We study social choice mechanisms in an implicit utilitarian framework with a metric constraint, where the goal is to minimize \textit{Distortion}, the worst case social cost of an ordinal mechanism relative to underlying cardinal utilities. We consider two additional desiderata: Constant sample complexity and Squared Distortion. Constant sample complexity means that the mechanism (potentially randomized) only uses a constant number of ordinal queries regardless of the number of voters and alternatives. Squared Distortion is a measure of variance of the Distortion of a randomized mechanism. Our primary contribution is the first social choice mechanism with constant sample complexity \textit{and} constant Squared Distortion (which also implies constant Distortion). We call the mechanism Random Referee, because it uses a random agent to compare two alternatives that are the favorites of two other random agents. We prove that the use of a comparison query is necessary: no mechanism that only elicits the top-k preferred alternatives of voters (for constant k) can have Squared Distortion that is sublinear in the number of alternatives. We also prove that unlike any top-k only mechanism, the Distortion of Random Referee meaningfully improves on benign metric spaces, using the Euclidean plane as a canonical example. Finally, among top-1 only mechanisms, we introduce Random Oligarchy. The mechanism asks just 3 queries and is essentially optimal among the class of such mechanisms with respect to Distortion. In summary, we demonstrate the surprising power of constant sample complexity mechanisms generally, and just three random voters in particular, to provide some of the best known results in the implicit utilitarian framework.

我们研究社会选择机制中的失真问题，提出了具有恒定样本复杂度和恒定失真度的社会选择机制Random Referee。我们证明了这种机制使用比较查询是必要的，通过正态指标证明了这种机制的失真最小，并介绍了一种只需要3个查询的机制Random Oligarchy。

带随机裁判的随机独裁者: 社会选择常数样本复杂度机制