The Algorithmic Learning Equations: Evolving Strategies in Dynamic Games

48 Pages Posted: 4 Aug 2022 Last revised: 24 Oct 2022

See all articles by Álvaro Cartea

Álvaro Cartea

University of Oxford; University of Oxford - Oxford-Man Institute of Quantitative Finance

Patrick Chang

University of Oxford - Oxford-Man Institute of Quantitative Finance

José Penalva

Universidad Carlos III, Madrid - Department of Business Administration

Harrison Waldon

University of Texas at Austin; University of Oxford - Oxford-Man Institute of Quantitative Finance

Date Written: July 28, 2022

Abstract

We introduce the algorithmic learning equations, a set of ordinary differential equations which characterizes the finite-time and asymptotic behavior of the stochastic interaction between state-dependent learning algorithms in dynamic games. Our framework allows for a variety of information and memory structures, including noisy, perfect, private, and public monitoring and for the possibility that players use distinct learning algorithms. We prove that play converges to a correlated equilibrium for a family of algorithms under correlated private signals. Finally, we apply our methodology in a repeated 2x2 prisoner's dilemma game with perfect monitoring. We show that algorithms can learn a reward-punishment mechanism to sustain tacit collusion. Additionally, we find that algorithms can also learn to coordinate in cycles of cooperation and defection.

Keywords: Stochastic Approximation, Learning in Games, Reinforcement Learning, Dynamic Games, Tacit Collusion, Artificial Intelligence

JEL Classification: C60, C72, C73, D83

Suggested Citation

Cartea, Álvaro and Chang, Patrick and Penalva, José and Waldon, Harrison, The Algorithmic Learning Equations: Evolving Strategies in Dynamic Games (July 28, 2022). Available at SSRN: https://ssrn.com/abstract=4175239 or http://dx.doi.org/10.2139/ssrn.4175239

Álvaro Cartea

University of Oxford ( email )

Mansfield Road
Oxford, Oxfordshire OX1 4AU
United Kingdom

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

Eagle House
Walton Well Road
Oxford, Oxfordshire OX2 6ED
United Kingdom

Patrick Chang (Contact Author)

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

Eagle House
Walton Well Road
Oxford, Oxfordshire OX2 6ED
United Kingdom

José Penalva

Universidad Carlos III, Madrid - Department of Business Administration ( email )

Calle Madrid 126
Getafe, 28903
Spain

Harrison Waldon

University of Texas at Austin ( email )

Texas

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

Eagle House
Walton Well Road
Oxford, Oxfordshire OX2 6ED
United Kingdom

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