Bayesian fictitious play in oligopoly : the case of risk-averse agents

A number of learning models have been suggested to analyze the repeated interaction of boundedly rational agents competing in oligopolistic markets. The agents form a model of the environment that they are competing in, which includes the market demand and price formation process, as well as their expectations of their rivals' actions. The agents update their model based on the observed output and price realizations and then choose their next period output levels according to an optimization criterion. In previous works, the global dynamics of price movement have been analyzed when risk-neutral agents maximize their expected rewards at each round. However, in many practical settings, agents may be concerned with the risk or uncertainty in their reward stream, in addition to the expected value of the future rewards. Learning in oligopoly models for the case of risk-averse agents has received much less attention. In this paper, we present a novel learning model that extends fictitious play learning to continuous strategy spaces where agents combine their prior beliefs with market price realizations in previous periods to learn the mean and the variance of the aggregate supply function of the rival firms in a Bayesian framework. Next, each firm maximizes a linear combination of the expected value of the profit and a penalty term for the variance of the returns. Specifically, each agent assumes that the aggregate supply of the remaining agents is sampled from a parametric distribution employing a normal-inverse gamma prior. We prove the convergence of the proposed dynamics and present simulation results to compare the proposed learning rule to the traditional best response dynamics.