Instability as insight: reinterpreting learning dynamics in repeated games through bounded rationality

The literature on repeated games has traditionally treated adaptive strategic behavior through frameworks such as reinforcement learning and evolutionary dynamics, both of which largely evaluate learning in terms of convergence to equilibrium or stochastic fluctuations around it. While analytically convenient, this perspective overlooks a fundamental feature of real multi-agent systems: agents operate under bounded cognition, finite computational resources, imperfect information, and continually changing strategic environments. As a result, their behavior is rarely perfectly stationary. This paper revisits Q-learning and replicator dynamics through the framework of bounded rationality, arguing that instability, oscillation, and partial convergence should not be viewed merely as failures of learning or artifacts of imperfect models. Rather, they are often natural consequences of adaptation under realistic constraints. Behaviors commonly dismissed as noise—including persistent cycling, metastability, and non-convergent trajectories—may instead contain information about strategic experimentation, exploration, and ongoing adaptation. Building on this perspective, we propose an initial operational framework for studying bounded-rational learning dynamics using tools from information theory and dynamical systems analysis, including entropy measures, Lyapunov exponents, information flow metrics, and continuous measures of rationality. These tools allow instability to be analyzed quantitatively rather than treated solely as a deviation from equilibrium. By shifting attention from equilibrium attainment to the structure of adaptation itself, this framework broadens the study of learning in repeated games and suggests new empirical, computational, and analytical approaches for investigating how agents learn within complex strategic environments. More generally, treating instability as a potentially informative feature rather than as residual noise may contribute to a richer mathematical understanding of adaptive flexibility in both artificial and natural systems.