Brown–von Neumann–Nash dynamics: The continuous strategy case

Abstract

In John Nash's proofs for the existence of (Nash) equilibria based on Brouwer's theorem, an iteration mapping is used. A continuoustime analogue of the same mapping has been studied even earlier by Brown and von Neumann. This differential equation has recently been suggested as a plausible boundedly rational learning process in games. In the current paper we study this Brown-von Neumann-Nash dynamics for the case of continuous strategy spaces. We show that for continuous payoff functions, the set of rest points of the dynamics coincides with the set of Nash equilibria of the underlying game. We also study the asymptotic stability properties of rest points. While strict Nash equilibria may be unstable, we identify sufficient conditions for local and global asymptotic stability which use concepts developed in evolutionary game theory.Journal of Economic Literature Classification Numbers: C70, 72.

Cumulative weighting optimization

“…Proof The outline of our proof follows [12] and [13]. The total variation norm on a σ -finite signed measure P over R + , B R + at time t is denoted by:…”

Independent random matching

“…Thus, games with continuous strategy spaces involve random matching with a continuum of types. Examples can be found in Sandholm (2001), Oechssler and Riedel (2002), Hofbauer et al (2008). In these games, the distribution of strategies in the population is given by a probability distribution on the strategy space S, written as τ.…”

“…Moreover, for a wide class of random matching models with a continuum population, it is not possible to capture the relevant attributes of all the agents in a finite type space. This is the case, e.g., for the models described in Cavalcanti and Puzzello (2010), Green and Zhou (2002), Lagos and Wright (2005), Molico (2006), Shi (1997), Zhu (2005), where there are no upper bounds on money holdings or money holdings are perfectly divisible, or those described in Hofbauer et al (2008), Oechssler and Riedel (2002), Sandholm (2001), van Veelen and Spreij (2009), where infinite strategy sets matter.…”

On the observational equivalence of random matching

“…12 Fundamental strategies prescribe that lower-storage-cost commodities are preferred to higher-storage-cost commodities unless the higher-storage-cost commodity is the agent's consumption good. Letting V t 21 denote the expected discounted utility for type 21 , we have…”

“…13 The aggregate law of motion for types 21 14 These expressions suggest that the expected payo¤ and the law of motion equations implicitly use a matching process that satis…es properties M2 and M3 generalized to the augmented notion of type including commodity holdings in addition to specialization types. 8 Essential in the sense that it is possible to support better allocations as equilibria with money than without money.…”