Cycles of cooperation and defection in imperfect learning

TL;DR

This paper models learning dynamics in the iterated prisoner's dilemma, revealing how imperfect sampling and stochastic effects can lead to persistent cycles of cooperation and defection, contrasting with traditional equilibrium predictions.

Contribution

It introduces an analytical framework to predict oscillatory behavior in learning processes due to imperfect sampling, expanding understanding of stochastic adaptation in strategic interactions.

Findings

Cooperative behavior can emerge when players discount distant past observations.

Stochastic sampling of opponents' strategies can cause sustained oscillations between cooperation and defection.

The developed analytical method predicts properties of these learning cycles.

Abstract

When people play a repeated game they usually try to anticipate their opponents' moves based on past observations, and then decide what action to take next. Behavioural economics studies the mechanisms by which strategic decisions are taken in these adaptive learning processes. We here investigate a model of learning the iterated prisoner's dilemma game. Players have the choice between three strategies, always defect (ALLD), always cooperate (ALLC) and tit-for-tat (TFT). The only strict Nash equilibrium in this situation is ALLD. When players learn to play this game convergence to the equilibrium is not guaranteed, for example we find cooperative behaviour if players discount observations in the distant past. When agents use small samples of observed moves to estimate their opponent's strategy the learning process is stochastic, and sustained oscillations between cooperation and defection can emerge. These cycles are similar to those found in stochastic evolutionary processes, but the origin of the noise sustaining the oscillations is different and lies in the imperfect sampling of the opponent's strategy. Based on a systematic expansion technique, we are able to predict the properties of these learning cycles, providing an analytical tool with which the outcome of more general stochastic adaptation processes can be characterised.