- Email: [email protected]
Benefits of intervention in spatial public goods games
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Luhe Yang a , Zhaojin Xu b , Lianzhong Zhang a , Duoxing Yang c a
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Keywords: Evolutionary game theory Public goods game Strategy intervention Fermi function
The anarchy driven by private punishment is inopportune and inappropriate in modern human societies. It is necessary for a person to appeal to a higher authority such as the police so that a society can maintain more stable. We introduce strategy intervention instead of punishment in spatial public goods game. Some defectors are forced to contribute to the common pool. We show how strategy intervention affects cooperation of a population. Interestingly, weak intervention restrains the impact of spatial reciprocity leading to a lower level of cooperation or even a full defection state. Such phenomena are in contrast to ordinary intuitions. Intervention is enforced by a higher authority which avoids the second-order problems. Furthermore, high synergy factor and proper intervention has a mutual impact on increasing group incomes. We highlight the importance of institutional intervention in a stable society. © 2018 Published by Elsevier B.V.
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Article history: Received 13 July 2018 Received in revised form 30 September 2018 Accepted 1 October 2018 Available online xxxx Communicated by C.R. Doering
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Introduction
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Although the conception of “Tragedy of commons” has been introduced by Hardin [1] for half a century, problems, including excessive deforestation, over-fishery in seas, and heavy pollutions in rivers and atmosphere, remain serious around the world. The common resources are equal for each individual to make use of without limiting conditions, a fact that a tragedy of resource exhaustion eventually kicks in. Public goods game (PGG) has attracted wide applications for decades in economics, biology and social sciences [2–6], which is proposed as a basic game theoretical model for studying interactions, mainly cooperative behaviors between irrelevant individuals. In the typical experiment of PGG, four players are taking part in the game, and are provided with 20 dollars individually. They decide how many dollars should be contributed to, and share the double amount of investments as outcomes of the common pool. Obviously, there exists an ideal situation, in which everyone gets a doubled payoff, only if they all contribute their total initial capitals to the common pool. In fact, when some players find that even without investments, they can still obtain lucrative incomes, they would rather choose to free ride on the other players’ contribution, which naturally leads to economic stalemate [7]. Without specific mechanisms introduced, cooperative behaviors among irrelevant individuals rarely emerge only by natural selec-
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tion [8]. For studying the interactions between individuals and how to enhance cooperation in the population, direct reciprocity [4,9], indirect reciprocity [10–12], kin selection [13], spatial selection [14] and multilevel selection [15,16] have been proposed. As described in strong reciprocity theory, a person would pursue fairness and justice rather than human rational and selfish intention costly even at a loss [7]. Such type of behavior is prescribed to “altruistic punishment” [17], which plays an important role in social cooperation. Due to the imitation from conditional cooperators to even a little few existing defectors, which decreases the level of cooperation [18], altruistic punishment become deterrent to free riders, and indirectly enhances cooperation in a population [19]. Especially for structured population, because of the limited interactions separating them from non-punishment players [20,21], altruistic punishment strategies can survive and spread. Although the costly behavior is interpreted as “negative emotion” or “impulsive negative reaction” brought from free riders [22,23], the explanation is incomplete. Moreover, it is possible for both cooperators and defectors (free riders) to have “negative motions”, arising more complicated problems such as “anti-social punishment” or rounds of punishment [24–28]. Relevant researches on altruistic behaviors have been carried out in evolutionary biology or neuropsychology [29–31] for decades. Some studies explored more effective punishment solutions to promote cooperation [32,33]. Dawkins believes that the genes of all creatures are selfish [34], which is the key of reproductive evolution. In terms of its selfish nature, gene can also evolve altruistic behaviors. Whereby, it still requires ultimate explanation
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School of Physics, Nankai University, Tianjin 300071, China School of Science, Tianjin University of Technology, Tianjin 300384, China Institute of Crustal Dynamics, China Earthquake Administration, Ministry of Emergency Management, Beijing 100085, China
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Benefits of intervention in spatial public goods games
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E-mail addresses: [email protected] (Z. Xu), [email protected] (L. Zhang). https://doi.org/10.1016/j.physleta.2018.10.001 0375-9601/© 2018 Published by Elsevier B.V.
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Methods
p C = r c (nC + β n I )/G − c ,
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The “self-adjusting rule” applied in Ref. [37] is more suitable for three-strategy voluntary public goods games (PGG), because players can make decisions referring to the payoffs of loners. But such dynamics is not pervasive and appropriate for two-strategy PGG without loners. Fermi function [5,14,39,51] has been acknowledged and verified to be effective dynamics of group interactions on structured population. Thus, preceding strategy update rules that have been commonly accepted are considered in this paper. In this case, we can figure out the significance of strategy intervention to our current research. For Monte Carlo simulations of public goods game, each player is confined to a site x on a 100 × 100 square lattice with periodic boundary conditions, which is the simplest networks for individuals to interact structurally [14]. Research in Ref. [45] demonstrates that group interactions can effectively link players on lattices. In our simulations, each individual has equal probability to cooperate (C ) or defect (D) as the initial conditions. An individual will be selected randomly to play with his 4 “von Neumann neighborhoods”, namely a single PGG. Such asynchronous updating is utilized, such that every individual has a chance to play once on average in each Monte Carlo Step (MCS). Actually, a player x should join in 5 single PGGs, which take place on site x and the other 4 neighbors’ site, thereby we take the sum of the five payoffs as the overall payoff Πsx of player x. After that, a player x randomly selects one of his four nearest neighbors, who acquires the payoff Πs y as player y on his own site. Whether player x choose to imitate the strategy of player y depends on a probability:
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Γ = 1/ 1 + exp (Πsx − Πs y )/ K ,
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Here Γ is given by the Fermi function. K denotes the amplitude of noise [46], and it is set to K = 0.5 for a player to imitate his neighbor’s choice more effectively to avoid choosing a worse strategy [47].
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From Eqs. (5), (6) and (7), it follows the average total payoff of the population P sum :
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Based on the imitation rule, intervention degree η (0 ≤ η ≤ 1) is introduced. Therefore, each individual has a probability η to be
here N sum = N C + N U + N I and N C , N U , N I represent the total number of C , D U and D I .
