We analyze the equilibria of two-person supergames consisting of the sequential play of a finite collection of stage games, where each stage game is strictly competitive and has two outcomes for each player. We show that in any Nash equilibrium of the supergame, play at each stage is a Nash equilibrium of the stage game provided players' preferences over certain outcomes in the supergame satisfy a weak monotonicity condition. Thus, equilibrium play in such supergames is invariant for a large class of preferences and, in particular, it does not depend on the players' attitudes toward risk. This enables us to conclude, for example, that O'Neill's (1985) experimental test of Nash equilibrium adequately controls for risk attitudes, despite the fact that the supergame obtained by repeating his two-outcome stage game has more than two outcomes. We also establish an invariance result for games with more than two players when the solution concept is subgame perfection.