An algorithm of game theory applied to the duopoly problem

In this note, an algorithm of Game Theory — Fictitious Play — is applied to a duopoly model: at each step of the algorithm, each duopolist chooses a quantity which maximizes his expected payoff, given the frequency distribution of his opponent's past choices. The algorithm is interpreted as a dynamic learning process and compared to Cournot's process. The successive pairs of quantities announced by the duopolists are proved to converge to a pair of non-cooperative equilibrium quantities. A more sophisticated version of the algorithm also has the same convergence properties.