Preference revelation games and strict cores of multiple-type housing market problems

We consider multiple-type housing market problems as introduced by Moulin (1995) and study the relationship between strict strong Nash equilibria and the strict core (two solution concepts that are defined in terms of the absence of weak blocking coalitions). We prove that for lexicographically separable preferences, the set of all strict strong Nash equilibrium outcomes of each preference revelation game that is induced by a strictly core stable mechanism is a subset of the strict core, but not vice versa, that is, there are strict core allocations that cannot be implemented in strict strong Nash equilibrium. This result is extended to a more general set of preference domains that satisfy strict core non-emptiness and a minimal preference domain richness assumption.