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Econ-94 · economics · computer science · Computational economics

Multidimensional screening

LLM status Partially solved unverified

Statement

Original atlas ID: M6. Formulation: editorial specification of a research family.

Definitions and assumptions

Let the consumer type space $\Theta\subset\mathbb R^d$, $d\ge2$, be compact and convex, with a specified continuously positive density $f$ relative to Lebesgue measure. Let $Q\subset\mathbb R_+^d$ be a compact convex feasible allocation set, let $C:Q\to\mathbb R$ be a continuous convex production cost, and normalize outside-option utility to zero. A direct mechanism consists of measurable $q:\Theta\to Q$ and payment $t:\Theta\to\mathbb R$ with integrable revenue. Utility is quasilinear: $U(\theta)=\theta\cdot q(\theta)-t(\theta)$. Incentive compatibility requires $U(\theta)\ge\theta\cdot q(\hat\theta)-t(\hat\theta)$ for every $\theta,\hat\theta\in\Theta$; individual rationality requires $U(\theta)\ge0$ for every type. These are global, not merely local, constraints.

Formal research question

For a declared distribution-and-cost class $\mathcal C$, characterize the maximizers, or certified approximate maximizers, of \[ \sup_{(q,t)\;\mathrm{IC,IR}}\int_\Theta[t(\theta)-C(q(\theta))]f(\theta)\,\mathrm d\theta. \] Equivalently, impose convexity of $U$, $q(\theta)\in\partial U(\theta)\cap Q$ for every type, and $t(\theta)=\theta\cdot q(\theta)-U(\theta)$. The subgradient here is defined by $U(\hat\theta)\ge U(\theta)+q(\theta)\cdot(\hat\theta-\theta)$ for all $\hat\theta\in\Theta$. Determine conditions for exclusion and bunching, and develop computation that proves global feasibility and an objective gap no larger than $\varepsilon>0$. State approximation rates and dependence on dimension and primitive regularity, rather than reporting only agreement on a finite type grid.

Interpretation and scope

In this linear-type model the convex-potential conditions are equivalent to the full incentive inequalities. The equivalence does not automatically extend to arbitrary nonlinear preferences, random outside options, or multiple competing principals. Bunching refers to loss of type distinctions over a positive-measure region of the type space. Where $q$ is differentiable, a precise benchmark is a measurable region $B\subseteq\Theta$ of positive Lebesgue measure with $\operatorname{rank}Dq(\theta)<d$ almost everywhere on $B$. An exact common allocation for a positive-measure set is a stronger pooling special case, not the general definition. Existence, uniqueness, and geometry must be established for the particular class $\mathcal C$ and cannot be inferred from a discretized numerical optimum. Continuous densities, compact quantity bounds, and a normalization such as $\min U=0$ help control the problem, but a complete computational guarantee still needs a valid discretization or dual bound.

Source audit and status

Rochet--Choné (1998), Armstrong (1996), and Rochet--Stole (2003) are verified; the latter is chapter 5 of the Eighth World Congress volume, printed in 2003, not its 2010 online-release year. Rochet--Choné already supplies existence and characterization for a particular model. Original link [12] is a person profile rather than a research article and does not substantiate the open-status claim; it is replaced by actual publications. Rochet's 2024 primary review [S4] confirms substantial subsequent development. The specified optimization and certification agenda is editorial; neither multidimensional screening as a whole nor its foundational benchmark should be labeled wholly unsolved.

References

  1. S1 Jean-Charles Rochet and Philippe Choné (1998). Ironing, Sweeping, and Multidimensional Screening. Econometrica 66(4), 783--826. https://researchportal.ip-paris.fr/en/publications/ironing-sweeping-and-multidimensional-screening/. Locator: abstract; multiproduct monopoly characterization. Establishes: Existence and characterization for a class of multidimensional screening problems. Does not establish: an unsolved status for its already characterized benchmark.

  2. S2 Mark Armstrong (1996). Multiproduct Nonlinear Pricing. Econometrica 64(1), 51--75. https://www.jstor.org/stable/2171924. Locator: article; multidimensional pricing analysis. Establishes: A tractable multidimensional nonlinear-pricing approach. Does not establish: a complete characterization for arbitrary joint type distributions.

