Stochastic Adaptive Learning With Committed Players in Games With Strict Nash Equilibria

Abstract

In this paper, we investigate the condition under which players of an adaptive learning model, overlapping those of stochastic fictitious play learning and experience-weighted attraction learning, learn to follow a logit quantal response equilibrium corresponding to a strict Nash equilibrium. In particular, we consider the situation in which a pair is randomly chosen from not only adaptive players but also committed players, who do not revise their behaviour and follow a fixed (mixed) action, to play a fixed normal form game in each period. When committed players follow a logit quantal response equilibrium corresponding to a strict Nash equilibrium, we show that if the probabilities of adaptive players facing other adaptive players are small enough, adaptive players learn to follow the equilibrium that committed players follow almost surely. Also, we show that if the probabilities are large enough, adaptive players of a more general adaptive learning model, overlapping those of payoff assessment learning and delta learning, may learn to follow an equilibrium different from the one that committed players follow with positive probability. Lastly, we also consider the case in which committed players do not exist and show that when players of the general adaptive learning model have enough experience and their behaviour is close enough to a logit quantal response equilibrium corresponding to a strict Nash equilibrium, then their behaviour converges to the equilibrium with probability close to one.

Keywords: Adaptive learning; stochastic fictitious play learning; experience-weighted attraction learning; quantal response equilibrium; stochastic approximation; equilibrium selection

JEL Classification: C72, D83