On the existence of a strictly strong Nash equilibrium under the student-optimal deferred acceptance algorithm

- We analyze a preference revelation game for students in the student-optimal DA algorithm.
- We show the existence of a strictly strong Nash equilibrium through a simple algorithm.
- The equilibrium outcome from our algorithm is the same matching as in the efficiency-adjusted deferred acceptance algorithm.
- In a one-to-one matching market, it is the student-optimal vNM stable matching.

This study 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 demonstrate the existence of a strictly strong Nash equilibrium in the preference revelation game through a simple algorithm that finds it. Specifically, (i) the equilibrium outcome from our algorithm is the same matching as in the efficiency-adjusted deferred acceptance algorithm and (ii) in a one-to-one matching market, it coincides with the student-optimal von Neumann-Morgenstern (vNM) stable matching. 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 exists a strictly strong Nash equilibrium under the college-optimal deferred acceptance algorithm if and only if the student-optimal stable matching is Pareto-efficient for students.