Algorithmic collusion under asynchronous price updating

This paper investigates the effect of asynchrony in agents' updates in the emergence of algorithmic collusion. We present a continuous-time model for algorithmic collusion in which two firms use $Q$-learning algorithms to set prices asynchronously in a Bertrand duopoly. The firms update their prices at times dictated by a Poisson clock. By controlling the extent of agents' asynchrony, we run extensive numerical experiments with three specifications of the algorithm to investigate the emergence of algorithmic collusion. The strength of collusion is measured by a standard collusion index, as well as by automatically detecting the reward-punishment schemes. This is done by recording a large number of algorithms' reactions to unilateral price cuts and comparing them with the reactions of untrained algorithms. Our findings indicate that asynchrony hampers collusion, especially when the algorithms are stateless. When they condition on their competitor's previous prices, the sensitivity of algorithmic collusion to asynchrony varies depending on the type of information they have access to. The implications of these results for the regulation of algorithmic pricing are discussed.