision rule) combined with a countable Measure.sum of $\beta^k$-scaled terms — a construction that stays entirely within Mathlib's existing measure-theoretic vocabulary without needing an Ionescu–Tulcea-style infinite product. A second, different difficulty is the state-space discretization section's grid interpolation, which presupposes a convex-combination structure ($x = \sum_k\lambda_kx_k$ for grid points $x_k$) on the state space that a general Borel space does not carry; this mission represents the grid operator and grid bounding function as data satisfying exactly the structural properties their two target theorems' own proofs use, rather than reconstructing the literal interpolation scheme — a deliberate, documented scope decision (see MODERATION_NOTES.md), not an approximation of either theorem's mathematical content.

Formalization scope

Every operator and value-function construction restates chunk 07a's own vocabulary (per this series' file-ownership convention, an independent copy in this chunk's own namespace), extended by the positive-model integrability bound $\varepsilon$, the occupation-measure/linear-program apparatus of §7.5.2, and the grid-approximation data of §7.5.3. The primal/dual optimal values $\mathrm{val}(P)$/$\mathrm{val}(D)$ are kept EReal-valued rather than real-valued specifically so that Theorem 7.5.6's own finiteness claims ($-\infty < \mathrm{val}(D)$, $\mathrm{val}(P) < \infty$) remain genuine, checkable content rather than being trivialized by a real-valued sInf/sSup's always-finite convention. Theorem 7.5.9's "optimal vertex" is stated via an explicit convex-combination (extreme-point) characterization using ENNReal weights, since Measure does not carry the module structure Mathlib's own Set.extremePoints requires.

Selected references

• N. Bäuerle and U. Rieder, Markov Decision Processes with Applications to Finance, Universitext, Springer, 2011. DOI: 10.1007/978-3-642-18324-9.
• R. A. Howard, Dynamic Programming and Markov Processes, MIT Press, 1960 (the policy improvement algorithm this section names after him).
• E. V. Denardo, "On linear programming in a Markov decision problem," Management Science, 1970 (the classical finite-state linear-programming formulation this section generalizes).
• W. J. Heilmann, "A note on the dual of a linear program with infinitely many constraints," cited by the book's own Remark 7.5.5 for the finitely-additive treatment the restricted dual $(D)$ over $M_b$ sidesteps.

Motivation

Shortest-path distances and network potentials are among the oldest objects in combinatorial optimization: a directed graph with arc lengths, its shortest-path distances, and the "feasible potentials" (vertex labels consistent with those lengths) underlie duality in min-cost flow, scheduling, and difference-constraint systems. Murota's Discrete Convex Analysis (SIAM, 2003) isolates the abstract structure behind these objects — distance functions satisfying the triangle inequality, and their associated sets of admissible potentials — and shows it is governed by exactly the same discrete-convexity machinery as submodular set functions: a one-to-one correspondence with a second family of well-behaved integer point sets, the L-convex sets. Where an M-convex set (chapter 4) is defined by an exchange axiom generalizing matroid base exchange, an L-convex set is defined by closure under coordinatewise lattice operations (∨, ∧) and translation by the all-ones vector — a genuinely different axiom system that nonetheless produces a parallel structural theory: hole-freeness, a polyhedral description via an induced distance function, and integral convexity.

Companion mission 05-lconvex-sets (Discrete Convex Analysis IV) covers this chapter's other half: the hole-free property (Theorem 5.2), the one-to-one correspondence between L-convex sets and integer-valued triangle-inequality distance functions (Theorem 5.5), the intersection properties (Theorem 5.7), and the chapter's discrete separation theorem (Theorem 5.9, its goal). This mission builds the vocabulary those results also need (redeclared here, since sibling drafts cannot yet import one another) and proves the results the chapter leaves for its second half: the fundamental facts connecting a distance function to its admissible potentials (Proposition 5.1), the two-way polyhedral correspondence's supporting propositions (5.3-5.4), Minkowski-sum convexity (Theorem 5.8), and — this mission's goal — the explicit description of an L-convex set's convex hull that establishes its integral convexity (Theorem 5.10).

