Evolution of Cooperation in Evolutionary Games for Heterogeneous Interactions

Commun. Theor. Phys. 57 (2012) 547 552 Vol. 57, No. 4, April 15, 2012 Evolution of Cooperation in Evolutionary Games for Heterogeneous Interactions QIAN Xiao-Lan ( ) 1, and YANG Jun-Zhong ( ) 2 1 School of Electronics and Information, Zhejiang University of Media and Communications, Hangzhou 310018, China 2 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China (Received October 21, 2011; revised manuscript received December 20, 2011) Abstract When a population structure is modelled as a square lattice, the cooperation may be improved for an evolutionary prisoner dilemma game or be inhibited for an evolutionary snowdrift game. In this work, we investigate cooperation in a population on a square lattice where the interaction among players contains both prisoner dilemma game and snowdrift game. The heterogeneity in interaction is introduced to the population in two different ways: the heterogenous character of interaction assigned to every player (HCP) or the heterogenous character of interaction assigned to every link between any two players (HCL). The resonant enhancement of cooperation in the case of HCP is observed while the resonant inhibition of cooperation in the case of HCL is prominent. The explanations on the enhancement or inhibition of cooperation are presented for these two cases. PACS numbers: 02.50.Le, 87.23.Kg, 89.75.Fb Key words: evolutionary games, heterogenous interactions, resonant enhancement and inhibition 1 Introduction Since Darwin, it has been widely accepted that natural selection favors egoists who try to maximize their individual benefits even at the expense of the group. [1] However, spontaneous cooperation in a population of selfish individuals is abundant in nature and in human society. Such a puzzle is often studied in the framework of evolutionary game theory adopting different 2 2 games such as prisoner s dilemma (PD) game and snowdrift (SD) game. [2 5] In a 2 2 game, players can either take a cooperation (C) strategy or a defection (D) strategy. The payoff (or fitness) of each player depends on the actions taken by the player and his opponent. Mutual cooperation results in a reward R for both players, whereas mutual defection results in a punishment P. In unilateral cooperation, the cooperator receives a suck s payoff S and the defector receives a temptation T. Different games are classified by the relative ranking of R, P, S, and T: the PD game with T > R > P > S and the SD game with T > R > S > P. [3 4] For an evolutionary game theory with P = 0 and R = 1 in a well-mixed population, the frequency of cooperators is described by the replicator equation ẋ = x(1 x)[(1 S T)x + S], [2,4] which leads to an extinction of cooperation in PD game and an equilibrium x = S/(S + T 1) in SD game. In recent years, several mechanisms to account for spontaneous cooperation among selfish individuals have been proposed. For example, kin selection, [6] reciprocity, [7] reputation, [8] voluntary participation [8] and structured populations. [10 19] Among these, the evolution of cooperation in structured populations has become a hot spot currently since the pioneering work of Nowak and May. [10] Nowak and May considered spatial evolutionary PD games where identical players located on a square lattice play games with their neighbors. With the rule that the most successful strategy is imitated, Nowak and May have shown that cooperation can be sustained through the way that cooperators form compact clusters to resist exploitation by defectors. Since then, evolutionary PD games on different types of population structures such as small world networks [11 12] and scale free networks [13 14] have been investigated. The enhancement of cooperation in different types of networks is determined by the efficiency of forming cooperator clusters. Especially, scale free network has been recognized as an extremely potent promotor for enhancing cooperation due to the hubs effects on cooperation. [13] However, the structured populations modelled by networks are not sufficient condition for enhancement of cooperation in evolutionary games. Hauert and Doebeli studied evolutionary SD games on a two-dimensional lattice and they found that, in comparison with that in the absence of structure, the presence of square lattice always depresses cooperation in the population. [20] In evolutionary SD games on a square lattice, cooperators always prefer to form small filament-like clusters, which is quite different from the compact cooperator clusters in PD games. Now one question arises: if both interactions (PD and SD games) exist simultaneously in the population on a square lattice, what will happen under the competition between Supported by Natural Science Foundation of China under Grant No. 11147112 E-mail: xlqianzumc@hotmail.com c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

