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Best-of-Both-Worlds Guarantees with Fairer Endings

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Pith's one-line read Randomized allocations can be almost envy-free in expectation while every realized outcome is EFX or EFX-with-charity, with algorithms for lexicographic, monotone, and subadditive valuations.

desk verdict Strong new BoBW results at EFX level, but Theorem 4 has an unproved monotonicity step that must be fixed. read the letter →

arxiv 2507.16209 v1 pith:H2WCTM4Q submitted 2025-07-22 cs.GT cs.DS

Telikepalli Kavitha , Surya Panchapakesan , Rohit Vaish , Vignesh Viswanathan , Jatin Yadav This is my paper · ORCID
classification cs.GTcs.DS MSC 91B32
keywords fairdivisionindivisiblegoodsbest-of-both-worldsex-antefairnessenvy-freenessuptoanygood(EFX)lexicographicpreferencesmonotoneandsubadditivevaluationsdependentrounding
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Fair division of indivisible goods has two competing fairness standards: randomized allocations that are fair in expectation (ex-ante) and deterministic allocations that are only approximately fair (ex-post). This paper asks whether one can combine both when the ex-post standard is the stronger envy-freeness up to any good (EFX), in which removing any single good from another agent's bundle removes the envy. Its central results are algorithmic: for lexicographic preferences, a polynomial-time lottery that is ex-ante $9/10$-envy-free and whose every realization is EFX and Pareto-optimal; for monotone valuations, a pseudopolynomial-time lottery that is ex-ante $1/2$-envy-free with EFX-with-charity in every realization; and for subadditive valuations, a pseudopolynomial-time lottery that is ex-ante $1/2$-proportional with EFX-with-bounded-charity in every realization. The paper also proves that the stronger ex-ante notion used in prior work, stochastic-dominance envy-freeness, cannot coexist with ex-post EFX even in the lexicographic case.

What carries the argument

Two mechanisms carry the results. For lexicographic preferences, the paper runs the simultaneous eating procedure for exactly one time unit; the goods still being eaten at the cutoff form the set of 'last consumed' goods, whose total mass $k$ is an integer. Those goods are merged into a single super-good, the resulting fractional matching is rounded (via the Birkhoff–von Neumann decomposition, or via dependent rounding when $k=2$), and the leftover goods are given as a tail to a uniformly random unenvied agent who received a last-consumed good. The guarantee is $3k/(3k+1)$-EF in general, exact EF when $k=1$, and $9/10$-EF for $k=2$ thanks to dependent rounding's negative correlation. For monotone and subadditive valuations, the central object is a randomized charity-swap loop: repeatedly choose an inclusion-minimal envied set of unallocated goods, pick a uniformly random agent who envies it, and swap that set with the agent's bundle. The equal opportunity of receiving each envied bundle gives the ex-ante $1/2$ guarantees, and an appended bounded-charity routine converts the ex-post support to EFX-with-bounded-charity.

What would settle it

Enumerate small subadditive instances, run the randomized charity-swap routine followed by the bounded-charity step, and record whether any agent's value for its own bundle decreases; a single such decrease would break the inequality chain used to prove the ex-ante guarantee of the subadditive result.

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Extended reading notes

Core claim

The central claim is that EFX-level ex-post fairness does not force giving up non-vacuous ex-ante fairness, once the ex-ante benchmark is relaxed suitably. For lexicographic preferences the construction is a one-unit run of the simultaneous eating procedure followed by decomposition (Birkhoff–von Neumann or dependent rounding) and a random tail assignment; it achieves $9/10$-EF in expectation and EFX plus Pareto optimality (no reallocation could improve one agent without hurting another) in every realization. For general monotone valuations the paper randomizes a charity-swap routine, selecting a uniformly random agent among those who envy a minimal envied set of unallocated goods at each step, which yields ex-ante $1/2$-EF and ex-post EFX-with-charity. For subadditive valuations the same randomized routine is followed by a bounded-charity step, giving ex-ante $1/2$-proportionality and ex-post EFX-with-bounded-charity (at most $n-1$ goods left unallocated). Together with a counterexample to ex-ante sd-EF with ex-post EFX, these constitute the first best-of-both-worlds guarantees at the EFX level for the three domains.

Load-bearing premise

The subadditive result assumes that the final bounded-charity step never lowers any agent's value for its own bundle; if that monotonicity fails, the argument that the expected allocation is half-proportional collapses.

