A Coupling Approach to Analyzing Games With Dynamic Environments

The theory of learning in games has extensively studied situations where agents respond dynamically to each other in a static environment by optimizing a fixed utility function. However, real-world environments evolve as a result of past agent choices. Unfortunately, the analysis techniques that enabled a rich characterization of the emergent behavior of games played in static environments fail to cope with games played in dynamic environments. To address this problem, 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. We obtain conditions under which the emergent behavior of a dynamic game can be characterized by performing the traditional analysis on a reference static environment game. As a case study, we pose a model of cyber threat intelligence sharing between firms, which features a dynamic environment with complex history dependence.