Average-case Analysis of the Assignment Problem with Independent Preferences

TL;DR

This paper proves that the average-case efficiency loss of the Random Priority mechanism in the assignment problem with independent preferences is bounded by a constant, improving previous bounds and showing small loss in most cases.

Contribution

It establishes a constant upper bound on the average-case approximation ratio for the assignment problem with independent preferences, extending prior results and providing new analytical tools.

Findings

The ratio is bounded by 1/μ for i.i.d. preference values.

The bound improves the previous 3.718 for uniform distributions.

In most instances, the efficiency loss is small.

Abstract

The fundamental assignment problem is in search of welfare maximization mechanisms to allocate items to agents when the private preferences over indivisible items are provided by self-interested agents. The mainstream mechanism \textit{Random Priority} is asymptotically the best mechanism for this purpose, when comparing its welfare to the optimal social welfare using the canonical \textit{worst-case approximation ratio}. Despite its popularity, the efficiency loss indicated by the worst-case ratio does not have a constant bound. Recently, [Deng, Gao, Zhang 2017] show that when the agents' preferences are drawn from a uniform distribution, its \textit{average-case approximation ratio} is upper bounded by 3.718. They left it as an open question of whether a constant ratio holds for general scenarios. In this paper, we offer an affirmative answer to this question by showing that the ratio is bounded by $1/\mu$ when the preference values are independent and identically distributed random variables, where $\mu$ is the expectation of the value distribution. This upper bound also improves the upper bound of 3.718 in [Deng, Gao, Zhang 2017] for the Uniform distribution. Moreover, under mild conditions, the ratio has a \textit{constant} bound for any independent random values. En route to these results, we develop powerful tools to show the insights that in most instances the efficiency loss is small.