Evolution of Cooperation in Continuous Prisoner s Dilemma Games on Barabasi Albert Networks with Degree-Dependent Guilt Mechanism

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1 Commun. Theor. Phys. 57 (2012) Vol. 57, No. 5, May 15, 2012 Evolution of Cooperation in Continuous Prisoner s Dilemma Games on Barabasi Albert Networks with Degree-Dependent Guilt Mechanism WANG Xian-Jia ( ), 1,2,4 QUAN Ji ( ), 2, and LIU Wei-Bing ( åï) 3,4 1 School of Economics and Management, Wuhan University, Wuhan , China 2 Institute of Systems Engineering, Wuhan University, Wuhan , China 3 School of Political Science and Public Management, Wuhan University, Wuhan , China 4 Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan , China (Received August 31, 2011; revised manuscript received Decenber 28, 2011) Abstract This paper studies the continuous prisoner s dilemma games (CPDG) on Barabasi Albert (BA) networks. In the model, each agent on a vertex of the networks makes an investment and interacts with all of his neighboring agents. Making an investment is costly, but which benefits its neighboring agents, where benefit and cost depend on the level of investment made. The payoff of each agent is given by the sum of payoffs it receives in its interactions with all its neighbors. Not only payoff, individual s guilty emotion in the games has also been considered. The negative guilty emotion produced in comparing with its neighbors can reduce the utility of individuals directly. We assume that the reduction amount depends on the individual s degree and a baseline level parameter. The group s cooperative level is characterized by the average investment of the population. Each player makes his investment in the next step based on a convex combination of the investment of his best neighbors in the last step, his best history strategies in the latest steps which number is controlled by a memory length parameter, and a uniformly distributed random number. Simulation results show that this degree-dependent guilt mechanism can promote the evolution of cooperation dramatically comparing with degree-independent guilt or no guilt cases. Imitation, memory, uncertainty coefficients and network structure also play determinant roles in the cooperation level of the population. All our results may shed some new light on studying the evolution of cooperation based on network reciprocity mechanisms. PACS numbers: kg, Le, k Key words: continuous prisoner s dilemma game, Barabasi Albert network, degree-dependent guilt, cooperation 1 Introduction Cooperation among individuals is common in various ecological and social systems. However, understanding how this altruistic behavior emerges in a situation where each individual is apparently tempted to defect in the context of Darwinian evolution remains a challenge so far. [1] In the game theory, this situation can be typically formulated as the prisoner s dilemma games (PDG). [2 3] In the standard PDG, the two players must simultaneously choose between the choices of cooperation (C) and defection (D). If both players cooperate they get more than if both defect, but defecting against a cooperator leads to the highest payoff, while cooperating with a defector leads to the lowest payoff. In terms of evolutionary game theory, [4 6] strategy D is the unique evolutionary stable strategy (ESS) of the game. So, without any mechanism for the evolution of cooperation, natural selection favors defection. To resolve the dilemma and explain the actually found altruistic, several mechanisms such as kin selection, [7] direct reciprocity, [2 3] indirect reciprocity, [8] network reciprocity, [9 12] group selection [13] and so on [14 17] have been proposed. Among the above mechanisms, network reciprocity, which extends the assumption of well-mixed population to heterogeneous population, has attracted the most attention. Heterogeneity implies that not everyone interacts equally likely with everyone else. This effect can be captured by using a network where individuals are locating on the vertices of the network, and individuals are constrained to play only with their immediate neighbors. Nowak et al. [9] pioneered the study of this spatial game model in the PDG. They found that a simple spatial structure can induce the emergence and persistence of cooperation. Complex networks provide a realistic and convenient framework to describe the population structure on which the evolution of cooperation can be studied. Evolutionary games on complex networks are significant extensions of traditional evolutionary game theory focusing on well- Supported by the National Natural Science Foundation of China under Grant Nos and and supported by Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) Corresponding author, quanji123@whu.