Robust Stochastic Stability in Dynamic and Reactive Environments

TL;DR

This paper introduces a probabilistic coupling framework to analyze stochastically stable states in dynamic games with feedback, exemplified by social precautions during an epidemic.

Contribution

It develops a novel probabilistic coupling approach to determine stability in games where utilities evolve with past actions, extending learning theory to feedback environments.

Findings

Maximally cautious social behavior can be stochastically stable in epidemic models.

The framework applies to games with environment feedback, broadening stability analysis.

Conditions for stability depend on feedback dynamics and utility changes.

Abstract

The theory of learning in games has extensively studied situations where agents respond dynamically to each other by optimizing a fixed utility function. However, in many settings of interest, agent utility functions themselves vary as a result of past agent choices. The ongoing COVID-19 pandemic provides an example: a highly prevalent virus may incentivize individuals to wear masks, but extensive adoption of mask-wearing reduces virus prevalence which in turn reduces individual incentives for mask-wearing. This paper develops a general framework using probabilistic coupling methods that can be used to derive the stochastically stable states of log-linear learning in certain games which feature such game-environment feedback. As a case study, we apply this framework to a simple dynamic game-theoretic model of social precautions in an epidemic and give conditions under which maximally cautious social behavior in this model is stochastically stable.