Editorial extensions

If this is right

  • For lexicographic preferences, instances now come with a polynomial-time lottery whose ex-ante fairness is $9/10$-EF while every realized allocation is EFX and Pareto-optimal, so the earlier EF1-level best-of-both-worlds guarantee is strengthened at the price of a small ex-ante loss.
  • Because ex-ante sd-EF is incompatible with ex-post EFX already in the lexicographic domain, any future theorem that targets EFX ex-post must either relax the ex-ante benchmark or replace sd-EF by a weaker ordinal notion.
  • For monotone valuations, the randomized charity-swap routine gives a best-of-both-worlds guarantee of ex-ante $1/2$-EF with ex-post EFX-with-charity, in a domain where purely EFX allocations are not known to exist.
  • For subadditive valuations, the same routine can be composed with a bounded-charity step to leave at most $n-1$ goods unallocated while retaining ex-ante $1/2$-proportionality; such allocations can then be completed to $1/2$-EFX and EF1 allocations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The constant $9/10$ arises from the exceptional case $k=2$: the same construction gives $3k/(3k+1)$-EF, which exceeds $9/10$ for all $k \ge 3$ and equals exact EF at $k=1$, so a different rounding scheme for the two-agent super-good case could plausibly raise the worst-case guarantee.
  • The randomized charity-swap loop is a template rather than a one-off algorithm: varying the distribution over envying agents, or the rule for choosing the minimal envied set, may yield better ex-ante factors or extend the construction beyond the monotone and subadditive classes considered here.
  • Composing the randomized charity-swap output with other deterministic fair-division routines, not only the bounded-charity one, would give a family of best-of-both-worlds guarantees with different trade-offs between the size of the unallocated pool and the ex-ante ratio.
  • The impossibility for sd-EF points toward a weaker ordinal benchmark, such as lexicographic sd-EF, as the right target for lexicographic preferences; if that notion is compatible with EFX, an exact ordinal best-of-both-worlds statement may exist.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper studies best-of-both-worlds (BoBW) guarantees in fair division of indivisible goods, aiming for ex-post EFX-type guarantees together with nontrivial ex-ante fairness. For lexicographic preferences it proves an impossibility result (Theorem 1): ex-ante sd-EF and ex-post EFX are incompatible. It then gives a polynomial-time algorithm (Theorem 2) achieving ex-ante 9/10-EF with ex-post EFX and Pareto optimality, using an early-terminated eating algorithm, Birkhoff-von Neumann decomposition, and, for the delicate k=2 case, dependent rounding. For monotone valuations it gives a pseudopolynomial-time algorithm (Theorem 3) achieving ex-ante 1/2-EF with ex-post EFX-with-charity, via a randomized version of the CKMS21 charity algorithm. For subadditive valuations it claims ex-ante 1/2-Prop with ex-post EFX-with-bounded-charity (Theorem 4), by post-processing the output of Algorithm 3 with the CKMS21 little charity algorithm.

Significance. If the theorems are correct, the paper makes a substantial contribution: it provides the first BoBW guarantees at the EFX level for lexicographic preferences, the first BoBW guarantee with EFX-with-charity for general monotone valuations, and the first with EFX-with-bounded-charity for subadditive valuations. The techniques are independently interesting: the randomized charity algorithm in Section 5.1 is simple and elegant, and its stochastic-dominance proof (Lemma 11) is sound and gives a genuine ex-ante 1/2-EF guarantee without any rounding. The lexicographic analysis is detailed, with a complete impossibility proof and a nontrivial dependent-rounding step for k=2. There are no fitted parameters or definitional shortcuts. The main caveat is that Theorem 4 relies on an unproved monotonicity assertion about the CKMS21 little charity algorithm, which is load-bearing for the ex-ante 1/2-Prop guarantee; this gap is localized and likely repairable, but it must be fixed before the result can be accepted.

