Best-of-Both-Worlds Fair Allocation of Indivisible and Mixed Goods

TL;DR

This paper develops algorithms for fair allocation of indivisible and mixed goods, ensuring fairness both before and after allocation, with guarantees like envy-freeness and EFX/EFM for different agent scenarios.

Contribution

It introduces polynomial-time randomized algorithms that achieve combined ex-ante and ex-post fairness guarantees for various agent and utility settings.

Findings

For two agents, polynomial-time algorithms achieve ex-ante envy-freeness and ex-post EFX/EFM.

For n agents with bi-valued utilities, allocations are ex-ante proportional and ex-post EFM.

Allocations are also ex-ante envy-free, ex-post EFX, and fractionally Pareto optimal.

Abstract

We study the problem of fairly allocating either a set of indivisible goods or a set of mixed divisible and indivisible goods (i.e., mixed goods) to agents with additive utilities, taking the best-of-both-worlds perspective of guaranteeing fairness properties both ex ante and ex post. The ex-post fairness notions considered in this paper are relaxations of envy-freeness, specifically, EFX for indivisible-goods allocation, and EFM for mixed-goods allocation. For two agents, we show that there is a polynomial-time randomized algorithm that achieves ex-ante envy-freeness and ex-post EFX / EFM simultaneously. For $n$ agents with bi-valued utilities, we show there exist randomized allocations that are (i) ex-ante proportional and ex-post EFM, and (ii) ex-ante envy-free, ex-post EFX, and ex-post fractionally Pareto optimal.