Learning to Play Nash in Deterministic Uncoupled Dynamics

This paper is concerned with the following problem. In a bounded rational game where players cannot be as super-rational as in Kalai and Leher (1993), are there simple adaptive heuristics or rules that can be used in order to secure convergence to Nash equilibria, or convergence only to a larger set designated by correlated equilibria? Are there games with uncoupled deterministic dynamics in discrete time that converge to Nash equilibrium or not? Young (2008) argues that if an adaptive learning rule follows three conditions – (i) it is uncoupled, (ii) each player’s choice of action depends solely on the frequency distribution of past play, and (iii) each player’s choice of action, conditional on the state, is deterministic – no such rule leads the players’ behavior to converge to Nash equilibrium. In this paper we present a counterexample, showing that there are simple adaptive rules that secure convergence, in fact fast convergence, in a fully deterministic and uncoupled game. We used the Cournot model with nonlinear costs and incomplete information for this purpose and also illustrate that this convergence can be achieved with or without any coordination of the players actions.