major comments (1)
  1. [5.2, Lemma 12 (proof of Theorem 4)] The inequality E[vi(Yi)] ≤ E[vi(Xi)] is asserted with the justification "since the agent utilities are non-decreasing" when the little charity algorithm of [CKMS21] is applied to the output Y of Algorithm 3. This monotonicity is load-bearing: it supplies the last inequality in the chain vi(M) ≤ E[vi(P)] + E[vi(Yi)] + Σ_{j≠i} E[vi(Yj)] ≤ 2n·E[vi(Yi)] ≤ 2n·E[vi(Xi)], from which the ex-ante 1/2-Prop guarantee of Theorem 4 follows. No lemma in [CKMS21] is cited for this property, and it does not follow merely from the fact that X is EFX-with-bounded-charity, because the transformation from Y to X could in principle reallocate goods and lower the utility of a fixed agent. The authors should either prove that the little charity algorithm weakly increases every agent's utility, or cite the specific lemma in [CKMS21] that establishes this. Without such a proof or citation, the ex-ante guarantee of Theorem 4 is unsupported.
minor comments (4)
  1. [4.4, Lemma 10] The statement of Lemma 10 contains a confusing condition: after restricting to g ∈ L ∪ U \ {g_i}, it then says "if g ≠ g_i", which is redundant, and the second bound is for g = g_i. Please rephrase the statement to distinguish clearly between the two cases.
  2. [Abstract and Theorem 4] Theorem 4 is stated as "There exists an algorithm with pseudopolynomial running time algorithm that..." with a duplicated word; this should be corrected to "pseudopolynomial-time algorithm".
  3. [References] References [FMNP24a] and [FMNP24b] appear to refer to the same EC 2024 paper by Feldman et al.; they should be merged into a single entry to avoid confusion.
  4. [5.1, Lemma 11] In the proof of Lemma 11, the equality Pr[k_{r0} = i | E] = 1/2 is stated without justification. It is true because k_r is uniform over H_r and conditioning on k_r ∈ {i,j} makes i and j symmetric, but the authors should spell this out, since the conditioning on the first significant iteration could otherwise appear to affect the distribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorems are derived from internal case analyses and independent black-box prior results, not from their own conclusions.

full rationale

I walked the derivation chains for Theorems 2, 3, and 4. For lexicographic preferences, the ex-post EFX+PO guarantee is imported from the external characterization of [HSVX21] (Propositions 3 and 4), while the ex-ante guarantee is proved by a fresh case analysis of the eating algorithm, B-vN decomposition, and dependent rounding (Theorems 6, 8, and 10; Lemmas 3 through 10). Nothing in these proofs defines the target guarantee in terms of itself, and no parameter is fitted and then reported as a prediction. For monotone valuations, Algorithm 3 is a randomized version of the known CKMS21 charity algorithm, but the ex-ante 1/2-EF guarantee is established by an approximate stochastic-dominance argument (Lemma 11) that does not assume its conclusion. For subadditive valuations, Theorem 4 calls the CKMS21 little charity algorithm as a black box; this is a legitimate use of an independent published result even though the author sets overlap. The only notable weakness is in Lemma 12, where the assertion "since the agent utilities are non-decreasing" is not proved or tied to a specific CKMS21 lemma; that is a possible correctness gap, but it is not circularity, because the assertion is not equivalent to the theorem being proved. The citations to [AFSV24], [FMNP24b], [GKPS06], and [HSVX21] are external evidence under the review rules. Overall, the derivation is self-contained modulo standard prior theorems, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The results rely on standard algorithmic theorems and on earlier results by the same authors in [HSVX21], [CKMS21], and [AFSV24]. No free parameters or invented entities are introduced. The weakest item is the unproved monotonicity assumption for the little charity algorithm in Lemma 12.

assumptions (8)
  • standard math Birkhoff-von Neumann decomposition (Proposition 1) and its rectangular generalization (Proposition 2) can be computed in polynomial time.
    Used to decompose fractional allocations into convex combinations of integral allocations in Algorithm 1.
  • standard math Dependent rounding theorem of Gandhi et al. (Theorem 7) provides integral allocations with marginal probabilities preserved and negative correlation.
    Used in Algorithm 2 for the k=2 case; the negative correlation property is essential for the 9/10-EF bound.
  • domain assumption For lexicographic preferences, an allocation is EFX iff every envied agent receives exactly one good (Proposition 3 from [HSVX21]).
    Used to establish ex-post EFX for the lexicographic algorithms.
  • domain assumption For lexicographic preferences, an allocation is EFX+PO iff it is induced by a sigma-unenvied picking sequence (Proposition 4 from [HSVX21]).
    Used to show ex-post EFX+PO of Algorithm 1 and Algorithm 2.
  • domain assumption There exists a pseudopolynomial-time algorithm that computes an EFX-with-charity allocation for monotone valuations, and the algorithm can be randomized with arbitrary choice of envying agent (from [CKMS21]).
    Algorithm 3 is claimed to reduce to this algorithm; the ex-post guarantee rests on this.
  • domain assumption There exists an EFX-with-bounded-charity algorithm (the little charity algorithm of [CKMS21]) that, starting from an EFX-with-charity allocation, yields a bounded-charity allocation and preserves non-decreasing utilities for all agents.
    Used in Algorithm 4 and Lemma 12; the non-decreasing utility part is not proved in this paper.
  • standard math The Probabilistic Serial (eating) algorithm satisfies anytime envy-freeness.
    Used to bound ex-ante envy in the lexicographic analysis.
  • standard math The fractional allocation of the eating algorithm run for one unit of time can be decomposed over picking sequences (Proposition 7, Appendix C).
    Used to prove ex-post EFX+PO of Algorithm 1 via Lemma 1.

