Correlated Equilibrium Under Costly Disobedience

In this paper, we extend Aumann's (1974) well-known solution of correlated equilibrium to allow for a cost of disobedience for each player. Calling the new solution costly correlated equilibrium (CCE), we derive the necessary and sufficient conditions under which the set of CCE strictly expands when the players' cost of disobedience is increased by the mediator in any finite normal-form game. These conditions imply that for any game that has a Nash equilibrium (NE) that is unpure, the set of CCE strictly expands with the addition of even arbitrarily small cost of disobedience, whereas for games that have a unique NE in pure strategies, the set of CCE stays the same unless the cost gets sufficiently high. We also study the welfare implications and changes in the value of mediation with exogenous cost changes. We find that strictly better social outcomes can be attained and the value of mediation cannot decrease with an increase in the cost level. We also illustrate how our model can be integrated with a cost-selection game where players non-cooperatively choose their costs of disobedience before mediation occurs. We show that there exist cost-selection games in which setting the cost of disobedience at zero is a strictly dominated strategy for each player as well as games this strategy becomes weakly dominant for everyone.