Possibility and Impossibility of Learning with Limited Behavior Rules

We consider boundedly rational learning processes in which players have a priori limited set of behavior rules. A behavior rule is a function from information to a stage-game action, which reflects the available information and one’s reasoning about how others act. Commonly used behavior rules include the adaptive rule and the conservative rule (inertia). Sophisticated players may use iterative best responses, called forward-looking behavior rules. The feasible behavior rules set the framework and limitations of learning processes. We investigate a general relationship between the set of feasible behavior rules and the properties of the long-run outcomes of any learning process restricted to use the feasible behavior rules. From very limited set of behavior rules to increasingly sophisticated ones, robust limit actions are what we call minimal weak-curb sets. In order to converge to a minimal weak-curb set of arbitrary stage game, it is sufficient that the players have one period memory and use one-step different behavior rules. For some classes of games where minimal weak-curb sets coincide with Nash equilibria, we have a global convergence to a Nash equilibrium under very limited set of behavior rules. For other games, however, there is a limit to learning. We show that for any finite set of behavior rules and finite memory length, there is a class of stage games whose unique Nash equilibrium cannot be reached from some initial states. More sophistication of reasoning does not mean better convergence in the sense that the process stays in a smaller set or finds a minimal curb set, which is a set-generalization of Nash equilibrium. The key to finding curb sets is not only the depth of thinking but also the stage game payoff structure. That is, smartness alone cannot find a rational action without the aid of a payoff incentive to explore actions. JEL classification number: C73.