The Spoils of Algorithmic Collusion: Profit Allocation Among Asymmetric Firms

TL;DR

This study uses Q-learning agents to simulate a two-firm, asymmetric Cournot duopoly and examines how output and profit sharing emerge when decisions are algorithmically driven. By calibrating seven degrees of asymmetry and several bargaining solutions, the authors show that asymmetry can increase total welfare for both consumers and producers relative to symmetric settings, and that algorithmic outcomes lie near the Pareto frontier, best described by equal relative gains. The static Nash equilibrium underestimates the impact of asymmetry on quantities but overestimates profits, while equal relative gains and Kalai-Smorodinsky provide the strongest explanations for observed allocations and their comparative statics. These findings challenge the belief that symmetry facilitates collusion and highlight important policy considerations as algorithms increasingly manage competitive decisions in digital markets.

Abstract

We study the propensity of independent algorithms to collude in repeated Cournot duopoly games. Specifically, we investigate the predictive power of different oligopoly and bargaining solutions regarding the effect of asymmetry between firms. We find that both consumers and firms can benefit from asymmetry. Algorithms produce more competitive outcomes when firms are symmetric, but less when they are very asymmetric. Although the static Nash equilibrium underestimates the effect on total quantity and overestimates the effect on profits, it delivers surprisingly accurate predictions in terms of total welfare. The best description of our results is provided by the equal relative gains solution. In particular, we find algorithms to agree on profits that are on or close to the Pareto frontier for all degrees of asymmetry. Our results suggest that the common belief that symmetric industries are more prone to collusion may no longer hold when algorithms increasingly drive managerial decisions.

Figures (14)

• Figure 1: Simulation results for different degrees of asymmetry across different outcome variables. The figure compares the simulation results to four theoretical benchmarks: Nash equilibrium, monopoly, alternating monopoly, and equal relative gains. For comparisons with additional theoretical benchmarks, see Figure \ref{['fig:res:1_all']} in Appendix \ref{['sec:additional-figures']}.
• Figure 2: Simulation results (comparative statics; symmetric case = 1) for different degrees of asymmetry across various outcome variables. The figure compares the simulation results to four theoretical benchmarks: Nash equilibrium, monopoly, alternating monopoly, and equal relative gains. For comparisons with additional theoretical benchmarks, see Figure \ref{['fig:res:comp:full']} in Appendix \ref{['sec:additional-figures']}.
• Figure 3: Pareto frontier and firm profits for our simulation results (black circles), equal relative gains (yellow squares), Nash equilibrium (purple squares), and min-max-disagreement (grey circles).
• Figure 4: Incentives to deviate based on one-period best response (moderate exploration)
• Figure B.1: Simulation results for different degrees of asymmetry (main results including high exploration; all theoretical predictions).