Stochastic Coalitional Better-Response Dynamics for Finite Games with Application to Network Formation Games

We consider a coalition formation among players, in an $n$-player strategic game, over infinite horizon. At each time a randomly selected coalition makes a joint deviation, from a current action profile to a new action profile, which is strictly beneficial for all the players belonging to the coalition.Such deviations define a stochastic coalitional better-response (CBR) dynamics. The stochastic CBR dynamics either converges to a $\cal{K}$-stable equilibrium or becomes stuck in a closed cycle.We also assume that at each time a selected coalition makes mistake in deviation with small probability. We prove that all $\cal{K}$-stable equilibria and all action profiles from closed cycles, having minimum stochastic potential, are stochastically stable. Similar statement holds for strict $\cal{K}$-stable equilibrium. We apply the stochastic CBR dynamics to the network formation games. We show that all strongly stable networks and closed cycles of networks are stochastically stable.