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where the investment of cooperators are set to unity (c = 1). We assume that the condition 1 < r < r < G reflecting the synergetic effects of cooperation. Note that when individuals imitate in the current round, both D U and D I are treated as defective behaviors due to the original choice before the intervention. Namely, D I is not an independent strategy. Additionally, each individual has a small probability μ to choose one of the available strategies (C , D) randomly ensuring that at least one individual takes random strategy in each round [49]. In our simulations, the value of mutation rate μ is small enough not to have an influence on the stability of cooperation. Usually, a simulation converges after approximate 104 MCSs for a 100 × 100 population. Here, we choose the averaged value of the last 1000 MCSs within total 6 × 104 ensuring more stable simulation results. The payoffs of caught defectors “I ” are counted in that of D-group, the average payoffs P C , P U , P I , P D of strategy C , D U , D I and D are given by
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selected by the system (a higher authority) after a strategy distribution of the population in each round. The defectors among the selected individuals are marked as “I ”, and they are forced to change their strategies (D I ): contribute a certain amount of investment β c (β ≥ 0) into common pool in current round and share the outcomes with cooperators and unselected defectors. Here, the coefficient β is defined as intervention intensity. The defectors who are not “caught” by the system are marked as “U” and still free ride to play strategy D U . A cooperator invests c into the common pool while a defector contributes nothing to free ride. Since punishment belongs to public goods [48], strategy intervention needs to cost as well. We propose r − r as the intervention cost, where r denotes origin synergy factor of the common pool and r the actual synergy factor. The intervention cost results from common pool, hereby individuals needn’t contact each other. It notes that the form of intervention cost can be diverse, essentially, what truly impacts the evolution of PGG is r , hereafter we focus r as a key parameter for further discussion. The payoff of player x in a single PGG depends on his and his neighbors’ strategies in a single PGG. When nC , nU , n I (nC + nU + n I = G = 5) represent the number of strategy C , D k (k = U , I ), respectively, their net payoffs p C , p U , p I take the following form:
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about the intrinsic motivation of altruistic behaviors. The anarchy driven by private punishment is rare in modern human societies, therefore, it is necessary for a person to appeal a higher authority in order to maintain a stable society [35–37]. An individual could choose to quit the competition to avoid being exploited with a small but fixed payoff [5]. But in many practical cases, individuals can’t escape but only participate in social activities. By virtue of theoretical analysis [35–41] or experiments [42,43], “pool punishment” has been revealed as a more realistic policy in public goods games. Such type of institutionalized punishment can be more efficient for enhancing cooperation than peer-punishment, which could be combined with participation [35] or conditional reward [36,41]. On the other hand, just as peer-punishment, a question remains considered [38,39,43] of whether “second-order” free riders should be sanctioned or not. As a recent research reported, successful individuals either act as partners or as rivals [44]. The rivals always put themselves first eventually leading to defection which is deterrent to social development. In this paper, based on stochastic imitation rules among individuals, we propose strategy intervention instead of punishment enforced by a central institute, and figure out how the mechanism impacts the cooperation among individuals.
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Fig. 1. Evolution of cooperation in spatial public goods game on the β –η parameter plane for (a) r = 3.5, (b) r = 4, (c) r = 4.2, (d) r = 4.5. Only D-strategy (D U + D I ) exists in Zone I while C -strategy exists in Zone II (full cooperation) and Zone III refers to the intermediate state (C + D U + D I ) of the population at dynamic equilibrium. The other parameters are μ = 0.0001 and K = 0.5. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
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Results
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A fact that no interactions exist among individuals is the biggest difference between intervention and punishment. Previous studies show that levels of cooperation could maintain the stability in structured population, as the synergy factor increases up to a threshold r ≈ 3.74 [47,50] based on Fermi function or r ≈ 3.90 [5] with replicator dynamics due to the network reciprocity. As depicted in Fig. 1, the frequency of cooperation changes nonmonotonously for different intermediate values of r , and strongly depends on the intervention degree η and intensity β . The patterns of intervention-based frequency are essentially different from that calculated by the probabilistic punishment [51,52]. It also shows in each panel results in the typical model without intervention (η = 0) considered. Intuitively, effects of the intervention and network reciprocity do not appear at low values of r , when the intervention (β or η ) is weak (Fig. 1a), such a fact that leads to full defection (Zone I). Subsequently, a full cooperation emerges (Zone II), as the intervention is strengthened forward a moderate level. It is illustrated that higher chance of detecting defection may result in lower cooperation level. Indeed, this is against our intuition, and similar counter-intuitive phenomena about mutation rate μ, competing strategies and renewable resources have been reported previously [53–55]. Visually, Zone III expands, while Zone I and Zone II narrow as r increases. The effect of spatial reciprocity becomes more apparent, and the intermediate state of strategies (C + D U + D I ) in the population (Zone III) “erodes” Zone I and Zone II. The spatial reciprocity can promote the levels of cooperation, and the intervention is deployed aiming to reduce the defective behaviors. Interestingly, the valleys of cooperation obviously remain in Zone I for higher value of r . Maybe it is not obvious to figure out the decrease of cooperation by colors, but we can see the valley and there must be a decreasing process between the non-intervention state (η = 0) and the valley. Preliminarily, results as shown in Fig. 1(b, c, d) demonstrate that the weak intervention leads to lower level of cooperation and higher level of defection than that without intervention, respectively. Although indistinct, the levels of cooperation remain unchanged when η = 0 in Fig. 1b. Actually, a certain num-
