An oscillating tragedy of the commons in replicator dynamics with game-environment feedback.
A tragedy of the commons occurs when individuals take actions to maximize their payoffs even as their combined payoff is less than the global maximum had the players coordinated. The originating example is that of overgrazing of common pasture lands. In game-theoretic treatments of this example, there is rarely consideration of how individual behavior subsequently modifies the commons and associated payoffs. Here, we generalize evolutionary game theory by proposing a class of replicator dynamics with feedback-evolving games in which environment-dependent payoffs and strategies coevolve. We initially apply our formulation to a system in which the payoffs favor unilateral defection and cooperation, given replete and depleted environments, respectively. Using this approach, we identify and characterize a class of dynamics: an oscillatory tragedy of the commons in which the system cycles between deplete and replete environmental states and cooperation and defection behavior states. We generalize the approach to consider outcomes given all possible rational choices of individual behavior in the depleted state when defection is favored in the replete state. In so doing, we find that incentivizing cooperation when others defect in the depleted state is necessary to avert the tragedy of the commons. In closing, we propose directions for the study of control and influence in games in which individual actions exert a substantive effect on the environmental state.
School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA 30332; jsweitz@gatech.edu.|School of Physics, Georgia Institute of Technology, Atlanta, GA 30332.
Joshua S
JS
Weitz
School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA 30332.|School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332.
Ceyhun
C
Eksin
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332.
Keith
K
Paarporn
School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA 30332.
Sam P
SP
Brown
School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA 30332.
William C
WC
Ratcliff
D005075:Biological Evolution; D003299:Cooperative Behavior; D030141:Feedback, Psychological; D005716:Game Theory; D006801:Humans; D017711:Nonlinear Dynamics
cooperation; environmental dynamics; evolutionary games; game theory; nonlinear dynamics