On the existence of a strictly strong Nash equilibrium under the student optimal DA algorithm

This paper analyzes a preference revelation game in the student optimal deferred acceptance algorithm in a college admission problem. We assume that each college’s true preferences are known publicly and analyze the strategic behavior of students. We show the existence of a strictly strong Nash equilibrium in the preference revelation game by providing a simple algorithm that finds it. In particular, (i) the equilibrium outcome by our algorithm is the same matching as in the efficiently-adjusted deferred acceptance algorithm (Kesten, 2010) and (ii) in a one-to-one matching market, it coincides with the student-optimal vNM stable matching (Ehlers (2007) and Wako (2010)). We also show that (i) when a strict core allocation in a housing market derived from a college admission market exists, it can be supported by a strictly strong Nash equilibrium and (ii) there may not exist a strictly strong Nash equilibrium under the college optimal deferred acceptance algorithm.