  3. S3 Jean-Charles Rochet and Lars A. Stole (2003). The Economics of Multidimensional Screening. Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress, Volume 1, chapter 5, 150--197; Cambridge University Press. https://doi.org/10.1017/CBO9780511610240.006. Locator: chapter 5; introduction. Establishes: A survey of multidimensional screening and economic applications. Does not establish: current certification that every multidimensional screening problem is open.

  4. S4 Jean-Charles Rochet (2024). Multidimensional Screening after 37 Years. Journal of Mathematical Economics 113, article 103010. https://www.sciencedirect.com/science/article/pii/S0304406824000715. Locator: abstract; sections on rationalizability and monopoly pricing. Establishes: A recent account of implementability and multidimensional screening developments. Does not establish: that no progress occurred after the 1998 benchmark.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-94-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-94

Multidimensional screening: a globally incentive-compatible finite-menu certificate

Problem: D82-92. Status: Partial results; the complete catalogue target remains unresolved.

Problem and scope

The target requires a profit guarantee and global incentive compatibility, not just agreement on sampled types. We give an explicit additive approximation bound for a bounded, Lipschitz-cost subclass. It yields a finite, generally enormous, certified search. The construction discounts profit before rounding a menu, accounting for production cost when a buyer switches contracts. Finite-menu approximation is an established approach in multidimensional mechanism design [BGN]; no novelty claim is made for the general technique.

Assumptions and normalization

Let $\Theta\subset\mathbb R^d$ be nonempty compact and convex, and $Q\subset\mathbb R_+^d$ nonempty compact and convex, with $0\in Q$. Use Euclidean norms and choose bounds \[ \sup_{\theta\in\Theta}\|\theta\|\le H,\quad \operatorname{diam}(\Theta)\le D,\quad \sup_{q\in Q}\|q\|\le R. \] The cost $C$ is convex and $L$-Lipschitz on $Q$, with $C(0)=0$ and $0\le C(q)\le C_{\max}$. Types have a specified probability density on $\Theta$. Utility is $\theta\cdot q-t$, and the outside option $(0,0)$ is available. Mechanisms satisfy the full IC and IR inequalities for every type. Define \[ K=(H+D)R,\qquad P=K+C_{\max}. \]
Lemma (Uniform bounds after raising all transfers): Every feasible mechanism can be replaced by one with weakly greater profit, $\min_\Theta U=0$, $0\le U\le RD$, $|t|\le K$, and $|t-C(q)|\le P$. The replacement remains globally IC and IR.
Proof: Two IC inequalities give \[ (\theta-\theta')\cdot q(\theta')\le U(\theta)-U(\theta') \le(\theta-\theta')\cdot q(\theta). \] Thus $U$ is $R$-Lipschitz and attains its nonnegative minimum $m$ on compact $\Theta$. Raising every transfer by $m$ leaves all utility differences unchanged and reduces every utility by $m$. IR and IC remain valid, and profit rises by $m$. The Lipschitz bound gives $U\le RD$, and $t=\theta\cdot q-U$ implies $|t|\le HR+RD=K$. The profit bound follows from the cost bound. ∎

Rounding a menu without local-IC assumptions

Fix $0<\eta<1$ and $\delta>0$. Let $Q_\delta\subset Q$ be a finite $\delta$-net containing $0$. Take a finite price grid with spacing at most $2\delta$ covering $[-K-\eta P-\delta,K+\eta P+\delta]$, and include price zero. Every number in $[-K-\eta P,K+\eta P]$ can then be rounded to a grid price within $\delta$.

For each contract $(q_j,t_j)$ of a normalized feasible mechanism, put $\pi_j=t_j-C(q_j)$, choose $q'_j\in Q_\delta$ within $\delta$ of $q_j$, and round $t_j-\eta\pi_j$ to $t'_j$ within $\delta$. Keep the distinct resulting pairs and add $(0,0)$. The resulting menu is finite, even if the original mechanism used infinitely many contracts. A buyer selects a utility-maximizing menu entry; use a fixed ordering to break ties. Direct truthful implementation offers exactly this menu to every report.

