Generalized matchings for preferences represented by simplest semiorder: stability and Pareto optimality

We consider an extension of the classical model of generalized Gale–Shapley matchings. The model describes a two-sided market: on one side, universities each of which has a restriction on the number of enrolled students; on the other side, applicants each of which can get a single place in the university. Both applicants and universities have preferences with respect to the desired distribution. We assume that each applicant constructs a linear order on the set of desired universities, and each university has preferences that are simplest semiorders. For this modification, we show that a stable matching always exists. Moreover, we formulate necessary and sufficient conditions for Pareto optimality of the stable matching.