Weighted Majority Games Have Asymptotic Value

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 f ∘ 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). In this paper we prove that all weighted majority games have asymptotic values. This result is then used further to show that if v is of the form v = f ∘ 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 bad 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, f ∘ μ has an asymptotic value. We also show that f ∘ μ does not necessarily have an asymptotic value when μ is a signed measure.