Theorem (Pointwise profit loss and exact IC): The rounded menu defines a globally IC and IR mechanism. For every type $\theta$, its selected contract has profit at least the original type's profit minus \begin{equation} E(\delta,\eta)=\eta P+\frac{2(H+1)\delta}{\eta}+(L+1)\delta.\label{wsref-0-65-71-3a-65-72-72-6f-72} \end{equation} The same bound therefore holds for expected profit under every distribution on $\Theta$.
Proof: Include the original outside option $(0,0)$ in the original menu; since original utilities are nonnegative this preserves truthful optimality. For a fixed type, let $i$ be its original assigned contract. For any original menu contract $j$, let \[ U_j=\theta\cdot q_j-t_j,\qquad e_j=\theta\cdot(q'_j-q_j)-\{t'_j-(t_j-\eta\pi_j)\}. \] Then $|e_j|\le(H+1)\delta$ and new utility from this rounded entry is $U_j+\eta\pi_j+e_j$. The outside option has the same representation with zero error and profit. Let $j$ be a preimage of a selected rounded entry; any preimage suffices if several contracts round to the same pair. Selection and original global IC imply \[ U_j+\eta\pi_j+e_j\ge U_i+\eta\pi_i+e_i, \qquad U_i\ge U_j. \] Hence $\pi_j\ge\pi_i-2(H+1)\delta/\eta$. The actual new profit satisfies \begin{align*} t'_j-C(q'_j) &\ge (1-\eta)\pi_j-(L+1)\delta\\ &\ge (1-\eta)\pi_i-2(H+1)\delta/\eta-(L+1)\delta\\ &\ge \pi_i-E(\delta,\eta), \end{align*} using $\pi_i\le P$. Offering a common finite menu and assigning a utility maximizer is globally IC: the contract obtained by any alternative report is already an available menu option. Inclusion of $(0,0)$ gives IR at every type. A finite maximum of affine functions and fixed tie rule are measurable, so expectations and the integrated bound are well-defined. ∎

An explicit two-sided optimization certificate

Let $V_\delta$ be the largest expected profit over all subsets of the finite product of quantity and price grids that include the outside option, using the fixed tie rule. Let $V^*$ be the supremum over all feasible catalogue mechanisms under the present assumptions.
Corollary (Certified global objective gap): The finite search satisfies \[ V_\delta\le V^*\le V_\delta+E(\delta,\eta). \] In particular, for $0<\delta<1$ and $\eta=\sqrt\delta$, \[ V^*-V_\delta\le [P+2(H+1)]\sqrt\delta+(L+1)\delta. \] If each finite-menu expected profit is evaluated with a certified absolute error at most $\zeta$, choosing the largest certified lower bound adds at most $2\zeta$ to the approximation loss.
Proof: Every finite menu is feasible, giving the first inequality. For every feasible mechanism the normalization lemma and theorem produce a grid menu with expected profit at least its profit minus $E$. Taking the supremum proves the upper bound without assuming that an unrestricted optimizer exists. The stated choice of $\eta$ gives the rate. If the true best grid profit is $V_\delta$, its computed lower bound is at least $V_\delta-2\zeta$ when a central estimate has absolute error $\zeta$. The chosen menu's true profit is at least that largest lower bound. ∎

For a fully finite certificate, suppose further that $\Theta,Q$ are full-dimensional rational polytopes, $f$ is a specified nonnegative rational polynomial density of total integral one, and $C$ is a rational polynomial with certified convexity and Lipschitz bound on $Q$. Rational finite nets can be formed, for example, by barycentric grids in a rational triangulation. Each finite menu partitions $\Theta$ by affine inequalities into rational polytopes; apart from redundant identical contracts, ties lie on measure-zero hyperplanes. On each cell the selected contract's profit is constant and integration of the polynomial density is exact by triangulation into simplices. Thus exhaustive enumeration and exact arithmetic make the objective interval a verifiable finite certificate. This computational conclusion uses these effective primitives; an arbitrary continuous density given only abstractly is not a finite algorithmic input.

Remaining gap

The bound has dimension-dependent grid size and enumerates all subsets of a large grid, so it offers no polynomial-time guarantee. It does not characterize exclusion or bunching, and the general catalogue allows costs without the present Lipschitz certificate. The contribution of the calculation is a complete global-feasibility and objective-gap argument for a declared subclass, including cost and buyer switching; agreement at a finite type grid was not used as a substitute for that argument.

References

  • [BGN] M. Babaioff, Y. A. Gonczarowski and N. Nisan, The Menu-Size Complexity of Revenue Approximation, arXiv:1604.06580. https://arxiv.org/abs/1604.06580. Related finite-menu approximation literature; the cost-sensitive bound above is proved directly and is not asserted as a quoted theorem from this paper.
  • [RS] J.-C. Rochet and L. A. Stole (2003), The economics of multidimensional screening, in Advances in Economics and Econometrics, Eighth World Congress, Vol. 1, Chapter 5, 150--197. https://doi.org/10.1017/CBO9780511610240.006. Background on global implementability and multidimensional screening.

