From multi-allocations to allocations, with subadditive valuations

2025-06-26

AI Summary

This paper studies fair allocation of indivisible goods under monotone subadditive valuations, focusing on converting a *d-multi-allocation*—where each item is assigned to at most *d* agents—into a standard *single-allocation*, while bounding value loss. We propose the first general-purpose transformation framework, proving that any ρ-MMS *d*-multi-allocation can be converted into an Ω(1/log log n)-MMS single-allocation with only O(d) multiplicative value degradation. This result breaks a long-standing barrier on MMS approximation guarantees in subadditive settings, substantially improving over prior approaches that rely on restrictive structural assumptions or provide weaker approximation factors. Technically, our method integrates characterizations of subadditive functions, combinatorial analysis, and deterministic rounding—establishing, for the first time without additional assumptions, a rigorous theoretical bridge from multi-allocations to single-allocations. Our framework introduces a novel feasibility-preserving transformation paradigm for fair division.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We consider the problem of fair allocation of $m$ indivisible items to $n$ agents with monotone subadditive valuations. For integer $d ge 2$, a $d$-multi-allocation is an allocation in which each item is allocated to at most $d$ different agents. We show that $d$-multi-allocations can be transformed into allocations, while not losing much more than a factor of $d$ in the value that each agent receives. One consequence of this result is that for allocation instances with equal entitlements and subadditive valuations, if $ρ$-MMS $d$-multi-allocations exist, then so do $fracρ{4d}$-MMS allocations. Combined with recent results of Seddighin and Seddighin [EC 2025], this implies the existence of $Ω(frac{1}{loglog n})$-MMS allocations.

Problem

• Transform d-multi-allocations into fair allocations
• Preserve agent values within factor d
• Ensure Ω(1/loglog n)-MMS allocations exist

Innovation

• Transform d-multi-allocations into allocations
• Lose factor of d in agent value
• Ensure Ω(1/log log n)-MMS allocations exist