Strongly consistent representations

In the preceding chapters we have studied representations of constitutions (effectivity functions) under a minimal stability condition, namely Nash consistency: for every admissible profile of preferences there should be at least one Nash equilibrium in the representing game. In Chapter 4 we have studied acceptable representations, meaning that all Nash equilibria are Pareto optimal. Another way to look at this is that not only single players do not wish to deviate, but also the grand coalition of all players has no incentive to jointly deviate to a different strategy profile. More generally, one may argue that in many interesting conflict situations preplay communication, direct or indirect, is possible – and sometimes unavoidable. This leads naturally to coordination of strategies by coalitions of players and may upset a given Nash equilibrium. To avoid this and maintain stability, we need to consider coalitional equilibrium concepts where coalitions have no profitable deviations. In this chapter we consider the strongest equilibrium concept, namely strong equilibrium. In a strong equilibrium no coalition S of players has a deviating S-tuple of strategies that is (strictly) profitable for each of its members – see Definition 5.2.1. A game form Γ is strongly consistent if for every profile of preferences R N the resulting game (Γ,R N ) has a strong equilibrium. The main goal in the chapter is to characterize effectivity functions that can be represented by strongly consistent game forms.