Near-Optimal Best-of-Both-Worlds Fairness for Few Agents

Abstract: We consider the problem of fair allocation of indivisible goods among agents with additive valuations, aiming for Best-of-Both-Worlds (BoBW) fairness: a distribution over allocations that is ex-ante fair, and additionally, it is supported only on deterministic allocations that are ex-post fair. We focus on BoBW for few agents, and our main result is the design of the first BoBW algorithms achieving near-optimal fairness for three agents. For three agents, we prove the existence of an ex-ante proportional distribution whose every allocation is Epistemic EFX (EEFX) and guarantees each agent at least $\tfrac{9}{10}$ of her MMS. As MMS allocations do not exist for three additive agents, in every allocation at least one agent might not be getting her MMS. To compensate such an agent, we also guarantee that if an agent is not getting her MMS then she is EFX-satisfied - giving her the strongest achievable envy-based guarantee. Additionally, using an FPTAS for near-MMS partitions, we present an FPTAS to compute a BoBW distribution preserving all envy-based guarantees, and also preserving all value-based guarantees up to $(1-\varepsilon)$. We further show that exact ex-ante proportionality can be restored when dropping EEFX. To do so, we first design, for two agents and any $\varepsilon > 0$, a Fully Polynomial-Time Approximation Scheme (FPTAS) that outputs a distribution which is ex-ante envy-free (and thus proportional) and ex-post envy-free up to any good (EFX), while guaranteeing each agent at least a $(1-\varepsilon)$-fraction of her maximin share (MMS). We then leverage this two-agent FPTAS algorithm as a subroutine to obtain, for three agents, the FPTAS guaranteeing exact ex-ante proportionality. We note that our result for two agents essentially matches the strongest fairness and efficiency guarantees achievable in polynomial time, and thus might be of independent interest.

• The paper constructs, for three agents, an ex-ante proportional distribution over at most six allocations that are EEFX, IMMX, and 9/10-MMS, with each agent falling below her MMS in at most one-third of outcomes.
• The paper introduces IMMX, requiring each agent to receive near-MMS value or be EFX-satisfied, and uses MMS partitions, role rotation, and epistemic fairness to combine envy-based and share-based guarantees.
• The paper provides FPTAS variants that restore exact ex-ante proportionality with approximately 9/10-MMS guarantees, while its two-agent algorithm achieves ex-ante envy-freeness, ex-post EFX, and near-optimal MMS value in polynomial time.

This paper studies Best-of-Both-Worlds (BoBW) fair division of indivisible goods among few agents with additive valuations, and establishes the first BoBW guarantees that are simultaneously near-optimal in both envy-based and share-based ex-post fairness (2602.14668). The central contribution is a construction for three agents of an ex-ante proportional distribution over at most six deterministic allocations, each of which is Epistemic EFX (EEFX), guarantees every agent at least $\tfrac{9}{10}$ of her maximin share (MMS), and satisfies a new fairness criterion the authors call IMMX: each agent either receives her MMS value or is EFX-satisfied. Complementing this, the paper gives an optimal polynomial-time BoBW result for two agents, and leverages it to obtain FPTAS variants for three agents that restore exact ex-ante proportionality.

Background and motivation

With indivisible goods, neither proportionality nor envy-freeness can always be guaranteed exactly, motivating relaxations such as EF1, EFX, and the MMS benchmark. MMS allocations fail to exist even for three additive agents, while EFX existence was established for three agents but remains open beyond. Epistemic EFX weakens EFX by requiring only that each agent's bundle can be embedded in some EFX allocation; EEFX allocations exist for all monotone valuations.

The BoBW paradigm seeks distributions that are fair ex-ante while being supported only on allocations that are fair ex-post. Prior work left substantial gaps even for few agents. Aziz and Freeman achieve ex-ante envy-freeness with ex-post EF1, but as the authors show via a diamonds-and-rocks example, such a distribution may guarantee each agent only a $1/n$ fraction of her MMS even when an ex-post envy-free allocation exists. Babaioff et al. achieve ex-ante proportionality with ex-post $\tfrac{1}{2}$-MMS, far below the best known existential bound of $\tfrac{11}{12}$-MMS for three agents, and their approach may violate even EF1. Garg and Sharma showed that a natural randomized variant of the Envy-Cycle Elimination algorithm fails to combine ex-ante proportionality with ex-post EEFX even for three agents. No prior BoBW result for more than two agents achieved any envy-based ex-post guarantee stronger than EF1 together with a near-optimal MMS fraction.

