Enhancement of cooperation by giving high-degree neighbors more help
aa r X i v : . [ phy s i c s . s o c - ph ] J u l Enhancement of cooperation by giving high-degreeneighbors more help
Han-Xin Yang , and Zhi-Xi Wu Department of Physics, Fuzhou University, Fuzhou 350116, China Center for Discrete Mathematics, Fuzhou University, Fujian, 350116, China Institute of Computational Physics and Complex Systems, Lanzhou University,Lanzhou, Gansu 730000, People’s Republic of ChinaE-mail: [email protected] E-mail: [email protected] Abstract.
In this paper, we study the effect of preferential assistance on cooperationin the donation game. Cooperators provide benefits to their neighbors at some costs.Defectors pay no cost and do not distribute any benefits. The total contribution ofa cooperator is fixed and he/she distributes his/her contribution unevenly to his/herneighbors. Each individual is assigned a weight that is the power of its degree, wherethe exponent α is an adjustable parameter. The amount that cooperator i contributesto a neighbor j is proportional to j ’s weight. Interestingly, we find that there exists anoptimal value of α (which is positive), leading to the highest cooperation level. Thisphenomenon indicates that, to enhance cooperation, individuals could give high-degreeneighbors more help, but only to a certain extent.PACS numbers: 02.50.Le, 89.75.Hc ONTENTS Contents1 Introduction 22 Model 33 Results and analysis 34 Conclusions 81. Introduction
Evolutionary game theory has been frequently employed as the theoretical frameworkto explain the emergence of cooperation among selfish individuals [1]. One of the mostused game model is the prisoner’s dilemma game (PDG) [2]. In PDG played by twoindividuals, each one simultaneously decides whether to cooperate or to defect. Theyboth receive R upon mutual cooperation and P upon mutual defection. If one cooperatesbut the other defects, the defector gets the payoff T , while the cooperator gains thepayoff S . The payoff rank for PDG is T > R > P > S . With the rapid developmentof network science, studies of PDG and other game models are implemented on variousnetworks [3, 4, 5, 6, 7, 8], including regular lattices [9, 10, 11, 12, 13, 14, 15, 16], randomgraphs [17, 18], small-world networks [19], scale-free networks [20, 21, 22, 23, 24, 25],multiplex networks [26, 27] and so on. For a given network, nodes represent individualsand links reflect social relationships. Individuals play the PDG with their directneighbors.An important special case of the PDG is the so-called donation game, where acooperator provides a benefit b to the other player at his/her cost c , with 0 < c < b . Adefector pays no cost and does not distribute any benefits. Thus, the payoff parametersin the donation game are T = b , R = b − c , P = 0, and S = − c . Ohtsuki et . al .discovered that natural selection favors cooperation if the benefit-to-cost b/c exceeds theaverage number of neighbors [28]. Allen et . al . provided a solution for weak selectionthat applies to any network and found that cooperation flourishes most in societies whichare based on strong pairwise ties [29]. Wu et . al . investigated impact of heterogeneousactivity and community structure on the donation game [30]. Hilbe et . al . showedthat in large, well-mixed populations, extortion strategies can play an important role,but only as catalyzers for cooperation and not as a long-term outcome [31]. Szolnokiand Perc found that extortion is evolutionarily stable in structured populations if thestrategy updating is governed by a myopic best response rule [32]. Xu et . al . discoveredthat extortion strategies can act as catalysts to promote the emergence of cooperationin structured populations via different mechanisms [33].In previous studies of the spatial donation game, a cooperator treats all neighborsequally and contributes to each neighbor with the same cost. However, in real life, anindividual usually has a preference for somebody and provides more benefit to him/her. ONTENTS i a weight k αi , where k i is i ’s degree and α is an adjustable parameter.We have found that, the cooperation level can be maximized at an optimal value of α .
2. Model
A cooperator i provides a benefit rc ij to a neighbor j at a cost c ij , where r is the benefit-to-cost ratio. For simplicity, we assume that r is the same for all pair interactions. Thetotal cost of cooperator i is c i = X jǫ Ω i c ij , (1)where the sum runs over all the direct neighbors of i (this set is indicated by Ω i ). Adefector pays no cost and does not distribute any benefits.Each individual i is assigned a weight k αi , where k i is i ’s degree and α is an adjustableparameter. For a fixed c i , cooperator i helps one of its neighbors j with a cost c ij proportional to j ’s weight c ij = c i k αj P lǫ Ω i k αl . (2)For α > < α = 0, a cooperator provides the same benefit to each neighbor.The payoff of individual i is given by M i = − c i s i + r X jǫ Ω i c ji s j , (3)where s i = 1 if i is a cooperator and s i = 0 if i is a defector. Note that the donation isnot symmetric, i.e., c ij = c ji .After each time step, all individuals synchronously update their strategies as follows.Each individual i randomly chooses a neighbor j and adopts j ’s strategy with theprobability [35] W ( s i ← s j ) = 11 + exp[( M i − M j ) /β ] , (4)where the parameter β ( >
0) characterizes noise to permit irrational choices. As thenoise β decreases, the individuals become more rational, i.e., they follow the strategiesof neighbors who have obtained higher payoffs with greater probabilities.
3. Results and analysis
We carry out our model in Barab´asi-Albert (BA) scale-free network [36] with size N = 5000. Without loss of generality, we assume that each cooperator pays the same ONTENTS c r = -0.1 = 0.5 = 1.5 Figure 1. (Color online) The fraction of cooperators ρ c as a function of the benefit-to-cost ratio r for different values of α . The average degree h k i = 10 and the noise β = 0.5. For each α , ρ c increases to 1 as r increases. total cost ( c i = 1 for any cooperator i ). Initially, the two strategies, cooperation anddefection, are randomly distributed among the individuals with the equal probability1/2. The equilibrium fraction of cooperators ρ c is obtained by averaging over the last10 Monte Carlo time steps from a total of 10 steps. Each data point results from 20different network realizations with 10 runs for each realization.Figure 1 shows the fraction of cooperators ρ c as a function of the benefit-to-costratio r for different values of α . One can see that for each value of α , ρ c increases to 1as r increases. For a small value of α (e.g., α = − . r isclose to 1. However, for a large value of α (e.g., α = 0.5 or α = 1.5), cooperators canstill survive, even if r = 1.Figure 2 shows the dependence of ρ c on α . One can see that for fixed values ofother parameters, there exists an optimal value of α (denoted as α opt ), leading to themaximal ρ c . The value of α opt is not fixed. From the insets of Fig. 2, one can see that α opt decreases as r increases, but increases as the average degree h k i or the noise β increases. Moreover, α opt is positive (around 0.6), indicating that cooperation can beoptimally enhanced if individuals give large-degree neighbors more help, but only to acertain extent.To explain the nonmonotonic behavior displayed in Fig. 2, we study an individual’spayoff as a function of its degree. The theoretical analysis is provided as follow.The cost that a cooperator j helps one of its neighbors i can be calculated as c ji = k αi P lǫ Ω j k αl = k αi k j P k max k l = k min P ( k l | k j ) k αl , (5) ONTENTS -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (b) < k > = 6 < k > = 10 < k > = 14 (c) = 0.1 = 0.5 = 1 c r = 1.9 r = 2.1 r = 2.3 (a) op t r op t < k > op t Figure 2. (Color online) The fraction of cooperators ρ c as a function of α for differentvalues of (a) the benefit-to-cost ratio r , (b) the average degree h k i and (c) the noise β , respectively. For (a), h k i = 10 and β = 0 .
