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Econ-161 · economics · Economics

Ex ante envy-freeness with ex post EF1 and Pareto optimality

LLM status Open unverified

Statement

ID: GT-061. Category: Fair division.  Kind: P. Status: Open in the cited source.

Definitions and mathematical statement

Let $N=\{1,\ldots,n\}$, $n\geq 3$, and let $M$ be a finite set of indivisible goods. For each $i\in N$, let \[ v_i(S)=\sum_{g\in S}a_{ig}\qquad(S\subseteq M),\qquad a_{ig}\geq 0. \] A complete allocation $A=(A_1,\ldots,A_n)$ is an ordered partition of $M$, with empty parts permitted. It is envy-free up to one good ($\mathrm{EF1}$) if, for every $i,j\in N$, \[ v_i(A_i)\geq v_i(A_j) \quad\text{or}\quad \exists g\in A_j:\ v_i(A_i)\geq v_i(A_j\setminus\{g\}). \] It is Pareto optimal if there is no complete allocation $B$ with $v_i(B_i)\geq v_i(A_i)$ for every $i$, and strict inequality for at least one $i$. Let $\mathcal A$ denote the finite set of complete allocations. A lottery is a probability vector $(\lambda_A)_{A\in\mathcal A}$, with $\lambda_A\geq0$ and $\sum_{A\in\mathcal A}\lambda_A=1$.
Question.
Does every such instance admit a lottery satisfying \[ \sum_{A\in\mathcal A}\lambda_A v_i(A_i) \ \geq\ \sum_{A\in\mathcal A}\lambda_A v_i(A_j) \qquad(i,j\in N), \] whose support is contained in the set of allocations that are both $\mathrm{EF1}$ and Pareto optimal?

Short English statement

Can randomization eliminate expected envy while every possible outcome remains EF1 and Pareto optimal?

Variants and scope boundaries

The goods are nonnegative and additive; this is not the mixed-goods-and-chores problem. Pareto optimality compares deterministic allocations. Fractional Pareto optimality is stronger and is not required here. The support condition concerns every positive-probability outcome.

Source relationship and status

[S1, Section 7] explicitly leaves this question open for $n\geq 3$. Its incompatibility results for stronger requirements do not settle the statement above.

References

[S1] Haris Aziz, Best-of-Both-Worlds Fairness and Pareto Optimality, 2026, arXiv:2608.08966v1. https://arxiv.org/pdf/2608.08966. Locators: Section 3 (definitions); Section 7, printed pp. 22--23 (the open question).

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-161-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-161

Envy-free lotteries: a separating certificate and a proportional-component construction

Problem ID: D63-158. Status: unresolved.

Target and scope

For at least three agents with nonnegative additive values, the target asks for a lottery that is envy-free in expectation and whose every supported allocation is EF1 and deterministically Pareto optimal. Aziz [Aziz, Section 7] leaves the general case open. We give an exact finite alternative identifying what a counterexample must certify, and a constructive positive result for independent components with proportional values. The construction actually has all-good EFX in every outcome. No claim of novelty is made for the proportional special case or finite separation.

Let $\mathcal G$ be the finite set of allocations that are both EF1 and Pareto optimal. For $i\ne j$ define \[ d_{ij}(A)=v_i(A_i)-v_i(A_j),\qquad d(A)\in{\mathbb{R}}^{q},\quad q=n(n-1). \] Checking membership in $\mathcal G$ by enumerating all complete allocations is finite, although generally exponential. The following result is a certificate reduction, not a polynomial-time algorithm.

An exact alternative and a small-support theorem

Theorem (Finite separation): Suppose $\mathcal G\ne\varnothing$. Exactly one of the following holds:
  1. there is a lottery on $\mathcal G$ with ${\mathbb{E}} d_{ij}(A)\ge0$ for all $i\ne j$;
  2. there are weights $w_{ij}\ge0$, not all zero, and $\gamma>0$ such that \[ \sum_{i\ne j}w_{ij}d_{ij}(A)\le-\gamma \quad\text{for every }A\in\mathcal G. \]
Whenever the first alternative holds it has a solution supported on at most $q+1$ allocations. For rational valuations a rational such lottery exists. If $\mathcal G=\varnothing$, no lottery exists trivially.
Proof: Let $K={\operatorname{conv}}\{d(A):A\in\mathcal G\}$ and let $O={\mathbb{R}}^q_{\ge0}$. A desired lottery exists precisely when $K\cap O\ne\varnothing$. A vector with the inequalities in the second alternative has nonnegative dot product with every point in $O$ but negative dot product with every point in $K$, so the alternatives cannot both hold.

If they are disjoint, minimize $\sum_\ell(\min\{x_\ell,0\})^2$ over compact $K$, and let $x^*$ be a minimizer. Set $y^*=(x^*)^+$ and $w=y^*-x^*\ge0$. Disjointness gives $w\ne0$. The objective is continuously differentiable with gradient $2(x-y)$ at $x=x^*$, where $y=x^+$. For every $z\in K$, the directional derivative along the segment toward $z$ is nonnegative, so $-w\cdot(z-x^*)\ge0$. As $w_\ell>0$ only where $x^*_\ell<0$, $w\cdot x^*=-\|w\|_2^2$. Therefore $w\cdot z\le-\|w\|_2^2$, proving the second alternative with $\gamma=\|w\|_2^2$.

