Equilibria under deferred acceptance: Dropping strategies, filled positions, and welfare

Abstract

We study many-to-one matching markets where hospitals have responsive preferences over students. We study the game induced by the student-optimal stable matching mechanism. We assume that students play their weakly dominant strategy of truth-telling. Roth and Sotomayor (1990) showed that equilibrium outcomes can be unstable. We prove that any stable matching is obtained in some equilibrium. We also show that the exhaustive class of dropping strategies does not necessarily generate the full set of equilibrium outcomes. Finally, we find that the 'rural hospital theorem' cannot be extended to the set of equilibrium outcomes and that welfare levels are in general unrelated to the set of stable matchings. Two important consequences are that, contrary to one-to-one matching markets, (a) filled positions depend on the equilibrium that is reached and (b) welfare levels are not bounded by the optimal stable matchings (with respect to the true preferences).

Stability of Equilibrium Outcomes under Deferred Acceptance: Acyclicity and Dropping Strategies

“…In light of this result, we assume throughout that students are truthful. It is worth noting that the assumption that students play their weakly dominant strategy of truth-telling is made in most of the literature, for example, Roth (1984b), Kojima and Pathak (2009), Ma (2010), andJaramillo, Kayı andKlijn (2013). Hospitals, on the other hand, can sometimes benefit from misrepresenting their preferences (see Roth and Sotomayor;1990, Example 4.1).…”

The outcome of competitive equilibrium rules in buyer–seller markets when the agents play strategically

“…Pérez-Castrillo and Sotomayor (2013) prove that buyers (respectively, sellers) do not have an incentive to misreport their valuation if the buyer-optimal (respectively, selleroptimal) competitive equilibrium is used by the designer in a one-to-many (respectively, many-to-one) buyer-seller market. 6 Papers analyzing the consequences of manipulation in marriage and the college admission models, that is, in models where there are no prices, include Gale andSotomayor (1985a, 1985b), Roth (1985), Roth and Sotomayor (1990), Sotomayor (2008), Kojima andPathak (2009), Ma (2010), Sotomayor (2012), and Jaramillo, Kayi, and Klijn (2013). this stage is interesting in itself because it also allows us to understand the buyers' behavior when the sellers' valuations are given, for example, because they are public knowledge or because sellers cannot manipulate them. We develop the analysis of this stage in two parts.…”

Equilibria of deferred acceptance with complete lists

“…In particular, there exist an equilibrium matching (the women-optimal stable matching) that weakly dominates all other equilibrium matchings from the women's point of view. However, for the more general many-to-one matching model such an equilibrium matching also does not need to exist (Jaramillo et al, 2013, Example 1).…”