Pricing with algorithms

Abstract

This paper studies Markov perfect equilibria in a repeated duopoly model where sellers choose algorithms. An algorithm is a mapping from the competitor's price to own price. Once set, the algorithms respond quickly. Customers arrive randomly and sellers can periodically revise their algorithms. The main results are that (i) for the simple two-price model with standard profit functions, monopoly pricing is the unique equilibrium outcome, and (ii) for general finite price grids, all equilibrium outcomes feature supra-competitive pricing. Sustenance of such collusion seems outside the scope of current antitrust laws for it does not involve any direct communication.

Keywords: algorithms, collusion, antitrust, bertrand, dynamic game