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Pith. "Pith review of Best-of-Both-Worlds Guarantees with Fairer Endings." pith.science (2026). https://pith.science/paper/H2WCTM4Q

@misc{pith2026250716209,
  author       = {Pith},
  title        = {Pith review of: Best-of-Both-Worlds Guarantees with Fairer Endings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2WCTM4Q}},
  note         = {Machine review of arXiv:2507.16209}
}
read the original abstract

Fair allocation of indivisible goods is a fundamental problem at the interface of economics and computer science. Traditional approaches focus either on randomized allocations that are fair in expectation or deterministic allocations that are approximately fair. Recent work reconciles both these approaches via best-of-both-worlds guarantees, wherein one seeks randomized allocations that are fair in expectation (ex-ante fair) while being supported on approximately fair allocations (ex-post fair). Prior work has shown that under additive valuations, there always exists a randomized allocation that is ex-ante stochastic-dominance envy-free (sd-EF) and ex-post envy-free up to one good (EF1). Our work is motivated by the goal of achieving stronger ex-post fairness guarantees such as envy-freeness up to any good (EFX) along with meaningful ex-ante guarantees. We make the following contributions: 1) We first consider lexicographic preferences, a subdomain of additive valuations where ex-post EFX allocations always exist and can be computed efficiently. On the negative side, we show that ex-ante sd-EF is fundamentally incompatible with ex-post EFX, prompting a relaxation of the ex-ante benchmark. We then present a poly. time algorithm that achieves ex-post EFX and PO together with ex-ante 9/10-EF. Our algorithm uses dependent rounding and leverages structural properties of EFX and PO allocations. 2)For monotone valuations, we study EFX-with-charity: a relaxation of EFX where some goods remain unallocated, with no agent envying the unallocated pool. We show that ex-post EFX-with-charity can be achieved alongside ex-ante 0.5-EF. 3)Finally, for subadditive valuations, we strengthen our previous ex-post guarantee to EFX-with-bounded-charity, where at most n-1 goods (n= no. of agents) remain unallocated, at the price of weakening the ex-ante guarantee to 0.5-proportionality.

Figures

Figures reproduced from arXiv: 2507.16209 by the authors.

Figure 1
Figure 1. A summary of our results for lexicographic preferences. The arrows denote logical impli￾cations between fairness notions. The property combinations known from prior work are shaded in gray, while the positive and negative results shown by us are in green and red, respectively. The above figure is inspired by [AFSV24, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A summary of our main results for subadditive valuations. The arrows denote logical implications between fairness notions. The property combinations known from prior work are shaded in gray, while the positive and negative results shown by us are in green/teal and red, respectively. The above figure is inspired by [AFSV24, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The blue goods are not in L while the red ones (they are the last consumed goods) are in L and the green goods are untouched. The figure on the right has a super-good formed by unifying the two red goods. Let G be the complete bipartite graph on vertex set N ∪S, where N is the set of agents and S is 4The algorithm of Feldman et al. [FMNP24b] also runs the eating algorithm for one unit of time. 5 [PITH_FULL_IMAGE:fi… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Eating timelines of agents i and j when x > y (left) and y > x (right). The dashed lines denote the time an agent spends on its last consumed good. Case (a): gj ̸= gi We first establish an upper bound on E[vi(Zj )]. The randomized bundle Zj of agent j consists of the f…
Figure 5
Figure 5. Figure 5: We consider permutations σ and its partner τ where the positions of agents i and j are swapped, while all other agents remain unchanged. Let ki and kj denote the positions of agents i and j in σ, respectively. If kj = n, then Aj and Ai could contain multiple goods. We …

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Best-of-Both-Worlds Fairness and Pareto Optimality

    cs.GT 2026-08 accept novelty 8.0 of 10

    A lottery over two-agent allocations can be simultaneously ex-ante envy-free, ex-post EFX, and ex-post Pareto optimal, but this becomes impossible for three agents with four goods.

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