Discovering Theorems in Game Theory: Two-Person Games with Unique Nash Equilibria

In this paper we provide a logical framework for using computers to discover theorems in (two-person finite) games in strategic form, and apply it to discover classes of games that have unique (pure-strategy) Nash equilibria. We consider all possible classes of games that can be expressed by a conjunction of two binary clauses, and our program re-discovered Kats and Thisse’s class of weakly unilaterally competitive two-person games, and came up with several other classes of games that have unique Nash equilibria. It also came up with new classes of strict games that have unique Nash equilibria, where a game is strict if for each player different profiles have different payoffs. Partly motivated by these findings, we also (manually) prove a result that says that a strict game has a unique Nash equilibrium iff it is best-response equivalent to a strictly competitive game.