Setting

Fix a finite ground set $V$. A distance function is a map $\gamma : V \times V \to \mathbb R \cup \{+\infty\}$ with $\gamma(v,v) = 0$; it may take negative finite values and need not be symmetric. It defines a directed graph $G_\gamma = (V, A_\gamma)$ with $A_\gamma = \{(u,v) : \gamma(u,v) < +\infty\}$, arc $(u,v)$ having length $\gamma(u,v)$. Write $\bar\gamma(u,v)$ for the shortest-path length from $u$ to $v$ in $G_\gamma$ ($+\infty$ if none exists); $\gamma$ is well defined ($\bar\gamma$ finite-valued wherever a path exists) exactly when $G_\gamma$ has no negative cycle. The triangle inequality $\gamma(v_1,v_2) + \gamma(v_2,v_3) \ge \gamma(v_1,v_3)$ defines the class $T[\mathbb R]$ (or $T[\mathbb Z]$ when integer-valued). A vector $p \in \mathbb R^V$ is an admissible potential of $\gamma$ if $p(v) - p(u) \le \gamma(u,v)$ for all $u \ne v$; write $D(\gamma)$ for the set of all such potentials.

A nonempty set $D \subseteq \mathbb Z^V$ is L-convex if it satisfies (SBS[Z]): $p, q \in D \implies p \vee q,\ p \wedge q \in D$ (coordinatewise max/min), and (TRS[Z]): $p \in D \implies p \pm \mathbf 1 \in D$. A set $S \subseteq \mathbb Z^V$ is integrally convex if every point of its convex hull $\overline S$ lies in the convex hull of $S$ restricted to that point's integral neighborhood $N(p) = \{y \in \mathbb Z^V : \lfloor p \rfloor \le y \le \lceil p \rceil\text{ coordinatewise}\}$ — a strong, local form of "no holes" saying every real point of the hull is explained by nearby integer points alone.

Formalization targets

Goal: integral convexity of L-convex sets

For an L-convex set $D \subseteq \mathbb Z^V$, writing $a = p - \lfloor p \rfloor$ for the fractional part of $p \in \mathbb R^V$, $\alpha_1 > \cdots > \alpha_m$ for the distinct nonzero values of $a$, and $U_i(p) = \{v : a(v) \ge \alpha_i\}$ (with $U_0 = \emptyset$):

$$\overline D = \{p \in \mathbb R^V : \lfloor p \rfloor + \chi_{U_i(p)} \in D\ \ (i = 0, 1, \ldots, m)\}, \qquad \text{hence } D \text{ is integrally convex}.$$

This is the weakest stable form available: it exhibits an explicit, finite set of at most $|V|+1$ integer witnesses for every point of the hull, which is what "integrally convex" asserts abstractly, rather than a numerical bound that a sharper construction could later shrink.

Supporting structural targets

Four further results build the correspondence this goal uses: the basic duality between a distance function's admissible potentials, its shortest-path closure, and negative-cycle freedom (Prop. 5.1); the induced-distance-function construction recovering a triangle-inequality distance function from any integer point set, and the convex hull of an L-convex set as its associated polyhedron (Prop. 5.3); the converse construction recovering an L-convex set from an integer-valued distance function (Prop. 5.4); and convexity in Minkowski sum (Thm. 5.8).

Significance

Theorem 5.10 is what makes "L-convex" a genuinely convex-analytic notion rather than a combinatorial curiosity: it shows the convex hull of an L-convex set is not merely a polyhedron (already known from the chapter's polyhedral-description results) but one with the strongest local integrality property discrete convex analysis considers, integral convexity — every real point's hull membership is certified by a small, explicitly constructed set of nearby lattice points, uniformly across the whole set. This is the L-convex counterpart of the corresponding M-convex fact (chapter 4's Theorem 4.24) and is used later in the book wherever L-convex functions (chapter 7) need their epigraphs' local structure. Proposition 5.1 is the combinatorial engine underneath: it is exactly the LP-duality statement between shortest paths and feasible potentials that appears, in various guises, throughout network flow theory, made precise here as the base case the L-convex correspondence rests on.

