Impartial Selection and the Power of Up to Two Choices

We study mechanisms that select members of a set of agents based on nominations by other members and that are impartial in the sense that agents cannot influence their own chance of selection. Prior work has shown that deterministic mechanisms for selecting any fixed number k of agents are severely limited and cannot extract a constant fraction of the nominations of the k most highly nominated agents. We prove here that this impossibility result can be circumvented by allowing the mechanism to sometimes but not always select fewer than k agents. This added flexibility also improves the performance of randomized mechanisms, for which we show a separation between mechanisms that make exactly two or up to two choices and give upper and lower bounds for mechanisms allowed more than two choices.