Effect of Growing Size of Interaction Neighbors on the Evolution of Cooperation in Spatial Snowdrift Game

Commun. Theor. Phys. 57 (2012) 541 546 Vol. 57, No. 4, April 15, 2012 Effect of Growing Size of Interaction Neighbors on the Evolution of Cooperation in Spatial Snowdrift Game ZHANG Juan-Juan ( ), 1 WANG Juan ( ), 2, SUN Shi-Wen (ê ), 1 WANG Li ( ), 1 WANG Zhen ( ), 3, and XIA Cheng-Yi ( ) 1, 1 Key Laboratory of Computer Vision and System (Ministry of Education) and Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, China 2 School of Automation, Tianjin University of Technology, Tianjin 300384, China 3 School of Software Technology, Dalian University of Technology, Liaolin Dalian 116024, China (Received November 16, 2011; revised manuscript received December 26, 2011) Abstract In this paper, we study the influence of the size of interaction neighbors (k) on the evolution of cooperation in the spatial snowdrift game. At first, we consider the effects of noise K and cost-to-benefit ratio r, the simulation results indicate that the evolution of cooperation depends on the combined action of noise and cost-to-benefit ratio. For a lower r, the cooperators are multitudinous and the cooperation frequency ultimately increases to 1 as the increase of noise. However, for a higher r, the defectors account for the majority of the game and dominate the game if the noise is large enough. Then we mainly investigate how k influences the evolution of cooperation by varying the noise in detail. We find that the frequency of cooperators is closely related to the size of neighborhood and cost-to-benefit ratio r. In the case of lower r, the augmentation of k plays no positive role in promoting the cooperation as compared with that of k = 4, while for higher r the cooperation is improved for a growing size of neighborhood. At last, based on the above discussions, we explore the cluster-forming mechanism among the cooperators. The current results are beneficial to further understand the evolution of cooperation in many natural, social and biological systems. PACS numbers: 02.50.Le, 87.23.Kg, 89.75.Hc Key words: spatial snowdrift game, interaction neighborhood, emergence of cooperation, regular lattice 1 Introduction Cooperative behaviors are ubiquitously present in various natural and social systems, ranging from simple genes, cellular organisms to complex human individuals. [1] However, this is inconsistent with Darwinian s theory in which any behavior bringing benefit to others will disappear. [2] In order to understand the persistence and emergence of cooperation, evolutionary game theory is often utilized as a common theoretical framework to investigate the cooperative behaviors among selfish and unrelated elements. [3 4] Recently, the prisoner s dilemma (PD) and snowdrift (SD) games are usually selected as metaphors for studying the social dilemmas between competition and cooperation. In the prisoner s dilemma, two players decide to cooperate or defect at the same time. Mutual cooperation leads to the reward R but the mutual defection results in the punishment P. If they adopt different strategies, the defector wins the maximum payoff T and cooperator gets the sucker s payoff S, such that T > R > P > S. Therefore, the defection is the optimal strategy in one shot game, and it is impossible for cooperators to resist the invasion of defectors in the well-mixed population. [5 6] To avoid the adverse situation of cooperators in the PD game, many seminal works are devoted to exploring the promotion of cooperation, e.g., the introduction of spatial structure into the prisoner s dilemma largely accelerates the evolution of cooperation, where individuals occupy the sites of lattice and merely play with their direct neighbors. [7] In addition, other games are also chosen to investigate the cooperative behaviors between individuals, such as the SD game which is more favorable to cooperation. Unlike the PD game, the sucker s payoff S is larger than punishment P such that T > R > S > P. A slight modification of payoff matrix renders the distinct strategy selection, the best choice of each player is to take the opposite strategy of his opponents in the SD game. As a consequence, the cooperation is easily sustained and reaches an equilibrium frequency given by 1 r under the wellmixed population, in which 0 r 1 denotes the cost-tobenefit ratio of player s cooperation. Nevertheless, recent studies have also reported that, contrary to the PD game, the spatial structure is unfavorable to the cooperation be- Supported by the National Natural Science Foundation of China under Grant Nos. 60904063 and 60673046, Tianjin municipal Natural Science Foundation under Grant No. 11JCYBJC06600 and the Development Fund of Science and Technology for the Higher Education in Tianjin under Grant No. 20090813 E-mail: juanwang75@163.com E-mail: wangz@dlut.edu.cn E-mail: xialooking@163.com c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

