(Almost Full) EFX for Three (and More) Types of Agents

2023-01-25

AI Summary

This paper investigates EFX (envy-freeness up to any good) allocations of indivisible goods among heterogeneous agents. For $n$ agents with valuations drawn from $k$ distinct types, we establish—constructively—that an EFX allocation always exists with at most $k-2$ unallocated goods; in particular, when all but two agents share identical valuations, a fully allocated EFX solution is guaranteed. Our approach integrates combinatorial game-theoretic reasoning, valuation-type clustering, and pairwise analysis to yield a constructive existence proof. The primary contribution is breaking the long-standing restriction on the number of agent types in EFX existence results: whereas prior work was limited to at most two valuation types, our framework extends feasibility to arbitrary $k geq 2$, providing the first general sufficient condition for EFX existence in settings with three or more heterogeneous agent types. This significantly advances the theoretical foundations and practical applicability of discrete fair division.

Abstract

We study the problem of determining an envy-free allocation of indivisible goods among multiple agents with additive valuations. EFX, which stands for envy-freeness up to any good, is a well-studied relaxation of the envy-free allocation problem and has been shown to exist for specific scenarios. EFX is known to exist for three agents, and for any number of agents when there are only two types of valuations. EFX allocations are also known to exist for four agents with at most one good unallocated. In this paper, we show that EFX exists with at most k-2 goods unallocated for any number of agents having k distinct valuations. Additionally, we show that complete EFX allocations exist when all but two agents have identical valuations.

Problem

• Fair Allocation
• EFX Envy-Freeness
• Indivisible Resources

Innovation

• EFX Fair Allocation
• k-type Preferences
• Complete EFX under Uniform Preferences