The Distortion of Prior-Independent b-Matching Mechanisms

TL;DR

This work studies prior-independent ordinal mechanisms for b-matching where agents have quotas and reveal only item rankings. It proves a fundamental $e/(e-1)$ distortion lower bound for one-to-one matching and designs the Random Survivors (RS) mechanism that matches this bound for all quotas and agent-specific, unknown UF distributions; it further reduces the per-instance distortion gap with the RSBS mechanism, while analyzing sequential and secretary variants (notably HQL) and establishing optimal guarantees within these restricted classes. The results show that simple, information-limited mechanisms can achieve near-optimal social welfare under diverse distributions, with Bayesian incentive compatibility preserved and extensibility to submodular valuations. The work advances understanding of distortion and distortion gaps in prior-independent settings and highlights practical mechanisms for fair, efficient resource allocation. Overall, the paper provides tight Bounds, practical mechanisms (RS, RSBS, HQL), and a rich set of directions for future research in prior-agnostic allocation problems.

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.

Figures (1)

• Figure 1: A plot of the bound on the distortion gap of algorithm RSBS at the RHS of Equation (\ref{['eq:distortion-gap-bound']}) as a function of the ratio $b_{max}/m$.

Theorems & Definitions (52)

• Theorem 1
• proof
• Lemma 2
• proof
• Claim 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof