  The theory of learning in games has extensively studied situations where
agents respond dynamically to each other in light of 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 has
highlighted the need to formulate and analyze such models which feature
game-environment feedback. For instance, 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. What is the interplay between epidemic severity and the behaviors
of a victim population? For initial answers, we develop 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. We then 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.