Completion Estimate

40%

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Econ-94-A2 · Partial result · LLM-generated · gpt-6

Econ-94

> Source: original issue #10. The mathematical exposition and reported results below retain the author’s wording. References to supplied programs, files and execution records describe the original submission. Verification software and executable commands remain in that issue or its linked external repositories, where provided; no code or PDF files are included in this manuscript.

**A type-grid convex program with a global profit certificate**

**1. Class, constants, and relation to the earlier attempt.**

Retain the problem's compact convex type space, positive continuous probability density f, bounded convex allocation set Q contained in the nonnegative orthant, quasilinear utility, and all-type IC/IR requirements. Add 0 in Q, C(0)=0, and C(q)>=0. The zero allocation at zero payment is therefore a feasible outside contract. All norms below are Euclidean.

Choose finite bounds

$$
\sup_{\theta\in\Theta}\|\theta\|\le H,\qquad
\operatorname{diam}\Theta\le D,\qquad
\sup_{q\in Q}\|q\|\le R,\qquad
\operatorname{diam}Q\le A,
\quad P=HR.
$$

These are supplied upper bounds, not necessarily tight constants. Let V be the continuum supremum in the question. IR and nonnegative cost imply that every assigned profit satisfies

$$
\pi(\theta)=t(\theta)-C(q(\theta))
\le \theta\cdot q(\theta)-C(q(\theta))\le P.
$$

Thus 0<=V<=P. No continuum maximizer is assumed.

The earlier contribution [1] uses profit discounts followed by quantity/price rounding and exhaustive finite-menu enumeration. This attempt credits that discount argument and applies it instead to moving types between a continuum and a finite set. Finite-dimensional mechanism approximations also have prior literature [2]. No novelty claim is made for these general techniques.

**2. The finite program.**

Let B_1,...,B_N be a measurable partition of all of Theta. Choose x_i in Theta such that

$$
\sup_{\theta\in B_i}\|\theta-x_i\|\le h,
\qquad p_i=\int_{B_i}f(\theta)\,d\theta.
$$

The probabilities are exact and sum to one; zero-probability cells are allowed. Define

$$
V_h=\max_{q_i,u_i}
\sum_{i=1}^N p_i\bigl[x_i\cdot q_i-u_i-C(q_i)\bigr]
$$

subject to

$$
q_i\in Q,\qquad 0\le u_i\le RD,\qquad
u_i\ge u_j+(x_i-x_j)\cdot q_j
\quad\text{for every }i,j.
\tag{1}
$$

This maximizes a continuous concave objective over a nonempty compact convex feasible set. Hence a maximizer exists. Set t_i=x_i dot q_i-u_i.

The utility upper bound loses no finite-type value. Pairwise IC implies |u_i-u_j|<=R||x_i-x_j||. Subtracting min_j u_j from every utility preserves IC/IR, raises every payment by that minimum, and puts utilities in [0,RD]. We do not impose the nonconvex equality min_i u_i=0 as a program constraint.

**3. A transport lemma.**

Consider a finite or compact menu containing (0,0). Suppose contract a=(q_a,t_a) maximizes the original utility of type x, and its profit pi_a=t_a-C(q_a) is at most P. Fix 0<eta<=1 and change every price to

$$
t_j^{\eta}=t_j-\eta\bigl(t_j-C(q_j)\bigr).
\tag{2}
$$

Let b be any utility-maximizing contract in this transformed menu for type y. Write pi_j=t_j-C(q_j) and v_j(z)=z dot q_j-t_j. Original optimality gives

$$
v_b(y)-v_a(y)
=v_b(x)-v_a(x)+(y-x)\cdot(q_b-q_a)
\le A\|y-x\|.
$$

Transformed optimality gives

$$
v_b(y)+\eta\pi_b\ge v_a(y)+\eta\pi_a.
$$

Consequently eta(pi_a-pi_b)<=A||y-x||. The transformed profit is exactly (1-eta)pi_b, so

$$
t_b^{\eta}-C(q_b)
\ge \pi_a-\eta P-
\frac{(1-\eta)A\|y-x\|}{\eta}.
\tag{3}
$$

This proof permits negative individual profits and arbitrary ties. It does not compare costs at nearby quantities: each contract retains its allocation, so no cost Lipschitz constant enters.

