This paper proposes a new approach to the study of economic problems that have hitherto been modeled as games with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies in the underlying problem. The authors' point of departure from the conventional approach is to view the sharing rules that resolve these indeterminacies as part of the solution rather than as part of the description of the model. A solution to the authors' model is a sharing rule, together with a profile of (mixed) strategies that satisfies the usual (Nash) best response criterion. Their main result is that such a solution always exists.