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.

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