5. For (b), r = 2 . β = 0 .
5. For(c), r = 1 . h k i = 10. For fixed values of other parameters, there exists anoptimal value of α (denoted as α opt ), leading to the maximal ρ c . The insets show thedependence of α opt on r , h k i and β respectively. where P ( k l | k j ) is the conditional probability that a node of degree k j has a neighbor ofdegree k l , k min and k max are the minimum and maximum node degrees of the network.Since BA networks have negligible degree-degree correlation [37], we have approximately P ( k l | k j ) = k l P ( k l ) / h k i , (6)where P ( k l ) is the degree distribution of BA networks. Substituting Eq. (6) into Eq.(5), we obtain c ji = k αi h k i k j h k α +1 i . (7)According to Eq. (3) and the mean-field theory, we can write the payoff M i as M i = − ρ c + rρ c X c ji = − ρ c + rρ c k i k max X k j = k min P ( k j | k i ) c ji . (8)Substituting Eqs. (6) and (7) into Eq. (8), we obtain M i = − ρ c + rρ c k α +1 i h k α +1 i . (9)From Eq. (9), one can find that for α > − < − α = −
1, individuals with different degreeclasses have the same payoffs.
ONTENTS = -2 = -1 = 0 = 1 M k k Figure 3. (Color online) The payoff M k as a function of degree k for different valuesof α . The benefit-to-cost ratio r = 40 and the average degree h k i = 20. All individualsare set to be cooperators. The data point are simulation results and the solid curvesare theoretical results from Eq. (9). The quantity h k α +1 i can be calculated as h k α +1 i R k max k min k α +1 P ( k ) dk , where the degreedistribution of BA networks is P ( k ) = 2 k min k − [37] and the maximum node degree ofthe network is about k max = k min √ N [38]. After calculating the integral, we obtain h k i = k min ln N and h k α +1 i = 2 k α +1 min ( N α − − / ( α −
1) (for α = 1). From Fig. 3, onecan see that the theoretical and numerical results are consistent.For very small values of α , lower-degree individuals gain more payoffs. Besides,the scale-free network is mainly composed of low-degree nodes. In this case, strategiesof low-degree individuals play important roles in the evolution of cooperation. Notethat cooperators gain less payoff than defectors in the same degree class. Thus, thewhole network will gradually fall into the state of full defection due to the presence ofabundant low-degree defectors. For large values of α , high-degree individuals (so-calledhubs) reap massive profits and gradually become cooperators [20, 21]. These hubsand some of their neighbors will form a cooperator cluster [39]. Within the cluster,cooperators can assist each other and the benefits of mutual cooperation outweigh thelosses against the outside defectors. However, for very large α , most individuals insidethe cooperator cluster gain nothing since almost all benefits are allocated to large-degreeindividuals. In this case, low-degree cooperators have negative payoffs since they haveto pay the cost of cooperation. On the contrary, the payoffs of defectors are positive.As a result, for very large α , the cooperator cluster is vulnerable to the invasion ofdefectors and become difficult to expand. Combining the results of the two limits of α ,the highest cooperation level should be achieved for some intermediate values of α . ONTENTS t = -1 = 0.5 = 4 c o f l o w - deg r ee node s c o f h i gh - deg r ee node s = -1 = 0.5 = 4 Figure 4. (Color online) Time series of the cooperator density ρ c for high-degree nodes(top panel) and low-degree nodes (bottom panel) respectively. The average degree h k i = 10, the benefit-to-cost ratio r = 2.1 and the noise β = 0.5. Without loss of generality,nodes with k >
40 ( k ≤
40) are divided into the high-degree (low-degree) class. For α = −
1, both kinds of nodes gradually become defectors. For α = 0 .
5, with time allhigh-degree nodes become cooperators and most low-degree nodes choose cooperation.For α = 4, although all high-degree nodes finally become cooperators, most low-degreenodes choose defection in the equilibrium state. To confirm the above analysis, we divide nodes into two classes: high-degree andlow-degree ones. Then we study the time evolution of the cooperation density for high-degree and low-degree nodes respectively. From Fig. 4, one can see that, for a smallvalue of α (e.g., α = − α (e.g., α = 0 . α = 4), the cooperator densityfor high-degree nodes increases to 1 while the cooperator density for low-degree nodesfirst decreases and then increases to a steady value. For α = 0 .
