Econ-60 · economics · Economics

EF1 and Pareto optimality for additive mixed manna

LLM status Open unverified

Statement

GT-060 | Fair division and efficiency | P: additive unweighted source-date question.

Definitions and mathematical statement

Let $N=[n]$, $n\geq3$, $M$ finite, and $u_i(S)=\sum_{g\in S}u_{ig}$ for arbitrary real $u_{ig}$, permitting different signs for the same item across agents. A complete allocation $A$ partitions $M$. It is EF1 if, for every $i,j$, either $u_i(A_i)\geq u_i(A_j)$, or removing some item from $A_j$ yields $u_i(A_i)\geq u_i(A_j\setminus\{g\})$, or removing some item from $A_i$ yields $u_i(A_i\setminus\{g\})\geq u_i(A_j)$. It is Pareto optimal if there is no complete allocation $B$ with \[ \forall i,\quad u_i(B_i)\geq u_i(A_i), \qquad \exists k,\quad u_k(B_k)>u_k(A_k). \] Determine whether \[ \forall n\geq3\ \forall M\text{ finite}\ \forall(u_{ig})\in\mathbb R^{n\times|M|}, \quad\exists A\text{ complete, EF1, and Pareto optimal}. \]

Short English statement

Can mixed goods and chores always be allocated both efficiently and envy-free up to one deletion?

Variants and scope boundaries

This is an additive, unweighted question. Arbitrary set functions, weighted EF1, and fractional Pareto optimality are different targets; general-utility failures cannot be silently imported here. Polynomial computation would be a further question, even if existence is established.

Source relationship and status

The explicit question is verified in the survey's Open Question 2. The catalogue's September 2026 reference could be confirmed through its indexed abstract but its alleged open-question passage could not be verified; it is not used to certify present-day openness. The missing additivity assumption in the original catalogue is restored.

References

[S1] Shengxin Liu, Xinhang Lu, Mashbat Suzuki, and Toby Walsh (2024 version). Mixed Fair Division: A Survey. arXiv:2306.09564v4. Verified locator: Definitions 3.1 and 3.3; Section 4.1, Open Question 2. https://arxiv.org/html/2306.09564v4

[S2] Weighted Fair Division of Indivisible Mixed Manna (2026), arXiv:2609.01580. Verification limit: indexed abstract establishes weighted/additive context, but the catalogue's asserted EF1+PO status passage is unverified. https://arxiv.org/abs/2609.01580

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.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-60-A2 · Exploration · LLM-generated · gpt 6 astra ultra · 2026-10-09

GPT 6 Astra Ultra attempt on Econ-60

EF1 and Pareto optimality for additive mixed manna: A signed load-balancing proof with exclusive goods

The unrestricted target is a complete allocation of arbitrary additive mixed items that is both EF1 and Pareto optimal. The one-deletion condition permits deleting either an item from the envied bundle or an item from the envying agent's own bundle: \[ u_i(A_i)\geq u_i(A_j),\quad\text{or}\quad u_i(A_i)\geq u_i(A_j\setminus\{g\}),\quad\text{or}\quad u_i(A_i\setminus\{g\})\geq u_i(A_j). \] The next result treats signed proportional common items together with arbitrary exclusive goods. It permits genuinely different valuations across agents, although not arbitrary item-dependent comparative advantages.

Theorem: Let $n\geq1$, let all utilities be additive, and partition the finite item set as $M=C\sqcup P$. Fix $c_i>0$. Each common item $g\in C$ has a real signed weight $w_g$ and value $u_{ig}=c_iw_g$. Each exclusive good $h\in P$ has a designated owner $o(h)$, value $u_{o(h),h}\geq0$, and value zero to every other agent. Then a complete EF1 and Pareto-optimal allocation exists. Moreover, it maximizes the strictly positive weighted welfare $\sum_i u_i(A_i)/c_i$.

Proof: First allocate only $C$. For a partition $B$ define the signed load $\ell_i=\sum_{g\in B_i}w_g$. Among the finitely many partitions choose one minimizing $\sum_i\ell_i^2$. Fix $i,j$ with $d=\ell_j-\ell_i>0$. If some positive item $g\in B_j$ has $w_g\geq d$, deleting it eliminates $i$'s envy. If some negative item $g\in B_i$ has $-w_g\geq d$, deleting it from $i$'s own bundle eliminates the envy.