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ber of D I -strategies exist in Zone I while η > 0, and individuals do not have to actively cooperate within a certain extent of β or η . The caught defectors become “passive cooperators” with the active cooperators replaced. Such “D U + D I ” state can reach a dynamic equilibrium, provided that the “C + D” state maintains at higher value of synergy factor. The evolutionary characteristics imply mutual effects of the network reciprocity and intervention. For further understanding the elementary invasion among different types of individuals, the same original homogeneous distribution of cooperators and defectors are set as the initial state (“Origin” in the leftmost panel of Fig. 2) of the spatio-temporal evolution for different levels of intervention and varying values of r . Correspondingly, time courses of the strategy fraction are located in the rightmost of each evolutionary panel (Fig. 2f–j). Results of a typical PGG for lower values of r are shown in Fig. 2 (from a1 to a3). It is clear that, without intervention (β = 0, η = 0), the group of defectors quickly invades C -group, and essentially leads to full defection of the population, while effects of the network reciprocity haven’t appeared. When r increase to r = 4 given weak intervention (β = 2, η = 0.25), it still yields a full defection state (D U + D I ) of the population (Fig. 2b1–b3). Compared with typical results (Fig. 2f), because of the functioning intervention, the invasion through D-group into C -group deteriorates (Fig. 2g). Such distinction indicates that the weak intervention can restrain effects of the spatial reciprocity, which result in a lower level of cooperation or even a full defection state of the population. By elevating the level of intervention (β = 2, η = 0.5) for r = 3.6, we contrarily obtain a full cooperation state of the population (Fig. 2c1–c3). The fraction of D-group slowly increases, then gradually declines until the individuals, who choose to defect vanish (Fig. 2h). Results (as shown in Figs. 1 and 2) suggest that only if β > 1 we can potentially achieve a full cooperation state in spatial PGG with a given proper intervention degree η even for a lower value of r . It is natural for cooperators to form clusters to fight against defective behaviors when adopting imitation rules [5,14]. When r increases to r = 4.2 with the weak intervention (β = 1, η = 0.25), C -group and D-group mutually invade, and the fraction of cooperators gradually decreases followed by the slowly increasing (Fig. 2i), and finally reaches a stable state. In comparison with the processes b: b1–b3, effects of the spatial reciprocity are enhanced, therefore, the level of cooperation maintains at a dynamic equilibrium (Fig. 2d1–d3). If we continue to strengthen the intervention levels then we obtain full cooperation (Fig. 2c3). The fraction of cooperators first rise and then drop toward a relatively stable state (Fig. 2j) which is opposite to the process Fig. 2i. It is found that small fluctuations occur at the equilibrium state (Fig. 2i, j) of the intermediate states (C + D U + D I ). We assume that, with full intervention degree (η = 1) considered, all defectors are forced to change their strategies, and there are only C -group and D I -group left. When β > 1, given that cooperators still make a contribution, they become “free riders” in contrast to the caught defectors, who invest more capital into the common pool. Such phenomena (C + D I ) is comparable to the typical model without intervention (C + D) at given values of r . The imitation of individuals depends on the payoffs of themselves and their neighbors. Actually, there exist mutual influences between the frequency and payoff of different types of players. Here, the aiming is to find the reason at that the frequency of cooperators changes with different levels of intervention. For facilitating the analysis, the fractions are overlaid with the corresponding average payoffs of strategy C , D U and D I (Fig. 3) at the given value of r = 4.2. At first step, we start a simulation without intervention (η = 0) until the fractions of cooperators and defectors are stable. Then, we change the value of η from 0 to 0.15 when β = 3. Because of more contributions from parts of caught defectors to the common pool, both cooperators and defectors ob-
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the same homogeneous distribution of strategies with equal number of cooperators (green) and defectors (red) as the initial condition (Origin) illustrated in the leftmost of the patterns. With intervention, D-strategies are divided into D U (orange) and D I (blue). 60000 MCs have been run in the simulations of Fig. 2 as well as that of Fig. 1. We hope to show the “process of mutual invasion” among different strategies from the “Origin” (a particular strategy distribution) to equilibrium state. Note that panels a3, b3, c3, d3, e3 in Fig. 2 show all stable states of the population. Since the invasion process evolves quickly, we amplified this process as illustrated in Fig. 2f–j, after where the strategies converge slowly till the equilibrium state. The other parameters are μ = 0.0001 and K = 0.5.
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Fig. 2. Spatial pattern formation and the corresponding time courses of fraction of strategies with different levels of intervention for (a: a1–a3) r = 3.6, β = 0, η = 0; (b: b1–b3) r = 4, β = 2, η = 0.25; (c: c1–c3) r = 3.6, β = 2, η = 0.5; (d: d1–d3) r = 4.2, β = 1, η = 0.25; (e: e1–e3) r = 4.2, β = 3.5, η = 0.9. All the simulations start from
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Fig. 3. Evolution of the fractions of strategies (with the same color settings as in Fig. 2) and the corresponding average payoffs over time with abrupt changes of intervention (β = 3, η = 0 → 0.15 → 0.5 → 0.9) for r = 4.2. The other parameters are μ = 0.0001 and K = 0.5.
tain more payoffs than those in the previous stage. Given that the increasing range of average payoffs are greater for D-group than that for C -group (L 1 > N 1 > M 1 ), thereby, the D-group dominates the invasion. Consequently, the fractions of D U and D I increase, whereas the fraction of C -strategies decreases fast until it reaches
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a full defection state. In this condition, only if the un-caught defectors (D U ) play with the caught ones (D I ), both sides can gain more payoffs. Otherwise, they get nothing without intervention for low value of r . Moreover, there exist more D U than D I when η = 0.15. The average payoffs of the two types of players are comparable.
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while the spatial reciprocity continues to predominate in the PGG. In some cases, the weak intervention (pink curve with inverted solid triangle) restrains the influence from spatial reciprocity, but it still obtains higher payoffs than that in the typical model. Actually, all of the selected points on the β –η parameter plane (with different levels of intervention) acquire more payoffs for the population, which results from the effect of intervention. Especially, it disagrees with our intuition that the excessive intervention reduces the number of cooperators more commonly than moderate level of intervention (Fig. 4a). Previous studies on PGG with punishment attempted to enhance cooperation based on a variety of mechanisms. But considering the effect of intervention, we propose that low levels of cooperation may not be worse than full cooperation state. Unlike the punishment, the intervention cost r − r comes from the outcome of common pool, and its numerical value equals to the decreased amount of synergy factor r. It is similar to the taxation system that all the individuals, who contribute either actively or passively, should share the cost. Furthermore, we take a certain level (β = 3, η = 0.75) of strong intervention (orange curve in Fig. 4b) subject to the prescribed value of = 5, if the intervention cost could be controlled low enough for the value of r within the “increment area” (r 4.25), the population could obtain more payoffs than that at a full cooperation state for r = 5 (gray solid line).
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Fig. 4. Fractions of cooperators and the corresponding average total payoffs of the population in dependence of r with different levels of intervention. The other parameters are μ = 0.0001 and K = 0.5. The blue curve with right-solid triangle and orange curve with solid rhombus denote the excessive intervention-dependent results, while the green curve with solid-triangle represents results with the moderate level of intervention.
