# A Note on Sophisticated vs. Simple Mixing in Two-Player Randomly Matched Games, with Thomas D. Jeitschko

In finite games, if any proportion of the players has a rationality bound such that it behaves predictively when indifferent between several alternatives (for instance, if the column player flips a coin whenever the utility of choosing left is equal to the utility of choosing right, even if this strategy is not part of a Nash Equilibrium) then, as long as this proportion is below some endogenous threshold, the game is stable in the sense that expected final payoffs for everyone are the same as in the perfect rationality case. Furthermore, it is not possible for more sophisticated players to take advantage of these naïve players. This result holds for continuous games of N players. The only assumption is that players cannot recognize the rationality level of their rivals so only the proportion of naïve players is publicly known; otherwise it is shown that equilibria does not generally exist in this proposed bounded rationality case.