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Due to copyright restrictions, the access to the full text of this article is only available via subscription.The prominent mechanism of the recent literature in the assignment problem is the probabilistic serial (PS). Under PS, the truthful (preference) proÖle always constitutes an ordinal Nash Equilibrium, inducing a random assignment that satisÖes the appealing ordinal e¢ ciency and envy-freeness properties. We show that both properties may fail to be satisÖed by a random assignment induced in an ordinal Nash Equilibrium where one or more agents are non-truthful. Worse still, the truthful proÖle may not constitute a Nash Equilibrium, and every non-truthful proÖle that constitutes a Nash Equilibrium may lead to a random assignment which is not ordinally e¢ cient, not even weakly envy-free, and which admits an ex-post ine¢ cient decomposition. A strong ordinal Nash Equilibrium may not exist, but when it exists, any proÖle that constitutes a strong ordinal Nash Equilibrium induces the random assignment induced under the truthful proÖle. The results of our equilibrium analysis of PS call for caution when implementing it in small assignment problems

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In a strategic form game a strategy profile is an equilibrium if no viable coalition of agents (or players) benefits (in the Pareto sense) from jointly changing their strategies. Weaker or stronger equilibrium notions can be defined by considering various restrictions on coalition formation. In a Nash equilibrium, for instance, the assumption is that viable coalitions are singletons, and in a super strong equilibrium, every coalition is viable. Restrictions on coalition formation can be justified by communication limitations, coordination problems or institutional constraints. In this paper, inspired by social structures in various real-life scenarios, we introduce certain restrictions on coalition formation, and on their basis we introduce a number of equilibrium notions. As an application we study our equilibrium notions in resource selection games (RSGs), and we present a complete set of existence and non-existence results for general RSGs and their important special cases.