None of these results are open — Murota presents them as, in his own words, "fundamental facts well known in network flow theory" (Proposition 5.1) systematized into the discrete convex analysis framework. What this mission contributes is a faithful, machine-checked formal statement of each, in the shared Lean vocabulary (LConvexSet, AdmissiblePotentials, ShortestDist) the rest of the Discrete Convex Analysis series can build on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The shortest-path closure $\bar\gamma$ is not a bookkeeping convenience but genuinely graph-theoretic content: proving Proposition 5.1 requires constructing an admissible potential from a shortest-path labeling and, conversely, deriving the negative-cycle-freeness of $G_\gamma$ from the mere existence of one admissible potential — a min-cost-flow-style LP duality argument, not a direct combinatorial check. Theorem 5.10's difficulty sits in a different place: the naive approach to "$D$ is integrally convex" would attempt an inductive argument peeling off one coordinate at a time, but the actual proof constructs a single, uniform family of $m+1$ witness points from the sorted fractional values of $p$ — a Carathéodory-style representation (Eq. (5.11)) that must simultaneously stay inside the integral neighborhood $N(p)$ and land in $D$ itself via the triangle inequality of $D$'s induced distance function, a construction with no one-coordinate-at-a-time shortcut.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; L-convex sets are Set (V → ℤ); distance functions are V → V → WithTop ℝ; admissible-potential sets are Set (V → ℝ). The shortest-path closure is formalized directly from finite walks (Fin (k+1) → V) rather than via a graph-library shortest-path predicate, matching the book's own construction. The Eq. (5.11) witnesses are built exactly as the book describes them — sorted distinct nonzero fractional values and their level sets — mirroring the Lovász-extension construction of the companion mission 20-ch04b-mconvexsets. No numeric constants are hard-coded anywhere in this mission (rule 7 is vacuous). The goal's explicit witness set (at most $|V|+1$ points) is not a trivializing special case: it holds for every L-convex set and every point of its hull, with no extra hypothesis narrowing the class. This mission's definitions (LConvexSet, AdmissiblePotentials, DistanceFunction, IsIntegrallyConvex) are redeclared from chunk 05-lconvex-sets (and, for IsIntegrallyConvex/IntegralNeighborhood, from chapter 3's own definitions) rather than imported, since sibling drafts in this series cannot yet reference one another; a later, published version of this book's namespace should consolidate them. Contributions completing any of the five sorrys are welcome; Proposition 5.1's LP-duality argument and the goal's Carathéodory-style construction are the two with the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• A. J. Hoffman, "On abstract dual linear programs," Naval Research Logistics Quarterly, 10 (1963), pp. 369-373 (feasible-potential duality in network flow theory).

Motivation

A competitive equilibrium is a list of production plans, consumption plans and prices at which every firm maximizes profit, every consumer maximizes utility within the budget, and no market has excess demand. Whether such prices exist at all is the consistency question behind general equilibrium theory, the welfare theorems, and applied equilibrium models used in policy analysis. Arrow and Debreu gave the first proof of existence for a model with production, private ownership and general convex preferences (Econometrica 22, 1954), using Debreu's existence theorem for abstract economies (PNAS 38, 1952).

Their Theorem I assumes that every consumer initially holds a positive amount of every commodity (Assumption IV.a). The authors call this "clearly unrealistic" (p. 280): a household does not hold every good, and most households own little beyond their labor. Theorem II, the subject of this mission, removes that assumption. It only asks that every consumer be able to supply some type of labor that is always productive of a commodity everyone desires. This is the version of the existence theorem that allows a wage-earner economy.

Timeline. Wald (1935–36) proved existence for special production models. Nash (1950) proved existence of equilibrium points for finite games, and Debreu (1952) extended it to abstract economies, in which each player's feasible set depends on the others' choices. Arrow and Debreu (1954) proved Theorems I and II. McKenzie's independent existence proof was published the same year (Econometrica 22, 1954).

Setting

There are $l$ commodities, $n$ producers and $m$ consumers; vectors live in $\mathbb R^l$ and $x\leqq y$ is componentwise. Producer $j$ has a production set $Y_j$. Consumer $i$ has a consumption set $X_i$, a utility $u_i$ on $X_i$, an endowment $\zeta_i$ and profit shares $\alpha_{ij}$. Write $Y=\sum_jY_j$, $X=\sum_iX_i$, $\zeta=\sum_i\zeta_i$, and let $P=\{p\geqq0,\ \sum_hp_h=1\}$ be the price simplex. A competitive equilibrium $(x_1^*,\dots,x_m^*,y_1^*,\dots,y_n^*,p^*)$ satisfies four conditions. Each $y_j^*$ maximizes $p^*\cdot y_j$ on $Y_j$. Each $x_i^*$ maximizes $u_i$ on $\{x_i\in X_i: p^*\cdot x_i\le p^*\cdot\zeta_i+\sum_j\alpha_{ij}p^*\cdot y_j^*\}$. The price vector satisfies $p^*\in P$. Finally $z^*=\sum x_i^*-\sum y_j^*-\zeta\leqq0$ and $p^*\cdot z^*=0$.