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We introduce the strategy intervention in spatial PGG and analyze its correlation with the evolutions of different types of strategies. When “pool punishment” and “peer punishment” coexist, they restrict each other and the latter is dominant which leading to promote the defection. This is different from our study where only institutional intervention functions [56]. Indeed, it is different from pool punishment that all the contributors no matter active or passive should bear the intervention cost. There is no punisher but only two optional strategies for players to choose in our model. This is a key distinction between intervention and pool punishment. With the intervention considered in PGG, cooperators do not have to worry about the revenge imposed by the punishment [25]. The network reciprocity and intervention have interplay on the evolutionary outcome of PGG in spatial lattice. According to our intuition, cooperation ought to be enhanced with the strengthened intervention, but the simulation results indicate that the weak intervention restrains the impact of spatial reciprocity leading to a lower level of cooperation, or even a full defection state of the population. Only if β > 1, it can possibly achieve a full cooperation state in spatial PGG with moderate value of η even at low value of r . The active cooperators become “free riders” over the caught defectors who invest more capital into the common pool when β > 1, which situation (C + D I ) is similar to the typical model without the intervention (C + D) for the same value of synergy factor. We note that the population can obtain more payoffs than that at full cooperation state only when the value of r is high enough. So the impact of network reciprocity resulting from large value of r on the group incomes can not be ignored. Qualitatively, only when the synergy factor is high enough and we control the intervention cost properly, can the population obtain more payoffs. It generates different payoffs for all types of groups forming evolutionary distribution of strategies in each round. Subsequently, the strategy with more payoffs can survive and spread, promoting the structured population toward a new equilibrium state. All the individuals, who actively or passively contribute, should share the intervention cost, which comes from the outcome of common pool, and its numerical value equals to the decreased
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Similarly, when η varies from 0.15 to 0.5, all the average payoffs for different types of players increase, with the increasing ranges described as M 2 > N 2 ≈ L 2 , which phenomena leads to the decline of D-group, until a full cooperation state is reached. Finally, we set the value of η as 0.9, and find a significant distinction of the increasing range between the three types of average payoffs (N 3 ≈ L 3 > M 3 ). It observes that the fractions remain constant at the beginning when the value of η changes, which looks unlike the previous stages. The main reason is that the random parameter μ is so small that an individual would probably be caught for high value of η , when they choose to defect. Only if amounts of defectors are not caught by the system, D-strategies could invade their neighbors and rapidly spread, which could result in a growth of D-group, until it reaches an equilibrium state. There are larger fluctuations in average payoffs of D U -group than those of D I -group due to the small proportion and uneven distribution of D U . It is seen from Fig. 1c that the intervention degree η and intensity β impacts the levels of cooperation in the similar way. It is found that, the intervention has effects on the distribution strategies in each round, which results in the distinction of average payoffs of different types of groups. Furthermore, the more payoffs are, the more potentially the strategies spread to reach a new equilibrium. It observes that the average payoffs irregularly fluctuate, when finite variations in fractions of defectors (cooperators) appear at a full cooperation (defection) state. Provided that the irregular fluctuations have no influences on the result analysis, we assume the payoffs of defectors (cooperators) be zero at the full cooperation (defection) state. As mentioned above, the network reciprocity does not function for a lower synergy factor. The weak intervention has little effects on the levels of cooperation for a lower value of r . However, the evolution of different strategies become more complex, when the value of r increases highly enough (Fig. 4). The black curve with solid blocks represents the level of cooperation in dependence on r without intervention. Although the fraction of cooperators remains in the same trend with weak intervention (red curve with solid circle), the relevant average total payoffs of the population are always higher than that of the typical model without intervention. The intervention has functioned on the outcome of the population,
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Acknowledgement
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This project is in part funded by the National Natural Science Foundation of China (NSFC) under Grant No. 41874113.
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References
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[44] C. Hilbe, K. Chatterjee, M.A. Nowak, Nature (2018), in press. [45] M. Perc, J. Gómez-Gardeñes, A. Szolnoki, L.M. Floría, Y. Moreno, J. R. Soc. Interface 10 (2013) 20120997. [46] G. Szabó, G. Fáth, Phys. Rep. 446 (2007) 97. [47] A. Szolnoki, M. Perc, G. Szabo, Phys. Rev. E 80 (2009) 056109. [48] T. Yamagishi, J. Pers. Soc. Psychol. 51 (1986) 110. [49] A. Traulsen, C. Hauert, H. De Silva, M.A. Nowak, K. Sigmund, Proc. Natl. Acad. Sci. USA 106 (2009) 709. [50] M. Perc, Eur. J. Phys. 38 (2017) 045801. [51] X. Chen, A. Szolnoki, M. Perc, New J. Phys. 16 (2014) 083016. [52] X. Chen, A. Szolnoki, M. Perc, Phys. Rev. E 92 (2015) 012819. [53] D. Helbing, A. Szolnoki, M. Perc, G. Szabó, Phys. Rev. E 81 (2010) 057104. [54] M. Perc, A. Szolnoki, Sci. Rep. 5 (2015) 11027. [55] X. Chen, A. Szolnoki, PLoS Comput. Biol. 14 (2018) e1006347. [56] A. Szolnoki, G. Szabó, L. Czakó, Phys. Rev. E 84 (2011) 046106. [57] T. Hobbes, Leviathan, Oxford University Press, Oxford, UK, 1909.
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G. Hardin, Science 162 (1986) 1243. C. Hauert, S. De Monte, J. Hofbauer, K. Sigmund, J. Theor. Biol. 218 (2002) 187. D.S. Wilson, E. Sober, Behav. Brain Sci. 17 (1994) 585. R.L. Trivers, Q. Rev. Biol. 46 (1971) 35. C. Hauert, S. De Monte, J. Hofbauer, K. Sigmund, Science 296 (2002) 1129. F. Berkes, D. Feeny, B. McCay, J.M. Acheson, Nature 340 (1989) 91. E. Fehr, S. Gachter, Nature 415 (2002) 137. D.G. Rand, M.A. Nowak, Trends Cogn. Sci. 17 (2013) 413. R. Axelrod, W.D. Hamilton, Science 211 (1981) 1390. M.A. Nowak, K. Sigmund, J. Theor. Biol. 194 (1998) 561. M.A. Nowak, K. Sigmund, Nature 393 (1998) 573. C. Wedekind, M. Milinski, Science 288 (2000) 850. W.D. Hamilton, Am. Nat. 97 (1963) 354. M.A. Nowak, R.M. May, Nature 359 (1992) 826. D.S. Wilson, Proc. Natl. Acad. Sci. USA 72 (1975) 143. A. Traulsen, M.A. Nowak, Proc. Natl. Acad. Sci. USA 103 (2006) 10952. A. Chaudhuri, Exp. Econ. 14 (2010) 47. J. Ledyard, Public goods: a survey of experimental research, in: J. Kagel, A. Roth (Eds.), Handbook of Experimental Economics, Princeton University Press, Princeton, USA, 1995.
