Acceptable representations

In Chapter 3 we have studied the existence of Nash consistent representations of effectivity functions. We have, in fact, shown that the same conditions that guarantee existence of Nash consistent representations also guarantee the existence of weakly acceptable representations, that is, representations that always admit also Pareto optimal Nash equilibria – see Corollaries 3.6.2 and 3.6.3. In this chapter we investigate a subset of the set of Nash consistent game forms, namely the set of acceptable game forms, where a game form is acceptable if: (i) it is Nash consistent and (ii) for every profile of preferences every Nash equilibrium outcome is Pareto optimal. Acceptable game forms were introduced in Hurwicz and Schmeidler (1978). One of the main results of this chapter is a complete characterization of the effectivity functions which can be represented by an acceptable game form. Assuming that the set of social states is a compact Hausdorff topological space and restricting ourselves to continuous preferences we obtain the following result: an effectivity function for at least three players has an acceptable representation if and only if: (i) it has a Nash consistent representation and (ii) no two disjoint coalitions can veto the same alternative. (This follows from Theorem 4.3.1 and Remark 4.6.2. A precise formulation of the latter condition is (4.7).) This result is easy to understand but the proof is quite involved. We outline it here.