Stable Voting Rules on the Edge of Optimal Metric Distortion

We prove the existence of a randomized voting rule with metric distortion at most $2.13713$, within $0.025$ of the lower bound of $2.11264$. Our rule comes from a generalization of stable $k$-lotteries developed in the context of committee selection. In contrast to prior work, our rule samples from a single distribution derived from a zero-sum game, without mixing between voting rules. Our result also gives sharp distortion bounds for stable $k$-lotteries, and in particular shows that stable $2$-lotteries have distortion $7\mathrm{/}3$, despite only relying on aggregate preferences over triples of candidates.