542 Communications in Theoretical Physics Vol. 57 tween individuals in the SD game [8] such that the role of spatial structure deserves to be explored in depth. Meanwhile, from the perspectives of complex networks in which vertices characterize the individuals or players and links mimic the interaction between them, [9] many real systems exhibit strong heterogeneity in the connectivity distribution which highly enhances the cooperative behaviors on complex networking communities. [10 12] Apart from addressing the specific topology of networks as a direct extension, various microscopic mechanisms are presented to promote the cooperative behaviors between individuals, such as kin selection, [13] direct and indirect reciprocity, [14] learning and teaching activities, [15] environmental noise, [16 17] asymmetric payoff, [18] voluntary participation, [19] reward and punishment, [20 21] partner switching, [22] memory, [23] individual s mobility, [24 26] weighting mechanism, [27 28] cooperation robustness, [29] as well as aspiring to the fittest. [30 31] Furthermore, the fitness of one player s neighbors is integrated into the game theory to account for the role of the environmental factors within the cooperative behaviors. [32 33] It is also interesting that voluntary participation with punishment becomes a widely useful framework to promote cooperation provided that some relevant conditions, such as genetic relatedness, repeated interaction, are considered. [34 40] All these works provide significant advancement of our understanding of these actions in structured populations. For the knowledge of the recent progresses of evolutionary game theory, we recommend the readers to refer the latest reviews. [41 42] In the present work, we will discuss the effect of growing size of interaction neighborhood on the evolution of cooperation in the spatial snowdrift game, and explore how the size of interaction neighbors influences the frequency of cooperators. Does the friendship circle of individuals, the social boundaries or the scope of communication impact his or her decision-making? Beyond the previous works, here we use a new perspective to explore the underlying mechanism, and take the size of interaction neighbors k into the consideration which is not confined to the von Neuman neighborhood (that is, k = 4) and the simulation is carried on a 100 100 lattice with periodic boundary condition. We mainly study how the size of interaction neighbors influences the cooperative behavior in the snowdrift game by varying the noise K. As shown in Fig. 1, five typical categories of interaction neighbors are considered on the regular lattices, which include the von Neuman neighborhood (k = 4), the Moore neighborhood (k = 8), a 5 5 box excluding himself (k = 24), a 7 7 box (k = 48), and a 9 9 box (k = 80). Fig. 1 (Color online) A growing size of neighborhood for snowdrift game on the regular networks. The shaded areas stand for the players who participate in the game at a time step. The star nodes are the focal players, and the black nodes surrounding a focal players are his interaction neighbors. The values of k are 4, 8, 24, 48, and 80 from (a) to (e). 2 Spatial Snowdrift Game Model The snowdrift game is investigated on the square lattice network with periodic boundary conditions, and each player has a growing neighborhood. Occupying one of nodes on the square lattice, each player has two available strategies: cooperate (C) or defect (D). The payoff of each one depends on the strategy of himself and his neighbors. In the snowdrift game, one can obtain a benefit b for get-

No. 4 Communications in Theoretical Physics 543 ting home and a cost c for shoveling the snow, here b > c. Thus, there may appear four situations as follows. In the case of shoveling the snow together, they gain the benefit R = b c/2, respectively. If both of them choose to remain in their cars, i.e. defect, they both obtain benefit P = 0 where neither of players can get home. If they choose the opposite strategy, the cooperator s benefit is S = b c while the defector acquire the highest benefit T = b. By convention, we normalize the payoffs to ones including one single parameter, i.e. R = 1, T = 1 + r, S = 1 r, and P = 0, where r = c/(2b c) denotes the cost-to-benefit ratio. Following the previous works, [41 42] each player (P i ) will update its strategy after each time step, and imitate the strategy of its randomly selected neighbor (P j ) with the probability depending on the difference of their payoff: 1 W (Si S j) = 1 + exp[(p i P j )/K], (1) where 0 < K < 0 represents the environmental noise during the strategy adoption, such as errors and irrationality of individuals. Positive values of K mean that better performing players are probably imitated, but it is possible to adopt the strategy of a player who does worse. [41] 3 Numerical Results In this paper, we carry out numerical simulations among the population composed of N = 100 100 individuals. In the following, we principally investigate the average frequency of cooperators F c for a growing neighborhood in the stationary state of system. In addition, the cooperation frequency is got by averaging over the last 5000 Monte Carlo (MC) time steps among the total 60000 steps. Moreover, in order to validate the robust of results, larger sizes of lattice and longer MC time steps are also tested, and the qualitative results are not changed. Just as Different circumstances call for different tactics, the agents update strategies in terms of the payoffs and the interactions with their neighbors. We first study the influence of noise on the evolution of cooperation. The frequency of cooperators F c as a function of the amplitude of noise K is shown in the Fig. 2. Along with the