Brown-von Neumann-Nash Dynamics:The Continous Strategy Case

In John Nash’s proofs for the existence of (Nash) equilibria basedon Brouwer’s theorem, an iteration mapping is used. A continuous—time analogue of the same mapping has been studied even earlier byBrown and von Neumann. This differential equation has recently beensuggested as a plausible boundedly rational learning process in games.In the current paper we study this Brown—von Neumann—Nash dynamicsfor the case of continuous strategy spaces. We show that forcontinuous payoff functions, the set of rest points of the dynamics coincideswith the set of Nash equilibria of the underlying game. We alsostudy the asymptotic stability properties of rest points. While strictNash equilibria may be unstable, we identify sufficient conditions forlocal and global asymptotic stability which use concepts developed inevolutionary game theory.....