5, low-degree nodesinside the cooperator cluster can gain enough payoffs to resist the invasion of defectors,leading to a high value of the cooperator density (about 0.8) in the stable state. For α = 4, low-degree nodes get little benefit and are vulnerable to the attack of defectors,resulting in a low cooperation level (about 0.3) in the equilibrium state.Next, we study the cooperator density ρ c ( k ) in the steady state as a function ofdegree k for different values of α . From Fig. 5, one can see that for a small value of ONTENTS
10 1000.30.40.50.60.70.80.91.0 c ( k ) k = -1 = -0.4 = 4 Figure 5. (Color online) The cooperator density ρ c ( k ) in the steady state as a functionof degree k for different values of α . For each value of α , the fraction of cooperatorsin the equilibrium state ρ c = 0.5, the average degree h k i = 6 and the noise β = 0.5.For α = -1, the benefit-to-cost ratio r = 3.95. For α = -0.4, r = 1.89. For α = 4, r = 1.78. For a very small values of α (e.g., α = -1), ρ c ( k ) is almost independent ofdegree k . For a large value of α (e.g., α = -0.4 or 4), high-degree nodes are occupiedby cooperators and ρ c ( k ) is minimized for medium-degree nodes. α (e.g., α = − ρ c ( k ) is almost the same for different values of k . For a larger valueof α (e.g., α = − . α = 4), almost all high-degree individuals become cooperatorswhile some low-degree individuals still choose defection. Here, we also find that ρ c ( k ) isminimized for medium-degree individuals. Such phenomenon has been observed in theweak prisoner’s dilemma game [40].In the above studies, we assume that each cooperator contributes the same total cost c i = 1. To validate the universality of the enhancement of cooperation by preferentialassistance, we consider a case in which the total cost of a cooperator is not a constantbut proportional to its degree, i.e., c i = k i . In this case, there also exists an optimalvalue of α , leading to the highest cooperation level, as shown in Fig. 6.
4. Conclusions
In conclusion, we have found that cooperation can be promoted when cooperatorscontribute more to high-degree neighbors, but only to some extent. In this case, high-degree individuals are proved to have high payoffs and act as cooperators. These hubsand some of their neighbors form a cooperator cluster, within which cooperators canassist each other and the benefits of mutual cooperation outweigh the losses againstdefectors. The above finding is robust with respect to different values of the benefit-
ONTENTS -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 c r = 4 r = 4.5 r = 5 Figure 6. (Color online) The fraction of cooperators ρ c as a function of α for differentvalues of the benefit-to-cost ratio r . The average degree h k i = 6 and the noise β =0.5. The total cost of each cooperator is equal to its degree, i.e., c i = k i . For each r , ρ c is maximized at an optimal value of α . to-cost ratio, different kinds of network structure, different levels of the noise to permitirrational choices, and different choices of the total cost of a cooperator.The heterogeneous resource allocation has also been considered in other kinds ofgame models such as the public goods game. Huang et . al . found that cooperation canbe enhanced if individuals invest more to smaller groups [41]. Meloni et . al . allowedindividuals to redistribute their contribution according to what they earned from thegiven group in previous rounds [42]. Their results showed that not only a Paretodistribution for the payoffs naturally emerges but also that if players do not investenough in one round they can act as defectors even if they are formally cooperators.Note that the donation game and the public goods game are based on pair interactionsand group interactions respectively. Together Refs. [41, 42] and our work offer a deeperunderstanding of the impact of the heterogeneous resource allocation on the evolutionof cooperation. Acknowledge
This work was supported by the National Science Foundation of China (GrantNos. 61773121, 61403083, 11575072 and 11475074), and the fundamental research fundsfor the central universities (Grant No. lzujbky-2017-172).
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