For support size, start with any feasible lottery. If more than $q+1$ weights are positive, the vectors $(1,d(A))$ on its support are linearly dependent. Choose nonzero coefficients $c_A$ with $\sum c_A=0$ and $\sum c_A d(A)=0$. The coefficients have both signs. Subtract $t c_A$ from the weights, where $t=\min_{c_A>0}\lambda_A/c_A$. Weights stay nonnegative, at least one vanishes, and the total weight and expected envy vector stay unchanged. Iterate.

For rational input, the original feasible set of lottery weights is a nonempty bounded polytope defined by rational inequalities. It has a vertex: minimize the coordinates successively over the polytope, retaining the minimizer face each time; after all coordinates the remaining face is a point. At a vertex a full-rank set of active constraints determines the point, which is rational by Gaussian elimination. Start the preceding support elimination with this rational lottery and use a rational linear dependence at every step. All resulting weights remain rational. ∎

The useful feature is the quantifier over all EF1--PO outcomes. Failure of a proposed rounding method, or a list of individually envious outcomes, does not constitute the second alternative.

A constructive positive class

Partition the agents into nonempty components $N_1,\ldots,N_r$ and the goods into $M_0,M_1,\ldots,M_r$. Every good in $M_0$ has value zero to everyone. For every $\ell\ge1$ assume weights $b_g\ge0$ on $M_\ell$ and positive scales $c_i$ for $i\in N_\ell$ such that \[ a_{ig}=c_i b_g\quad(i\in N_\ell,g\in M_\ell),\qquad a_{ig}=0\quad(i\notin N_\ell,g\in M_\ell). \] Zero weights within $M_\ell$ may be moved to $M_0$. Thus every remaining good in a nonzero component is valued positively by all its agents.

Within component $\ell$, sort its goods by nonincreasing $b_g$ and place each in a currently least-loaded one of $|N_\ell|$ initially empty bundles, where load means the sum of $b_g$. Break ties deterministically. Independently in each component, uniformly permute these bundles among its agents. Finally put all globally zero goods in any one bundle of minimum load in its component, or, equivalently, include them in that component's greedy construction before permuting. If there are no goods outside $M_0$, choose any component and perform this last operation there.

Lemma (Descending greedy partitions): For any nonnegative item weights, the descending least-load rule produces bundles $B_1,\ldots,B_k$ satisfying \[ b(B_i)\ge b(B_j\setminus\{g\})\quad \text{for all }i,j\text{ and every }g\in B_j. \]
Proof: If $B_j$ is empty there is nothing to check. Let $g_*$ be its last assigned good. Just before that assignment its load was at most the current load of every other bundle, hence at most every other bundle's final load. Because the goods were processed in nonincreasing order, $b_{g_*}\le b_g$ for every $g\in B_j$. Thus $b(B_j\setminus\{g\})\le b(B_j)-b_{g_*}\le b(B_i)$. This also handles zero goods: if its last good has weight zero, that bundle's final load is itself at most every other final load. ∎
Theorem (Proportional-component lottery): Every instance of the displayed component form admits a lottery that is ex ante envy-free and ex post Pareto optimal and all-good EFX, hence EF1. For rational input, its ex ante marginal assignment probabilities can be computed in polynomial time, and an outcome can be sampled in expected polynomial time using unbiased random bits. It also has a lottery representation with at most $n(n-1)+1$ supported outcomes.
Proof: Fix an agent $i\in N_\ell$. Against another agent in $N_\ell$, Lemma Lemma (Descending greedy partitions), multiplied by $c_i$, gives all-good EFX for every permutation. Goods from other components are valued zero by $i$, so her comparison with their bundles has right side zero. Globally zero goods were added through the greedy zero-weight step in one component; this preserves the lemma's all-good conclusion there and affects no other valuation. Thus every outcome is EFX and EF1.

Under a uniform component permutation, each component agent receives each of its bundles with probability $1/|N_\ell|$. Agent $i$ values her own expected bundle and every other member's expected bundle at the same number $c_i b(M_\ell)/|N_\ell|$. Bundles outside her component have expected value zero. Hence all expected envy inequalities hold.

For efficiency assign agent $i$ welfare weight $1/c_i>0$. Each good in $M_\ell$ contributes at most $b_g$ to the weighted welfare of any complete allocation, since outside-component holders value it zero; goods in $M_0$ contribute zero. Every constructed outcome attains the sum of these upper bounds because goods are assigned within their component. A Pareto improvement would strictly increase this positively weighted welfare, which is impossible. This proves deterministic Pareto optimality.

Sorting and rational load comparisons have polynomial cost. A uniform random permutation can be sampled in expected polynomial bit time by successive uniform choices with rejection sampling. Each good in a component is marginally assigned uniformly within that component, giving the marginal probabilities directly. Theorem Theorem (Finite separation) gives the small-support existence assertion; we do not claim that enumerating all support outcomes for its compression is polynomial in the input size. ∎

The efficiency proof uses a common supporting welfare functional. For general heterogeneous valuations such a functional need not be compatible with every fairness-preserving permutation.

The unresolved step

The full problem would follow if, for arbitrary additive valuations, every nonnegative array $w$ admitted an EF1--PO outcome with $\sum w_{ij}d_{ij}(A)\ge0$. Theorem Theorem (Finite separation) makes that statement exactly sufficient and necessary when the good-outcome set is nonempty. The proportional-component construction proves it only on the stated restricted domain. No general inequality establishing it, and no negative certificate contradicting it, is obtained here.

The cited HTML source was checked on 9 October 2026. Its Section 7 expressly distinguishes the unresolved $n\ge3$ EF1 question from stronger EFX or fractional-efficiency variants. The proof here uses deterministic PO as the target does. No finite experiment is used to infer existence.

References

Completion Estimate

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