548 Communications in Theoretical Physics Vol. 57 forming compact cooperator clusters by individuals playing PD game and forming dendritic skeleton by individuals playing SD game? Actually, the simultaneously existence of the interactions of PD and SD games indicates the presence of one type of heterogeneity in the population. In similar works, the effects of the heterogeneity on cooperation have been intensively investigated. [5,21 29] Especially, Perc et al. found that, by keeping the rank of T > R > P > S unchanged, the introduction of noise to the payoff matrix can improve cooperation and result in a resonance-type behavior of cooperation. [21 23] Then, it will be interesting to ask whether the heterogeneity induced by the presence of both PD and SD games in the population could lead to certain resonance-type behaviors? To answer these questions, in this work, we investigate the evolution of cooperation in the population located on a square lattice when the interaction among players contains both PD and SD games. 2 Model We consider a square lattice with periodic boundary conditions where each player occupies a node. The evolution of the system consists of two stages in each generation. In the first stage, each player interacts with his nearest neighbors by playing game. The total payoff P of every player in each generation is the sum over all interactions he participates. In the second stage, each player (denote by i) updates his strategy by comparing his payoff P i with that of another player (denoted by j) randomly chosen from his neighbors. If P j > P i, the player i learns the strategy of the player j with a probability proportional to (P j P i )/4(T S). Different from the ordinary evolutionary game theory on a lattice where interactions among players are limited to a given game, both PD game and SD game are simultaneously present in the population in this work. The heterogeneity in the interaction among players is introduced to the system in two different ways. In the first situation, the population is divided into two groups: one with players performing PD game and the other with players performing SD game, and the interactions between any two players are represented by an asymmetric bimatrix game. [21 26] We name this situation as HCP (Heterogeneous character of interaction assigned to players). In HCP, if a player is assigned by a PD (or SD) character, he will accumulate his payoff according to the payoff matrix for PD (or SD) game. The schematic diagram for HCP is shown in Fig. 1(a). The payoff gained by the focal player denoted as A in the figure is 2(R sd +S sd ). If the strategy of the focal player A is changed to be defection, then his payoff will be 2(T sd + P sd ). In the second situation, the links on the lattice are divided into two groups: links in one group are assigned by PD game where the two players at the ends of the links play PD game, and the links in the other group are assigned by SD game where the two players at the ends of the links play SD game. We call this situation as HCL where the interactions between any two players are represented by a symmetric payoff matrix. Figure 1(b) shows the schematic diagram for this situation. The payoff acquired by the focal player A in Fig. 1(b) is S sd +R sd +S pd +R pd. Similarly, the payoff of the focal player A when his strategy becomes defection is P sd + T sd + P pd + T pd. Fig. 1 (a) The schematic diagram for the HCP situation. The colors of red, black, yellow and grey represent PD cooperator, PD defector, SD cooperator and SD defector, respectively. (b) The schematic diagram for the HCL situation. the red node for cooperator and the black for defector. The yellow connection for PD interaction and the blue for SD interaction.

No. 4 Communications in Theoretical Physics 549 Throughout this work, the dimension of the lattice is set to be L = 100. For PD game, we adopt parameterization introduced by Nowak and May [10] such that R pd = 1 and P pd = S pd = 0. For SD game, we set S sd = 2 T sd, P sd = 0 and R sd = 1. Both T pd and T sd are set to be controlling parameters. Another important parameter in our models is the fraction of players adopting SD game in the situation of HCP or the fraction of links adopting SD game in the situation of HCL, which is denoted by v. When v = 0, the evolutionary PD game on a lattice is recovered for both situations of HCP and HCL. On the contrary, v = 1 denotes the evolutionary SD game on a lattice. During simulations, the different characters are initially randomly assigned to players (or links) for the situation of HCP (or HCL) and then kept unchanged. And each player initially takes cooperation or defection with equal probability. Then the system evolves according to the rules proposed above. To characterize how v influences cooperation, we monitor the frequency of cooperators, x(v), which is obtained by averaging instantaneous x over a time interval t = 10 3 after a transient process of t = 5 10 4. Clearly, x(0) recovers the cooperator frequency x pd for the evolutionary PD game on the lattice and x(1) the cooperator frequency x sd for the evolutionary SD game. 3 Results and Analysis To get a general view, we investigate the relationship between the cooperator frequency x(v) and v in the range of T sd from 1.1 to 1.8 for T pd at T pd = 1.02, 1.08, 1.12. The results are shown in Fig. 2 from which several features can be revealed. The first important feature is that enhancement of x(v) is always present in the case of HCP [See Figs. 2(a) 2(c)]. In an ordinary view, x(v) will change monotonically from x pd to x sd with the increase of v. However, Figs. 