Singular Games have Asymptotic Values

The asymptotic value of a game v with a continuum of players is defined whenever all the sequences of Shapley values of finite games that “approximate” v have the same limit. In this paper we prove that if v is defined by v(S) = f(μ(S)), where μ is a nonatomic probability measure and f is a function of bounded variation on [0, 1] that is continuous at 0 and at 1, then v has an asymptotic value. This had previously been known only when v is absolutely continuous. Thus, for example, our result implies that the nonatomic majority voting game, defined by v(S) = 0 or 1 according as μ(S) ≤ 1/2 or μ(S) > 1/2, has an asymptotic value. We also apply our result to show that other games of interest in economics and political science have asymptotic values, and adduce an example to show that the result cannot be extended to functions f that are not of bounded variation.