The assumptions of Theorem II are as follows. I: production sets are closed, convex and contain $0$; $Y\cap\Omega=\{0\}$ (no output without input); $Y\cap(-Y)=\{0\}$ (no reversible production). II: each $X_i$ is closed, convex and bounded below. III: $u_i$ is continuous, has no satiation point, and satisfies $u_i(tx+(1-t)x')>u_i(x')$ whenever $u_i(x)>u_i(x')$ and $0<t<1$. IV.b: shares are nonnegative and sum to one for each firm. Two sets of commodities are defined from the data. The set $\mathcal D$ contains the commodities always desired by every consumer: from any $x_i\in X_i$, adding some positive amount of the commodity stays in $X_i$ and raises $u_i$. The set $\mathcal P$ contains the types of productive labor: for every $y\in Y$, (a) $y_h\le0$, and (b) some $y'\in Y$ satisfies $y'_{h'}\ge y_{h'}$ for all $h'\ne h$ and $y'_{h''}>y_{h''}$ for some $h''\in\mathcal D$. The remaining assumptions are:

• IV′.a: each consumer has some $x_i\in X_i$ with $x_i\leqq\zeta_i$ and $x_{hi}<\zeta_{hi}$ for some $h\in\mathcal P$;
• V: some $x\in X$ and $y\in Y$ satisfy $x_h<y_h+\zeta_h$ for every $h$;
• VI: $\mathcal D\ne\emptyset$;
• VII: $\mathcal P\ne\emptyset$.

Formalization targets

Goal: Theorem II (§4.5, p. 281)

$$\text{Assumptions I–III, IV}',\ \text{V–VII}\ \Longrightarrow\ \exists\,(x^*,y^*,p^*)\ \text{satisfying Conditions 1–4.}$$

Milestones (§5, pp. 282–287)

They follow the paper's proof. Let $\pi=|\mathcal P|$ and $P^\varepsilon=\{p\in P: p_h\ge\varepsilon\ \forall h\in\mathcal P\}$ for $0<\varepsilon\le1/(2\pi)$. Let $E^\varepsilon$ be the abstract economy in which consumers maximize utility under budget constraints, producers maximize profit, and a market participant chooses $p\in P^\varepsilon$ to maximize $p\cdot z$. The milestones are:

1. §5.0 (1). On $P^\varepsilon$ every consumer can spend strictly less than $p\cdot\zeta_i$.
2. §5.1.1 (5). Equilibrium points of $E^\varepsilon$ satisfy $x^*-y^*\leqq\zeta'$ for a vector $\zeta'$ independent of $\varepsilon$.
3. §5.2.0. The attainable sets relative to $\zeta'$ are bounded.
4. §5.2.1. The truncated economy $\tilde E^\varepsilon$ has an equilibrium point.
5. §5.2.2 (3)–(5). An equilibrium point of $\tilde E^\varepsilon$ is one of $E^\varepsilon$.
6. §5.3.0 (2). If $p^*_h>\varepsilon$ for all $h\in\mathcal P$, the point is a competitive equilibrium.
7. §5.3.2 (1). Limits of equilibrium points as $\varepsilon\to0$ are quasi-equilibria for consumers.
8. §5.3.4 (3). If the floors bind, some desired commodity has limit price $0$.
9. §5.3.4 (6). If the floors bind, limit consumption minimizes expenditure over $X_i$.
10. §5.3.5. For some $\varepsilon$ the floor does not bind.

Significance

Theorem II is the existence theorem for a competitive economy in which consumers may own nothing but their labor. It shows that the survival assumption IV.a can be traded for conditions on labor, desirability and the possibility of an overall excess supply. Section 5.3.3 of the paper also isolates the quasi-equilibrium, in which utility maximization under the budget is replaced by cost minimization at a given utility level. That notion is used in later existence and welfare arguments.

