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 environmentdependent 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.evolutionary games | game theory | cooperation | nonlinear dynamics | environmental dynamics G ame theory is based on the principle that individuals make rational decisions regarding their choice of actions given suitable incentives (1, 2). In practice, the incentives are represented as strategy-dependent payoffs. Evolutionary game theory extends game-theoretic principles to model dynamic changes in the frequency of strategists (3). Replicator dynamics is one commonly used framework for such models. In replicator dynamics, the frequencies of strategies change as a function of the social makeup of the community (4-6). For example, in a snowdrift game (also known as a hawk-dove game), individuals defect when cooperators are common but cooperate when cooperators are rare (2). As a result, cooperation is predicted to be maintained among a fraction of the community (4, 6). In contrast, in the prisoner's dilemma (PD), individuals are incentivized to defect irrespective of the fraction of cooperators. This leads to domination by defectors (6, 7).Here, we are interested in a different kind of evolutionary game in which individual action modifies both the social makeup and environmental context for subsequent actions. Strategydependent feedback occurs across scales from microbes to humans in public good games and in commons' dilemmas (8-11). Among microbes, feedback may arise due to fixation of inorganic nutrients given depleted organic nutrient availability (12, 13), the production of extracellular nutrient-scavenging enzymes like siderophores (14-16) or enzymes like inver...
In a simple susceptible-infected-recovered (SIR) model, the initial speed at which infected cases increase is indicative of the long-term trajectory of the outbreak. Yet during real-world outbreaks, individuals may modify their behavior and take preventative steps to reduce infection risk. As a consequence, the relationship between the initial rate of spread and the final case count may become tenuous. Here, we evaluate this hypothesis by comparing the dynamics arising from a simple SIR epidemic model with those from a modified SIR model in which individuals reduce contacts as a function of the current or cumulative number of cases. Dynamics with behavior change exhibit significantly reduced final case counts even though the initial speed of disease spread is nearly identical for both of the models. We show that this difference in final size projections depends critically in the behavior change of individuals. These results also provide a rationale for integrating behavior change into iterative forecast models. Hence, we propose to use a Kalman filter to update models with and without behavior change as part of iterative forecasts. When the ground truth outbreak includes behavior change, sequential predictions using a simple SIR model perform poorly despite repeated observations while predictions using the modified SIR model are able to correct for initial forecast errors. These findings highlight the value of incorporating behavior change into baseline epidemic and dynamic forecast models.
We study an SIS epidemic process over a static contact network where the nodes have partial information about the epidemic state. They react by limiting their interactions with their neighbors when they believe the epidemic is currently prevalent. A node's awareness is weighted by the fraction of infected neighbors in their social network, and a global broadcast of the fraction of infected nodes in the entire network. The dynamics of the benchmark (no awareness) and awareness models are described by discrete-time Markov chains, from which mean-field approximations (MFA) are derived. The states of the MFA are interpreted as the nodes' probabilities of being infected. We show a sufficient condition for existence of a "metastable", or endemic, state of the awareness model coincides with that of the benchmark model. Furthermore, we use a coupling technique to give a full stochastic comparison analysis between the two chains, which serves as a probabilistic analogue to the MFA analysis. In particular, we show that adding awareness reduces the expectation of any epidemic metric on the space of sample paths, e.g. eradication time or total infections. We characterize the reduction in expectations in terms of the coupling distribution. In simulations, we evaluate the effect social distancing has on contact networks from different random graph families (geometric, Erdős-Renyi, and scale-free random networks).
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 over-grazing 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 new 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 new directions for the study of control and influence in games in which individual actions exert a substantive effect on the environmental state.Game theory is based on the principle that individuals make rational decisions regarding their choice of actions given suitable incentives [1, 2]. In practice, the incentives are represented as strategy-dependent payoffs. Evolutionary game theory extends game theoretic principles to model dynamic changes in the frequency of strategists [3]. Replicator dynamics is one commonly used framework for such models. In replicator dynamics, the frequencies of strategies change as a function of the social makeup of the community [4][5][6]. For example in a snowdrift game (also known as a hawk-dove game), individuals defect when cooperators are common but cooperate when cooperators are rare [2]. As a result, cooperation is predicted to be maintained amongst a fraction of the community [4, 6]. Whereas, in the prisoner's dilemma individuals are incentivized to defect irrespective of the fraction of cooperators. This leads to domination by defectors [6, 7].Here, we are interested in a different kind of evolutionary game in which individual action modifies both the social makeup and environmental context for subsequent actions. Strategy-dependent feedback occurs across scales from microbes to humans in public good games and in commons' dilemmas [8][9][10][11]. Amongst microbes, feedback may arise due to fixation of inorganic nutrients given depleted organic nutrient availability [12, 13], the production of extracellular nutrientscavenging enzymes like siderophores [14][15][16] or enzymes like invertase that hydrolyze diffusible products [17], and * Electronic address: jsweit...
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