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Econ-147 · economics · Economics

EFX and Pareto optimality for personalized bi-valued goods

LLM status Open unverified

Statement

ECON-003 | P | Explicit published open question, at source date

Definitions and formal target

For every agent $i$, fix two positive numbers $a_i<b_i$; each good $g$ has $v_{ig}\in\{a_i,b_i\}$. Utilities are additive. EFX requires $v_i(A_i)\geq v_i(A_j\setminus\{g\})$ for every $i,j$ and $g\in A_j$. A complete allocation $A$ is Pareto optimal (PO) when no complete allocation $B$ satisfies $v_i(B_i)\geq v_i(A_i)$ for all $i$, strictly for at least one.

\[ \forall n,M,(a_i,b_i),(v_{ig}),\qquad \exists A\in\operatorname{EFX}(v)\cap\operatorname{PO}(v)? \] Both sets refer to integral, complete allocations; no fractional reassignment is allowed in the PO comparison.

Clarifications and required assumptions

The numbers $a_i,b_i$ may differ across agents, but are constant across items for a given agent. Thus "two types of goods", meaning two item value vectors shared across agents, is a different restriction. The main quantifiers range over all finite $n\geq2$, all finite $M$, all positive pairs $a_i<b_i$, and all matrices respecting them. PO refers to every complete integral partition, including partitions that fail EFX.

Variants and changes of scope

Common ratio. If $b_i/a_i=r$ for every $i$, positive row scaling converts the domain into common bi-valued values $\{1,r\}$; this is a restricted benchmark. Zero low value. $a_i=0$ changes the domain and the EFX/EFX$_0$ distinction. Fractional PO. Permit comparison with $x_{ig}\geq0$, $\sum_i x_{ig}=1$, and utilities $\sum_gv_{ig}x_{ig}$; this strengthens integral PO. Algorithm. Ask for polynomial time in $n,m$ and the rational input length, separately from existence.

Audit conclusion and source relationship

The draft's later status citation is genuine, but [S1] is now the direct paper that poses the joint EFX--PO question. [S2] corroborates it subsequently. Existence of EFX alone does not answer the joint question. The main domain retains strictly positive personalized values.

References and exact source roles

[S1] Jiarong Jin and Biaoshuai Tao (2025). On Pareto-Optimal and Fair Allocations with Personalized Bi-Valued Utilities. arXiv:2507.18251v1; author preprint.

Verified locator / source role: Introduction, personalized bi-valued domain; concluding discussion of EFX and PO. Direct source for the question, rather than only a later paper mentioning it.

https://arxiv.org/html/2507.18251v1

[S2] Wentao He and Biaoshuai Tao (2026). EFX for Additive Chores: Nonexistence, Pareto Incompatibility, and Bi-Valued Existence. arXiv:2606.08872v2, 9 July 2026; original submission 7 June 2026. Author preprint.

Verified locator / source role: Section 6 (open questions); Section 2 (definitions). This version is newer than the draft's v1 citation.

https://arxiv.org/html/2606.08872v2

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-147-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-147

Certificates and constructive subcases for personalized bi-valued EFX and efficiency

Status: Unresolved. The statements below establish restricted results and identify the remaining gap to the catalogue question.

Abstract

We prove a price certificate that implies both EFX and fractional Pareto optimality, and construct an allocation meeting it for a separated-ratio two-type subclass with suitable divisibility. We also prove EFX and integral Pareto optimality whenever the number of goods does not exceed the number of agents. Personalized ratios remain unrestricted in the latter result. Neither subcase establishes the general existence conjecture.

Short target and definitions

There are $n\ge2$ agents and $m$ indivisible goods. Agent $i$ has additive values $v_{ig}\in\{a_i,b_i\}$ with $0<a_i<b_i$. A complete allocation is a partition $(A_1,\ldots,A_n)$. It is EFX if, for all $i,j$ and every $g\in A_j$, \[ v_i(A_i)\ge v_i(A_j\setminus\{g\}). \] It is Pareto optimal (PO) if no complete integral allocation weakly improves every utility and strictly improves one. Fractional PO allows fractional competitors and is stronger. The catalogue asks for EFX and PO for every personalized bi-valued instance. The following certificate applies more generally to strictly positive additive valuations.