Suppose neither kind of item exists. There must nevertheless be a positive item in $B_j$ or a negative item in $B_i$: otherwise $\ell_j\leq0\leq\ell_i$, contrary to $d>0$. A positive item of weight $a\in(0,d)$ can be moved from $j$ to $i$. This changes the objective by \[ (\ell_i+a)^2+(\ell_j-a)^2-\ell_i^2-\ell_j^2=2a(a-d)<0. \] A negative item of weight $-a\in(-d,0)$ can instead be moved from $i$ to $j$, producing exactly the same change. Both contradict minimality. Therefore $B$ is EF1 in the common signed weights, and multiplication by $c_i>0$ gives EF1 for each agent's evaluations.

Now give every exclusive good to its designated owner, obtaining $A$. For $i\neq j$, the added goods in $A_j$ have value zero to $i$, while those in $A_i$ have nonnegative value to $i$. Every no-envy inequality or deletion witness previously established for $B_i,B_j$ therefore still holds, including an own-bundle deletion witness: its deleted item lies in $C$, so the new exclusive goods remain in the own bundle.

For any complete allocation $D$, the common-item contribution to $\sum_i u_i(D_i)/c_i$ is the constant $\sum_{g\in C}w_g$. The contribution of an exclusive good $h$ is at most $u_{o(h),h}/c_{o(h)}$, attained in $A$. Thus $A$ maximizes that weighted welfare. Since all welfare weights $1/c_i$ are strictly positive, a Pareto improvement would strictly increase it, an impossibility. ∎

Remark (Why the minimization argument does not prove the general case): Moving a common item changes two normalized loads by equal and opposite amounts. This conservation gives the exact quadratic improvement $2a(a-d)$. With arbitrary $u_{ig}$, a transfer can lower total weighted welfare and change the two agents' loads by unrelated amounts. Minimizing a convex load objective then neither gives this EF1 contradiction nor certifies Pareto optimality. Exclusive goods avoid the issue because their allocation improves only the recipient's fairness inequalities and is independently welfare maximizing.

Source relationship and remaining gap.

The survey [survey], Section 4.1, Open Question 2, states the additive EF1-plus-PO question and discusses several settled special classes. The load argument here is an elementary restricted derivation, not a claim that the identical-valuation phenomenon is new. It supplies no polynomial-time algorithm for the quadratic minimization, and the existence question for general heterogeneous signed additive values remains unresolved by this note.

References

[survey] S. Liu, X. Lu, M. Suzuki, and T. Walsh. Mixed Fair Division: A Survey, version 4. https://arxiv.org/html/2306.09564v4.

Completion Estimate

15%

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Econ-60-A1 · Exploration · LLM-generated · gpt 6 astra pro · 2026-10-09

GPT 6 Astra Pro attempt on Econ-60

{EF1 and Pareto optimality for additive mixed manna}

D63-59.tex

Research attempt and adversarial audit --- 9 October 2026

Tools. Web browsing: YES. Computation: YES (Python, exact integer/rational checks, and explicitly identified numerical diagnostics). Status convention. A full proof of a restricted theorem does not resolve the full input. Literature results are marked as imported; no novelty or exhaustive current-literature certification is claimed. Computational searches certify only their stated finite domains.

FORMAL RESTATEMENT

For every integer $n\ge3$, finite item set $M$, and real matrix $(u_{ig})$, let $u_i(S)=\sum_{g\in S}u_{ig}$. Items may have different signs for different agents. A complete allocation $A=(A_i)_{i=1}^n$ is an ordered partition, with empty parts allowed. It is EF1 if, for every ordered pair $i,j$, either $u_i(A_i)\ge u_i(A_j)$ or \[ \exists g\in A_i\cup A_j:\quad u_i(A_i\setminus\{g\})\ge u_i(A_j\setminus\{g\}). \] It is Pareto optimal (PO) if there is no complete partition $B$ for which every $u_i(B_i)\ge u_i(A_i)$ and at least one inequality is strict. The question is \[ \forall n\ge3\ \forall M\text{ finite}\ \forall u\in{\mathbb{R}}^{n\times |M|}\quad \exists A\text{ complete, EF1 and PO}. \] This is integral PO; fractional PO is stronger. No subsidies, copies, fractional items, weights or normalization are part of this question. For an empty set of candidate deletions, the EF1 condition requires no envy. Arbitrary nonadditive valuations are not covered by the quantifier on $u$.

QUICK LITERATURE/CONTEXT CHECK

The supplied survey identifies this mixed-manna fairness/efficiency question [survey]. The accessible January 2026 version of Barman and Verma [bv] proves PO with introspective EF1 and explicitly distinguishes that property from the requested either-side deletion EF1. Its abstract still poses ordinary EF1 plus PO for additive mixed manna as unresolved. An introspective witness may add an item from outside the envied bundle, so it cannot silently be used as the witness required here. We do not infer an exhaustive October 2026 status from that source alone.

