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

In this paper, we investigate the condition under which players of an adaptive learning model, including the one of stochastic fictitious play learning, learn to follow a logit quantal response equilibrium corresponding to a strict Nash equilibrium, the one which approaches the Nash equilibrium as the noise term approaches zero. In particular, we consider the situation in which adaptive players face a fixed normal form game repeatedly and may interact with committed players, who do not revise their behaviour and follow a specific strategy in each period. When committed players follow a logit quantal response equilibrium corresponding to a strict Nash equilibrium and the probability of each adaptive player interacting with committed players is large enough, we show that adaptive players learn to follow the equilibrium that committed players follow almost surely; we also show that any strict Nash equilibrium can be logit quantal response equilibrium approachable so that we can pick any strict Nash equilibrium for committed players to follow. Also, we show that when the probability of interacting with committed players is small, players of a more general adaptive learning model, including the ones of stochastic fictitious play learning, payoff assessment learning and delta learning may learn to follow an equilibrium different from the one committed players follow with positive probability. Lastly, we also consider the case in which there do not exist committed players 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.