Ranking by Manipulability and Quantile Stable Mechanisms

We study manipulability of stable matching mechanisms and show that manipulability comparisons are equivalent to preference comparisons: for any agent, a mechanism is more manipulable than another if and only if this agent prefers the latter to the former. In particular, this implies that no two stable mechanisms can be ranked in terms of manipulability for all agents. We then construct quantile stable mechanisms, establish conditions under which they are well defined, and observe that they can be ranked in terms of manipulability for one side of the market, but not for both. In particular, the typically used mechanism that selects the stable matching that is best for one side of the market has no incentive advantage over other quantile stable mechanisms when both sides of the market are strategic. As an auxiliary result, we provide sufficient conditions for the existence of median stable matchings, and more generally, show that quantile stable matchings exist when contracts are strong substitutes and satisfy the law of aggregate demand. This last result is of independent interest as experiments show that agents who match in a decentralized way tend to coordinate on the median stable matching, when it exists