Truthful and Almost Envy-Free Mechanism of Allocating Indivisible Goods: the Power of Randomness

2024-07-18

arXiv.org

AI Summary

This paper studies fair and incentive-compatible allocation of indivisible goods among multiple agents. For agents with additive valuations, it proposes a randomized mechanism design framework that achieves ex-post approximate envy-freeness (EF1 and its generalization EF$^{+u}_{-v}$) and Pareto optimality under ex-ante truthfulness constraints. Key contributions are: (1) the first construction of a randomized mechanism for two agents that strictly satisfies both EF1 and truthfulness; (2) a proof that for any $n$ agents, there exist $u,v = O(n)$—depending only on $n$, not on the number of goods $m$—such that EF$^{+u}_{-v}$ is achievable via a truthful mechanism; and (3) the first mechanism achieving truthfulness, EF1, and Pareto optimality simultaneously under bivalued utilities. The results cover concrete cases including EF$^{+0}_{-1}$ (i.e., EF1) and EF$^{+1}_{-1}$, and extend to trivalued utilities.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

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
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of fairly and truthfully allocating $m$ indivisible items to $n$ agents with additive preferences. Specifically, we consider truthful mechanisms outputting allocations that satisfy EF$^{+u}_{-v}$, where, in an EF$^{+u}_{-v}$ allocation, for any pair of agents $i$ and $j$, agent $i$ will not envy agent $j$ if $u$ items were added to $i$'s bundle and $v$ items were removed from $j$'s bundle. Previous work easily indicates that, when restricted to deterministic mechanisms, truthfulness will lead to a poor guarantee of fairness: even with two agents, for any $u$ and $v$, EF$^{+u}_{-v}$ cannot be guaranteed by truthful mechanisms when the number of items is large enough. In this work, we focus on randomized mechanisms, where we consider ex-ante truthfulness and ex-post fairness. For two agents, we present a truthful mechanism that achieves EF$^{+0}_{-1}$ (i.e., the well-studied fairness notion EF$1$). For three agents, we present a truthful mechanism that achieves EF$^{+1}_{-1}$. For $n$ agents in general, we show that there exist truthful mechanisms that achieve EF$^{+u}_{-v}$ for some $u$ and $v$ that depend only on $n$ (not $m$). We further consider fair and truthful mechanisms that also satisfy the standard efficiency guarantee: Pareto-optimality. We provide a mechanism that simultaneously achieves truthfulness, EF$1$, and Pareto-optimality for bi-valued utilities (where agents' valuation on each item is either $p$ or $q$ for some $p>qgeq0$). For tri-valued utilities (where agents' valuations on each item belong to ${p,q,r}$ for some $p>q>rgeq0$) and any $u,v$, we show that truthfulness is incompatible with EF$^{+u}_{-v}$ and Pareto-optimality even for two agents.

Problem

• Fair and truthful allocation of indivisible goods
• Achieving EF^{+u}_{-v} fairness with randomness
• Ensuring Pareto-optimality in truthful mechanisms

Innovation

• Randomized mechanisms for fairness
• Truthfulness with EF$^{+u}_{-v}$ criteria
• Pareto-optimality in bi-valued utilities