The theorem has been proved since 1954; this mission does not reopen it. The work here is the machine-checked proof. The companion mission on Theorem I formalizes the shared model and Debreu's lemma. As of September 2026 neither theorem has a Lean formalization on the platform, and Mathlib contains no general equilibrium theory.

Difficulty

The obvious approach reuses the proof of Theorem I: build the abstract economy of consumers, producers and a price-choosing participant, and apply Debreu's lemma. That fails at the boundary of the price simplex. Without IV.a, a consumer's cheapest point in $X_i$ can cost as much as the endowment at some prices, so the budget correspondence is not continuous there and the lemma does not apply. The paper therefore keeps prices of productive labor at least $\varepsilon$ and must then show that the floor does not bind for some $\varepsilon$. That is a limit argument as $\varepsilon\to0$ which uses Assumptions V, VI and VII together, and each of the milestones 7–9 is a step of it. Debreu's lemma itself needs a Kakutani-type fixed point theorem for correspondences, which Mathlib does not provide.

Formalization scope

Commodity vectors are Fin l → ℝ. Consumers are indexed by Fin m and producers by Fin n. The inner product is ⬝ᵥ. The paper's $x<y$ is strict in every component and is written componentwise, never as Lean's < on functions. $\mathcal D$ and $\mathcal P$ are computed from the economy, not supplied as parameters. Utilities are total functions, but every assumption on $u_i$ quantifies over $X_i$ only. "Maximizes" is membership plus an inequality against every feasible alternative; no supremum is used. $E$, $E^\varepsilon$ and $\tilde E^\varepsilon$ are built by one constructor over the players Fin m ⊕ Fin n ⊕ Unit. The vector $\zeta'$ takes the lower bounds $\xi_i$ of Assumption II as an explicit argument. The milestones of §5.3 are stated for the limit of a sequence of equilibrium points, which is how the paper constructs them. Assumption V is dropped from every milestone except §5.3.5 and the goal. With V, the case assumption of §5.3.1 is contradictory and those milestones would hold vacuously.

A trivializing formalization is ruled out as follows. The assumptions are satisfiable with IV.a failing: a sorry-free check covers two goods, one consumer who owns nothing and can only supply labor, and one firm turning labor into the desired good. The goal therefore does not hold vacuously.

Needed infrastructure includes a Kakutani fixed point theorem or Debreu's lemma, compactness of truncated action sets, and sequential compactness arguments in $\mathbb R^l$. AGT.brouwer_fixed_point is on the platform and can serve as a starting point. The lemma is reusable well beyond this mission. Contributions to any milestone, to the lemma, or to the boundedness results shared with Theorem I are welcome.

Selected references

• K. J. Arrow and G. Debreu, Existence of an Equilibrium for a Competitive Economy, Econometrica 22(3), 265–290, 1954. https://doi.org/10.2307/1907353
• G. Debreu, A Social Equilibrium Existence Theorem, Proceedings of the National Academy of Sciences 38(10), 886–893, 1952. https://doi.org/10.1073/pnas.38.10.886
• L. W. McKenzie, On Equilibrium in Graham's Model of World Trade and Other Competitive Systems, Econometrica 22(2), 147–161, 1954. https://doi.org/10.2307/1907352
• J. F. Nash, Equilibrium Points in n-Person Games, Proceedings of the National Academy of Sciences 36(1), 48–49, 1950. https://doi.org/10.1073/pnas.36.1.48

Motivation

Submodular set functions are the combinatorial stand-in for convexity: a function $\rho : 2^V \to \mathbb R$ on the subsets of a finite ground set $V$ is submodular if $\rho(X) + \rho(Y) \ge \rho(X \cup Y) + \rho(X \cap Y)$, and this single diminishing-returns inequality drives an enormous range of combinatorial optimization — matroid rank functions, graph cut capacities, entropy, coverage functions, and the max-flow min-cut theorem all arise as special or dual cases (Edmonds 1970; Lovász 1983; Fujishige 2005). M-convex sets are the "vector" incarnation of the same idea: subsets $B$ of $\mathbb Z^V$ satisfying an exchange axiom that generalizes the basis-exchange property of matroids to sets of integer points lying on a common hyperplane. Murota's Discrete Convex Analysis (SIAM, 2003) develops both sides of this correspondence and proves they coincide exactly: M-convex sets are precisely the integer points of the base polyhedra of integer-valued submodular functions. This mission covers the second half of that development — the structural theory (integrality, holes, Minkowski sums) that turns the correspondence into a working calculus, and its capstone, a discrete separation theorem for two disjoint M-convex sets whose separating hyperplane is forced to have $\{0,1\}$- or $\{0,-1\}$-valued coefficients.