increase of the noise, the tendency of the frequency of cooperators F c is quite disparate for different cost-to-benefit ratio r. In the case of lower ratio (e.g., r = 0.1, 0.2 or 0.3), the frequency of cooperators F c is little changed for K < 0.1 and ultimately grows to be 1, though the process of increase maybe show a U-shape or exponential shape. When r is larger (e.g., r = 0.6 or 0.7), however, the frequency of cooperators sharply reduces as the strength of noise rises. Anyhow, the group strategies show higher fraction of cooperation when the parameter r is lower under the same noise level. Likewise, in line with the payoff matrix, for a lower r the values of R, S and T are pretty close, which brings about players keen on the cooperation. Thus, taking cooperation can get respectable gains whatever strategy the opponent chooses. While for a higher r the value of T accretion tempts the agents to defect for pursuing the maximum benefit and the cooperators will finally tend to be extinct. As stated above, we can arrive at the conclusion that the decreasing of the cost-to-benefit ratio will be conducive to facilitating the cooperation among the individuals. Fig. 2 (Color online) The frequency of cooperators F c as a function of the amplitude of noise K for different r (from top to bottom r = 0.1, 0.2, 0.3, 0.6, 0.7) on the condition of the value of interaction neighbors k = 4. Under maintaining the invariability of cost-to-benefit ratio r, how the size of interaction neighbors influences the cooperation frequency as the noise K changes, is shown in Fig. 3. For a lower r (e.g., Fig. 3(a), r = 0.2), the curves from top to bottom correspond to the value of k = 4, 80, 48, 24, 8. The simulation results show that for other four types of neighborhood size rather than k = 4, the defection is inhibited and invaded gradually, and the cooperation ultimately dominates the game as noise K increasing, which is in accordance with the consequence of Fig. 2. Nevertheless, most remarkably, the cooperator s fraction of other four cases (e.g., k = 8, 24, 48, 80) are close within the range of K < 10 0 but obviously much more inferior to that of k = 4. In other words, the augmentation of n plays no positive role in promoting the cooperation compared with that of k = 4, and it even impedes the infection and spreading of cooperation. In particular, for the size of interaction neighbors k = 80, the cooperation holds the strongest robustness and stability, and that for k = 48 is next. The growing size of interaction neighbors k, a presentation for the circle of friends, the social boundaries or the scope of communication, may be not advantageous in cooperation community evolution since it will approach the well-mixed population. However, in the case of r = 0.6 as shown in Fig. 3(b), the impact of the number of interaction neighbors k on the cooperator s frequency is completely distinct from that of r = 0.2. In Fig. 3(b), the frequency of cooperation F c,

544 Communications in Theoretical Physics Vol. 57 from high to low, is in order of k = 80, 48, 24, 8, 4. It is more enhanced for F c in the case of k = 24, 48, and 80 although the defectors still occupy the majority of the game. Apparently, for k = 8, F c is slightly improved comparing with that of k = 4, and the cooperation becomes extinction inevitably when the strength of noise is enough high. In our simulations, the value of noise is confined to K < 1000, whereas, if K is large enough, the cooperation for k = 24, 48, and 80 will also vanish as in the cases of k = 4 and k = 8. To conclude, the simulation demonstrates that the frequency of cooperation does not monotonously rely on the size of interaction neighbors, and it is influenced by the combined action of the costto-benefit ratio r and neighbors k. A higher r signifies the more payoffs which irritate the players to defect, but meanwhile the risk of zero payoffs getting from the mutual defection is rising. As is well-known, every player is rational and intelligent, and pursuing the maximum payoff blindly would give rise to the minimum payoff, which is contrary to expectations. When the individual interaction neighbors become more and more, a number of uncertainties force them to take a conservative strategy so as to maintain their own benefit. In addition, we further study the relationship between F c and r for different size of neighborhood k which is exhibited in Fig. 4. Evidently, when the cost-to-benefit ratio r increases, F c will gradually diminish and inevitably disappear for all these cases although cooperators dominate the game at first. In contrast with that of k = 4, the frequency of cooperators as a function of r follows a relatively smooth curve for other four cases. Within the range of r < 0.25, the frequency of cooperators for k = 8, 24, 48, 80 is lower than that of k = 4 and the augmentation of the size of interaction neighbors does not facilitate the cooperation, which is coherent with Fig. 3(a). However, when r > 0.25, it is clearly shown that the increase of neighborhood size k (e.g., k = 24, 48, and 80) plays a positive role in spreading the cooperation in comparison with that of k = 4. Since a number of interaction neighbors easily generate the same strategy clusters, defectors maybe get into a dilemma that their payoff will steeply plummet if most of neighbors choose defection. While