A useful price certificate

For positive prices $p_g$ put $p(B)=\sum_{g\in B}p_g$ and $\alpha_i=\max_g v_{ig}/p_g$ when $m>0$.
Lemma (Price-EFX plus maximum bang per buck): Suppose every assigned good $g\in A_i$ satisfies $v_{ig}=\alpha_i p_g$. Suppose also that for every $i,j$ and $g\in A_j$, \[ p(A_i)\ge p(A_j)-p_g. \] Then the allocation is EFX and fractionally Pareto optimal.
Proof: Every value satisfies $v_{ih}\le\alpha_i p_h$, whereas allocated values attain equality. Therefore \[ v_i(A_j\setminus\{g\})\le\alpha_i[p(A_j)-p_g] \le\alpha_i p(A_i)=v_i(A_i), \] proving EFX. For any feasible fractional allocation $x$, \[ \sum_i{\sum_g v_{ig}x_{ig}\over\alpha_i} \le\sum_{i,g}p_gx_{ig}=\sum_g p_g =\sum_i{v_i(A_i)\over\alpha_i}. \] All $\alpha_i$ are positive. A fractional Pareto improvement would strictly increase the weighted sum on the left relative to the final expression, a contradiction. ∎
Equal spending is a stronger, convenient sufficient condition for the price inequality. Finding an integral allocation satisfying the certificate is the substantive obstacle; the lemma does not assert such an allocation always exists.

A separated-ratio construction

Suppose goods fall into common types $H,L$: every agent values each good of $H$ at $b_i$ and each good of $L$ at $a_i$. This alignment is an added restriction; arbitrary personalized bi-valued matrices need not have common types. Partition agents into two nonempty sets $I_H,I_L$. Write $r_i=b_i/a_i>1$.
Theorem (An explicit EFX and fractional-PO subcase): Suppose positive integers $q_H,q_L$ satisfy \[ |H|=q_H|I_H|,\qquad |L|=q_L|I_L|,\qquad \max_{i\in I_L}r_i\le q_L/q_H\le\min_{i\in I_H}r_i. \] Give each agent in $I_H$ any $q_H$ goods from $H$, and each agent in $I_L$ any $q_L$ goods from $L$. The resulting complete allocation is envy-free and fractionally Pareto optimal, and consequently EFX and PO.
Proof: Set $p_g=q_L/q_H$ on $H$ and $p_g=1$ on $L$. Every bundle has price $q_L$. For $i\in I_H$, the ratio assumption gives $b_i/p_H\ge a_i/p_L$, so allocated high goods maximize bang per buck. For $i\in I_L$, it gives $a_i/p_L\ge b_i/p_H$, so allocated low goods do so. The lemma proves fractional PO and EFX. In fact, for every pair $i,j$, $v_i(A_j)\le\alpha_i p(A_j)=\alpha_i p(A_i)=v_i(A_i)$, which is full envy-freeness. Divisibility permits the specified partition of each type of goods. ∎
For example, two agents with low values 1 and high values 3 and 2, respectively, and two common high goods plus five common low goods satisfy the theorem with $q_H=2,q_L=5$. Their assigned utilities are 6 and 5. The first values the other bundle at 5, while the second values the first bundle at 4. The ratios differ, so this example is not a common-ratio rescaling.

A second subcase without aligned types

Theorem (At most one good per agent): For any strictly positive additive valuations, if $m\le n$, an EFX and integral-PO allocation exists and is computable by maximum-weight bipartite matching. In particular the result applies to arbitrary personalized bi-valued matrices in this size regime.
Proof: If $m=0$ the empty allocation suffices. Otherwise choose an injection assigning every good to a distinct agent and maximizing the sum of the assigned values; this is a maximum-weight matching covering the goods. Every bundle has size zero or one, so removing any good from another bundle leaves value zero. Nonnegative own values prove EFX.

Suppose an integral allocation Pareto improved this matching. Its $m$ original recipients all have strictly positive initial utility. Each must still receive at least one good. As only $m$ goods exist, each receives exactly one and every other agent receives none. The improving allocation is consequently another injection with the same recipient set. Its sum of values is strictly larger, contradicting optimality of the matching among all injections. Standard finite bipartite matching optimization is polynomial in the rational input length. The proof uses integrality of the competitor and does not establish fractional PO for this construction. ∎

A normalization boundary

Proposition: Positive rescaling of each agent's utilities preserves EFX and both notions of PO. Such rescaling converts personalized pairs $\{a_i,b_i\}$ into a common pair $\{1,r\}$ for all agents if and only if all ratios $b_i/a_i$ are the same.
Proof: Each EFX or Pareto inequality compares utilities of the same agent, so multiplication by a positive factor preserves it and preserves strictness. Scaling the low value to one forces scale $1/a_i$, making the high value $b_i/a_i$. Equality to a common $r$ is therefore necessary and sufficient. ∎
The unrestricted question allows $m>n$, unaligned high-value sets, and arbitrary different ratios. The price lemma does not provide existence there; divisibility and separation in the construction may both fail. Removing those restrictions while maintaining integral EFX and efficiency is the remaining gap. No counterexample to general existence has been established here.

References

  • [JT] J. Jin and B. Tao (2025), On Pareto-Optimal and Fair Allocations with Personalized Bi-Valued Utilities, arXiv:2507.18251v1. https://arxiv.org/html/2507.18251v1. Defines the personalized domain and poses the joint EFX--PO question. The certificate and restricted constructions above are proved directly; no novelty claim is made for price-based efficiency arguments.

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