Fair Division of Lotteries

We study fair division of uncertain items, where the utility of each item is a random variable whose realization is observed only after the allocation is determined. This uncertainty creates a distinction between ex-ante fairness, evaluated before utilities are realized, and ex-post fairness, evaluated after realization. We examine to what extent these two notions are compatible. We first identify conditions under which ex-ante envy-free allocations maximize ex-post envy-freeness probability. These include two-agent instances where utilities toward the same item follow the same distribution from a convolution-power family. For an arbitrary number of agents, we establish analogous optimality results when the common generator is non-atomic or log-concave. Beyond convolution-power families, we derive approximation guarantees for ex-ante envy-free allocations relative to the optimal ex-post envy-freeness probability under log-concave or symmetric utilities. We also study the computational complexity of evaluating ex-post envy-freeness probabilities, establishing hardness results as well as pseudo-polynomial-time algorithms.