Optimal control policies for evolutionary dynamics with environmental feedback

We study a dynamical model of a population of cooperators and defectors whose actions have long-term consequences on environmental "commons" - what we term the "resource". Cooperators contribute to restoring the resource whereas defectors degrade it. The population dynamics evolve according to a replicator equation coupled with an environmental state. Our goal is to identify methods of influencing the population with the objective to maximize accumulation of the resource. In particular, we consider strategies that modify individual-level incentives. We then extend the model to incorporate a public opinion state that imperfectly tracks the true environmental state, and study strategies that influence opinion. We formulate optimal control problems and solve them using numerical techniques to characterize locally optimal control policies for three problem formulations: 1) control of incentives, and control of opinions through 2) propaganda-like strategies and 3) awareness campaigns. We show numerically that the resulting controllers in all formulations achieve the objective, albeit with an unintended consequence. The resulting dynamics include cycles between low and high resource states - a dynamical regime termed an "oscillating tragedy of the commons". This outcome may have desirable average properties, but includes risks to resource depletion. Our findings suggest the need for new approaches to controlling coupled population-environment dynamics.Comment: Initial submission version to CDC 201

The Impact of Irrational Behaviours in the Optional Prisoner's Dilemma with Game-Environment Feedback

In the optional prisoner's dilemma (OPD), players can choose to cooperate and defect as usual, but can also abstain as a third possible strategy. This strategy models the players' participation in the game and is a relevant aspect in many settings, e.g. social networks or opinion dynamics where abstention is an option during an election. In this paper, we provide a formulation of the OPD where we consider irrational behaviours in the population inspired by prospect theory. Prospect theory has gained increasing popularity in recent times thanks to its ability to capture aspects such as reference dependence or loss aversion which are common in human behaviour. This element is original in our formulation of the game and is incorporated in our framework through pairwise comparison dynamics. Recently, the impact of the environment has been studied in the form of feedback on the population dynamics. Another element of novelty in our work is the extension of the game-environment feedback to the OPD in two forms of dynamics, the replicator and the pairwise comparison. The contribution of this paper is threefold. First, we propose a modelling framework where prospect theory is used to capture irrational behaviours in an evolutionary game with game-environment feedback. Second, we carry out the stability analysis of the system equilibria and discuss the oscillating behaviours arising from the game-environment feedback. Finally, we extend our previous results to the OPD and we discuss the main differences between the model resulting from the replicator dynamics and the one resulting from the pairwise comparison dynamics.Comment: 12 pages, 5 figure