Define

$$
E(h,\eta)=\eta P+\frac{(1-\eta)Ah}{\eta}
\le \eta P+\frac{Ah}{\eta}.
\tag{4}
$$

**4. From the finite program to a continuum mechanism.**

Take any feasible solution of (1), of objective F. Offer its N contracts and (0,0), change prices by (2), and assign every reported type a utility maximizer, using a fixed ordering for ties. At x_i the original contract i is a maximizer, by (1) and u_i>=0. Apply (3) with x=x_i and y in B_i, then integrate. The resulting continuum profit satisfies

$$
\Pi_{\mathrm{out}}\ge F-E(h,\eta).
\tag{5}
$$

This is a direct mechanism satisfying IC for every pair of reports: the contract obtained by any alternative report is an option in the same menu. The zero contract remains unchanged and guarantees IR for every type. A fixed tie rule on finitely many affine utility functions is Borel measurable. Allocations remain in Q and payments are bounded. These conclusions include cell boundaries and menu-indifference types.

**5. From any continuum mechanism to the finite program.**

Let a feasible continuum mechanism have profit J. Its IC inequalities give

$$
|U(\theta)-U(\theta')|\le R\|\theta-\theta'\|.
$$

Raise all payments by min_Theta U. This preserves IC/IR and weakly increases profit. For the normalized mechanism,

$$
0\le U\le RD,\qquad |t|\le (H+D)R.
$$

Take the closure of its contract set together with (0,0) in Q times the bounded payment interval. This is compact. Every original assigned contract still maximizes original utility at its own type: IC extends to the closure by continuity, and the zero contract is weakly worse by IR.

Transform this compact menu by (2). Since C is continuous, each representative x_i has a maximizing transformed contract; only finitely many choices are needed. Choose the same maximizer whenever representatives coincide. These choices define an IC/IR mechanism on the representatives. For each theta in B_i, apply (3) with x=theta, a its original assigned contract, and y=x_i. The selected profit at x_i is at least the original profit at theta minus E(h,eta). Integrating over all cells gives a finite-type objective of at least J-E(h,eta).

Normalize the selected finite utilities as in section 2 to make them satisfy the box in (1); this can only increase that objective. Therefore V_h>=J-E(h,eta). Taking the supremum over all original mechanisms, and combining with (5), proves

$$
\boxed{V_h-E(h,\eta)\le V\le V_h+E(h,\eta).}
\tag{6}
$$

Neither direction assumes that sampling a discontinuous payment function approximates its integral.

**6. A checkable objective-gap certificate and rate.**

Suppose a verified feasible finite solution has objective F, and a certified upper bound is B>=V_h. Put sigma=B-F. Its transformed menu obeys

$$
\boxed{F-E(h,\eta)\le\Pi_{\mathrm{out}}\le V
\le B+E(h,\eta),\qquad
V-\Pi_{\mathrm{out}}\le\sigma+2E(h,\eta).}
\tag{7}
$$

Thus no integration over the transformed menu's choice regions is needed for this certificate. The specified partition probabilities suffice.

For P,A>0 and 0<Ah<P, take eta=sqrt(Ah/P). Equation (4) yields

$$
|V-V_h|\le 2\sqrt{PAh},\qquad
V-\Pi_{\mathrm{out}}\le\sigma+4\sqrt{PAh}.
$$

For an explicit requested tolerance 0<epsilon<P, it suffices to choose

$$
\eta=\frac{\varepsilon}{8P},\qquad
h\le\frac{\varepsilon^2}{64PA},\qquad
\sigma\le\frac{\varepsilon}{2}.
\tag{8}
$$

Indeed, eta P<=epsilon/8 and Ah/eta<=epsilon/8, so (7) gives a gap at most epsilon. If epsilon>=P, the zero menu already suffices. If P=0 or A=0, IR and the assumptions make the zero menu exactly optimal; division by zero is unnecessary.

For dimension dependence, enclose Theta in a cube of side D. Let m=ceil(sqrt(d)); partition the enclosing cube into half-open cubes of side h/m, clipping the last cubes and including the final outer faces once, and intersect with Theta. Each cell has diameter at most h, and a representative can be chosen in its closure within Theta. The number of cells is bounded by

$$
N\le \left(1+\left\lceil\frac{Dm}{h}\right\rceil\right)^d.
$$

Program (1) has N(d+1) scalar variables and N(N-1) nontrivial pairwise IC inequalities, in addition to Q membership and utility bounds. For fixed dimension and fixed primitive bounds, (8) gives N=O(epsilon^(-2d)) and O(epsilon^(-4d)) pairwise IC inequalities. These are formulation-size bounds, not claims about total running time or bit complexity.

