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