The prevalence of chaotic dynamics in games with many players

TL;DR

This paper investigates how adaptive learning in multi-player games often leads to complex behaviors like chaos, especially as the number of players and actions increases, challenging the assumption of stable equilibria.

Contribution

It introduces a generating-functional approach to analyze the convergence of learning dynamics and demonstrates that chaos becomes prevalent in large, complex games.

Findings

Learning dynamics often exhibit chaos in large games.

The stable fixed point region shrinks as the number of players increases.

Complex non-equilibrium behaviors are likely common in multi-player games with many actions.

Abstract

We study adaptive learning in a typical p-player game. The payoffs of the games are randomly generated and then held fixed. The strategies of the players evolve through time as the players learn. The trajectories in the strategy space display a range of qualitatively different behaviors, with attractors that include unique fixed points, multiple fixed points, limit cycles and chaos. In the limit where the game is complicated, in the sense that the players can take many possible actions, we use a generating-functional approach to establish the parameter range in which learning dynamics converge to a stable fixed point. The size of this region goes to zero as the number of players goes to infinity, suggesting that complex non-equilibrium behavior, exemplified by chaos, may be the norm for complicated games with many players.