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 898 Communications in Theoretical Physics Vol. 57 mixed populations. The effect of network structure on the cooperation of the population in the PDG has attracted much attention in these years and varieties of surprising conclusions have been found. [18 28] Now it is accepted that PDG on heterogeneous complex networks tend to promote cooperation. The network topology is proved to be a critical issue in cooperation emergence. It should be noted that the aforementioned research all use the standard PDG as a metaphor for studying the evolution of cooperation, in which players can either cooperate or defect at each time. But in real situations, cooperation is almost never all or nothing. This is the motivation for the continuous prisoner s dilemma games (CPDG). [29 35] In the CPDG, the cooperation is based on the concept of investment: an act which is costly, but which benefits other individuals, where benefit and cost depend on the level of investment made. The essential problem of cooperation in the CPDG remains. The CPDG can be viewed as a generation of the standard PDG in which any level of investment can be made. As we know, Refs. [29 35] only studied the CPDG and the evolution of cooperation either through reciprocal altruism or using lattice networks. How is about the cooperation level of CPDG on complex networks? The answer is unknown. It is worth while to mentioning that the strategy update rules in the PDG on networks include imitation of the best neighbors, [18] probability based imitation rules between two connected individuals [19 20,22 26] and the Moran process based rules; [10 11] whereas in the CPDG, the strategy update rules are different owing to the continuous strategy space. Previous works in the CPDG usually consider the imitation and mutation rule: [31,33,35] each individual in the next generation adopts the strategy associated with the individual in its local neighborhood (including the individual itself) that has the highest payoff, but allows occasional mutation. In this paper, we study CPDG on Barabasi Albert (BA) scale-free networks, in which each agent on a vertex of the networks makes an investment and interacts with all of his neighboring agents. Making an investment is costly, but which benefits its neighbors, where benefit and cost depend on the level of investment made. The payoff of each individual is given by the sum of payoffs it receives in its interactions with all its neighbors. Unlike the aforementioned research, we also consider individual s guilty emotion in the games. The negative guilty emotion produced in comparing with its neighbors can reduce the utility of individuals directly. We assume that the reduction amount depends on the individual s degree and a baseline level parameter. Not only imitation and mutation are considered in the strategy update rules in our model, we also add memory effects. Each player makes his investment in the next step based on a convex combination of the investments of his best neighbors in the last step, his best history strategies in the latest steps which number is controlled by a memory length parameter, and a uniformly distributed random number. The combination parameters are called imitation coefficient, memory coefficient, and uncertainty coefficient respectively. The cooperative level is characterized by the average investment of the population. We investigate the co-effect of these parameters and the BA network structure on the cooperation level of the population. The rest of the paper is organized as follows. In the next section, we describe the model of the evolutionary games and the strategy update rule used in this work. The simulation results and discussions are given in Sec. 3. And the paper is concluded by the last section. 2 Model The model consists of a set of N individuals connecting through a network. Individuals are located on the vertices of the network which defines each agent s neighborhood in which they can interact. At each time step, each individual makes an investment and interacts with all of his neighboring agents. Making an investment I involves a cost C(I) to the donor and brings a benefit B(I) to the individual who is the recipient, where both the cost and benefit depend on the investment level I. As standard assumptions, the benefit function B( ) and the cost function C( ) are all increasing with B(0) = C(0) = 0. Specifically, we use the nonlinear benefit function B(I) = a(1 exp( b I)) (a, b > 0) [31 33] and the nonlinear cost function C(I) = c 0 I 2 (c 0 > 0) [32] in this paper. Unlike the standard PDG in which the players can either cooperate or defect; the strategy space in the CPDG is infinite and individuals can exhibit variable degrees of cooperation. The payoff of each individual is given by the sum of payoffs it receives in its interactions with all its neighbors. The total payoff S agent i will get from the CPDG at time t when he invests I i (t) can be described as follows: S(I i (t)) = (B(I j (t)) C(I i (t))) = B(I j (t)) Ω i C(I i (t)). (1) Here I j (t) is the investment of agent j at time t. Ω i is the set of i s neighboring agents. Ω i is the number of elements in set Ω i. In our model, we consider the guilty emotion of individuals in the games. For two directly connected agents i and j, if the investment of individual i is less than his neighbor j, which lead to a less benefit B(I i ) i brings to j than the benefit B(I j ) j brings to i, agent i will suffer a guilty emotion. We suppose the guilty emotion of individuals can be added and will have a direct effect on the utility of individuals. Considering this negative emotion,

