Efficiency of Truthful and Symmetric Mechanisms in One-sided Matching

TL;DR

This paper analyzes the efficiency of truthful, symmetric mechanisms in one-sided matching problems with different preference models, providing bounds on social welfare and connecting to online bipartite matching.

Contribution

It introduces new bounds for social welfare in truthful, symmetric mechanisms and links one-sided matching to online bipartite matching for mechanism design.

Findings

Truthful, symmetric, optimal mechanisms exist with intractable mechanisms.

Random Serial Dictatorship guarantees at least one-third of maximum social welfare.

Mechanisms can achieve 0.69 of optimal welfare under certain assumptions.

Abstract

We study the efficiency (in terms of social welfare) of truthful and symmetric mechanisms in one-sided matching problems with {\em dichotomous preferences} and {\em normalized von Neumann-Morgenstern preferences}. We are particularly interested in the well-known {\em Random Serial Dictatorship} mechanism. For dichotomous preferences, we first show that truthful, symmetric and optimal mechanisms exist if intractable mechanisms are allowed. We then provide a connection to online bipartite matching. Using this connection, it is possible to design truthful, symmetric and tractable mechanisms that extract 0.69 of the maximum social welfare, which works under assumption that agents are not adversarial. Without this assumption, we show that Random Serial Dictatorship always returns an assignment in which the expected social welfare is at least a third of the maximum social welfare. For normalized von Neumann-Morgenstern preferences, we show that Random Serial Dictatorship always returns an assignment in which the expected social welfare is at least $\frac{1}{e}\frac{\nu(\opt)^2}{n}$, where $\nu(\opt)$ is the maximum social welfare and $n$ is the number of both agents and items. On the hardness side, we show that no truthful mechanism can achieve a social welfare better than $\frac{\nu(\opt)^2}{n}$.