A new fairness notion: IMMX

The paper introduces the IMMX criterion: an allocation is IMMX if every agent either receives at least $(1-\varepsilon)$ of her MMS or is EFX-satisfied by the allocation. The motivation is twofold. First, since MMS allocations do not always exist for three agents, some agent may inevitably fall short of her MMS, and the strongest feasible envy-based compensation for that agent is EFX-satisfaction (envy-freeness would imply receiving the proportional share, hence the MMS). Second, an agent whose bundle already exceeds her MMS has little grounds for complaint even if she envies another bundle up to one good; the authors illustrate this with a five-item instance where the unique non-EFX-satisfied agent receives value 100 against an MMS of 2. Since EFX-satisfaction implies at least $\tfrac{4}{7}$-MMS, IMMX provides a robust floor on share-based guarantees. Existence of IMMX allocations for $n \geq 4$ is posed as an open question, since both MMS existence fails there and EFX existence is unresolved.

Near-optimal BoBW existence for three agents

The main theorem establishes that for any three additive valuations there exists a distribution $\mu$ over at most six allocations such that $\mu$ is ex-ante proportional, and every supporting allocation simultaneously satisfies: one agent is EFX-satisfied and receives her proportional share; one agent is EFX-satisfied and receives at least $\tfrac{9}{10}$ of her MMS; and one agent is EEFX-satisfied and receives at least her full MMS. Consequently every supporting allocation is EEFX, $1/n$0-MMS, and IMMX, and each agent falls below her MMS with probability at most $1/n$1 — essentially best possible given that some agent must fall short in every allocation when MMS allocations fail to exist.

The $1/n$2 figure is near-optimal from several angles. It substantially improves the previous polynomial-time guarantee of $1/n$3-MMS, approaches the best known existential approximation of $1/n$4-MMS for three agents, and sits within $1/n$5 of the hardness upper bound of $1/n$6-MMS. Ex-ante proportionality itself cannot be improved, since no distribution can give every agent more than her proportional share under identical valuations. Moreover, the analysis of the MMS approximation is tight: the authors exhibit an instance where their algorithm yields exactly $1/n$7-MMS for some agent.

The construction exploits the epistemic nature of EEFX. Each agent acts once as "divider," computing an MMS partition refined into an MMS-EFX partition via a Realloc procedure adapted from Plaut and Roughgarden's leximin-style argument, which converts any partition into an EFX one without decreasing the minimum bundle value. Because EEFX constraints are invariant to how the remaining bundles are repartitioned, the divider's fairness is secured regardless of what happens to the other bundles. The other two agents then take subdivider and chooser roles across paired allocations: the chooser receives a top-valued bundle (hence above-proportional value and EFX satisfaction), while the subdivider repartitions two bundles and is guaranteed $1/n$8-MMS with EFX satisfaction, using an analysis drawn from Feige and Norkin's coarse atomic partial search argument. Averaging over the three role assignments yields exact ex-ante proportionality, since each non-divider agent accumulates across the pair the combined value of the top two bundles of her own partition, which is at least $1/n$9.

A notable structural feature is that all guarantees hold robustly for every agent independently of others' reports, and equal treatment of equals holds ex-ante. Additionally, every supporting allocation guarantees each agent her MXS share (a consequence of EEFX implying MXS) and her residual maximin share (RMMS); the paper shows RMMS and $\tfrac{1}{2}$0-MMS are incomparable guarantees, so the construction delivers the maximum of the two.

Computational considerations

The existential construction requires exact MMS partitions, which are NP-hard to compute. The paper handles this in several ways. Given oracles for MMS partitions, the entire distribution is computable in polynomial time using at most fifteen oracle calls. When item values are integers bounded by a polynomial in the number of goods, setting the FPTAS error parameter small enough recovers exact MMS partitions in polynomial time.

For general inputs, replacing exact partitions with $\tfrac{1}{2}$1-approximate ones via an FPTAS preserves all envy-based guarantees (EEFX, EFX satisfaction) and scales all share-based guarantees by $\tfrac{1}{2}$2, yielding an FPTAS with $\tfrac{1}{2}$3-proportionality ex-ante and $\tfrac{1}{2}$4-MMS ex-post. However, the authors demonstrate concretely that naive substitution breaks exact ex-ante proportionality: approximate cut-and-choose can leave the cutting agent with expected value strictly below half the total, and can also destroy EFX satisfaction; similarly, identical agents using different approximation algorithms can end up with ex-ante envy.