Companion mission 04-mconvex-sets (Discrete Convex Analysis III) covers the same chapter's foundational results: the equivalence of the exchange-axiom variants, the one-to-one correspondence between M-convex sets and integer submodular functions (Theorem 4.15), Edmonds's intersection theorem (Theorem 4.18), and Frank's discrete separation theorem for submodular/ supermodular pairs (Theorem 4.17). This mission builds on that vocabulary (redeclared here, since draft missions in the same series cannot yet import one another) and proves the results the chapter leaves for its second half.

Setting

Fix a finite ground set $V$. A vector $x \in \mathbb Z^V$ assigns an integer $x(v)$ to each $v \in V$; write $x(X) = \sum_{v \in X} x(v)$ for $X \subseteq V$. For $x, y \in \mathbb Z^V$, the positive support $\operatorname{supp}^+(x-y) = \{v : x(v) > y(v)\}$ and negative support $\operatorname{supp}^-(x-y) = \{v : x(v) < y(v)\}$ record where $x$ exceeds, and falls short of, $y$. A nonempty set $B \subseteq \mathbb Z^V$ is M-convex if it satisfies the exchange axiom (B-EXC[Z]): for all $x, y \in B$ and $u \in \operatorname{supp}^+(x-y)$, some $v \in \operatorname{supp}^-(x-y)$ has both $x - \chi_u + \chi_v \in B$ and $y + \chi_u - \chi_v \in B$, where $\chi_u$ is the characteristic vector of $u$.

A set function $\rho : 2^V \to \mathbb R \cup \{+\infty\}$ with $\rho(\emptyset) = 0$ and $\rho(V) < +\infty$ is submodular (the class $S[\mathbb R]$, or $S[\mathbb Z]$ when integer-valued) if $\rho(X) + \rho(Y) \ge \rho(X \cup Y) + \rho(X \cap Y)$ for all $X, Y$. Its base polyhedron is $B(\rho) = \{x \in \mathbb R^V : x(X) \le \rho(X)\ (\forall X),\ x(V) = \rho(V)\}$. The Lovász extension $\hat\rho : \mathbb R^V \to \mathbb R \cup \{\pm\infty\}$ linearly interpolates $\rho$ off $\{0,1\}^V$: sorting the distinct values of $p \in \mathbb R^V$ as $\hat p_1 > \cdots > \hat p_m$ and setting $U_i = \{v : p(v) \ge \hat p_i\}$, it is $\hat\rho(p) = \sum_{i=1}^{m-1}(\hat p_i - \hat p_{i+1})\rho(U_i) + \hat p_m \rho(U_m)$.

Formalization targets

Goal: discrete separation for M-convex sets

$$B_1 \cap B_2 = \emptyset \implies \exists\, p^* \in \{0,1\}^V \cup \{0,-1\}^V,\quad \inf_{x \in B_1}\langle p^*, x\rangle - \sup_{x \in B_2}\langle p^*, x\rangle \ge 1,$$

for M-convex sets $B_1, B_2 \subseteq \mathbb Z^V$ (Theorem 4.21). This is the weakest stable form of the result — it asserts only the existence of a combinatorially special separator, not any bound tied to $|V|$ or a particular construction, so it is not invalidated by a sharper algorithm for finding $p^*$.

Supporting structural targets

Eleven further results build the calculus this goal rests on: the hyperplane property of M-convex sets (Prop. 4.1), an equivalent one-sided exchange axiom (Prop. 4.2), nonemptiness and the support-function identity for $B(\rho)$ (Props. 4.4-4.5), integrality of $B(\rho)$ for integer-valued $\rho$ (Prop. 4.6), the hole-free property identifying an M-convex set with the integer points of its own convex hull (Thm. 4.12), the two-way polyhedral description of M-convex sets via induced submodular functions (Props. 4.13-4.14), the equivalence of submodularity with convexity of the Lovász extension (Thm. 4.16, due to Lovász), integrality of the intersection of M-convex sets (Thm. 4.22), and Minkowski-sum identities for base polyhedra and M-convex sets (Thm. 4.23).

