Best Response Dynamics and Neural Networks

We consider a population of players in a setting that allows to analyze local as well as global interaction. Using the formalism of automata networks we show that best response is a special case of a biased majority (minority) imitation. For a population of best response players we first discuss the known properties of the deterministic dynamics as a preparation and reference for stochastic dynamics. The stochastic dynamics of the system will always have a stationary distribution. It turns out that in a special case of asynchronous updating and logistic noise this distribution is of Boltzmann type. We further show that with a Boltzmann distribution, the long-run equilibria are associated with a minimum of a cost function defined in the paper. Comparison of our results with the existing literature suggests robustness of the previous long-run equilibrium results. In the case of differentiation games, we demonstrate the sensivity of long-run equilibrium to the choice of interaction structure. ∗Department of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA. Email: [email protected]. †Bios Group L.P., 317 Paseo de Peralta, Santa Fe, NM 87501. Email: [email protected].