The impossibility of definitive solutions for some games

In his dissertation of 1950, Nash based his concept of solution to a game on the principles that “a rational prediction should be unique, that the players should be able to deduce and make use of it.” In this paper, we address the issue of when such definitive solutions are possible. We assume player rationality at least as strong as Aumann’s rationality. By formalizing Nash’s reasoning, we show that any justified definitive solution to a game is a Nash equilibrium, hence games without Nash equilibria cannot have definitive solutions under any notion of rationality. However, each strategic game with Nash equilibria admits a justified definitive solution under some notion of rationality. For Aumann’s rationality, no game with two or more Nash equilibria can have a definitive solution whereas some games with a unique Nash equilibrium have definitive solutions and some do not.