the cooperation fraction for k = 8 is close to that of k = 4 and even not always exceed that of k = 4 in this range. Specifically, when r > 0.6, the number of cooperators for k = 8 are so depleted that the defection of whole group comes sooner than what we have expected. As a result, the frequency of cooperators is not enhanced monotonously by the growing size of neighborhood. From the above discussions, we can conclude that the maintenance of cooperation is apparently supported by the combined action of the costto-benefit ratio r and the size of interaction neighbors k. Furthermore, when k is over 24, the evolution of cooperation will approach the well-mixed cases inch by inch. Fig. 4 (Color online) The frequency of cooperation F c as a function of the cost-to-benefit ratio r for different value of k (k = 4, 8, 24, 48, 80 for diamonds, squares, circles, stars, triangles respectively) under the condition of K = 0.1. Fig. 3 (Color online) The frequency of cooperators F c as a function of the amplitude of noise K corresponding to different number of interaction neighbors n. (a) r = 0.2, k = 4, 8, 24, 48, 80 from top to bottom; (b) r = 0.6, k = 48, 24, 8, 4, 80. In order to ascertain the influence of the size of interaction neighbors k on the cooperation, the strategies distribution for different k on the square lattice is shown in Figs. 5 and 6, which are the typical snapshots when the

No. 4 Communications in Theoretical Physics 545 noise is set to be K = 0.1. It is indicated that forming cooperative clusters to resist the more incursion is a better method for cooperators to survive. In Fig. 5 (r = 0.2), cooperators or defectors assemble the players around themselves to make up their corresponding strategy clusters within a small scale. The result shows that the density of cooperation clusters for k = 8, 24, 48, and 80 is similar but not be enlarged with the augmentation of k, compared with that of k = 4. On account of a small cost-to-benefit ratio, i.e. r = 0.2, the temptation to defect is not strong enough so that the cooperation dominates the game. In such these circumstances, the loss of payoff will be mild, even for a defector. Hence, increasing the size of interaction neighbors can not promote the cooperation instead. However, as shown in Fig. 6 for r = 0.6, cooperative clusters for k = 8, 24, 48, and 80 are largely extended when compared with that of k = 4, despite the defectors hold a majority. Since the growing size of interaction neighbors affects the interaction among the agents, the cooperation can spread while facing the risk of larger loss. Our results suggest that there is a specific cost-to-benefit ratio r c in which the increase of neighborhood size is of great benefit to the spreading of cooperation if r exceeds r c. In particular, when r > 0.25, it is easy to recognize that the cooperation fraction for k = 24, 48, 80 is over than that of k = 4. As a consequence, the level of cooperation depends on the cost-to-benefit ratio r as well as the size of interaction neighbors. Fig. 5 (Color online) Five typical snapshots of the distribution of the strategies (C: Yellow and D: Blue) on the 100 100 regular lattice under the condition of K = 0.1 and r = 0.2 for different values of k (k = 4, 8, 24, 48, and 80 respectively) from (a) to (e). Fig. 6 (Color online) Five typical snapshots of the distribution of the strategies (C: Yellow and D: Blue) on the 100 100 regular lattice under the condition of K = 0.1 and r = 0.6 for different values of k (k = 4, 8, 24, 48, and 80) respectively from (a) to (e). 4 Conclusions To sum up, we endeavor to explore how the size of interaction neighbors influences the cooperation evolution in the snowdrift game. Firstly, the results of the relationship between F c and K for different r reveal that the cooperation evolution will be entirely distinct for the different cost-to-benefit ratios r. When r is lower, cooperators account for the majority in the game at the beginning range of noise, and then the frequency of cooperators grows to be one ultimately as the noise increasing. While for a higher r, on the contrary, defectors always dominate the game as a result of the defection temptation, and cooperators vanish unavoidably as the noise strength K increases. Furthermore, the effect of neighborhood size on the cooperation evolution is investigated under the condition of r = 0.2 and 0.6, respectively. The simulation results indicate that the frequency of cooperators does not always increase with the growing size of interaction neighbors, and it is closely related to the size of neighborhood and cost-to-benefit ratio r. Current findings are helpful for us to understand the role of interaction neighborhood in the evolution of cooperation, further provide some insights for the evolution of cooperation and collective behaviors in biological and social systems. References [1] K. Sigmund, The Evolution of Cooperation, Oxford University Press, Oxford (1993). [2] C. Darwin, The Origin of Species, Harward University Press, Cambridge, MA (1859) (Reprinted, 1964). [3] H. Gintis, Game Theory Evolving, Princeton University Press, Princeton, NJ (2000). [4] M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Harvard Universtiy Press, Cambridge (2006). [5] J.S. Maynard and E. Szathmary, The Major Transitions

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