Abstract
In real situations, the value of public goods will be reduced or even lost because of external factors or for intrinsic reasons. In this work, we investigate the evolution of cooperation by considering the effect of depreciation of public goods in spatial public goods games on a square lattice. It is assumed that each individual gains full advantage if the number of the cooperators nc within a group centered on that individual equals or exceeds the critical mass (CM). Otherwise, there is depreciation of the public goods, which is realized by rescaling the multiplication factor r to (nc/CM)r. It is shown that the emergence of cooperation is remarkably promoted for CM > 1 even at small values of r, and a global cooperative level is achieved at an intermediate value of CM = 4 at a small r. We further study the effect of depreciation of public goods on different topologies of a regular lattice, and find that the system always reaches global cooperation at a moderate value of CM = G − 1 regardless of whether or not there exist overlapping triangle structures on the regular lattice, where G is the group size of the associated regular lattice.
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1. Introduction
Cooperation is ubiquitous in human and animal societies [1, 2], and the persistence of cooperation in selfish individuals is a challenge faced by the scientists who often resort to evolutionary game theory [3–6], across the fields from sociology and ecology to economics [7]. The prisoner's dilemma game (PDG) [8, 9], through pairwise interaction, and the public goods game (PGG) [10], through group interaction, are widely used as metaphors for investigating cooperative behaviors in social dilemmas. In a typical PGG played by N individuals, each cooperator contributes an amount to the public pool, while defectors contribute nothing. The total contribution is multiplied by a factor r, and redistributed equally to each participant. Thus defectors bear no cost when collecting identical benefits to cooperators, which ultimately results in widespread defection in a well-mixed population for r < N [11]. However, this theoretical prediction is at odds with experimental findings [12]; accordingly several mechanisms and theoretical supplements to the classical public goods game are proposed for describing the persistence of cooperative behavior, e.g., repeated interaction, direct reciprocity, punishment [12–14], spatially structured populations [15–17] and voluntary participation [18–20].
In the public goods games, the multiplication factor can represent the value of public goods returning to the public; a larger value of the multiplication factor enables a better return. In real situations, the value of the public goods will be reduced or even lost due to some external factors or for intrinsic reasons. For example, on a pastureland open to the public, herders are free to put their cattle on the pasture. If the number of cattle herded is under the maximum allowed by the pasture, each herder can enjoy the full advantage of the pasture; otherwise, the quality of pasture is destroyed, and the value returned to the herders will decrease. In this work, we investigate the effect of depreciation of the public goods on the evolution of cooperation in spatial public goods games where critical mass [21–27] is introduced as the inherent property of the public goods which is measured by the fraction of cooperators in a group.
The paper is organized as follows. A description of the model is proposed and it is discussed in detail in section 2. In section 3, numerical simulations and the corresponding analysis are presented. Finally, conclusions are drawn in section 4.
2. Models
We study the public goods game on a square lattice with periodic boundary conditions. Each player has only two strategies, namely cooperation and defection, and interacts only with its nearest four neighbors, so the group size is G = 5. Here each individual collects payoff only from one group centered on that individual. Considering the depreciated effect of the public goods, a critical mass (CM) is introduced, and the benefit function Px of the player x is defined as
Equation (1)
where is the normalized multiplication factor, and G is the group size centered on x. sx is x' s strategy (sx = 1 for a cooperator, 0 for a defector), and ncx is the number of the cooperators in this group. Here 1 ≤ CM ≤ G denotes the value of the critical mass. It is seen that player x (no matter whether a cooperator or a defector) will get the full payoff if the number of cooperators within a group equals or exceeds the threshold. Otherwise, the multiplication factor is reduced to (nc/CM)r, and the payoff of each player follows the traditional form of the benefit function.
Following previous works [9], [28–32], after each time step, a player x randomly selects one neighbor y and the probability with which x adopts the strategy of y depending on the difference of their total payoffs:
Equation (2)
where κ denotes the amplitude of the noise level. Here we set κ = 0.1.
3. Simulation and analysis
Simulations are carried out for a population of N = 200 × 200 individuals with the synchronous updating rule on a square lattice. Initially, the two strategies of cooperation (C) and defection (D) are randomly distributed with the equal probability of 1/2. The key quantity for characterizing the cooperative behavior is the density of cooperators ρc, which is defined as the fraction of cooperators in the whole population. In all simulations, ρc is obtained by averaging over the last 5000 Monte Carlo (MC) time steps of the total 35 000. Each data point results from an average of 30 realizations.
