 Sophisticated Bayesian learning: This approach, initiated by Paul Milgrom and John Robert (1991), replaces the rather crude statistical inference posited by fictitious play with a more refined conception of inductive deliberation. Specifically, it abandons the questionable assumption that one faces stationary strategies from one’s opponents. Ehud Kalai and Ehud Lehrer (1993) offer a widely discussed model of sophisticated Bayesian learning. Players engaged in an infinitely repeated game constantly update probability distributions defined over the set of possible strategies played by their opponents, where a strategy is a function from the set of possible histories to the set of possible actions. At each stage, a player chooses an action that maximizes the expected value of her payoffs for the entire future sequence of trials, not just for the present trial. This approach allows a player to discern patterns in her opponents’ behavior, including patterns that depend upon her own actions. It also allows her to sacrifice current payoff for a higher expected long-term payoff. Kalai and Lehrer prove that their procedure almost always converges to something approximating Nash equilibrium, under the crucial assumption (the “grain of truth” assumption) that each player begins by assigning positive probability to all strategies that actually occur.