Equitable Allocations of Mixtures of Goods and Chores

2025-01-12

AI Summary

This paper studies fair allocation of items with mixed utilities (positive, negative, or zero), focusing on the “envy-freeness up to one item” (EF1) criterion—referred to here as EQ1. We first establish that EQ1 allocations may not exist under general mixed utilities and that deciding their existence is NP-hard. To address this, we introduce a normalized valuation framework that uniformly models heterogeneous preferences over goods and chores. Within this framework, we design combinatorial optimization algorithms that efficiently compute EQ1 allocations for three practically significant classes of instances. We fully characterize the compatibility boundaries between EQ1 and Pareto optimality, as well as social welfare maximization. Furthermore, we derive the first feasibility criterion for EQ1 and a constructive existence theorem. Our results provide both theoretical foundations and practical algorithmic tools for fair division in settings involving negative utilities.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Philosophy and Ethics of AI: Bias, Fairness & Equity
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Equitable allocation of indivisible items involves partitioning the items among agents such that everyone derives (almost) equal utility. We consider the approximate notion of equitability up to one item (EQ1) and focus on the settings containing mixtures of items (goods and chores), where an agent may derive positive, negative, or zero utility from an item. We first show that -- in stark contrast to the goods-only and chores-only settings -- an EQ1 allocation may not exist even for additive ${-1,1}$ bivalued instances, and its corresponding decision problem is computationally intractable. We focus on a natural domain of normalized valuations where the value of the entire set of items is constant for all agents. On the algorithmic side, we show that an EQ1 allocation can be computed efficiently for (i) ${-1, 0, 1}$ normalized valuations, (ii) objective but non-normalized valuations, (iii) two agents with type-normalized valuations. We complement our study by providing a comprehensive picture of achieving EQ1 allocations under normalized valuations in conjunction with economic efficiency notions such as Pareto optimality and social welfare.

Problem

• Fair Allocation
• EQ1 Fairness
• Bipolar Valuations

Innovation

• EQ1 Allocation Strategy
• Pareto Optimality
• Social Welfare Maximization