Size versus Truncation Robustness in the Assignment Problem

Abstract

We study the size of matchings (expected number of agents matched to some objects) generated by random mechanisms in the assignment problem. We show that no mechanism that satisfies two weak axioms, weak truncation robustness and weak regularity, achieves an approximation ratio better than 1 - 1/e = 63.2%. This result indicates that it is impossible to achieve a matching size larger than 63.2% of the maximum feasible size in the worst case, as long as agents' preferences over objects are private information. Our result indicates that the random serial dictatorship mechanism and probabilistic serial mechanism (which indeed has an approximation ratio of 1 - 1/e) have the best approximation ratio among a broad class of mechanisms.