On MMS, APS and XOS

Abstract: We consider allocations of a set of $m$ indivisible goods to $n$ agents of equal entitlements that have valuations from the class XOS. A previous sequence of works showed allocations that obtain an $α$-approximation for the maximin share (MMS), for values of $α$ that gradually approach $\frac{1}{4}$ from below (the currently known ratio is $\frac{4}{17}$). In this work we attempt to obtain ratios better than $\frac{1}{4}$, and manage to do so for sufficiently large $n$. Our methodology is to first investigate the gap between the anyprice share (APS) and the MMS when all agents have the same XOS valuations, for which we design an allocation algorithm and prove that each agent receives at least $α> \frac{11}{40}$ times the APS. Then, we derive inspiration from this algorithm, and modify it so that it applies also when agents have different XOS valuations. Using this modified version, we show that for some sufficiently large $n_0$, there is an $α$-MMS allocation (in fact, an $α$-APS allocation) for every $n \geq n_0$.

• The paper introduces a novel potential-based and randomized method that surpasses the previous 1/4 MMS guarantee for XOS valuations.
• It employs a greedy allocation combined with martingale concentration techniques to ensure fairness even with non-identical valuations.
• The limiting MMS/APS ratio converges to approximately 0.2768 as the number of agents grows, improving theoretical guarantees for large markets.

Improved Approximate MMS and APS Allocations for XOS Valuations

Problem Statement and Fairness Frameworks

The paper considers the allocation of $m$ indivisible goods among $n$ agents with equal entitlements, where each agent’s valuation is XOS (fractionally subadditive). The core fairness concept is the Maximin Share (MMS) guarantee: for any agent $i$, MMS is the maximum value she can guarantee herself if she divides the goods into $n$ bundles and receives the least-valued bundle. Since MMS allocations (where every agent attains at least her MMS) may not always exist, especially for general, non-additive valuations, attention turns to $\alpha$-MMS allocations: allocations where each agent attains at least $\alpha$ times her MMS value.

A related, weaker relaxation is the Anyprice Share (APS), which leverages fractional allocations or price-based constraints; the APS is always at least the MMS. Existence of $\alpha$-APS allocations thus implies existence of $\alpha$-MMS allocations for the same $\alpha$.

Summary of Prior State-of-the-Art

Prior to this work, for XOS agents, the leading existential guarantees were:

• $\frac{1}{4}$-MMS allocations provably exist for all $n$0
• This ratio was matched for APS allocations
• The main proof technique is the sequential allocation approach, giving away bundles of high value items, then recursively handling the remainder

Substantially improving the approximation ratio beyond $n$1 has been elusive. Moreover, current techniques for both MMS and APS share similar limitations, appearing to hit a natural barrier at $n$2.

Methodological Contributions

This work develops a distinct algorithmic and analytical framework, advancing both existential results and algorithmic understanding for large $n$3:

1. Identical XOS Valuations—APS/MMS Gap Analysis: The authors first tackle the special case where all agents have the identical XOS valuation. They show, via a greedy allocation algorithm coupled with a carefully defined “potential function” (as a weighted sum over the APS fractional partition), that the minimal guaranteed approximation (minimum-over-agent value over all steps) can be pushed strictly above $n$4 as $n$5. The analysis relies on tracing how this potential function decreases with each allocation, upper bounding the per-step item reallocation “probabilities” via combinatorial arguments and LP duality.
2. General (Non-Identical) Valuations for Large $n$6:

The main technical challenge is that APS fractional partitions differ for each agent, and thus removing an item for one agent can drastically reduce the potential for others. The authors circumvent this by: - Developing a randomized allocation protocol, analyzing martingale concentration over the per-agent potential functions along the allocation process. - Handling problematic cases via a “stealing” mechanism: if a bundle allocated to one agent causes substantial value loss to another, the latter can “steal” a subset, ensuring the total devaluation is tightly controlled. - Addressing the presence of large items (with value at least $n$7). The combinatorics of large items are handled via structural lemmas on matchings in bipartite graphs, partitioning agents and items so that large items can be allocated directly, and reducing the remainder to the small item setting.

