The asymptotic value of a game v with a continuum set of players, I, is defined whenever all the sequences of the Shapley values of finite games that ‘approximate’ v have the same limit. A weighted majority game is a game of the form fℝoslash;μ where μ is a positive measure and f(x)=1 if x≥q and f(x)=0 otherwise, and q is a real number, 0<q<μ(I). The paper proves that all weighted majority games have asymptotic values. This result is then used further to show that if v is of the form v=fℝoslash;μ, where μ is a 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 f is absolutely continuous, or when μ has at most finitely many atoms or when μ is purely atomic. Thus, the essential novelty is that even when μ has countably many atoms and a nonatomic part, fℝoslash;μ has an asymptotic value. The paper also shows that fℝoslash;μ does not necessarily have an asymptotic value when μ is a signed measure.
