Climate collective risk dilemma with feedback of real-time temperatures

Although the change of global mean temperature may be too small to be considered in a short time, the long-term accumulation without control can bring out unpredictable disasters [1–5]. In fact, various phenomena of climatic anomalies have been observed, although it is only a $0.75\ ^{\circ}\text{C}$ rise of global mean temperature compared with that of about 100 years ago. Therefore, every little increase of temperature may reduce the fortune of everyone in the long run. It is a sensible requirement that to prevent global warming is an extremely urgent issue. Otherwise, the ever-rising temperature may cause countless disasters, such as polar ice melting, sea level rise and loss of bio-diversity. This task is so huge that it cannot be done without the cooperation among all the nations [6–10].

Previous game-theoretic works usually model the climate change as public goods game (PGG) or collective risk game [7–20]. In those models, global warming is depicted as a sudden event, which may happen with a given probability when the collective target is not met. Based on these models, one of the main results is that high risk can significantly promote cooperation. Global warming, however, is not an event which happens suddenly. The temperature changes day by day. Therefore, the probability of risk changes with time. In reality, the environment, which we particularly refer to the temperature, is the direct control target. Thus the dynamics of temperature should be modeled explicitly. Motivated by this, we propose a model with the dynamics of human behaviour and temperature coupled. Based on the model, we investigate: i) whether the temperature can be stabilized to a relatively safe level in the long run; ii) how the temperature dynamics influences individual's decision-making.

We assume that individuals are engaged in a threshold public goods game. Herein the temperature and the individual's strategy evolve simultaneously. When only a few individuals cooperate, the temperature rises. The individuals' payoffs are gradually discounted by the ever-rising temperature. In contrast with previous models, the temperature is continuously changing during the human strategy updating process in our model. We study the interplay between the cooperation level and the real-time temperature change during the evolutionary process via simulation aided by analytical approximation.

We consider a well-mixed population with finite individuals of size N. In particular, each individual represents a country or international organization in the global public goods game. Each individual k initially has a single unit of money, and it is assigned a strategy $S_{k}\in\{C, D\}$. Here C and D represent cooperation and defection, respectively. Players simultaneously decide whether to contribute their money to the common pool. The D individual shares nothing, which leads to a single unit eventually to itself. The C players contribute part of their fortune, $c\ (0<c<1)$. All the contributions in the account are totalled, then multiplied by a gain-factor $r\ (1<r<N)$, and equally distributed to all the individuals, irrespective of whether they are cooperators or not (see fig. 1).

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In this model, except for producing direct payoffs, public goods also account for preventing potential environmental risks, such as global warming. We introduce the temperature $(T)$ as a real-time environmental variable. The amount of public goods from the game, $c\,r\,i$, will influence the temperature, where i is the number of cooperators in the population. When the public goods do not reach a threshold limit, θ, the temperature will rise; otherwise, the temperature may fall and finally maintain at a reference level, T_R. We assume that the function takes effect when $T(t)>T_{R}$, and denote $\Delta T=T(t)-T_R$. This suggests that the most successful control of anthropogenic GHG emission will help maintain the temperature at the most optimum situation, T_R (the universally approved cosy temperature is about $14\ ^{\circ}\text{C}$), but not any lower than such level. We take a simplified example of temperature change function as shown in eq. (1), to depict its relationship with the amount of public goods. The function is simple but effectively portrays some certain typical characters of climate change:

$$\begin{eqnarray} T(t+1)=\begin{cases} T(t)\!+\![e^{\alpha(\theta-c\,r\,i)}\!-\!1] T(t)^\beta, &T(t)>T_R, \\ T_R,&\text{otherwise}, \end{cases} \end{eqnarray} \tag{ 1 }$$

where T(t) is the temperature of current time t, $T(t+1)$ is that of the next time step, α and β are constant coefficients which affect the rising rate of the temperature.

Owing to the feedback effect, the decision-making process in the game is sensitive to temperature changes. The update of individual's strategy is synchronous with the temperature change. The payoff of each individual is not only dependent on the interaction with counterparts, but also intervened by the rising temperature. The higher the current temperature is, the more the payoffs discounted. We denote the payoffs of cooperators and defectors, respectively, as follows:

$$\begin{eqnarray} \pi_{C}(i)&=\frac{1+\frac{c\,r\,i}{N}-c}{e^{(\gamma\,\Delta T)}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \pi_{D}(i)&=\frac{1+\frac{c\,r\,i}{N}}{e^{(\gamma\,\Delta T)}},\end{eqnarray} \tag{ 3 }$$

where γ is a constant coefficient, which denotes the decrease rate of payoffs owing to the temperature rise. For bigger γ, the rising temperature will remarkably lessen the payoffs. For smaller γ, the change of temperature has little effect on the payoffs. In particular, $\gamma=0$ can be deemed as a limit case with no feedback. In such case, only the individuals' behaviours influence the temperature but the temperature does not affect the individuals' behaviours.

