Level-r Model with Adaptive and Sophisticated Learning

In standard models of iterative thinking, there is a fixed rule hierarchy and every player chooses a fixed rule level. Nonequilibrium behavior emerges when some players do not perform enough thinking steps. Existing approaches however are inherently static. In this paper, we generalize models of iterative thinking to incorporate adaptive and sophisticated learning. Our model has three key features. First, the rule hierarchy is dynamic, i.e., the action that corresponds to each rule level can evolve over time depending on historical game plays. Second, players’ rule levels are dynamic. Specifically, players update beliefs about opponents’ rule levels in each round and change their rule level in order to maximize payoff. Third, our model accommodates a continuous rule hierarchy, so that every possible observed action can be directly interpreted as a real-numbered rule level r. The proposed model unifies and generalizes two seemingly distinct streams of nonequilibrium models (level-k and belief learning models) and as a consequence nests several well-known nonequilibrium models as special cases. When both the rule hierarchy and players’ rule levels are fixed, we have a static level-r model (which generalizes the standard level-k model). When only players’ rule levels are fixed, our model reduces to a static level-r model with dynamic rule hierarchy and captures adaptive learning. When only the rule hierarchy is fixed, our model reduces to a dynamic level-r model and captures sophisticated learning. Since our model always converges to the iterative dominance solution, it can serve as a model of the equilibration process. Using experimental data on p-beauty contests, we show that our model describes subjects’ dynamic behavior better than all its special cases. In addition, we collect new experimental data on a generalized price matching game and the estimation results show that it is crucial to allow for both adaptive and sophisticated learning in predicting dynamic choice behaviors across games. (JEL A12, A13, C72, D63)