1. Analytic Approach and Concentration: For large $n$8, the stepwise analysis of potential drop can be approximated by a differential equation, yielding a limiting value for the attainable ratio $n$9. The authors show that as $i$0 increases, the limiting $i$1 approaches approximately $i$2, governed by an explicit transcendental equation.
2. Algorithmic Implementation: The allocation algorithms are polynomial-time assuming the availability of value and "anyprice" oracles (the latter arising from the alternative, price-based APS definition and implementable via the ellipsoid method), despite the underlying problem being APX-hard via value queries alone.

Main Theorems and Performance Guarantees

The central formal claims are:

• Existence of $i$3-MMS and $i$4-APS Allocations for All Large $i$5:

There exists $i$6 and $i$7 such that for $i$8, there is an allocation where every agent gets at least $i$9 times her MMS (resp. APS). The value $n$0 improves with $n$1, and converges to a precise analytic value satisfying

$n$2

which numerically yields $n$3.

The proofs show that, as $n$4, the potential function can be kept above $n$5 through the entire allocation process, for both the identical and nonidentical valuation cases.

• Tight Multiplicative Gap Between APS and MMS for XOS:

For all $n$6 and XOS $n$7, $n$8, with the above $n$9.

• Algorithmic Properties:

The algorithms are randomized and, given access to value and anyprice oracles, run in expected polynomial time. The method is constructive except for the inherent query complexity obstacles dictated by the hardness of computing MMS/APS for XOS via value queries alone.

Numerical Implications, Bounds, and Contrasts

• The improvement over $\alpha$0 is explicit: $\alpha$1, and for massive $\alpha$2 can reach approximately $\alpha$3.
• Achieving $\alpha$4 requires $\alpha$5, situating these results firmly in the large-market regime.
• The previous proof techniques are fundamentally unable to beat the $\alpha$6-MMS/APX bound, whereas the new analysis leverages potential-based recursion and stochastic process concentration, generalizing greedy principles to the fractional (APS) context in ways not accessible before.

Theoretical and Practical Implications

• Theoretical Implications:

This result provides a new upper bound on the best-possible fraction of MMS or APS achievable under general XOS valuations as the number of agents grows. It also underscores the subtle gap between analyses based solely on partitioning versus those exploiting refined probabilistic and potential-based arguments in the space of subadditive combinatorial valuations.

The implications extend to a tight understanding of the approximation landscape: while matching lower bounds for XOS do not exist beyond the $\alpha$7 barrier, this work demonstrates that there is provable slack above $\alpha$8.

• Algorithmic Significance:

The approach introduces a martingale-based method for allocation under independence constraints, with high-probability guarantees rather than worst-case bounds, an advancement compared to deterministic methods. The "stealing" mechanism for bundles ensures robust value preservation for all agents in the face of highly diverse and overlapping valuations.

• Practical Aspects:

The algorithms are implementable in large-scale settings where anyprice queries are accessible (e.g., when price-based oracle access is realizable), and thus have implications for automated resource division in digital and economic multi-agent systems where XOS preferences arise.

• Wider Applicability and Future Research:

The methods and structural results for the handling of large items and matching-based allocation partitions could find applications in other combinatorial allocation settings. The martingale concentration toolkit developed may inspire new existence proofs in fair division and beyond, particularly for other valuation hierarchies and extended notions of fairness.

Conclusion

This paper surpasses previous existential and constructive guarantees for fair allocation of indivisible goods to XOS agents, specifically for the large-agent regime. By developing new potential-based and stochastic analytical tools, it establishes a strictly higher approximation threshold than was previously thought attainable for MMS and APS allocations. These results reveal both the limitations of classical greedy/partitioning approaches and the added value of probabilistic process analysis in the context of combinatorial fair division. Future directions include strengthening the bounds for moderate $\alpha$9, extending the techniques to broader valuation classes, and further bridging the gap between existential, algorithmic, and hardness results for fair division.