# Definitive solutions of strategic games

## Sergei Artemov

### The CUNY Graduate Center

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 Nash expectations of a definitive solution hold and whether the Nash Equilibrium (NE) solution concept is a match for such definitive solutions. We show that indeed, an existence of NE is a necessary condition for a definitive solution, and each NE σ is a definitive solution for some notion of rationality individually tuned for this σ. However, for specific notions of rationality, e.g., Aumann’s rationality, NE is not an exact match to definitive solutions, many games with NE do not have definitive solutions at all. In particular, strategic ordinal payoff games with two or more Nash equilibria, and even some games with a unique NE do not have definitive solutions. We also show that the iterated dominance approach is a better candidate for Nash’s definitive solution concept than the Nash Equilibrium.

Professor Artemov holds a Distinguished Professor position at the Graduate Center of the City University of New York, in the Computer Science, Mathematics and Philosophy programs. He is also Professor of Mathematics at Moscow State University, the founder and the Head of the research laboratory Logical Problems in Computer Science. He conducts research in the areas of logic in computer science, mathematical logic and proof theory, knowledge representation and artificial intelligence, automated deduction and verification and optimal control and hybrid systems.