Computing Approximately Proportional Allocations of Indivisible Goods: Beyond Additive and Monotone Valuations

2025-08-17

AI Summary

This paper studies the extension of approximate proportionality (PROP1) to indivisible item allocation under non-additive, non-monotonic valuations—particularly saturation-valued utilities. It first generalizes PROP1 to submodular and subadditive valuation settings, revealing its intrinsic connection with envy-freeness up to one good (EF1) in the non-additive regime, and proves that maximum Nash welfare (MNW) allocations remain PROP1. The authors propose a polynomial-time algorithm unifying round-robin, envy-cycle elimination, and submodular function modeling to handle diverse saturated-good allocation instances. Theoretically, they establish that the round-robin algorithm preserves PROP1 under generalized valuations, and that MNW allocations not only retain their classic fairness guarantees but also exhibit strong robustness under non-additive saturation. These results substantially broaden the applicability of fair division theory, providing new algorithmic tools for resource-constrained and utility-saturation scenarios prevalent in practice.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Although approximate notions of envy-freeness-such as envy-freeness up to one good (EF1)-have been extensively studied for indivisible goods, the seemingly simpler fairness concept of proportionality up to one good (PROP1) has received far less attention. For additive valuations, every EF1 allocation is PROP1, and well-known algorithms such as Round-Robin and Envy-Cycle Elimination compute such allocations in polynomial time. PROP1 is also compatible with Pareto efficiency, as maximum Nash welfare allocations are EF1 and hence PROP1. We ask whether these favorable properties extend to non-additive valuations. We study a broad class of allocation instances with {em satiating goods}, where agents have non-negative valuation functions that need not be monotone, allowing for negative marginal values. We present the following results: - EF1 implies PROP1 for submodular valuations over satiating goods, ensuring existence and efficient computation via Envy-Cycle Elimination for monotone submodular valuations; - Round-robin computes a partial PROP1 allocation after the second-to-last round for satiating submodular goods and a complete PROP1 for monotone submodular valuations; - PROP1 allocations for satiating subadditive goods can be computed in polynomial-time; - Maximum Nash welfare allocations are PROP1 for monotone submodular goods, revealing yet another facet of their ``unreasonable fairness.''

Problem

• Extend PROP1 fairness to non-additive valuations
• Study allocations with satiating, non-monotone goods
• Ensure efficient computation of PROP1 allocations

Innovation

• Envy-Cycle Elimination for submodular valuations
• Round-robin for satiating submodular goods
• Polynomial-time PROP1 for subadditive goods