On Best-of-Both-Worlds Fairness via Sum-of-Variances Minimization

Abstract

We consider the problem of fairly allocating a set of indivisible goods among agents with additive valuations. Ex-ante fairness (proportionality) can trivially be obtained by giving all goods to a random agent. Yet, such an allocation is very unfair ex-post. This has motivated the Best-of-Both-Worlds (BoBW) approach, seeking a randomized allocation that is ex-ante proportional and is supported only on ex-post fair allocations (e.g., on allocations that are envy-free-up-to-one-good (EF1), or give some constant fraction of the maximin share (MMS)). It is commonly pointed out that the distribution that allocates all goods to one agent at random fails to be ex-post fair as it ignores the variances of the values of the agents. We examine the approach of trying to mitigate this problem by minimizing the sum-of-variances of the values of the agents, subject to ex-ante proportionality. We study the ex-post fairness properties of the resulting distributions. In support of this approach, observe that such an optimization will indeed deterministically output a proportional allocation if such exists. We show that when valuations are identical, this approach indeed guarantees fairness ex-post: all allocations in the support are envy-free-up-to-any-good (EFX), and thus guarantee every agent at least 4/7 of her maximin share (but not her full MMS). On the negative side, we show that this approach completely fails when valuations are not identical: even in the simplest setting of only two agents and two goods, when the additive valuations are not identical, there is positive probability of allocating both goods to the same agent. Thus, the supporting ex-post allocation might not even be EF1, and might not give an agent any constant fraction of her MMS. Finally, we present similar negative results for other natural minimization objectives that are based on variances.