Figure 1(a) shows the variation of ρc with r for different values of the critical mass (CM) where the solid squares represent the situation in the original version in which accumulated benefit is proportional to the fraction of cooperators in a group. It is shown that compared with the original version, the emergence of cooperation is remarkably promoted for CM > 1. To be precise, a moderate fraction of cooperators can prevail even at small values of the multiplication factor r for CM = 2, 3, and a global cooperative level is achieved earlier than that in the original version when CM = 4, 5. Besides, it is worth noting that no cooperation emerges at CM = 1, and cooperation sustains a moderate level invariant with r for the values of CM = 2, 3. Moreover, the increase in the overall cooperation for the intermediate r cannot be sustained if CM becomes too high, which suggests that there exists an optimal CM for the evolution of cooperation. As shown in figures 1(b) and (c), a highest cooperative level is achieved at an intermediate value of CM for r = 2, 3 respectively. In particular, it induces the global cooperative level when CM = 4 at r = 3.
Figure 1.
Figure 1. (a) Cooperation density ρc as a function of multiplication factor r for different values of the critical mass (CM); ((b), (c)) variation of ρc with CM at the values of r = 3, 2, respectively.
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Before we go on studying, it is necessary to analyze the implicit function of this mechanism in the evolution of cooperation. According to equation (1), the individuals can be divided into four types: cooperators A and defectors B whose groups meet the condition of nc ≥ CM centered on them, and cooperators AA and defectors BB whose groups do not meet the condition. Since individuals A and B always get the full payoff from the public goods games, the mechanism has two effects on the evolution of cooperation: on one hand, individuals A contribute to the formation of cooperative clusters and the spread of cooperation; on the other hand, individuals B have a negative effect on cooperation, since they enhance the ability of defectors to exploit the cooperators—the smaller CM, the stronger this ability. Accordingly, the evolution of cooperative behavior is affected by these two aspects of this mechanism. On the basis of the above analysis, we explain the results shown in figure 1: when CM = 1, the exploitation ability of defectors is strongest, and the cooperators can be easily exploited as long as there exists one cooperator. Indeed, spatial reciprocity cannot work for CM = 1, since there is no advantage for cooperators to aggregate without any extra payoff. So defectors dominate, and no cooperators survive; for CM = 2 and 3, the exploitation ability becomes weaker; there exists a balance in the competition between the cooperators and defectors. Cooperative clusters can be formed, but cooperators on the boundary are not firm enough to spread the cooperative behavior in the population because of the strong ability of exploitation of the defectors. When CM = 4, the ability of exploitation for the defectors B is very weak; cooperators can just defeat defectors absolutely with increase of r. As for CM = 5, individuals B become extinct; global cooperation can be achieved as r increases.
In order to understand the results better, we study the time evolution of the boundary payoff and the probability of transition between cooperators and defectors at r = 3.0 for different values of the critical mass in figure 2 in which WCD indicates the transition probability of cooperators changing into defectors, which is defined as WCD = NCD/NC, where NCD is the number of cooperators changing into defectors among the cooperators, and NC is the number of these cooperators in the whole population. One can see that in figures 2(a) and (d) at CM = 3, the boundary payoff of cooperators PC-boun is larger than PD-boun, and the transition probability is WDC > WCD, which indicates that cooperator clusters can be formed, and the system maintains a dynamic balance after it enters a stable state. In figures 2(b) and (e) at CM = 4, we have for the boundary payoffs PC-boun > PD-boun, and WCD decreases with t whereas WDC increases as t evolves. It is represented that cooperator clusters can be formed, and cooperation will spread among the whole population. In figures 2(c) and (f) at CM = 5, PC-boun < PD-boun and WDC < WCD. Besides, WCD increases with t, whereas WDC decreases with t, which indicates that cooperator clusters cannot be formed, and defectors will dominate the whole population.
Figure 2.
Figure 2. Time evolution of boundary payoffs P-boun and the probability of transition WT between cooperators and defectors for different values of critical mass at r = 3.0: ((a), (d)) CM = 3, ((b), (e)) CM = 4, ((c), (f)) CM = 5. PC-boun (PD-boun) indicates boundary payoff of cooperators (defectors), and WCD (WDC) represents the transition probability of cooperators (defectors) changing into defectors (cooperators).
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Finally, we investigate the effect of this mechanism on different types of lattice. Figure 3 shows four topologies of regular lattice: honeycomb lattice, square lattice, triangle lattice and a square lattice extension with eight nearest neighbors, where the honeycomb lattice and square lattice are not provided with the structures of overlapping triangles. Figure 4 shows the variation of cooperation density ρc with the critical mass for the above four lattices; it is presented that global cooperation is always achieved at CM = G − 1 regardless of whether or not the regular lattice is equipped with overlapping triangles. It is proved that the topology of the regular lattice has no significance under this mechanism; similar results and relevant details are also discussed in [33].
