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...

The COVID-19 pandemic has caused more than 1,000,000 reported deaths globally, of which more than 200,000 have been reported in the United States as of October 1, 2020. Public health interventions have had significant impacts in reducing transmission and in averting even more deaths. Nonetheless, in many jurisdictions, the decline of cases and fatalities after apparent epidemic peaks has not been rapid. Instead, the asymmetric decline in cases appears, in most cases, to be consistent with plateau- or shoulder-like phenomena—a qualitative observation reinforced by a symmetry analysis of US state-level fatality data. Here we explore a model of fatality-driven awareness in which individual protective measures increase with death rates. In this model, fast increases to the peak are often followed by plateaus, shoulders, and lag-driven oscillations. The asymmetric shape of model-predicted incidence and fatality curves is consistent with observations from many jurisdictions. Yet, in contrast to model predictions, we find that population-level mobility metrics usually increased from low levels before fatalities reached an initial peak. We show that incorporating fatigue and long-term behavior change can reconcile the apparent premature relaxation of mobility reductions and help understand when post-peak dynamics are likely to lead to a resurgence of cases.

Individuals change their behavior during an epidemic in response to whether they and/or those they interact with are healthy or sick. Healthy individuals may utilize protective measures to avoid contracting a disease. Sick individuals may utilize preemptive measures to avoid spreading a disease. Yet, in practice both protective and preemptive changes in behavior come with costs. This paper proposes a stochastic network disease game model that captures the self-interests of individuals during the spread of a susceptible-infected-susceptible disease. In this model, individuals strategically modify their behavior based on current disease conditions. These reactions influence disease spread. We show that there is a critical level of concern, i.e., empathy, by the sick individuals above which disease is eradicated rapidly. Furthermore, we find that risk averse behavior by the healthy individuals cannot eradicate the disease without the preemptive measures of the sick individuals. Empathy is more effective than risk-aversion because when infectious individuals change behavior, they reduce all of their potential infections, whereas when healthy individuals change behavior, they reduce only a small portion of potential infections. This imbalance in the role played by the response of the infected versus the susceptible individuals on disease eradication affords critical policy insights.

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Consumer demand profiles and fluctuating renewable power generation are two main sources of uncertainty in matching demand and supply. This paper proposes a model of the electricity market that captures the uncertainties on both the operator and user sides. The system operator (SO) implements a temporal linear pricing strategy that depends on real-time demand and renewable generation in the considered period combining real-time pricing with time-of-use pricing. The announced pricing strategy sets up a noncooperative game of incomplete information among the users with heterogeneous, but correlated consumption preferences. An explicit characterization of the optimal user behavior using the Bayesian Nash equilibrium solution concept is derived. This explicit characterization allows the SO to derive pricing policies that influence demand to serve practical objectives, such as minimizing peak-to-average ratio or attaining a desired rate of return. Numerical experiments show that the pricing policies yield close to optimal welfare values while improving these practical objectives.Index Terms-Demand response management (DRM), game theory, renewable energy.