Conditional Random Dictatorship and the Value of a Bayesian Cooperative Game

We present and analyze a theory of cooperative bargaining under asymmetric information, based on the equity principles of the conditional random dictatorship—choosing with equal probability one of the individuals to act as a dictator under the participation constraint of the other players. Our approach leads to a unified bargaining solution that has as special cases the Shapley value in transferable utility games and the Maschler-Owen consistent value in non-transferable utility games. This solution is shown to be the smallest possible set satisfying two axioms: one of these axioms states what the solution should be for the class of problems where conditional random dictatorship yields an efficient outcome, and the other axiom relates the solutions of a problem to the solutions of its extensions.