Stable Matching Games

20^th May 2020, 13:00
Rida Laraki
University of Liverpool

Abstract

In 1962, Gale and Shapley introduced a matching problem between two sets of distinct agents M and W (men/women, buyers/sellers, workers/firms, students/universities, doctors/hospitals), who need to be matched by taking into account that each side has a preference on the other side. They defined a matching as stable if no unmatched couple can Pareto improve by forming couple and proved its existence using a "deferred-acceptance" algorithm. Our article offers a new extension of this model by assuming that each matched couple obtains its payoff as the outcome of a strategic-form game that may be couple-specific. A matching, together with a strategy profile, is externally stable if no unmatched couple can find a strategy profile in their game that Pareto improves their previous payoffs. It is internally stable if no individual agent can, by deviating and changing her strategy inside her couple, increase her payoff without breaking the external stability of the couple (e.g. her partner has a better outside option). This is a constrained Nash equilibrium condition. By combining a deferred acceptance algorithm with a new algorithm, we prove the existence of an externally and internally stable matching for several classes of games, including zero-sum games, potential games, and infinitely repeated games. Finally, we show that our model extends matching with monetary transfers (Shapley and Shubik, Gale and Demange) and matching with contracts (Milgrom and Hatfield).