The evolution of strategies is described as follows. We take the imitation process with explorations as the strategy updating rule [21–27]. An individual A is chosen randomly. It switches to the other strategy with the exploration probability μ. With probability $1-\mu$, it imitates another randomly chosen individual B with a probability based on the payoff difference, i.e., $1/[1+e^{-\omega(\pi_{B}-\pi_{A})}]$. Here $\pi_{A}$ and $\pi_{B}$ are the payoffs of individuals A and B, and ω denotes the selection intensity. For $\omega\rightarrow0$, an individual imitates the strategy of others almost randomly, which is deemed to be "weak selection". For ω → ∞, a more successful player is always imitated, which can be seen as "strong selection". Realizing that selection intensity can dramatically change the evolutionary outcome [28], we discuss a wide range of selection intensities. In particular, we concentrate on the moderate selection intensity, with which human beings might adjust their strategies [27].

First of all, it is found that the temperature can be stabilized at a relatively safe level by our human beings in the long run. The evolutionary results are shown in fig. 2, therein the temperature and cooperation level are both stable, though fluctuations and sudden rising occur from time to time. With the change of temperature, the individuals adjust their behaviours and switch their strategies correspondingly. The typical microscopic dynamics of temperature is depicted in fig. 3. At the reference temperature, the individuals try to pursue higher short-term benefits, then they are more likely to defect. The consequence is the temperature rising with the decline of the cooperation level. Once the temperature rises, the payoffs of individuals decline, which induces them to regulate their behaviours in the following decision-making process. When the cooperation level rises and the public goods exceed the threshold, the temperature will descend towards the reference, otherwise it will continuously rise. The more the public goods exceed the threshold, the more effectively the temperature is regulated. The cyclical recurring of such fluctuation and regulation process can be frequently observed.

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In our model, cooperative behaviours can emerge although defection is the dominant strategy in the public goods game. By introducing the temperature feedback mechanism, the average abundance of the cooperation strategy is stable (see fig. 2(B)). Rational individuals naturally tend to take free-ride and expect others to contribute to the common pool. However it always leads to the case that public goods fail the threshold, where individuals' payoffs decline owing to the temperature rising. While the gap between the payoffs of C and D players is narrowed, the inferior strategy C has more opportunity than in the reference temperature situation. Compared with PGG without temperature feedback, our feedback model always shows better temperature control and cooperation improvement (see fig. 4).

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The evolutionary results are dependent on the change rate of temperature, α (see fig. 5). It can be seen that the mean temperature is relatively stable compared with that without control. With the increase of α, the temperature is more sensitive to the public goods, and it is challenging to control the temperature in this case. This challenge, however, promotes cooperation. This may hint that the more risk, the more advantage for the cooperation. In particular, a large change rate implies the sudden change of the climate, which previous game-theoretical models often assume [11–20].

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The threshold θ and contribution rate c have an impact on the evolution of the temperature and cooperation level (see fig. 6). On the one hand, for the average temperature, it always maintains a close value near the reference temperature under smaller threshold. While for higher thresholds, the average temperature shows a rising trend. High threshold means that to stabilize the temperature at a relatively safe level, the population should maintain at a long-term high cooperation level. Besides, a high ratio of contribution also causes the rise of mean temperature. The requirement that individuals should contribute more of their properties is harder to be maintained for a long time, which leads to the fluctuation of the cooperation level, further causing the rise of temperature. This suggests that the climate change should be controlled as soon as possible, before it exceeds the level which can be afforded.

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On the other hand, for the cooperation level, high threshold promotes cooperation (see fig. 6(B)). A similar conclusion that risk promotes cooperation has already been revealed in previous papers modeling threshold PGG or collective risk games [12,19,29,30]. However, a lower ratio of contribution promotes cooperation, which depicts the case where cooperators only contribute a small part of their fortune.

In addition, we study the effects of various selection intensities on the evolutionary results (see fig. 7). Compared with the cases under weak selection, the cooperation levels decline under stronger selection. While the average temperature may show a rising trend under stronger selection, it is still relatively stable compared with that without control. Therefore, the results in the feedback model are robust for a wide range of ω. In particular, as is suggested by the former behavioural experiments, the human's selection intensity is around 0.1 [27].