To restore exact ex-ante proportionality in polynomial time, the paper develops a five-stage FPTAS that drops the EEFX requirement. Agents first select, among all three computed approximate partitions, the one offering the best worst-case-value guarantee; the three-agent construction runs on these selected partitions; a two-agent subroutine repairs cut-and-choose errors; and two adoption stages allow agents to replace their derived allocations with permutations of certificates offering better worst-case guarantees. The resulting distribution is exactly ex-ante proportional, and every supporting allocation guarantees one agent her proportional share with EFX satisfaction, one agent either $\tfrac{1}{2}$5-MMS or EFX satisfaction with at least $\tfrac{1}{2}$6-MMS, and one agent at least $\tfrac{1}{2}$7-MMS. This strictly dominates the prior state of the art of ex-ante proportionality with only $\tfrac{1}{2}$8-MMS and no envy guarantee.

Optimal poly-time BoBW for two agents

As a building block — and of independent interest — the paper presents an FPTAS for two additive agents outputting a distribution over at most two allocations that is ex-ante envy-free (hence proportional), ex-post EFX, and guarantees each agent at least $\tfrac{1}{2}$9 of her MMS ex-post. This matches the strongest feasible guarantees in all three dimensions simultaneously, and the $\tfrac{11}{12}$0 factor is optimal for polynomial time unless P = NP, since computing the MMS is NP-hard. The classical randomized cut-and-choose protocol achieves these guarantees with exact MMS but is not polynomial-time, and simulating it with approximate partitions provably fails to preserve ex-ante envy-freeness or ex-post EFX.

The algorithm modifies Algorithm 3 of Bu et al. in two ways: it seeds the local search with agent-supplied bipartitions rather than the trivial $\tfrac{11}{12}$1 split, and it adds a guard so that an agent adopts the other agent's candidate partition only when it strictly improves her minimum bundle value. The authors prove that prior polynomial-time BoBW algorithms — Bu et al.'s Algorithm 3 and Garg–Sharma's ECEG2 — each lose a constant fraction of the MMS: explicit instances show Algorithm 3 can deliver only $\tfrac{11}{12}$2 of the MMS (with four goods and identical valuations) and ECEG2 only $\tfrac{11}{12}$3. The modified algorithm eliminates this loss.

Verifiability

The paper addresses deployability through efficient verification. The support size of six allows each agent to compute her expected value explicitly and verify ex-ante proportionality, and agents can jointly perform the randomization themselves. For EEFX — whose direct verification is not known to be polynomial-time — the algorithm outputs per-agent certificates: partitions witnessing that the agent's bundle EFX-dominates the alternatives, checkable in polynomial time. Share-based guarantees are verifiable using standard FPTAS approximations of the MMS value. This contrasts with earlier BoBW constructions whose support grows with the number of items.

Limitations and open questions

The paper is candid about its boundaries. The main existential result is not polynomial-time without MMS oracles, and the polynomial-time variant sacrifices EEFX to recover exact ex-ante proportionality. The $\tfrac{11}{12}$4-MMS analysis is tight for the proposed construction, leaving open whether a better fraction (up to the $\tfrac{11}{12}$5 upper bound, or the $\tfrac{11}{12}$6 existential benchmark) is achievable alongside EEFX and ex-ante proportionality. The ex-ante guarantee is proportionality, not envy-freeness; strengthening it while retaining near-optimal ex-post guarantees remains open, as does achieving BoBW with exact EFX rather than EEFX for three agents. Extending the framework beyond three agents is identified as the principal challenge, where even EFX existence is unresolved; the authors suggest EEFX-based BoBW as the realistic target, which would imply at least $\tfrac{11}{12}$7-MMS ex-post. Whether IMMX allocations exist for $\tfrac{11}{12}$8 additive agents is likewise open.

Conclusion

This paper closes much of the gap between existing BoBW guarantees and the natural fairness benchmarks for few agents. For three agents it proves the existence of ex-ante proportional distributions supported entirely on EEFX, $\tfrac{11}{12}$9-MMS, IMMX allocations — the first BoBW result combining a strong envy-based ex-post guarantee with a near-optimal share-based one for more than two agents — and complements this with FPTAS algorithms achieving exact ex-ante proportionality and $(1-\varepsilon)$0-MMS in polynomial time. Its two-agent FPTAS attains the strongest feasible combination of ex-ante envy-freeness, ex-post EFX, and $(1-\varepsilon)$1-MMS, resolving the two-agent case essentially optimally. The techniques — exploiting the invariance of epistemic fairness under repartitioning, together with adoption mechanisms for approximate partitions — provide a template that may extend to larger agent populations, though the paper leaves the central questions of EFX-based BoBW and guarantees beyond three agents open.