Sizhong Lan 1 1 China Mobile Research Institute, Beijing 100053, China a) (Dated: September 24, 2019) It had been proved that every non-cooperative game had a Nash equilibrium point. Although many existing algorithms are capable of finding equilibrium points, it is still unclear what force is driving the players to them in the real world. We show that, the players' immediately and constantly pursuing profitable strategies is sufficient for the game to evolve towards equilibrium point, and meanwhile the game needs minimum information exchange among players and no mediation from beyond players. Accordingly, we suggest that in reality the tendency towards Nash equilibrium could be more pervasive and irresistible than expected. Technically, the players' pursuit of profitable strategies gives rise to a sequence of adjusted strategies for our study its approximation to the true equilibrium point. And the sequence can be nicely visualized as a clear path towards an equilibrium point. Our theory has limitations in optimizing the accuracy of equilibrium point approximation. I. INTRODUCTION In 1951 John Nash proved that every non-cooperative game has an equilibrium point 1 by using Brouwer's fixed point theorem 2 . Nash's proof is existential for equilibrium point and yet non-constructive for finding one.

In this article,  I propose a framework for machine theorem discovery and illustrate its use in discovering state invariants in planning domains and properties about Nash equilibria in game theory. I also discuss its potential use in program verification in software engineering. The main message of the article is that many AI problems can and should be formulated as machine theorem discovery tasks.

We consider all possible games that have unique PNE payoffs. Our starting point is the classes of games that can be expressed by a conjunction class of two-person strictly competitive games. We first formulate of two binary clauses, and our program rediscovered the notions of games, strictly competitive games and Kats and Thisse's class of weakly unilaterally PNEs in first-order logic. Under our formulation, a class of competitive two-person games, and came games corresponds to a first-order sentence. In particular, the up with several other classes of games that have sentence that corresponds to the class of strictly competitive unique pure Nash equilibrium payoffs. It also came games is a conjunction of two binary clauses with all variables up with new classes of strict games that have unique universally quantified. So we implemented a program pure Nash equilibria, where a game is strict if for that examines all these universally quantified conjunctions of both player different profiles have different payoffs.