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It is worth noting that the dynamics is driven not only by selection, but also by mutation. To assess how the temperature dynamics influences the cooperation level, we study the gradient of selection of cooperators [23,31,32]:

$$\begin{eqnarray} &&g_C(x)=(1-\mu)(1-x)x\frac{-\omega}{2e^{(\gamma\,\Delta T)}}c+\mu(1-2x), \end{eqnarray} \tag{ 4 }$$

where x denotes the fraction of cooperators. It is obtained under the assumption that the temperature dynamics is much faster than the dynamics of the strategy updating, thus $\Delta T$ can be fixed in the selection gradient. Further, the root of such gradient of selection can be approximately obtained: $x^{*}=\mu/[\frac{\omega}{2}\frac{1}{e^{(\gamma\,\Delta T)}}c]$. Based on this, fig. 8 shows that to what extent selection affects the evolution of cooperation. We find that for non-weak selection, the theoretical results are similarly in agreement with simulation results, thus selection acts as a main force for the evolutionary process. For weak selection, however, the mutation plays a key role. Actually, weak selection itself means the selection of individuals is comparatively random. In this case, the decision-making is not rational and is independent of the payoff differences between players.

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Furthermore, the gradient of selection of cooperators is closely related with the change of temperature. When the temperature deviates a little from the reference one, namely $\Delta T\rightarrow0$, the effect of selection is much greater than that due to mutation. With the rise of temperature, the influence of random exploration becomes notable. Since in this case the payoff difference between cooperators and defectors is obviously narrowed, the effect of selection is no longer prominent. Once the temperature is restored to the reference level, the selection process still has the most impact on the evolutionary process.

As is shown in fig. 4, a model with temperature feedback always shows a better temperature control and cooperation improvement than the PGG models without feedback under various mutation rates. Hence the change of mutation rate does not alter such advantage of the feedback model, although it influences the quantitative relationship to a certain extent.

One of the key assumptions in our model is that if the temperature rises, the individuals' fortune will be discounted. When the public goods fail to reach the threshold, all the players will face the punishment by losing part of their payoffs. Only if the cooperators meet the collective target, may their fortune be kept. This can be deemed as punishment from the environment. It is to be noted that this punishment has two characteristics: i) the punisher is outside the group; ii) the entire population is punished. Thus it differs from peer punishment [33–40], in which the punisher is inside the group, and most often only defectors are punished.

Actually, by introducing feedback between temperature and collective behaviour, the individuals not only play games with their counterparts, but also interact with the environment. The players' strategies and the temperature co-evolve simultaneously. The real-time temperature changes will be fed back to payoffs calculation, then influence the individuals' decision-making, which in return rehabilitates the temperature towards the reference. When the temperature is around the reference value, the individuals' payoffs are discounted little, thus selection plays the key role. Cooperators are less benefited and will decrease in number. Yet those few cooperators contribute a small amount of the public goods, which leads to an increase in the temperature. This will reduce the payoffs of both cooperators and defectors, thus they are close in payoff. Effectively, defectors will adopt the cooperators' strategy more often by random drift. Thus the cooperators will increase in number. At the same time, the temperature decreases to the reference value owing to the large amount of public goods by the abundance in cooperation. Then it closes the circle. What we find is that this cycle dynamics promotes cooperation and stabilizes the temperature to a low level.

Compared with former climate game models [7–20], our model explicitly captures the dynamics of temperature as well as the player's behaviour updating. By introducing a temperature dynamics, the long-term effect of the individuals' decision-making on the environment can be depicted. At the same time, the gradual interaction between the temperature and the behaviours can be revealed through their coevolution. Our model offers a framework to facilitate the coevolution process where the temperature is assumed to be closely related with the amount of public goods collected by the population. This restricts to the assumption that the climate change is directly affected by humankind and can be controlled if suitable measures are adopted, while it is not exactly clear how the temperature evolves with the amount of collective public goods [41]. If the accurate quantified relationship can be represented, in addition with the mechanism we put forward, it may obliterate the limits of our simplified model to better predict the future.

As to improve the performance of climate control, the international society needs to introduce some mechanisms or institutions to help promote the global cooperation in protecting the environment. A straightforward way could be assessing the environment change into the economic cost. Once people realize that they are actually paying for the change of temperature, such realness of short-term losses would obviously attract more attention to protecting the environment, and more apt to influence the strategy decision-making. The long-term climate control may appear to have a promising future.

We thank the anonymous referees for their constructive and insightful comments. This work was supported by the National Natural Science Foundation of China (NSFC) Grants No. 61020106005 and No. 61375120. BW gratefully acknowledges sponsorship from the Max-Planck Society.