An Impossibility Theorem in Matching Problems

This paper studies the possibility of strategy-proof rules yielding satisfactory solutions to matching problems. Alcalde and Barberá (1994) show that efficient and individually rational matching rules are manipulable in the one-to-one matching model. We pursue the possibility of strategy-proof matching rules by relaxing efficiency to the weaker condition of respect for unanimity. Our first result is positive. We prove that a strategy-proof rule exists that is individually rational and respects unanimity. However, this rule is unreasonable in the sense that a pair of agents who are the best for each other are matched on only rare occasions. In order to explore the possibility of better matching rules, we introduce the natural condition of "respect for pairwise unanimity." Respect for pairwise unanimity states that a pair of agents who are the best for each other should be matched, and an agent wishing to stay single should stay single. Our second result is negative. We prove that no strategy-proof rule exists that respects pairwise unanimity. This result implies Roth (1982) showing that stable rules are manipulable. We then extend this to the many-to-one matching model.