Best of Both Worlds Guarantees for Equitable Allocations

TL;DR

This paper investigates the existence and computation of randomized fair allocations that are equitable in expectation and satisfy a fairness condition in every deterministic outcome, providing comprehensive theoretical and algorithmic results.

Contribution

It characterizes the conditions for existence of 'Best of Both Worlds' equitable allocations and offers algorithms for their computation, especially for two agents and binary valuations.

Findings

Ex ante EQ and ex post EQ1 allocations always exist for two agents.

Deciding existence of such allocations is NP-complete for three or more agents.

Efficient algorithms exist for binary valuations and a fixed number of agents.

Abstract

Equitability is a well-studied fairness notion in fair division, where an allocation is equitable if all agents receive equal utility from their allocation. For indivisible items, an exactly equitable allocation may not exist, and a natural relaxation is EQ1, which stipulates that any inequitability should be resolved by the removal of a single item. In this paper, we study equitability in the context of randomized allocations. Specifically, we aim to achieve equitability in expectation (ex ante EQ) and require that each deterministic outcome in the support satisfies ex post EQ1. Such an allocation is commonly known as a `Best of Both Worlds' allocation, and has been studied, e.g., for envy-freeness and MMS. We characterize the existence of such allocations using a geometric condition on linear combinations of EQ1 allocations, and use this to give comprehensive results on both existence and computation. For two agents, we show that ex ante EQ and ex post EQ1 allocations always exist and can be computed in polynomial time. For three or more agents, however, such allocations may not exist. We prove that deciding existence of such allocations is strongly NP-complete in general, and weakly NP-complete even for three agents. We also present a pseudo-polynomial time algorithm for a constant number of agents. We show that when agents have binary valuations, best of both worlds allocations that additionally satisfy welfare guarantees exist and are efficiently computable.