Exactly and strongly consistent representations of effectivity functions

As argued in the introductory section of the previous chapter it is important to find robust voting procedures for constitutions. In our approach, ‘robust’ means ‘exactly and strongly consistent’ (ESC): the game induced by the social choice function should have a strong equilibrium for each profile of preferences which, moreover, results in the same outcome as truthful voting. In this chapter we extend the study of ESC representations to general effectivity functions. We start, in Section 10.2, by generalizing the definition of a feasible elimination procedure to arbitrary effectivity functions: the generalization uses blocking coalitions instead of blocking coefficients. An effectivity function will be called ‘elimination stable’ if it admits at least one feasible elimination procedure (f.e.p.) for each profile of linear orderings (strict preferences). We show that if the effectivity function is maximal, stable, and elimination stable, then it has an ESC representation. In fact, each selection from the maximal alternatives – alternatives that result from an f.e.p. – is an ESC representation.