3 No. 5 Communications in Theoretical Physics 899 individual utility or fitness f can be described as follows: f(i i (t)) =S(I i (t)) r i max(b(i j (t)) B(I i (t)), 0). (2) Here r i is the guilt parameter of agent i. If r i is a constant, all the individuals have the same guilty emotion. In reality, star people which have large degrees may have stronger guilty emotion for the consideration of face-saving or some other public effects. It is particularly natural to suppose that individual s guilty emotion is depending on his degree, with r i = g( Ω i ). The function g( ) is increasing and we use the logarithmic function in this paper. r i = g( Ω i ) = r 0 ln Ω i. (3) Here r 0 > 0 is a parameter that can be tuned, the value of which denotes the baseline level of guilt in the population. During the evolutionary processes of the games, every agent is allowed to learn from the strategy associated with the individual in its local neighborhood (including the individual itself) that has the highest fitness in the last step. In addition, we assume that agents have a memory length k that they can remember their best strategies (which lead to the highest fitness) in the most recent k steps. Also considering some uncertainty in the environment, the investment of agent i at time t + 1 can be described as follows: where I i (t + 1) = α I j0 (t) + β I i (h 0 ) + ε Ĩ, (4) j 0 = argmax f(i j (t)), (5) h 0 = arg max f(i i(h)). (6) max(0,t k+1) h t Here α, β, and ε denote imitation coefficient, memory coefficient, and uncertainty coefficient respectively. They subject to 0 < α, β, ε < 1 and α + β + ε = 1. Ĩ is a random variable that obeys the uniform distribution, i.e., Ĩ U(0, I max ) and I max is the upper amount one would invest, I max = Arg(B(I) = C(I)). (7) I>0 I j0 (t) is the investment of i s best neighbor which has the highest fitness in time t. I i (h 0 ) is the investment of agent i that he gets the highest fitness in the most recent k steps when in time t, and k denotes the memory length. In this paper, we investigate the co-effect of imitation coefficient α, memory coefficient β, uncertainty coefficient ε, memory length k, baseline level of guilt r 0 and the network structure on the average investment (the level of cooperation) of the population. 3 Simulation Results and Discussion The simulations are carried out on the BA scale-free networks. There are N = 1000 agents located on the nodes of the networks. The key quantity for characterizing the cooperative behavior of the system is the average investment of the population. The BA scale-free network is built with a standard algorithm. Initially, a small fully connected graph with m 0 nodes is built. And then, at each time step, a new node is added to the network till N = Each newly introduced node also adds a given number of m edges to the network by connecting m existing nodes according to the probability proportionately to the node degree of the nodes on the network. In our simulations, we set m 0 = 10, and the default value of variable m is m = 5 if we do not otherwise specify. Initially, each agent invests amount that uniformly distributed in the interval [0, I max ]. The average investment of the population is obtained by averaging over 1000 generations after a transient time of generations. Each data is averaged by 10 runs on 10 different networks. A synchronous update rule is adopted here. In our simulations, we set parameters a = 8, b = 1 in the benefit function and parameter c 0 = 0.7 in the cost function. [31,33] The default values of the variables of memory length and uncertainty coefficient are k = 3 and ε = 0.05 respectively, if we do not otherwise specify. In order to make the simulation processes more simple, but at the same time can show the conclusions to the readers, all these default values are the comparative results of several simulation experiments. It needs to be emphasized that concerning the simulation graphs, although all the simulation results are shown as some continuous curves, we only get the simulation results of some discrete points which are finally connected with line segments of every two adjacent points one by one. So, in the discrete points we take, the results are the statistical results, but in the middle of these points, the results are not. Figure 1 gives the average investment of the population as a function of the baseline level of guilt r 0 for different values of imitation coefficient α given fixed m = 5, ε = 0.05, and k = 3. As shown in the figure, all curves are increasing which imply that the guilty emotion of individuals can promote the cooperation level of the population. What is connotative but more interesting, the curves appear approximately first convex, and then concave. For too small or too large values of r 0, the cooperation level increases slowly as r 0 increases; but for intermediate values of r 0, the cooperation level increases rapidly as r 0 increases. The effect of the imitation coefficient α on the system s cooperation level can also be shown by the figure. Six values of α = 0.2, 0.3,..., 0.7 are considered respectively. The corresponding six curves depict that smaller values of α will lower the cooperation level no matter what values of r 0. Nevertheless, the effect is obvious when r 0 is small; and is gradually waning as r 0 increases. When r 0 is

