An Impossibility Theorem for Von Neumann-Morgenstern Solutions

Recently two game theoretic interpretations of social choice procedures have been offered. First, Wilson (1970) and Plott, (1974) suggested that, for each environment, the value of a choice function might constitute a "solution" or stable set that could arise from the play of some underlying cooperative game. In this view, and important problem is to determine if and under what conditions a given solution concept (or notion of stability) can, for some game, characterize the behavior of a given social choice function. Secondly, social choice functions have been interpreted as collections of equilibria of an underlying noncooperative game (see Gibbard (1973), Peleg (1978), Maskin (1977), and Ferejohn and Grether (1979). In this framework, one major problem is to determine for a given equilibrium correspondence of a suitably chosen noncooperative game. A closely related problem is to determine which noncooperative games possess nonempty equilibrium correspondences of various sorts. In this paper, we pursue a cooperative game-theoretic interpretation of social choice. And in particular we show that, if a social choice function arises as a Von Neumann Morgenstern solution in each environment, then it is essentially oligarchical in exactly the same sense that "core" selecting choice functions are oligarchic. The conditions under which this conclusion is obtained are, in fact, slightly more restrictive than those for the results on core selecting choice functions but are still weak enough that our result applies to almost any commonly occurring voting scale.