In recent years game theorists and experimental economists have focused a great deal of attention on the question of how people learn when repeatedly playing a simple matrix game. While some, investigators focus on reinforcement learning in which people learn by looking back at their experience and seeing what has been successful in the past, others, focus on belief learning and look to the past to update beliefs about their opponent's future action. Still others, select the best features of both of these models (among other things) in an approach that has proven to be remarkable successful.
In all of this research, however, there is an assumption that while past actions and payoffs are observable, beliefs are unobservable and therefore must be represented by proxies and inferred. The research proposed here attempts to take advantage of laboratory techniques by directly eliciting the beliefs of subjects in our experiments using a "proper scoring rule" which provides subjects with an incentive to report their beliefs truthfully. As a result, this proposed research presents, an investigation of belief learning in which all relevant variables are observable; i.e., it studies belief learning with real beliefs.,
In a just completed paper on two-by-two constant-sum games with real beliefs, the principal investigators, come to two broad conclusions. First, under the assumption that people best-respond to some beliefs, they find that stated beliefs fit the data better than the other beliefs studied (Cournot, fictitious play and more generally, the class of gamma-weighted empirical distributions). Second, under the assumption that people choose strategies via a logistic belief learning rule generated by some beliefs, we again find that stated beliefs fit the data the best.
The research proposed here extends our previous work in four different directions. First we propose to investigate whether our data can better be explained by a reinforcement learning model. This comparison is interesting since we intend to compare the performance of "the best" reinforcement learning model to our stated belief learning model, a model which has out-performed all of the other history based belief learning models we have compared it to. In other words, we intend to compare the best reinforcement learning model to the best belief learning model.
Second, using our elicited beliefs, we intend to see if, in the repeated play of matrix games, subjects converge to an "equilibrium in beliefs" in which each subject believes their opponent to behave as dictated by the equilibrium of the theory. This will allow an additional test of the veracity of game theoretical equilibrium notions since relying on actions alone to support the theory can be misleading. In short, what may appear to be disequilibrium with respect to actions may still be an equilibrium with respect to beliefs.
Third, we construct new theoretical models of belief formation and test them using the data generated by our previous and proposed experiments.
Finally, we will expand the set of games we investigate beyond the two-person constant sum games to determine how the belief formation process changes as we change the normal form of the game subjects play.