2(a) 2(c) show that, with the increase of v, the cooperator frequency x(v) first increases from x pd and then decreases to x sd after a maximum of x(v) is reached. The promotion of x(v) with v is strongly dependent on the value of x sd and x pd. Furthermore, when x pd is fixed, the optimal fraction v op shifts to a lower value with the decrease of x sd. The promotion of x(v) disappears when x sd is so low that v op = 0. Similarly, the promotion of x(v) disappears either when x sd is so high that v op = 1. The second important feature revealed by Fig. 2 is the possible inhibition of cooperation in the situation of HCL. As shown in Figs. 2(d) 2(f), x(v) becomes lower than both x pd and x sd in a certain range of v when x pd and x sd become sufficiently low. The range of v supporting the inhibition of cooperation increases with decreases of x pd and of x sd. To be mentioned, the enhancement of cooperation can also be observed in the case of HCL provided that both x pd and x sd are high enough. 3.1 Cooperation Promotion in HCP In this subsection, we investigate how the enhancement of cooperation shown in Figs. 2(a) 2(c) is produced. Since the enhancement of cooperation is dependent on x pd and x sd, for example whether x(0) increases or decreases with v depends on x sd with a given x pd and the enhancement of cooperation is prominent when x pd x sd, it is helpful to consider how cooperation is enhanced when both x pd and x sd are around zero. For this aim, we first let T pd = 1.135 and T sd = 1.665 where x pd = 0 and x sd = 0. The results are presented in Fig. 3(a) where the enhancement of cooperation is clear. However, the behavior of x(v) at v = 0 and v = 1 are quite different. For example, x(v) increases once v deviates from v = 0, while x(v) keeps at zero when v is away from v = 1 and it rises only when v is lower than a certain threshold. Furthermore, we present in Fig. 3(a) the results for T pd = 1.125 and T sd = 1.655 where both x pd and x sd are just a little above zero (x pd 0.02 and x sd 0.02). Clearly, x(v) shows a similar behavior. That is, the introduction of small amount of PD players into a population of SD players does not enhance cooperation as the introduction of small amount of SD players into population of PD players does. To see what happens when x(v) is elevated from zero, we choose two parameter sets, which are denoted as A and B in Fig. 3(a) and plot snapshots for them in Figs. 3(b) and 3(c), respectively. As shown in Fig. 3(b) where small amount of SD players are distributed sporadically among PD players, a single SD cooperator may survive when facing the invasion of defectors and the existence of a PD cooperator depends on whether it is the neighbor of a SD cooperator. The patterns revealed by Fig. 3(b) can be understood as the following. Suppose that one SD cooperator is in a sea of PD defectors. In this local environment, the payoff of the SD cooperator is 4S sd, and that of his neighboring PD defectors is T pd. Thus, once the condition S sd > T pd /4 is satisfied, the SD cooperator is stable against the exploitation of surrounding defectors and he may turn his neighboring defectors into cooperators. Therefore, the enhancement of cooperation at around v = 0 roots in the fact that isolated SD cooperators may act as a source of cooperation. Consequently, the slope of x(v) at v = 0 will be proportional to the fraction of SD players v. To be mentioned, the slope of x(v) at v = 0 also depends on the time what PD players in the neighborhood of SD cooperators spend on the strategy of cooperation. Since the transition rate for a player to change his strategy is proportional to the payoff difference between him and his opponent, the average time staying on cooperation for a PD neighbor of SD cooperators is proportional to S sd, which leads that slope of x(v) at v = 0 decreases when x sd decreases. However, with the

550 Communications in Theoretical Physics Vol. 57 increase of v, SD players will not be isolated from each others anymore. The strengthened interaction among SD players will depress the role of SD cooperators as a motor of cooperation and leads to the fall of x(v). For example, the SD cooperator will be turned into a defector when an SD defector becomes his neighbor. Figure 3(c) shows the snapshot for the parameter set B where x(v) is a little above zero. Interestingly, we find the same mechanism behind the enhancement of cooperation as that for v around zero: SD cooperators act as a motor of cooperation. Since the mechanism requires SD cooperators to be surrounded by PD players, which is satisfied only when the fraction of PD players is sufficiently large, we observe the onset of cooperation only at sufficiently large 1 v in Fig. 3(a). Fig. 2 For different T sd, the cooperator frequencies x(v) against v at T pd = 1.02, 1.08, 1.12 in the situation of HCP are shown in (a) (c) and that in the situation of HCL are shown in (d) (f). Fig. 3 (a) The cooperator frequencies x(v) against v in the HCP situation are plotted for different sets of T sd and T pd. The black curve is for T pd = 1.125 and T sd = 1.655 while the red curve for T pd = 1.135 and T sd = 1.665. (b) (c) The snapshots of the strategy pattern after transient process at points A and B denoted in the plot (a).

