On Perfectly Coalition-proof Nash Equilibria

The most prominent solution in the theory of noncooperative games is the Nash equilibrium. However, there are some fundamental problems with the interpretation and application of it that led to many refinements (see, e.g., Selten, 1975). The following difficulty concerning the application of Nash equilibria has been pointed out in Bernheim, Peleg, and Whinston [1987]. Let g = (∑ 1, …, ∑ n ; h 1, …, h n ) be a game in strategic form and assume that g has two or more equilibria. In order to choose an equilibrium point the players have to meet and coordinate their strategies. Thus, from a theoretical point of view, preplay communication is needed for the implementation of Nash equilibria. However, one may argue that the players of g may use a theory that selects a unique equilibrium for each game in order to avoid preplay communication (see Harsanyi, 1975). This is possible only at a theoretical level, because in real life games there are almost always possibilities for direct or indirect communication, and there are no means to prevent coalitions of players from coordinating their strategies. In summary, in important classes of noncooperative environments preplay communication is unavoidable and, therefore, coalitions may coordinate their strategies and deviate from an equilibrium.