Cooperative equilibria of finite games with incomplete information

Recently, Askoura et al. (2013) proved the nonemptiness of the α-core of a finite Bayesian game GR with Young measure strategies and nonatomic type spaces, without requiring that the expected payoffs be concave. Under the same hypotheses as theirs, we demonstrate that Scarf's method (1971) works with some adjustments to prove the nonemptiness of the α-core of a similar game GM with pure strategies. We prove that the nonemptiness of the α-core of a GM is equivalent to that of its associated characteristic form game GMC, that the core of GMC and hence the α-core of a GM is nonempty, and that the nonemptiness of the α-core of a GM is equivalent to that of a GR, which clearly implies the result of Askoura et al. (2013). Our proofs hinge on an iterated version of Lyapunov's theorem for Young measures to purify partially as well as fully Young measure strategies in an expected payoff function, which is a main methodological contribution of this paper.