No. 4 Communications in Theoretical Physics 551 3.2 Cooperation Inhabitation in HCL Now we consider the inhibition of cooperation in the case of HCL. As shown in Fig. 2(d) 2(f), the inhibition of cooperation occurs only when both x pd and x sd are low. For simplicity, we suppose that both x pd and x sd are sufficiently low. Bearing on mind that R sd = R pd = 1 and P sd = P pd = S pd = 0, it is clear that the replacement of a PD (or SD) interaction by an SD (or PD) interaction will not contribute to the variation of cooperation if the original PD interaction links two cooperators or two defectors. Therefore, how a typical pair of cooperator and defector reacts to the change of the type of interaction between them is critical to understand the inhibition of cooperation. We first consider the case with v close to 0 where only small amount of PD interaction is replaced by SD interaction. Consider an event that the interaction in a typical pair of cooperator and defector is changed from PD to SD interaction. The cooperator receives a payoff of S sd = 2 T sd and the defector a payoff of T sd after the event, [30] that is, the payoff difference between the defector and the cooperator is around 2T sd 2. Comparing with the payoff difference T pd S pd = T pd between the defector and the cooperator before the event, we know that the cooperator in this typical pair with SD interaction may become a defector with a larger probability than that with a PD interaction if T sd > 1 + T pd /2 is satisfied. Otherwise, if T sd < 1 + T pd /2, the defector in the pair with SD interaction is more likely to be turned into a cooperator. Therefore at v close to zero, the inhibition of cooperation is observed for small T sd and cooperation is improved if T pd is sufficiently large, which is in agreement with the numerical results in Fig. 2(d) 2(f). Furthermore, we consider the case with v close to 1 where only small amount of SD interactions are replaced by PD interactions. It is important to note that the cooperators always tend to form filament-like cluster in evolutionary SD games on a lattice. That is, a cooperator always has two defector neighbors for v close to 1 provided that the cooperator is not at the ends of filament. Therefore, it is necessary to consider two typical pairs of cooperator and defector, for example pair A and pair B, but with the same cooperator. In this group of two defectors and one cooperator, the cooperator acquires 2S sd and the defectors T sd before the change the type of interaction. After that the interaction in the pair A is changed to a PD interaction, the payoff for the cooperator is S sd +S pd = S sd, the defector in the pair A gets T pd and the defector in the pair B T sd. Following the above analysis, we know that the payoff difference between the defector and cooperator in the pair A changes from T sd 2S sd to T pd S sd and that in the pair B from T sd 2S sd to T sd S sd. Now under the condition for the inhibition of cooperation at v 0, the change of SD to PD interaction in the pair A increases the possibility of turning the defector in the pair A into a cooperator. On the other hand, the change of SD to PD interaction in the pair A always increases the possibility that the cooperator follows the defector in pair B. Combining the reactions of the pair A and the pair B to the change of interaction from SD to PD game together, we have the condition for the inhibition of cooperation at v 1 as T sd < (4 + T pd )/3 through some simple algebraic calculations. In a comparison with Fig. 2(d) 2(f), we find that the analysis is in agreement with the numerical simulations. Especially, based on the condition for the inhibition of cooperation for v close to 1, we can know immediately the reason that x(v) behaves differently at v 1 for T sd = 1.62 in Figs. 2(d) 2(f). 4 Conclusion In conclusion, in the presence of heterogeneity in interaction, we have studied the evolution of cooperation in evolutionary games on a 2D lattice. The heterogeneity is introduced in two different ways: the heterogenous character of interaction assigned to players (HCP) and the heterogenous character assigned to links between players (HCL). The interaction is represented by an asymmetric bimatrix game in the situation of HCP while, in the situation of HCL, the neighboring players play either PD or SD game depending on the interaction assigned to the link between them. We find a strong enhancement of cooperation in HCP and an inhabitation of cooperation in HCL. 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