Rationalizable Foresight Dynamics: Evolution and Rationalizability

This paper considers a adjustment process in a society with a continuum of agents. Each agent takes an action upon entry and commits to it until he is replaced by his successor at a stochastic point in time. In this society, rationality is common knowledge, but beliefs may not be coordinated with each other. A rationalizable foresight path is a feasible path of action distribution along which every agent takes an action that maximizes his expected discounted payoff against another path which is in turn a rationalizable foresight path. An action distribution is accessible from another distribution under rationalizable foresight if there exists a rationalizable foresight path from the latter to the former. An action distribution is said to be a stable state under rationalizable foresight if no rationalizable foresight path departs from the distribution. A set of action distributions is said to be a stable set under rationalizable if it is closed under accessibility and any two elements of the set are mutually accessible. Stable sets under rationalizable foresight always exist. These concepts are compared with the corresponding concepts under perfect foresight. Every stabel state under rationalizable foresight is shown to be stable under perfect foresight. But the converse is not true. An example is provided to illustrate that the stability under rationalizable foresight gives a sharper prediction than under perfect foresight.