On the Nash Stability in the Hedonic Coalition Formation Games

This paper studies the Nash stability in hedonic coalition formation games. We address the following issue: for a general problem formulation, is there any utility allocation method ensuring a Nash-stable partition? We propose the definition of the Nash-stable core and we analyze the conditions for having a non-empty Nash-stable core. More precisely, we prove that using relaxed efficiency in utility sharing allows to ensure a non empty Nash-stable core. Then, a decentralized algorithm called Nash stability establisher is proposed for finding the Nash stability in a game whenever at least one exists. The problem of finding the Nash stability is formulated as a non-cooperative game. In the proposed approach, during each round, each player determines its strategy in its turn according to a random round-robin scheduler. We prove that the algorithm converges to an equilibrium if it exists, which is the indicator of the Nash stability.