Stochastic adaptive learning with committed players in games with strict Nash equilibria

We investigate the convergence properties of an adaptive learning model that overlaps those of stochastic fictitious play learning and experience-weighted attraction learning in normal form games with strict Nash equilibria. In particular, we consider the case in which adaptive players play a game against not only other adaptive players but also committed players, who do not revise their behaviour but follow a fixed (strict Nash equilibrium or corresponding logit quantal response equilibrium) action. We then provide conditions under which the adaptive learning process, the choice probability profile of adaptive players, almost surely converges to the logit quantal response equilibrium that committed players follow. We also provide conditions under which the adaptive learning process of a more general adaptive learning model which overlaps those of payoff assessment learning and delta learning converges to a logit quantal response equilibrium different from the equilibrium that committed players follow with positive probability.