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[19] F. Guala, Behav. Brain Sci. 35 (2012) 1. [20] C.T. Bergstrom, D. Helbing, A. Szolnoki, M. Perc, G. Szabó, PLoS Comput. Biol. 6 (2010) e1000758. [21] D. Helbing, A. Szolnoki, M. Perc, G. Szabó, New J. Phys. 12 (2010) 083005. [22] E. Fehr, S. Gächter, Am. Econ. Rev. 90 (2000) 980. [23] A.G. Sanfey, J.K. Rilling, J.A. Aronson, L.E. Nystrom, J.D. Cohen, Science 300 (2003) 1755. [24] L. Denant-Boemont, D. Masclet, C.N. Noussair, Econ. Theory 33 (2007) 145. [25] B. Hermann, C. Thöni, S. Gächter, Science 319 (2008) 1362. [26] N. Nikiforakis, J. Public Econ. 92 (2008) 91. [27] M.A. Janssen, C. Bushman, J. Theor. Biol. 254 (2008) 541. [28] D.G. Rand, J.J. Armao, M. Nakamaru, H. Ohtsuki, J. Theor. Biol. 265 (2010) 624. [29] S.A. West, C.E. Mouden, A. Gardner, Evol. Hum. Behav. 32 (2011) 231. [30] P. Barclay, Behav. Brain Sci. 35 (2012) 16. [31] D. Quervain, U. Fischbacher, V. Treyer, M. Schellhammer, U. Schnyder, A. Buck, E. Fehr, Science 305 (2004) 1254. [32] H. Yang, X. Chen, Appl. Math. Comput. 316 (2018) 460. [33] Q. Wang, N. He, X. Chen, Appl. Math. Comput. 328 (2018) 162. [34] R. Dawkins, The Selfish Gene, Oxford University Press, Oxford, UK, 1976. [35] T. Sasaki, A. Brannstrom, U. Dieckmann, K. Sigmund, Proc. Natl. Acad. Sci. USA 109 (2012) 1165. [36] X. Chen, T. Sasaki, A. Brannstrom, U. Dieckmann, J. R. Soc. Interface 12 (2011) 20140935. [37] L. Yang, Z. Xu, L. Zhang, D. Yang, Europhys. Lett. 122 (2018) 10003. [38] A. Szolnoki, G. Szabo, M. Perc, Phys. Rev. E 83 (2011) 036101. [39] M. Perc, Sci. Rep. 2 (2012) 344. [40] S. Schoenmakers, C. Hilbe, B. Blasius, A. Traulsen, J. Theor. Biol. 356 (2014) 36. [41] T. Sasaki, S. Uchida, X. Chen, Sci. Rep. 5 (2015) 8917. [42] A. Traulsen, T. Röhla, M. Milinski, Proc. R. Soc. B 279 (2012) 3716. [43] H. Ozono, Y. Kamijo, K. Shimizu, Sci. Rep. 7 (2017) 14379.
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amount of original synergy factor r. What actually impact the evolution is r , therefore, the population can acquire more payoff than that at full cooperation state, by limiting the intervention cost within a reasonable range. Passive cooperators (D I ) share more cost if they invest more capital (β > 1) than that of active cooperators (C ). The caught defectors are forced to participate and contribute, rather than their payoffs being cut down directly with punishment. In the real world, citizens are taxed to establish a higher central authority (e.g. the police) to make the decision of intervention or punishment, only in this way can a society be more stable than the anarchy driven by the private punishment. Just as Hobbes described in his Leviathan [57], in a stable society, we establish a central authority not for all to achieve the best, but for all to prevent the worst. In summary, a proper intervention can either enhance cooperation or increase the total payoffs of the population, which should be helpful for utilizing common resources and governing communities.
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Keywords: Evolutionary game theory Public goods game Strategy intervention Fermi function
The anarchy driven by private punishment is inopportune and inappropriate in modern human societies. It is necessary for a person to appeal to a higher authority such as the police so that a society can maintain more stable. We introduce strategy intervention instead of punishment in spatial public goods game. Some defectors are forced to contribute to the common pool. We show how strategy intervention affects cooperation of a population. Interestingly, weak intervention restrains the impact of spatial reciprocity leading to a lower level of cooperation or even a full defection state. Such phenomena are in contrast to ordinary intuitions. Intervention is enforced by a higher authority which avoids the second-order problems. Furthermore, high synergy factor and proper intervention has a mutual impact on increasing group incomes. We highlight the importance of institutional intervention in a stable society. © 2018 Published by Elsevier B.V.
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Article history: Received 13 July 2018 Received in revised form 30 September 2018 Accepted 1 October 2018 Available online xxxx Communicated by C.R. Doering
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Introduction
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Although the conception of “Tragedy of commons” has been introduced by Hardin [1] for half a century, problems, including excessive deforestation, over-fishery in seas, and heavy pollutions in rivers and atmosphere, remain serious around the world. The common resources are equal for each individual to make use of without limiting conditions, a fact that a tragedy of resource exhaustion eventually kicks in. Public goods game (PGG) has attracted wide applications for decades in economics, biology and social sciences [2–6], which is proposed as a basic game theoretical model for studying interactions, mainly cooperative behaviors between irrelevant individuals. In the typical experiment of PGG, four players are taking part in the game, and are provided with 20 dollars individually. They decide how many dollars should be contributed to, and share the double amount of investments as outcomes of the common pool. Obviously, there exists an ideal situation, in which everyone gets a doubled payoff, only if they all contribute their total initial capitals to the common pool. In fact, when some players find that even without investments, they can still obtain lucrative incomes, they would rather choose to free ride on the other players’ contribution, which naturally leads to economic stalemate [7]. Without specific mechanisms introduced, cooperative behaviors among irrelevant individuals rarely emerge only by natural selec-
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tion [8]. For studying the interactions between individuals and how to enhance cooperation in the population, direct reciprocity [4,9], indirect reciprocity [10–12], kin selection [13], spatial selection [14] and multilevel selection [15,16] have been proposed. As described in strong reciprocity theory, a person would pursue fairness and justice rather than human rational and selfish intention costly even at a loss [7]. Such type of behavior is prescribed to “altruistic punishment” [17], which plays an important role in social cooperation. Due to the imitation from conditional cooperators to even a little few existing defectors, which decreases the level of cooperation [18], altruistic punishment become deterrent to free riders, and indirectly enhances cooperation in a population [19]. Especially for structured population, because of the limited interactions separating them from non-punishment players [20,21], altruistic punishment strategies can survive and spread. Although the costly behavior is interpreted as “negative emotion” or “impulsive negative reaction” brought from free riders [22,23], the explanation is incomplete. Moreover, it is possible for both cooperators and defectors (free riders) to have “negative motions”, arising more complicated problems such as “anti-social punishment” or rounds of punishment [24–28]. Relevant researches on altruistic behaviors have been carried out in evolutionary biology or neuropsychology [29–31] for decades. Some studies explored more effective punishment solutions to promote cooperation [32,33]. Dawkins believes that the genes of all creatures are selfish [34], which is the key of reproductive evolution. In terms of its selfish nature, gene can also evolve altruistic behaviors. Whereby, it still requires ultimate explanation
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School of Physics, Nankai University, Tianjin 300071, China School of Science, Tianjin University of Technology, Tianjin 300384, China Institute of Crustal Dynamics, China Earthquake Administration, Ministry of Emergency Management, Beijing 100085, China
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Benefits of intervention in spatial public goods games
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E-mail addresses: [email protected] (Z. Xu), [email protected] (L. Zhang). https://doi.org/10.1016/j.physleta.2018.10.001 0375-9601/© 2018 Published by Elsevier B.V.
