We study metric distortion in one-sided matching. Given $n$ agents and $n$ items in an unknown metric space, our goal is to match agents to items so that the total distance between matched pairs is small. However, instead of direct access to the distances in the metric, we are only given each agent's ranking of the items by distance. What is the smallest worst-case approximation factor (called the distortion) that a mechanism can guarantee in expectation? Despite significant interest over the last decade, the best-known guarantee has been the $O(n)$ distortion of Random Serial Dictatorship since the beginning. We give a matching mechanism with distortion $O(n^{\log_4 3})$, the first sublinear upper bound. The key idea is to use a partial matching derived from a truncated version of either Random Serial Dictatorship or the Probabilistic Serial rule. We then complete this matching using a new, general-purpose cluster completion algorithm that ensures unmatched agents receive nearby items. Along the way, we pin down the optimal distortion of Random Serial Dictatorship and Probabilistic Serial up to logarithmic and constant factors, respectively, and give improved lower bounds for truthful mechanisms.
We prove the existence of a randomized voting rule with metric distortion at most $2.1248$, within $0.013$ of the lower bound of $2.11264$. Our rule comes from a generalization of \emph{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. In particular we show that stable $2$-lotteries have distortion $7/3$, despite only relying on aggregate preferences over triples of candidates, and that stable $2$-lotteries are optimal in this class of voting rules.
Sepideh Mahabadi, Prasanna Ramakrishnan, Jakub Tarnawski
We consider the problem of allocating indivisible goods to agents with additive valuations, under two classic notions of fairness and efficiency: envy-freeness up to any good (EFX) and Pareto optimality. While some of the earliest work on EFX showed that Pareto optimal EFX allocations may not exist if agents can have zero value for some goods, it has remained open whether the same is true when all values are positive. We present a simple instance in which every good has positive value to every agent, but no Pareto optimal EFX allocation exists. We then generalize to a family of instances showing that Pareto optimal $α$-EFX allocations may not exist for any $α> 2/3$.