Best of Both Worlds: Ex-Ante and Ex-Post Fairness in Resource Allocation

We study the problem of allocating indivisible goods among agents with additive valuations. When randomization is allowed, it is possible to achieve compelling notions of fairness such as envy-freeness, which states that no agent should prefer any other agent's allocation to her own. When allocations must be deterministic, achieving exact fairness is impossible but approximate notions such as envy-freeness up to one good can be guaranteed. Our goal in this work is to achieve both simultaneously, by constructing a randomized allocation that is exactly fair ex-ante and approximately fair ex-post. The key question we address is whether ex-ante envy-freeness can be achieved in combination with ex-post envy-freeness up to one good. We settle this positively by designing an efficient algorithm that achieves both properties simultaneously. If we additionally require economic efficiency, we obtain an impossibility result. However, we show that economic efficiency and ex-ante envy-freeness can be simultaneously achieved if we slightly relax our ex-post fairness guarantee. On our way, we characterize the well-known Maximum Nash Welfare allocation rule in terms of a recently introduced fairness guarantee that applies to groups of agents, not just individuals.