Passively-Strictly Strong Nash Equilibrium in a Preference Revelation Game under the Student-Optimal Deferred Acceptance Algorithm

Abstract

We revisit a college admission market and a related preference revelation game under the student-optimal deferred acceptance algorithm (SODA). Previous research has demonstrated the existence of a strictly strong Nash equilibrium (SSN) based on either an iterative deferred acceptance algorithm (DA-SSN) or the core of a corresponding house allocation problem (Core-SSN). We propose a new equilibrium concept called passively-strictly strong Nash equilibrium (P-SSN). It rules out a kind of deviation called passively weak deviation which includes students who were threatened to deviate. Then we show two preliminary existence results about P-SSN. (i) If the DA-SSN and the Core-SSN are not equivalent, then neither of them is a P-SSN. (ii) If the matching determined by the DA-SSN satisfies a property called irrelevance of low-tier agents, then the DA-SSN is also a P-SSN.