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Stochastic Coalitional Better-response Dynamics and Strong Nash Equilibrium

We consider coalition formation among players in an n-player finite strategic game over infinite horizon. At each time a randomly formed coalition makes a joint deviation from a current action profile such that at new action profile all players from the coalition are strictly benefited. Such deviations define a coalitional better-response (CBR) dynamics that is in general stochastic. The CBR dynamics either converges to a strong Nash equilibrium or stucks in a closed cycle. We also assume that at each time a selected coalition makes mistake in deviation with small probability that add mutations (perturbations) into CBR dynamics. We prove that all strong Nash equilibria and closed cycles are stochastically stable, i.e., they are selected by perturbed CBR dynamics as mutations vanish. Similar statement holds for strict strong Nash equilibrium. We apply CBR dynamics to the network formation games and we prove that all strongly stable networks and closed cycles are stochastically stable.