The concept of Nash equilibrium is the basic analytic tool of non-cooperative Game Theory. All applications of non-cooperative games, which include a sizable fraction of the work in economic theory, invariably rely on Nash equilibria. A typical application consists of modeling a competitive situation as a game and then narrowing down the set of possible outcomes by applying the equilibrium conditions. If the remaining properties all share a certain property, then that property is viewed as a consequence of the analysis. This project is concerned with two aspects of the above-described process. First, there is the issue of equilibrium "refinement", i.e., the formulation of conditions which "sensible" equilibria presumably must satisfy. Equilibrium refinements have been motivated by two main ideas. One is "strategic stability" or robustness in the face of logical calculations that the players may perform, and the other is "evolutionary stability", or robustness in the face of a dynamic process of "survival of the fittest". Second, there is the issue of the modeling itself: Are the conclusions from equilibrium analysis distorted by the simplifications required in real-life situations? If so, what should be done? Specifically, the project reformulates the definition of strategic stability so that it can be applied in a more direct manner; axiomatizes the solution concept; and derives additional properties satisfied by the solution concept. It extends the definition of evolutionary stability used by biologists to general games and studies its relationship with strategic stability. As regards the problem of modeling, the project examines ways of relaxing assumptions of common knowledge and of unlimited complexity in players' strategies. Finally, the two themes - of equilibrium refinement and of the sensitivity of equilibrium analysis to the formulation of models - are combined in order to provide a practical analysis of real competitive interactions. These are fundamental and extremely difficult problems. The results of this research should provide new, powerful analytic tools for economic theory.