4 900 Communications in Theoretical Physics Vol. 57 large enough, the effect of α on the system s cooperation level is not obvious r 0 α=0.2 α=0.3 α=0.4 α=0.5 α=0.6 α=0.7 Fig. 1 The average investment of the population vs. baseline level of guilt r 0 with different values of imitation coefficient α, given fixed values of m = 5, ε = 0.05, and k = 3. The value of r 0 is restricted in the interval [0, 1.2]. In order to study the effect of memory length k on the system s cooperation level, we choose three small values of α to confirm large values of memory coefficient β, otherwise the results do not make sense. For each value of α, a pair of memory length k = 3 and 9 are used respectively. m = 5, ε = 0.05 are fixed in this simulation. Figure 2 gives the corresponding results. It is shown in the figure that the curve corresponding to k = 3 is always above the curve corresponding to k = 9 in all three cases of α = 0.2, 0.3, 0.4 when other parameters are fixed and this effect is particularly obvious when r 0 is small α=0.2, k=3 α=0.2, k=9 α=0.3, k=3 α=0.3, k=9 α=0.4, k=3 α=0.4, k= r 0 Fig. 2 The average investment of the population vs. baseline level of guilt r 0 with different values of imitation coefficient and memory length (α, k), given fixed values of m = 5 and ε = The value of r 0 is restricted in the interval [0, 1]. Figures 1 and 2 reveal that memory including both memory coefficient and memory length in our model plays a negative role in the cooperation behavior of the CPDG on the BA networks. The effect of uncertainty coefficient ε on the system s cooperation level has also been investigated. Figure 3 gives the average investment of the population as a function of ε for different values of r 0 given fixed m = 5, α = 0.3 and k = 3. Six values of r 0 = 0, 0.4, 0.6, 0.8, 1, 1.5 are considered respectively. It is worthy to mention that r 0 = 0 corresponds to the case of no guilty emotion. The corresponding six curves depict that the increase of ε can promote the cooperation level when r 0 is not too large; but when r 0 is large enough, ε has little effect on the cooperation level of the population. We also note that for small values of r 0 (in our simulations, r 0 = 0, 0.4), there is a very short interval corresponding to small values of ε, the cooperation level decreases as ε increases r 0 /0 r 0 /0 4 r 0 /1 0 r 0 /1 5 r 0 /0 6 r 0 /0 8 α= ε Fig. 3 The average investment of the population vs. uncertainty coefficient ε with different values of baseline level of guilt r 0, given fixed values of m = 5, α = 0.3, and k = 3. The value of ε is restricted in the interval (0,0.45]. In the BA scale-free networks, increasing m will increase the average degree and also the heterogeneity of networks. As it has been shown that the increase of the guilt level can promote the cooperation, thus the effect of increasing the average degree of networks will promote the cooperation level of the population when r 0 is large enough. But how about small values of r 0 and the heterogeneity effect on the cooperation level, and then what is the overall effects of increasing m in our model? Figure 4 gives the results of the average investment of the population as a function of r 0 for different values of m given fixed α = 0.5, ε = 0.05, and k = 3. Four values of m = 4, 6, 8, 10 are considered. When r 0 = 0, which corresponds to the no guilty emotion case, large values of m result in small values of cooperation level. As r 0 increases, the situation gradually changes. There is a turning point corresponding to r 0 = 0.4 in the figure, and beyond this point, large values of m will result in large values of cooperation level. So, increasing the average degree of the networks has two side effects on the cooperation in our model. One side

5 No. 5 Communications in Theoretical Physics 901 is the increase of the guilty emotion when average degree of the network increases, which will promote the cooperation level. The other side is the increase of density of interactions between individuals, which will inhibit the cooperation level. When r 0 is large enough, positive effect exceeding the negative role, then increasing m favors the evolution of cooperation. corresponding to degree-dependent guilt are always above the curves corresponding to degree-independent guilt in all cases and for any values of 0.2 α 0.7. Thus, the degree-dependent guilt mechanism is an effective mechanism for the evolution of cooperation. (a) r 0 =0.2; ε=0.05; m=5; k=3 α=0.7 α= a/ r 0 m/4 m/6 m/8 m/10 Fig. 4 The average investment of the population vs. baseline level of guilt r 0 with different values of network parameter m, given fixed values of α = 0.5, ε = 0.05, and k = 3. The value of r 0 is restricted in the interval [0, 2] Degree-dependent guilty, r 0=0.6 Degree-independent guilty, r 0=1.38 Degree-dependent guilty, r 