Coevolutionary dynamics via adaptive feedback in collectiverisk social dilemma game
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The paper provides a valuable, indepth mathematical analysis of the coevolutionary dynamics resulting from a coupling of players' strategies and (collective) risk, as well as illustrative numerical simulations of the system's trajectories for different starting conditions. It is therefore a solid contribution to our understanding of how cooperation can be sustained when there is feedback between individual decisions and the global risk of disaster. This paper will be of interest to scientists working on mathematical biology/ecology, and more generally various aspects of human decisionmaking, the interplay between human decisions and the environment, and public goods provision.
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Abstract
Human society and natural environment form a complex giant ecosystem, where human activities not only lead to the change of environmental states, but also react to them. By using collectiverisk social dilemma game, some studies have already revealed that individual contributions and the risk of future losses are inextricably linked. These works, however, often use an idealistic assumption that the risk is constant and not affected by individual behaviors. We here develop a coevolutionary game approach that captures the coupled dynamics of cooperation and risk. In particular, the level of contributions in a population affects the state of risk, while the risk in turn influences individuals’ behavioral decisionmaking. Importantly, we explore two representative feedback forms describing the possible effect of strategy on risk, namely, linear and exponential feedbacks. We find that cooperation can be maintained in the population by keeping at a certain fraction or forming an evolutionary oscillation with risk, independently of the feedback type. However, such evolutionary outcome depends on the initial state. Taken together, a twoway coupling between collective actions and risk is essential to avoid the tragedy of the commons. More importantly, a critical starting portion of cooperators and risk level is what we really need for guiding the evolution toward a desired direction.
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eLife assessment
The paper provides a valuable, indepth mathematical analysis of the coevolutionary dynamics resulting from a coupling of players' strategies and (collective) risk, as well as illustrative numerical simulations of the system's trajectories for different starting conditions. It is therefore a solid contribution to our understanding of how cooperation can be sustained when there is feedback between individual decisions and the global risk of disaster. This paper will be of interest to scientists working on mathematical biology/ecology, and more generally various aspects of human decisionmaking, the interplay between human decisions and the environment, and public goods provision.

Reviewer #1 (Public Review):
This is a quite nice work equipped with healthy scientific substance underpinned by a solid mathematical approach.
The authors based on a PGG with the threshold; M (that ranges; 1 < M < N, where N indicates the game size), whether cooperation bringing fruit or not, in which, according to the commonly used parameterization, b and c mean the cooperation fruit and the cost for cooperation. As a kernel in their model, they presumed that an individual will lose his endowment (cooperation fruit in this context) with a probability r, which represents the risk level of collective failure (Eqs. (1 & 2)). Let alone, they presumed a wellmixed and infinite motherpopulation to ensure their analytical formulation and analysis, and to apply the replicator dynamics. Subsequently, they presumed the coevolution of …
Reviewer #1 (Public Review):
This is a quite nice work equipped with healthy scientific substance underpinned by a solid mathematical approach.
The authors based on a PGG with the threshold; M (that ranges; 1 < M < N, where N indicates the game size), whether cooperation bringing fruit or not, in which, according to the commonly used parameterization, b and c mean the cooperation fruit and the cost for cooperation. As a kernel in their model, they presumed that an individual will lose his endowment (cooperation fruit in this context) with a probability r, which represents the risk level of collective failure (Eqs. (1 & 2)). Let alone, they presumed a wellmixed and infinite motherpopulation to ensure their analytical formulation and analysis, and to apply the replicator dynamics. Subsequently, they presumed the coevolution of cooperation fraction; x, and risk level; r, by introducing another dynamical system for r, of which the general form is defined by Eq. (3).
For a downtoearth discussion, they presumed two types of concrete forms for nonlinear function; U(x,r). Both types premise the socalled logistic type form; containing r*(1  r). One is whattheycalled Linear; Eq. (5). Another is Eq. (7), called Exponential. Up to here, all the modeling approach is well depicted and quite understandable.
By exploring some numerical results backed by their theoretical ground, the authors got phase diagram (Figs. 3 and 5); whether a coevolutionary destiny evaluated by (x,r) being absorbed by the dominance of unwilling (less cooperative) situation (say, Ddominant); (0,1), or by bistable equilibrium (either better state or Ddominant depending on an initial condition) along u (parameter appeared in the dynamical equation for r) and c/b (roughly speaking; it implies dilemma strength).
The result seems interesting and conceivable. As a rough sketch, the two types of U(x,r) seem less different. But the higher absorbing point of (x,r) out of the two cases of bistable equilibria is mutually different (yellow region). The authors deliberately illustrated the timeseries of properties and trajectory of (x,r) in some representative cases in Figs. 4 and 6.
As a whole, I really evaluate this work as impressive.

Reviewer #2 (Public Review):
Liu, Chen and Szolnoki investigated the coupled dynamics of individual cooperation level and collective risk (i.e. the probability of future loss of all endowment). Their model encapsulates the assumption that not only does risk affect individual decisionmaking, but that there is also feedback between individual strategies, i.e. the level of individual contributions, and the level of risk. The authors investigate two main forms of this feedback, considering strategies linearly affecting the evolution of risk as well as nonlinear (exponential) feedback. They mathematically analyze both these dynamical systems, identifying the fixed points, parametrized by the enhancement rate of defection u and the cost/benefit ratio of cooperation, and analyzing the stability of these points. The results of this systematic …
Reviewer #2 (Public Review):
Liu, Chen and Szolnoki investigated the coupled dynamics of individual cooperation level and collective risk (i.e. the probability of future loss of all endowment). Their model encapsulates the assumption that not only does risk affect individual decisionmaking, but that there is also feedback between individual strategies, i.e. the level of individual contributions, and the level of risk. The authors investigate two main forms of this feedback, considering strategies linearly affecting the evolution of risk as well as nonlinear (exponential) feedback. They mathematically analyze both these dynamical systems, identifying the fixed points, parametrized by the enhancement rate of defection u and the cost/benefit ratio of cooperation, and analyzing the stability of these points. The results of this systematic analysis show that, while the undesirable equilibrium state of full defection and high risk is always stable independent of the form of the feedback, the coevolutionary dynamics can exhibit a wide range of behaviors. In particular, depending on the initial conditions (frequency of cooperators), sustainable cooperation levels can be reached. This can happen by convergence to a stable fixed point with positive cooperation rates; additionally, the authors also prove that a Hopf bifurcation can take place in the system, such that a stable limit cycle with persistent oscillations in strategy and risk state can appear. Interestingly, the evolutionary outcomes do not depend significantly on the character of the feedback between strategy and risk. These theoretical results are supplemented by representative numerical examples, visualizing the phase plane and temporal dynamics of cooperation and risk for particular initial conditions and parameters.
The main conclusions of the paper are fully supported by the results, as they are directly derived from the comprehensive mathematical analysis of the coevolutionary dynamics and do not rely on external data. Additionally, the stability analysis is clean and the comprehensive numerical examples deepen the reader's understanding. Another strength of the paper is the fact that the considered model is complex enough to be able to still represent somewhat realistic settings while being simple enough to rigorously analyze. One particularly interesting finding is the fact that the exact form of the risk feedback function or its speed does not play a very significant role in the outcome of the dynamics.
The paper hence adds to the literature on the coevolution of environment and strategies in a productive way and will be of interest to various research communities in mathematical biology/ecology and decisionmaking.
