On the Structure and Diversity of Rational Beliefs

The formation of expectations and probability beliefs has played a central role in the formulation of sequential and other dynamic equilibria. The specification of “rational” expectations in either economics or game theory requires agents to possess extraordinary information and knowledge about the underlying structure of the economy or the game. It is usually hard to conceive how agents come to possess such information and knowledge. The recent response to this problem has been to formulate dynamic processes of learning which aim to show how agents learn what they know when formulating their beliefs. The problem is that this research has not solved the initial problem. Without engaging in a full-scale survey of the results of the recent effort, we think it is accurate to say that there are examples worked out where complete learning does take place. However, in general, the learning approach has not been able to provide a satisfactory justification for agents to be fully knowledgeable rationally expecting agents. This conclusion has a counterpart in the statistical literature where a spirited debate has been taking place on the problem of “Bayes consistency” (see Diaconis and Freedman [1986] for an excellent recent survey). This is the problem of ensuring the convergence of posterior distributions to the mass-point distribution at the true parameter. We note that “Bayes consistency” may fail even when the statistician is able to conduct independent, repeated controlled experiments. A learning economic agent cannot obtain independent observations and must be content with the actual data generated by the system.