Convergence to multiple price equilibria in algorithmic collusion

Abstract

Simulation studies of algorithmic collusion often summarise results through average prices and profts. Such summaries conceal the variety and distribution of distinct prices, cycles, and collusive regimes. This limitation is salient given that game theory predicts multiple equilibria, implying a distribution rather than a single outcome. This paper examines the heterogeneity hidden by aggregate metrics in a sequential pricing duopoly with Q learning agents. Using an established simulation framework for studying actions of pricing algorithms, the simulations cover three memory confgurations: symmetric one period, one period versus none, and one period versus two period memory. K means cluster analysis of steady state prices reveals a spectrum: monopoly focal price equilibria, collusive focal price equilibria at non monopoly grid points and incomplete Edgeworth cycles with supra competitive averages. These regimes are consistent with the multiplicity of the Markov perfect equilibria in the game theory literature. In a symmetric baseline, the monopoly outcome is a minority. Memory asymmetry eliminates or reduces cycling and increases the share of fxed price outcomes, stabilising rather than breaking collusion. Aggregate numbers collapse qualitatively distinct results into a single fgure; cluster analysis uncovers the hidden heterogeneity behind those averages and shows that algorithmic coordination spans a spectrum of regimes rather than a single outcome. For the competition policy, instruments that distinguish between pricing regimes are more informative than price level benchmarks alone.