Learning to Collude in a Pricing Duopoly

We construct a price algorithm based on simultaneous-perturbation stochastic-approximation (SPSA) and show that, if implemented collaboratively by two price-setting firms in a duopoly, they will learn to form a cartel: their prices will converge to those that maximize the firms’ joint profit in case this is profitable for both firms, and to a Nash equilibrium otherwise. In addition, if the competitor is not willing to collaborate but behaves according to a reaction function, we show that the prices generated by our algorithm converge to a best-response to the competitor’s price. This is done without communication or explicit signaling, so that implementation of the algorithm would be legal under current competition law. Our analysis shows that collusion by self-learning price algorithms is in theory possible.