For discontinuous games Simon and Zame introduced a new equilibrium notion by allowing payoff indeterminacy in the form of endogenous sharing rules. It is based on a scheme by which the discontinuous game is approximated by a sequence of discrete games. In this paper we propose a new approximation by means of a sequence of continuous games. This offers more precision and has the advantage of allowing the use of purification methods. As examples show, this scheme can also be used to directly compute equilibria in the sense of Simon and Zame.