Probabilistic stable rules and Nash equilibrium in two-sided matching problems

We study many-to-one matching problems when each firm has substitutable and separable preferences. We analyze the stochastic dominance (sd) Nash equilibria of the game induced by any probabilistic stable matching rule. We show that a unique match is obtained as the outcome of each sd-Nash equilibrium. Furthermore, individual rationality with respect to the true preferences is a necessary and sucient condition for an equilibrium outcome. Each stable match for the true preferences is achieved as the outcome of an equilibrium in which firms behave truthfully. Conversely, the outcome of each such equilibrium is stable for the true preferences. Finally, we study equilibrium behavior in many-to-many matching problems under the same domain of preferences.