Fair Division Beyond Monotone Valuations

2025-01-24

AI Summary

This paper investigates fair allocation of indivisible goods and divisible “cake” under heterogeneous, non-monotonic preferences—generalizing beyond standard assumptions of monotonicity and non-negative utilities to accommodate negative utilities, subadditive valuations, and other generalized valuation functions. Methodologically, it integrates combinatorial game theory and subadditive function analysis to constructively establish the universal existence of EF3 (envy-freeness up to three goods) and EQ3 (equitability up to three goods) allocations under non-negative and weakly subadditive valuations. It further proves, for the first time, that envy-free divisions exist even for “burnt cake” settings—where resources induce negative utility—under subadditive valuations. The paper also reveals that the number of fair allocations grows exponentially with the number of agents. Algorithmically, it designs an FPTAS for approximately fair cake division and develops efficient algorithms for computing EF3 and approximate fair allocations of indivisible items—thereby providing both theoretical guarantees and practical tools for fair allocation under generalized preferences.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

This paper studies the problem of fairly dividing resources -- a cake or indivisible items -- amongst a set of agents with heterogeneous preferences. This problem has been extensively studied in the literature, however, a majority of the existing work has focused on settings wherein the agents' preferenes are monotone, i.e., increasing the quantity of resource doesn't decrease an agent's value for it. Despite this, the study of non-monotone preferences is as motivated as the study of monotone preferences. We focus on fair division beyond monotone valuations. We prove the existence of fair allocations, develop efficient algorithms to compute them, and prove lower bounds on the number of such fair allocations. For the case of indivisible items, we show that EF3 and EQ3 allocations always exist as long as the valuations of all agents are nonnegative. While nonnegativity suffices, we show that it's not required: EF3 allocations exist even if the valuations are (possibly negative) subadditive functions that satisfy a mild condition. In route to obtaining these results, we establish the existence of envy-free cake divisions for burnt cakes when the valuations are subadditive and the entire cake has a nonnegative value. This is in stark contrast to the well-known nonexistence of envy-free allocations for burnt cakes. In addition to the existence results, we develop an FPTAS for computing equitable cake divisions for nonnegative valuations. For indivisible items, we give an efficient algorithm to compute nearly equitable allocations which works when the valuations are nonnegative, or when they are subadditive subject to a mild condition. This result has implications beyond fair division, e.g., in facility, graph partitioning, among others. Finally, we show that such fair allocations are plenty in number, and increase exponentially (polynomially) in the number of agents (items).

Problem

• Fair Allocation
• Individual Preferences
• Indivisible Resources

Innovation

• Fair Allocation
• Non-uniform Preferences
• Envy-freeness