  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 real situations, the strategic environment varies as a
result of past agent choices. Unfortunately, the analysis techniques that
enabled a rich characterization of the emergent behavior in static environment
games fail to cope with dynamic environment games. To address this, we develop
a general framework using probabilistic couplings to extend the analysis of
static environment games to dynamic ones. Using this approach, we obtain
sufficient conditions under which traditional characterizations of Nash
equilibria with best response dynamics and stochastic stability with log-linear
learning can be extended to dynamic environment games. As a case study, we pose
a model of cyber threat intelligence sharing between firms and a simple dynamic
game-theoretic model of social precautions in an epidemic, both of which
feature dynamic environments. For both examples, we obtain conditions under
which the emergent behavior is characterized in the dynamic game by performing
the traditional analysis on a reference static environment game.
