  The replicator equation in evolutionary game theory describes the change in a
population's behaviors over time given suitable incentives. It arises when
individuals make decisions using a simple learning process - imitation. A
recent emerging framework builds upon this standard model by incorporating
game-environment feedback, in which the population's actions affect a shared
environment, and in turn, the changing environment shapes incentives for future
behaviors. In this paper, we investigate game-environment feedback when
individuals instead use a boundedly rational learning rule known as logit
learning. We characterize the resulting system's complete set of fixed points
and their local stability properties, and how the level of rationality
determines overall environmental outcomes in comparison to imitative learning
rules. We identify a large parameter space for which logit learning exhibits a
wide range of dynamics as the rationality parameter is increased from low to
high. Notably, we identify a bifurcation point at which the system exhibits
stable limit cycles. When the population is highly rational, the limit cycle
collapses and a tragedy of the commons becomes stable.