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The “self-adjusting rule” applied in Ref. [37] is more suitable for three-strategy voluntary public goods games (PGG), because players can make decisions referring to the payoffs of loners. But such dynamics is not pervasive and appropriate for two-strategy PGG without loners. Fermi function [5,14,39,51] has been acknowledged and verified to be effective dynamics of group interactions on structured population. Thus, preceding strategy update rules that have been commonly accepted are considered in this paper. In this case, we can figure out the significance of strategy intervention to our current research. For Monte Carlo simulations of public goods game, each player is confined to a site x on a 100 × 100 square lattice with periodic boundary conditions, which is the simplest networks for individuals to interact structurally [14]. Research in Ref. [45] demonstrates that group interactions can effectively link players on lattices. In our simulations, each individual has equal probability to cooperate (C ) or defect (D) as the initial conditions. An individual will be selected randomly to play with his 4 “von Neumann neighborhoods”, namely a single PGG. Such asynchronous updating is utilized, such that every individual has a chance to play once on average in each Monte Carlo Step (MCS). Actually, a player x should join in 5 single PGGs, which take place on site x and the other 4 neighbors’ site, thereby we take the sum of the five payoffs as the overall payoff Πsx of player x. After that, a player x randomly selects one of his four nearest neighbors, who acquires the payoff Πs y as player y on his own site. Whether player x choose to imitate the strategy of player y depends on a probability:
p I = r c (nC + β n I )/G − β c ,
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Γ = 1/ 1 + exp (Πsx − Πs y )/ K ,
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Here Γ is given by the Fermi function. K denotes the amplitude of noise [46], and it is set to K = 0.5 for a player to imitate his neighbor’s choice more effectively to avoid choosing a worse strategy [47].
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Based on the imitation rule, intervention degree η (0 ≤ η ≤ 1) is introduced. Therefore, each individual has a probability η to be
here N sum = N C + N U + N I and N C , N U , N I represent the total number of C , D U and D I .
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where the investment of cooperators are set to unity (c = 1). We assume that the condition 1 < r < r < G reflecting the synergetic effects of cooperation. Note that when individuals imitate in the current round, both D U and D I are treated as defective behaviors due to the original choice before the intervention. Namely, D I is not an independent strategy. Additionally, each individual has a small probability μ to choose one of the available strategies (C , D) randomly ensuring that at least one individual takes random strategy in each round [49]. In our simulations, the value of mutation rate μ is small enough not to have an influence on the stability of cooperation. Usually, a simulation converges after approximate 104 MCSs for a 100 × 100 population. Here, we choose the averaged value of the last 1000 MCSs within total 6 × 104 ensuring more stable simulation results. The payoffs of caught defectors “I ” are counted in that of D-group, the average payoffs P C , P U , P I , P D of strategy C , D U , D I and D are given by
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selected by the system (a higher authority) after a strategy distribution of the population in each round. The defectors among the selected individuals are marked as “I ”, and they are forced to change their strategies (D I ): contribute a certain amount of investment β c (β ≥ 0) into common pool in current round and share the outcomes with cooperators and unselected defectors. Here, the coefficient β is defined as intervention intensity. The defectors who are not “caught” by the system are marked as “U” and still free ride to play strategy D U . A cooperator invests c into the common pool while a defector contributes nothing to free ride. Since punishment belongs to public goods [48], strategy intervention needs to cost as well. We propose r − r as the intervention cost, where r denotes origin synergy factor of the common pool and r the actual synergy factor. The intervention cost results from common pool, hereby individuals needn’t contact each other. It notes that the form of intervention cost can be diverse, essentially, what truly impacts the evolution of PGG is r , hereafter we focus r as a key parameter for further discussion. The payoff of player x in a single PGG depends on his and his neighbors’ strategies in a single PGG. When nC , nU , n I (nC + nU + n I = G = 5) represent the number of strategy C , D k (k = U , I ), respectively, their net payoffs p C , p U , p I take the following form:
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about the intrinsic motivation of altruistic behaviors. The anarchy driven by private punishment is rare in modern human societies, therefore, it is necessary for a person to appeal a higher authority in order to maintain a stable society [35–37]. An individual could choose to quit the competition to avoid being exploited with a small but fixed payoff [5]. But in many practical cases, individuals can’t escape but only participate in social activities. By virtue of theoretical analysis [35–41] or experiments [42,43], “pool punishment” has been revealed as a more realistic policy in public goods games. Such type of institutionalized punishment can be more efficient for enhancing cooperation than peer-punishment, which could be combined with participation [35] or conditional reward [36,41]. On the other hand, just as peer-punishment, a question remains considered [38,39,43] of whether “second-order” free riders should be sanctioned or not. As a recent research reported, successful individuals either act as partners or as rivals [44]. The rivals always put themselves first eventually leading to defection which is deterrent to social development. In this paper, based on stochastic imitation rules among individuals, we propose strategy intervention instead of punishment enforced by a central institute, and figure out how the mechanism impacts the cooperation among individuals.