0=0.8 Degree-independent guilty, r 0=1.84 Degree-dependent guilty, r 0=1.0 Degree-independent guilty, r 0= α Fig. 5 The average investment of the population vs. imitation coefficient α with different values of r 0 in the degree-dependent guilt case and the corresponding r i in the degree-independent guilt case (r i = r 0 ln Ω i, where Ω i is the average degree of the network), given fixed values of m = 5, ε = 0.05, and k = 3. The value of α is restricted in the interval [0.2, 0.7] ,000 10,400 10, (b) Iteration number (c) Iteration number r 0 =0.6; ε=0.05; m=5; k=3 Degree-dependent guilty, r 0 =1 Degree-independent guilty, r i =2 α=0.4; m=5; ε=0.05; k=3 α=0.7 α= Iteration number In order to illustrate the effectiveness of this degreedependent guilt mechanism for the evolution of cooperation, we compare the case of degree-independent guilt where r i is constant with the case of degree-dependent guilt with r i = r 0 ln Ω i. Figure 5 gives the corresponding results. m = 5, ε = 0.05, k = 3 are fixed in the experiments. When m = 5, the average degrees of networks Ω i are approximately 10 and ln , so we let r i = 2.3 r 0 in the case of degree-independent guilt to compare. Three values of r 0 = 0.6, 0.8, 1 are considered respectively. Simulation results show that the curves Fig. 6 The evolution of average investment with time in CPDG on the BA networks with different values of α and r 0 (or r i), given fixed ε = 0.05, m = 5, k = 3. (a) r 0 = 0.2, α = 0.2, and 0.7; (b) r 0 = 0.6, α = 0.2, and 0.7; (c) α = 0.4, r 0 = 1, and r i = 2. Why the introduction of the individual guilt will greatly facilitate the cooperation level of the population and why the degree-dependent guilt will promote the cooperation greater in the BA networks? We can give some intuitive explanation. In fact, if one individual shows uncooperative, i.e., he invests very little compared with its

6 902 Communications in Theoretical Physics Vol. 57 neighbors, and he will suffer a negative guilty emotion, which can reduce the utility of the individual directly. Thus, when the guilt parameter r 0 (or r i ) is too large, noncooperative strategy is no longer the optimal for him and he will invest more in the next step. So the introduction of the individual guilt will greatly facilitate the cooperation level of the population. If the guilt level is depending on the individual s degree, a node with a larger degree has a greater sense of guilt and he will suffer more when he shows uncooperative. In the BA networks, the nodes are heterogeneous and some central nodes have larger degrees compared with other nodes. For these individuals, they are more willing to cooperate. The cooperation of the central nodes will lead to more individuals opt for cooperation. So the degree-dependent guilt will promote the cooperation greater in the BA networks. Figure 6 shows the evolutionary processes of average investment with time for different values of α and r 0 (or r i ) when ε, m, and k are fixed. The iteration processes a generation of times and we only give the result of the last 1000 times. As is obviously shown in Fig. 6(a) and 6(b), a larger imitation coefficient (α = 0.7) will have a larger variance of the average investment than that of a smaller imitation coefficient (α = 0.2) does when other parameters are fixed; and a larger value of r 0 (r 0 = 0.6) will have a smaller variance of the average investment than that of a smaller value of r 0 (r 0 = 0.2) does when other parameters are fixed. Figure 6(c) shows the evolutionary processes for the case of degree-dependent guilt (r 0 = 1) and the case of degree-independent guilt (r i = 2). 4 Conclusions In summary, we have studied the effect of guilty emotion of individuals on the cooperation behavior of the population in the evolutionary CPDG. Individuals are connected by the BA scale-free networks. The negative guilty emotion produced in comparing with its neighbors can reduce the utility of individuals directly. The reduction amount depends on the individual s degree and a parameter r 0 denoting the baseline level of guilt. We also study the effects of imitation, memory, environmental uncertainty and the average degree of the BA networks on the cooperation level of the population. We get the following results: (i) the introduction of the guilty emotion of individuals will greatly promote cooperation in the population; (ii) degree-dependent guilt will promote cooperation even more compared with the degree-independent guilt in the BA networks; (iii) imitation coefficient plays a positive role of cooperation, but this role is little when r 0 is large enough; (iv) both memory coefficient and memory length play negative roles of cooperation, but the roles are little when r 0 is large enough; (v) uncertainty coefficient plays a positive role of cooperation in most cases, but this role is little when r 0 is large enough; (vi) average degree of the network plays a negative role of cooperation when r 0 is small, whereas plays a positive role of cooperation when r 0 is large. We also consider the case that individuals are connected by the WS networks. 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