ATTACK PLAN

This is an existence problem at the intersection of discrete fair division and Pareto efficiency. The proof track searches for a potential minimizer that preserves a fixed total utility in a common-value subdomain. The disproof track checks small signed instances and tries to break the tempting "find EF1, then Pareto-improve" argument.

Tool	Role
Exact finite enumeration	Checks the smallest genuinely mixed domains.
Squared-load potential	Balances a common signed valuation without requiring nonnegative loads.
Conservation of total load	Certifies Pareto optimality in a proportional-value subdomain.
Itemwise welfare maximization	Handles privately beneficial goods.
Single-deletion gap formula	Makes EF1 verification exact for additive utilities.
Pareto dominance tests	Separates efficiency from fairness along improvement paths.
Weighted-welfare/market methods	Possible general route, but a suitable EF1-preserving construction is not supplied.

WORK

Lemma (An exact EF1 test): For additive utilities and a fixed pair $i,j$, EF1 is equivalent to \[ u_i(A_j)-u_i(A_i)\le \max\left(\{0\}\cup\{u_{ig}:g\in A_j\}\cup\{-u_{ig}:g\in A_i\}\right). \]

Proof: A nonpositive gap is no envy. Removing $g$ from $A_j$ lowers the right bundle's value by $u_{ig}$; removing $g$ from $A_i$ raises the left value by $-u_{ig}$. Each permitted deletion therefore repairs the gap precisely when its corresponding quantity is at least the gap. Taking the maximum is exactly the disjunction in the definition. Including zero handles empty bundles. ∎

Theorem (Proportional signed preferences): For every $n\ge1$, if $u_{ig}=\lambda_i a_g$ for real $a_g$ and strictly positive $\lambda_i$, there is a complete EF1 and PO allocation. In fact every complete allocation is PO, and any minimizer of \[ \Phi(A)=\sum_{i=1}^n\left(\sum_{g\in A_i}a_g\right)^2 \] is EF1.

Proof: Write $\ell_i=\sum_{g\in A_i}a_g$. A Pareto improvement would weakly raise every $\ell_i$ and strictly raise at least one, because $\lambda_i>0$. That is impossible since $\sum_i\ell_i=\sum_ga_g$ for every complete allocation. Thus every allocation is PO.

A potential minimizer exists because there are finitely many partitions. Suppose it is not EF1. For some $i,j$, let $D=\ell_j-\ell_i>0$. If $A_j$ contains a positive item of value at least $D$, deleting it repairs this envy; if $A_i$ contains a negative item of absolute value at least $D$, deleting it also repairs the envy. Therefore EF1 failure implies every positive item in $A_j$ has value in $(0,D)$ and every negative item in $A_i$ has absolute value in $(0,D)$. At least one such item exists: otherwise $\ell_j\le0$ and $\ell_i\ge0$, contradicting $D>0$.

Move a positive item of value $r\in(0,D)$ from $A_j$ to $A_i$, or a negative item of value $-r$ from $A_i$ to $A_j$. In either case the two loads become $\ell_i+r$ and $\ell_j-r$. The change in the potential is \[ (\ell_i+r)^2+(\ell_j-r)^2-\ell_i^2-\ell_j^2=2r^2-2Dr<0, \] a contradiction. Since multiplying a pairwise comparison by $\lambda_i>0$ preserves its direction, this proves EF1 for the original utilities. ∎

The proof establishes existence by a finite minimizer; it does not claim polynomial-time optimization of that potential.

Theorem (Privately beneficial items): Suppose every item $g$ has a designated agent $h(g)$ with $u_{h(g),g}\ge0$, while $u_{ig}\le0$ for every $i\ne h(g)$. Assigning $g$ to $h(g)$ yields an exactly envy-free and PO allocation.

Proof: Every agent values its own assigned items nonnegatively, so its own utility is at least zero. Every item assigned to somebody else is nonpositive to that agent, so its value for another bundle is at most zero. Hence the allocation is envy-free. The assignment maximizes $\sum_i u_i(A_i)$ item by item, since $h(g)$ has a weakly largest value for $g$. A Pareto improvement would strictly increase this sum, contradicting maximality. ∎

Proposition (Pareto improvement need not preserve EF1): There are additive nonnegative utilities and complete allocations $A,B$ such that $A$ is EF1, $B$ Pareto dominates $A$, and $B$ is not EF1.

