The Distortion of Prior-Independent b-Matching Mechanisms

TL;DR

This paper analyzes the performance limits of ordinal mechanisms in m-item, n-agent matching problems under stochastic preferences, establishing a lower bound and proposing mechanisms that achieve near-optimal distortion and distortion gap.

Contribution

It introduces the first tight bounds on the distortion of ordinal mechanisms under stochastic preferences and proposes mechanisms that attain these bounds.

Findings

No ordinal mechanism can beat a distortion of approximately 1.582.

Proposed mechanisms achieve the optimal distortion of e/(e-1).

A mechanism with a near-optimal distortion gap of 1.076 is developed.

Abstract

In a setting where $m$ items need to be partitioned among $n$ agents, we evaluate the performance of mechanisms that take as input each agent's \emph{ordinal preferences}, i.e., their ranking of the items from most- to least-preferred. The standard measure for evaluating ordinal mechanisms is the \emph{distortion}, and the vast majority of the literature on distortion has focused on worst-case analysis, leading to some overly pessimistic results. We instead evaluate the distortion of mechanisms with respect to their expected performance when the agents' preferences are generated stochastically. We first show that no ordinal mechanism can achieve a distortion better than $e/(e-1)\approx 1.582$, even if each agent needs to receive exactly one item (i.e., $m=n$) and every agent's values for different items are drawn i.i.d.\ from the same known distribution. We then complement this negative result by proposing an ordinal mechanism that achieves the optimal distortion of $e/(e-1)$ even if each agent's values are drawn from an agent-specific distribution that is unknown to the mechanism. To further refine our analysis, we also optimize the \emph{distortion gap}, i.e., the extent to which an ordinal mechanism approximates the optimal distortion possible for the instance at hand, and we propose a mechanism with a near-optimal distortion gap of $1.076$. Finally, we also evaluate the distortion and distortion gap of simple mechanisms that have a one-pass structure.