Games with endogenous transition probabilities and endogenous stage payoffs (or ETP–ESP games for short) are stochastic games in which both the transition probabilities and the payoffs at any stage are continuous functions of the relative frequencies of all past action combinations chosen. We present methods to compute large sets of jointly-convergent pure-strategy rewards in two-player ETP–ESP games with communicating states under the limiting average reward criterion. Such sets are useful in determining feasible rewards in a game, and instrumental in obtaining the set of (Nash) equilibrium rewards.