Significance

The discrete separation theorem is what makes M-convexity discrete rather than merely a polyhedral fact: ordinary separation of two disjoint convex sets by a hyperplane is classical, but here the separator is forced into $\{0,1\}^V \cup \{0,-1\}^V$ — a purely combinatorial object — with no loss of strength. This is the mechanism behind integrality results across combinatorial optimization (e.g., that the intersection of two integral base polyhedra is integral, Theorem 4.22, used pervasively in matroid intersection and submodular flow algorithms). The structural results (holes, Minkowski sums, the Lovász-extension convexity equivalence) are the working toolkit every later use of M-convexity in the book — proximity theorems for M-convex functions (chunks 06+), the discrete conjugacy theorem, submodular flows — draws on without restating.

None of these results are open: Murota attributes the exchange-axiom theory to the matroid and submodular-function literature it systematizes, citing Edmonds, Frank, and Lovász by name for the specific theorems. What this mission produces is a machine-checked formal statement of each result exactly as the book states it, in a shared Lean vocabulary (ExchangeAxiomB, BasePolyhedron, LovaszExtension) that the rest of the Discrete Convex Analysis series builds on; no result here has a prior formalization on the platform (see Formalization scope).

Difficulty

The separation theorem is not proved by convex separation directly — the whole point is that the naive proof (apply the ordinary hyperplane separation theorem to the convex hulls of $B_1, B_2$, then argue the separator can be taken $\{0,1\}$-valued) does not go through, because convex separation alone gives no control over the separator's coefficients. The book instead derives it from Edmonds's intersection theorem (Theorem 4.18, chunk 04-mconvex-sets) applied to a submodular/supermodular pair built from $B_1, B_2$'s associated set functions (Theorem 4.15), routed through Frank's discrete separation theorem (Theorem 4.17) — a genuine two-step reduction, not a direct argument. A second, independent difficulty sits in the supporting results: the hole-free property (Theorem 4.12) requires an explicit induction reducing an arbitrary convex combination representing an integer point to a single element of $B$, a combinatorial exchange argument with no shortcut through general polyhedral theory.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; M-convex sets are Set (V → ℤ); submodular/supermodular functions are Finset V → WithTop ℝ / WithBot ℝ; base polyhedra are Set (V → ℝ). The Lovász extension is formalized directly from the book's own sorted-values construction (SortedValues, LevelSet, Eq. (4.4)-(4.6)), not via an equivalent closed form. Since WithTop ℝ carries no Module ℝ structure, convexity for Theorem 4.16 is stated via a bespoke nonnegative-scalar action (ScalarWithTop) rather than Mathlib's ConvexOn — this changes no mathematical content, only its packaging (see MODERATION_NOTES.md). No numeric constants are hard-coded anywhere in this mission (rule 7 is vacuous). The goal's hypothesis (ExchangeAxiomB plus Nonempty on each $B_i$) is exactly the book's own definition of M-convexity — no weaker substitute (e.g. requiring a specific $\rho$ witness in the hypothesis rather than deriving one, or dropping the $\{0,1\}/\{0,-1\}$ constraint on $p^*$ in favor of a generic separator) would be faithful, and both trivializations are ruled out by construction. This mission's definitions (ExchangeAxiomB, BasePolyhedron, SubmodularSetFunction, LovaszExtension) are redeclared from chunk 04-mconvex-sets rather than imported, since sibling drafts in this series cannot yet reference one another; a later, published version of this book's namespace should consolidate them. Contributions completing any of the twelve sorrys are welcome; the hole-free property (Theorem 4.12) and the goal are the two with the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• J. Edmonds, "Submodular functions, matroids, and certain polyhedra," in Combinatorial Structures and Their Applications, 1970, pp. 69-87.
• A. Frank, "An algorithm for submodular functions on graphs," Annals of Discrete Mathematics, 16 (1982), pp. 97-120.
• L. Lovász, "Submodular functions and convexity," in Mathematical Programming: The State of the Art, Springer, 1983, pp. 235-257.