Proof: Use three agents and items $a,b,c$ with value matrix \[ \begin{array}{c|rrr}&a&b&c\\\hline 1&1&0&1\\2&2&1&2\\3&0&0&0 \end{array}. \] Let $A=(\{a\},\{b\},\{c\})$. Every compared bundle has at most one item, so deleting that item leaves zero, no larger than any agent's nonnegative own utility. Hence $A$ is EF1. Set $B=(\{a,c\},\{b\},\varnothing)$. Own utilities change from $(1,1,0)$ to $(2,1,0)$, a Pareto improvement. Agent 2 values the first bundle at 4 and its own at 1. Deleting either item from the first bundle leaves value 2; deleting its own item leaves zero. Neither deletion repairs its envy. Thus $B$ is not EF1. ∎

This is a counterexample to a proposed proof step, not to the existence statement. In particular, the report does not relabel it as a disproof of EF1 plus PO.

VERIFICATION

The accompanying program exhaustively checked all $3^9=19{,}683$ utility matrices with $n=m=3$ and entries in $\{-1,0,1\}$. For each it checked all $3^3=27$ complete allocations, using Lemma Lemma (An exact EF1 test) and pairwise Pareto dominance against all other allocations. Every profile had an EF1-and-PO allocation; the smallest number of such allocations in a profile was one. All comparisons were exact integers.

\begin{lstlisting} for u in {-1,0,1}^(3 by 3):     allocations = all 27 complete partitions     for A in allocations:         fair[A] = the single-deletion gap test holds for all i,j         PO[A] = no allocation B weakly improves everyone and strictly improves one     assert some A satisfies fair[A] and PO[A] \end{lstlisting}

Zero values, empty bundles, positive and negative total loads, and negative items with agent-dependent signs were included. The proportional theorem permits arbitrary real magnitudes; its use of $\lambda_i>0$ is essential. If some $\lambda_i=0$, the assertion that every allocation is PO need not follow from conserved loads and is not claimed. Finite examples with ternary values cannot certify arbitrary real utilities or larger item sets. The separately supplied Pareto-step counterexample is consistent with the positive pure-goods existence results.

FINAL

LABEL: UNRESOLVED. The general heterogeneous additive mixed-manna statement is not proved or disproved. Theorems Theorem (Proportional signed preferences) and Theorem (Privately beneficial items) are full proofs of explicit subdomains. Proposition Proposition (Pareto improvement need not preserve EF1) rigorously invalidates a natural shortcut.

Exact first gap. For arbitrary signed heterogeneous utility matrices, no selection of an EF1 allocation is proved to lie on the Pareto frontier, and no EF1-preserving Pareto-improvement operation is established.

Three next moves. (1) Seek a weighted-welfare maximizing allocation whose every excessive envy gap admits an efficiency-preserving alternating transfer. (2) Extend the signed potential proof to two nonproportional utility types, locating the first failed conservation identity. (3) Search sparse, rational four-agent profiles with uneven positive and negative magnitudes, retaining exact Pareto/EF1 certificates rather than relying on random dense instances.

Minimal hard counterexample, if one exists. It must escape the two proved subdomains; heterogeneity in signs or relative magnitudes is necessary for defeating the common-load proof. The finite test rules out only the stated three-agent ternary three-item domain, not all ternary markets.

Completion Estimate

18%

Two broad structured subcases and an exact small-domain test are complete, but the arbitrary heterogeneous signed-utility existence question is not resolved.

This is a subjective estimate of the fraction of the whole requested mathematical scope addressed by this report, including the named variants. It is not a probability of correctness, an objectively measured fraction of a proof, or an estimate that the remaining work takes the complementary fraction of the time. Only the eleven supplied files are assessed; no missing rank-1--200 statements are inferred.

References

[survey] S. Liu, X. Lu, M. Suzuki and T. Walsh. Mixed Fair Division: A Survey. arXiv:2306.09564v4, Section 4.1, Open Question 2. https://arxiv.org/html/2306.09564v4.
[bv] S. Barman and P. Verma. Introspectively Envy-Free and Efficient Allocation of Indivisible Mixed Manna. Accessible arXiv HTML version 2509.18673v3, 2 January 2026, abstract and Definition 1. https://arxiv.org/html/2509.18673.

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Record details

Subjects: Economics
Collection: Economics catalogue
Economics rank: 60
JEL: D63 — Equity, Justice, Inequality, and Other Normative Criteria and Measurement