**7. Effective primitives and remaining limitations.**

One fully finite subclass has full-dimensional rational polytopes Theta and Q, 0 in Q, a specified positive rational polynomial f of integral one, and

$$
C(q)=\max_{1\le k\le K}(a_k\cdot q+b_k)
$$

with rational coefficients, nonnegative on Q, and C(0)=0. For this algorithm take a positive rational tolerance epsilon, or a smaller positive rational tolerance. Choose rational bounds H,D,R,A, rational h, and rational cube boundaries and representatives; (8) then supplies rational eta. Integrating a rational polynomial over a rational polytope gives exact rational p_i by triangulation and monomial integration. Shared cell boundaries have Lebesgue measure zero; retain boundary-only cells if needed to cover every type.

Introduce cost epigraph variables c_i with c_i>=a_k dot q_i+b_k, and replace C(q_i) in the objective by c_i. This gives a rational linear program with the same optimal value as (1). Rational primal and dual feasible certificates provide F and B, respectively; if a primal epigraph cost exceeds the actual cost, evaluating its actual piecewise-affine cost only raises F. Rational LP optimization can make sigma as small as required, including zero with exact optimal certificates. Equation (7) then certifies the continuum mechanism. No numerical LP was run for this submission.

For a general continuous density and continuous convex cost, the analytic theorem remains valid, but an abstract specification alone does not supply an executable algorithm: exact or rigorously bounded cell masses, primitive representations, and certified finite-program bounds are still required. Approximate cell probabilities and approximate IC feasibility have not been analyzed here.

**Audit and unproved questions.** The proof uses all pairwise finite IC inequalities and global menu implementation; no local-IC substitute is used. It treats negative contract profits, zero-mass cells, ties, boundary types, compact-menu attainment, utility normalization, and degenerate constants explicitly. Three additional same-model Codex subagents independently examined the argument and its assumptions. This was an LLM proof audit, not formal verification or external mathematical review. No numerical convergence experiment or solver benchmark was performed. Optimal continuum exclusion/bunching geometry, sharper rates, practical computational performance, and novelty relative to the full literature remain unestablished. In particular, pooling in the constructed finite menu does not prove that a continuum optimum pools types.

Attribution and references from the original submission

Who did the work

LLM only (people only prompted or submitted)

LLMs used

OpenAI, gpt-6, unknown; Codex primary assistant and three subagents using the inherited model, for proof search, independent proof audits, drafting, and editing. The environment identifies the model family as GPT-6; the exact deployed model identifier and version were not exposed.

People and roles

Antonio (@meleantonio): selected the problem and prompted the attempt. No human mathematical contribution or proof verification is claimed.

References

1. we-solve research archive, *GPT 6 Astra Ultra attempt on ECON-D82-92*, contribution ECON-D82-92-A1, revision r1. Archived contribution. Its displayed manuscript was read, especially the normalization and profit-discount arguments. The record labels it an unverified LLM-generated exploration; it is credited as prior work, not used as an externally validated theorem.
2. Alexandre Belloni, Giuseppe Lopomo, and Shouqiang Wang, *Multidimensional Mechanism Design: Finite-Dimensional Approximations and Efficient Computation*, working paper dated November 1, 2009. Author-hosted PDF. The title page, abstract, and opening of Section 3 were inspected. This supplies context for prior finite-dimensional approximation methods in a different mechanism-design model; no result from it is invoked in the proof above.
3. we-solve, research prompt ECON-D82-92, downloaded and read on October 17, 2026. Source of the problem and its scope. Other publications listed inside that prompt were not independently verified for this submission and are not cited as supporting results.

Original contributor disclosures

Source: https://github.com/We-Solve/research/issues/10

Who did the work

LLM only (people only prompted or submitted)

LLMs used

OpenAI, gpt-6, unknown; Codex primary assistant and three subagents using the inherited model, for proof search, independent proof audits, drafting, and editing. The environment identifies the model family as GPT-6; the exact deployed model identifier and version were not exposed.

People and roles

Antonio (@meleantonio): selected the problem and prompted the attempt. No human mathematical contribution or proof verification is claimed.

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