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Fig. 1. Evolution of cooperation in spatial public goods game on the β –η parameter plane for (a) r = 3.5, (b) r = 4, (c) r = 4.2, (d) r = 4.5. Only D-strategy (D U + D I ) exists in Zone I while C -strategy exists in Zone II (full cooperation) and Zone III refers to the intermediate state (C + D U + D I ) of the population at dynamic equilibrium. The other parameters are μ = 0.0001 and K = 0.5. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
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A fact that no interactions exist among individuals is the biggest difference between intervention and punishment. Previous studies show that levels of cooperation could maintain the stability in structured population, as the synergy factor increases up to a threshold r ≈ 3.74 [47,50] based on Fermi function or r ≈ 3.90 [5] with replicator dynamics due to the network reciprocity. As depicted in Fig. 1, the frequency of cooperation changes nonmonotonously for different intermediate values of r , and strongly depends on the intervention degree η and intensity β . The patterns of intervention-based frequency are essentially different from that calculated by the probabilistic punishment [51,52]. It also shows in each panel results in the typical model without intervention (η = 0) considered. Intuitively, effects of the intervention and network reciprocity do not appear at low values of r , when the intervention (β or η ) is weak (Fig. 1a), such a fact that leads to full defection (Zone I). Subsequently, a full cooperation emerges (Zone II), as the intervention is strengthened forward a moderate level. It is illustrated that higher chance of detecting defection may result in lower cooperation level. Indeed, this is against our intuition, and similar counter-intuitive phenomena about mutation rate μ, competing strategies and renewable resources have been reported previously [53–55]. Visually, Zone III expands, while Zone I and Zone II narrow as r increases. The effect of spatial reciprocity becomes more apparent, and the intermediate state of strategies (C + D U + D I ) in the population (Zone III) “erodes” Zone I and Zone II. The spatial reciprocity can promote the levels of cooperation, and the intervention is deployed aiming to reduce the defective behaviors. Interestingly, the valleys of cooperation obviously remain in Zone I for higher value of r . Maybe it is not obvious to figure out the decrease of cooperation by colors, but we can see the valley and there must be a decreasing process between the non-intervention state (η = 0) and the valley. Preliminarily, results as shown in Fig. 1(b, c, d) demonstrate that the weak intervention leads to lower level of cooperation and higher level of defection than that without intervention, respectively. Although indistinct, the levels of cooperation remain unchanged when η = 0 in Fig. 1b. Actually, a certain num-
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ber of D I -strategies exist in Zone I while η > 0, and individuals do not have to actively cooperate within a certain extent of β or η . The caught defectors become “passive cooperators” with the active cooperators replaced. Such “D U + D I ” state can reach a dynamic equilibrium, provided that the “C + D” state maintains at higher value of synergy factor. The evolutionary characteristics imply mutual effects of the network reciprocity and intervention. For further understanding the elementary invasion among different types of individuals, the same original homogeneous distribution of cooperators and defectors are set as the initial state (“Origin” in the leftmost panel of Fig. 2) of the spatio-temporal evolution for different levels of intervention and varying values of r . Correspondingly, time courses of the strategy fraction are located in the rightmost of each evolutionary panel (Fig. 2f–j). Results of a typical PGG for lower values of r are shown in Fig. 2 (from a1 to a3). It is clear that, without intervention (β = 0, η = 0), the group of defectors quickly invades C -group, and essentially leads to full defection of the population, while effects of the network reciprocity haven’t appeared. When r increase to r = 4 given weak intervention (β = 2, η = 0.25), it still yields a full defection state (D U + D I ) of the population (Fig. 2b1–b3). Compared with typical results (Fig. 2f), because of the functioning intervention, the invasion through D-group into C -group deteriorates (Fig. 2g). Such distinction indicates that the weak intervention can restrain effects of the spatial reciprocity, which result in a lower level of cooperation or even a full defection state of the population. By elevating the level of intervention (β = 2, η = 0.5) for r = 3.6, we contrarily obtain a full cooperation state of the population (Fig. 2c1–c3). The fraction of D-group slowly increases, then gradually declines until the individuals, who choose to defect vanish (Fig. 2h). Results (as shown in Figs. 1 and 2) suggest that only if β > 1 we can potentially achieve a full cooperation state in spatial PGG with a given proper intervention degree η even for a lower value of r . It is natural for cooperators to form clusters to fight against defective behaviors when adopting imitation rules [5,14]. When r increases to r = 4.2 with the weak intervention (β = 1, η = 0.25), C -group and D-group mutually invade, and the fraction of cooperators gradually decreases followed by the slowly increasing (Fig. 2i), and finally reaches a stable state. In comparison with the processes b: b1–b3, effects of the spatial reciprocity are enhanced, therefore, the level of cooperation maintains at a dynamic equilibrium (Fig. 2d1–d3). If we continue to strengthen the intervention levels then we obtain full cooperation (Fig. 2c3). The fraction of cooperators first rise and then drop toward a relatively stable state (Fig. 2j) which is opposite to the process Fig. 2i. It is found that small fluctuations occur at the equilibrium state (Fig. 2i, j) of the intermediate states (C + D U + D I ). We assume that, with full intervention degree (η = 1) considered, all defectors are forced to change their strategies, and there are only C -group and D I -group left. When β > 1, given that cooperators still make a contribution, they become “free riders” in contrast to the caught defectors, who invest more capital into the common pool. Such phenomena (C + D I ) is comparable to the typical model without intervention (C + D) at given values of r . The imitation of individuals depends on the payoffs of themselves and their neighbors. Actually, there exist mutual influences between the frequency and payoff of different types of players. Here, the aiming is to find the reason at that the frequency of cooperators changes with different levels of intervention. For facilitating the analysis, the fractions are overlaid with the corresponding average payoffs of strategy C , D U and D I (Fig. 3) at the given value of r = 4.2. At first step, we start a simulation without intervention (η = 0) until the fractions of cooperators and defectors are stable. Then, we change the value of η from 0 to 0.15 when β = 3. Because of more contributions from parts of caught defectors to the common pool, both cooperators and defectors ob-
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the same homogeneous distribution of strategies with equal number of cooperators (green) and defectors (red) as the initial condition (Origin) illustrated in the leftmost of the patterns. With intervention, D-strategies are divided into D U (orange) and D I (blue). 60000 MCs have been run in the simulations of Fig. 2 as well as that of Fig. 1. We hope to show the “process of mutual invasion” among different strategies from the “Origin” (a particular strategy distribution) to equilibrium state. Note that panels a3, b3, c3, d3, e3 in Fig. 2 show all stable states of the population. Since the invasion process evolves quickly, we amplified this process as illustrated in Fig. 2f–j, after where the strategies converge slowly till the equilibrium state. The other parameters are μ = 0.0001 and K = 0.5.
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Fig. 2. Spatial pattern formation and the corresponding time courses of fraction of strategies with different levels of intervention for (a: a1–a3) r = 3.6, β = 0, η = 0; (b: b1–b3) r = 4, β = 2, η = 0.25; (c: c1–c3) r = 3.6, β = 2, η = 0.5; (d: d1–d3) r = 4.2, β = 1, η = 0.25; (e: e1–e3) r = 4.2, β = 3.5, η = 0.9. All the simulations start from
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Fig. 3. Evolution of the fractions of strategies (with the same color settings as in Fig. 2) and the corresponding average payoffs over time with abrupt changes of intervention (β = 3, η = 0 → 0.15 → 0.5 → 0.9) for r = 4.2. The other parameters are μ = 0.0001 and K = 0.5.
tain more payoffs than those in the previous stage. Given that the increasing range of average payoffs are greater for D-group than that for C -group (L 1 > N 1 > M 1 ), thereby, the D-group dominates the invasion. Consequently, the fractions of D U and D I increase, whereas the fraction of C -strategies decreases fast until it reaches
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a full defection state. In this condition, only if the un-caught defectors (D U ) play with the caught ones (D I ), both sides can gain more payoffs. Otherwise, they get nothing without intervention for low value of r . Moreover, there exist more D U than D I when η = 0.15. The average payoffs of the two types of players are comparable.
