Impartial Selection with Additive Guarantees via Iterated Deletion

Abstract

Impartial selection is the selection of an individual from a group based on nominations by other members of the group, in such a way that individuals cannot influence their own chance of selection. For this problem, we give a deterministic mechanism with an additive performance guarantee of $O(n^{(1+\hat{I}º)/2})$ in a setting with $n$ individuals where each individual casts $O(n^{\hat{I}}º)$ nominations, where $\hat{I}º\in [0,1]$. For $\hat{I}º=0$, i.e., when each individual casts at most a constant number of nominations, this bound is $O(\sqrt{n})$. This matches the best-known guarantee for randomized mechanisms and a single nomination. For $\hat{I}º=1$ the bound is $O(n)$. This is trivial, as even a mechanism that never selects provides an additive guarantee of $n-1$. We show, however, that it is also best possible: for every deterministic impartial mechanism there exists a situation in which some individual is nominated by every other individual and the mechanism either does not select or selects an individual not nominated by anyone.