The Fan minimax inequality implies the Nash equilibrium theorem

Approximating Nash Equilibrium Via Multilinear Minimax

On the one hand we state {\it Nash equilibrium} (NE) as a formal theorem on multilinear forms and give a pedagogically simple proof, free of game theory terminology. On the other hand, inspired by this formalism, we prove a {\it multilinear minimax theorem}, a generalization of von Neumann’s bilinear minimax theorem. Next, we relate the two theorems by proving that the solution of a multilinear...

Generalizations of the Nash Equilibrium Theorem in the KKM Theory

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. Th...

Construction of bipotentials and a minimax theorem of Fan

The bipotential theory is based on an extension of Fenchel’s inequality (see section 1 and example 1 in section 2). Despite several powerful applications (frictional contact [6], non-associated Drucker-Prager model [1], or Lemaitre plastic ductile damage law [2], to cite a few), the bipotentials don’t have yet a complete mathematical treatment. This is a second paper on the mathematics of the b...

A general structure theorem for the Nash equilibrium correspondence

I consider n–person normal form games where the strategy set of every player is a non–empty compact convex subset of Euclidean space, and the payoff function of player i is continuous and concave in player i’s own strategies. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. In this setting we demonstrat...

The fan theorem