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while the spatial reciprocity continues to predominate in the PGG. In some cases, the weak intervention (pink curve with inverted solid triangle) restrains the influence from spatial reciprocity, but it still obtains higher payoffs than that in the typical model. Actually, all of the selected points on the β –η parameter plane (with different levels of intervention) acquire more payoffs for the population, which results from the effect of intervention. Especially, it disagrees with our intuition that the excessive intervention reduces the number of cooperators more commonly than moderate level of intervention (Fig. 4a). Previous studies on PGG with punishment attempted to enhance cooperation based on a variety of mechanisms. But considering the effect of intervention, we propose that low levels of cooperation may not be worse than full cooperation state. Unlike the punishment, the intervention cost r − r comes from the outcome of common pool, and its numerical value equals to the decreased amount of synergy factor r. It is similar to the taxation system that all the individuals, who contribute either actively or passively, should share the cost. Furthermore, we take a certain level (β = 3, η = 0.75) of strong intervention (orange curve in Fig. 4b) subject to the prescribed value of = 5, if the intervention cost could be controlled low enough for the value of r within the “increment area” (r 4.25), the population could obtain more payoffs than that at a full cooperation state for r = 5 (gray solid line).
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Fig. 4. Fractions of cooperators and the corresponding average total payoffs of the population in dependence of r with different levels of intervention. The other parameters are μ = 0.0001 and K = 0.5. The blue curve with right-solid triangle and orange curve with solid rhombus denote the excessive intervention-dependent results, while the green curve with solid-triangle represents results with the moderate level of intervention.
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Discussion
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We introduce the strategy intervention in spatial PGG and analyze its correlation with the evolutions of different types of strategies. When “pool punishment” and “peer punishment” coexist, they restrict each other and the latter is dominant which leading to promote the defection. This is different from our study where only institutional intervention functions [56]. Indeed, it is different from pool punishment that all the contributors no matter active or passive should bear the intervention cost. There is no punisher but only two optional strategies for players to choose in our model. This is a key distinction between intervention and pool punishment. With the intervention considered in PGG, cooperators do not have to worry about the revenge imposed by the punishment [25]. The network reciprocity and intervention have interplay on the evolutionary outcome of PGG in spatial lattice. According to our intuition, cooperation ought to be enhanced with the strengthened intervention, but the simulation results indicate that the weak intervention restrains the impact of spatial reciprocity leading to a lower level of cooperation, or even a full defection state of the population. Only if β > 1, it can possibly achieve a full cooperation state in spatial PGG with moderate value of η even at low value of r . The active cooperators become “free riders” over the caught defectors who invest more capital into the common pool when β > 1, which situation (C + D I ) is similar to the typical model without the intervention (C + D) for the same value of synergy factor. We note that the population can obtain more payoffs than that at full cooperation state only when the value of r is high enough. So the impact of network reciprocity resulting from large value of r on the group incomes can not be ignored. Qualitatively, only when the synergy factor is high enough and we control the intervention cost properly, can the population obtain more payoffs. It generates different payoffs for all types of groups forming evolutionary distribution of strategies in each round. Subsequently, the strategy with more payoffs can survive and spread, promoting the structured population toward a new equilibrium state. All the individuals, who actively or passively contribute, should share the intervention cost, which comes from the outcome of common pool, and its numerical value equals to the decreased
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Similarly, when η varies from 0.15 to 0.5, all the average payoffs for different types of players increase, with the increasing ranges described as M 2 > N 2 ≈ L 2 , which phenomena leads to the decline of D-group, until a full cooperation state is reached. Finally, we set the value of η as 0.9, and find a significant distinction of the increasing range between the three types of average payoffs (N 3 ≈ L 3 > M 3 ). It observes that the fractions remain constant at the beginning when the value of η changes, which looks unlike the previous stages. The main reason is that the random parameter μ is so small that an individual would probably be caught for high value of η , when they choose to defect. Only if amounts of defectors are not caught by the system, D-strategies could invade their neighbors and rapidly spread, which could result in a growth of D-group, until it reaches an equilibrium state. There are larger fluctuations in average payoffs of D U -group than those of D I -group due to the small proportion and uneven distribution of D U . It is seen from Fig. 1c that the intervention degree η and intensity β impacts the levels of cooperation in the similar way. It is found that, the intervention has effects on the distribution strategies in each round, which results in the distinction of average payoffs of different types of groups. Furthermore, the more payoffs are, the more potentially the strategies spread to reach a new equilibrium. It observes that the average payoffs irregularly fluctuate, when finite variations in fractions of defectors (cooperators) appear at a full cooperation (defection) state. Provided that the irregular fluctuations have no influences on the result analysis, we assume the payoffs of defectors (cooperators) be zero at the full cooperation (defection) state. As mentioned above, the network reciprocity does not function for a lower synergy factor. The weak intervention has little effects on the levels of cooperation for a lower value of r . However, the evolution of different strategies become more complex, when the value of r increases highly enough (Fig. 4). The black curve with solid blocks represents the level of cooperation in dependence on r without intervention. Although the fraction of cooperators remains in the same trend with weak intervention (red curve with solid circle), the relevant average total payoffs of the population are always higher than that of the typical model without intervention. The intervention has functioned on the outcome of the population,
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Acknowledgement
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This project is in part funded by the National Natural Science Foundation of China (NSFC) under Grant No. 41874113.
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amount of original synergy factor r. What actually impact the evolution is r , therefore, the population can acquire more payoff than that at full cooperation state, by limiting the intervention cost within a reasonable range. Passive cooperators (D I ) share more cost if they invest more capital (β > 1) than that of active cooperators (C ). The caught defectors are forced to participate and contribute, rather than their payoffs being cut down directly with punishment. In the real world, citizens are taxed to establish a higher central authority (e.g. the police) to make the decision of intervention or punishment, only in this way can a society be more stable than the anarchy driven by the private punishment. Just as Hobbes described in his Leviathan [57], in a stable society, we establish a central authority not for all to achieve the best, but for all to prevent the worst. In summary, a proper intervention can either enhance cooperation or increase the total payoffs of the population, which should be helpful for utilizing common resources and governing communities.
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