Cooperation with both synergistic and local interactions can be worse than each alone
aa r X i v : . [ phy s i c s . b i o - ph ] A p r Cooperation with both synergistic and local interactions can beworse than each alone
Aming Li , ∗ , Bin Wu , † , and Long Wang , ‡ December 14, 2013
1. Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems,College of Engineering, Peking University, 100871 Beijing, China2. Research Group for Evolutionary Theory, Max-Planck-Institute for Evolutionary Biology,August-Thienemann-Str. 2, 24306 Pl¨on, Germany
Abstract
Cooperation is ubiquitous ranging from multicellular organisms to human societies. Popula-tion structures indicating individuals’ limited interaction ranges are crucial to understand thisissue. But it is still at large to what extend multiple interactions involving nonlinearity in payoffplay a role on cooperation in structured populations. Here we show a rule, which determines theemergence and stabilization of cooperation, under multiple discounted, linear, and synergisticinteractions. The rule is validated by simulations in homogenous and heterogenous structuredpopulations. We find that the more neighbors there are the harder for cooperation to evolve formultiple interactions with linearity and discounting. For synergistic scenario, however, distinctfrom its pairwise counterpart, moderate number of neighbors can be the worst, indicating thatsynergistic interactions work with strangers but not with neighbors. Our results suggest thatthe combination of different factors which promotes cooperation alone can be worse than thatwith every single factor.
The particulars of why and how cooperation evolves have perplexed evolutionary biologists andsociologists enduringly [1, 2, 3, 4]. A cooperator takes an altruistic action which supplies a benefit, b , for another individual at a cost, c , while a defector does nothing. One of the main tasksof evolutionary theory is to explain how and why cooperation is present. Evolutionary gametheory provides a powerful platform to understand the evolution of cooperation in unstructuredpopulations, with the replicator equation in infinite populations [3] and stochastic dynamics infinite populations [5, 6, 7].Recently the assumption of a well mixed population is removed, and the population allowsindividuals to interact locally [8, 9]. Typically networks are adopted to depict such populationstructure, since it is simple in definition, while complex in property [8, 10]. The nodes of thenetwork represent individuals, while the edges denote connections in between [11, 12]. In this way, ∗ [email protected] † [email protected] ‡ [email protected] We consider a finite population located on a graph of size N . Individuals are assigned to thenodes of the graph, whereas social ties between them are represented by the edges [11, 12]. Everyindividual has k neighbors. As illustrated in Fig. 1, players participate in the public goods gameorganized by themselves and their neighbors [46], that is to say, each player participates in k + 12 C D D D C D C C D D C D C D C D C D C D C b dc C C D C a Figure 1: (Color online) Illustration of updating on a network. We show a network with size N = 22and every player has k = 3 neighbors here. Cooperator C and defector D (D ) are neighbors ofthe selected cooperator C with updating. Both of C and D (D ) have neighbors with strategyof cooperation or defection except C , which are called C and D , C (C ) and D (D ),respectively. C and D also have neighbors C , C adopting strategy of cooperation andD , D with defection, where both of a and b mean 1, 2, or 3. Each player organizes a publicgoods game with all of its k neighbors. Thus each individual participates in k + 1 public goodsgames of size k + 1 [46]. As an example, for the payoff of C , all players marked within the dashedcurve are relevant. The payoff comes from all games C participates in, where one game (shadedin blue) held by C , that is, part ( a ), and the other three (shaded in red) held by C , C , andD , that is, part ( b ), ( c ), and ( d ).public goods games of size n = k + 1.For the public goods game, the first cooperator contributes a benefit b while the j th (1 ≤ j ≤ n )cooperator contributes bδ j − to the common pool. Every cooperator pays the same cost c . Defectorsexploit the group by reaping benefits without paying anything. The accumulated benefits aredistributed equally to all the n players in the group irrespective of their behaviors. Thus, defectorsand cooperators receive the following payoffs P D ( i ) = bn (1 + δ + δ + · · · + δ i − ) = bn − δ i − δP C ( i ) = P D ( i ) − c (1)where i is the number of cooperators within the group. Here δ > < δ <
1) or synergy ( δ >
1) factor. As δ = 1, it degenerates to the linear public goods gamewith P D ( i ) = rci/n , where r = b/c is the multiplication factor.After playing the public goods game, the payoff P of every player is transformed into fitness f by fitness mapping [5, 47]. Here we adopt the linear fitness which consists of baseline fitness andthe payoffs arising from games [5, 12], i.e., f = 1 − w + wP where w varying from 0 to 1 is theintensity of selection. For w →
0, the selection is weak. It means that the game is merely one ofmany factors which contribute to the entire fitness of an individual [5, 12].As to the updating rule, the “death-birth” (DB) process [4] is employed. Within the process,a player in a population is randomly selected to die at each time step, and then all neighbors ofthe focused player, with probability proportional to their individual fitness, compete for the vacantsite. 3 F i x a t i on p r obab ili t y , C R a ndo m g r a ph R a ndo m r e gu l a r g r a ph a k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10 F i x a t i on p r obab ili t y , C b c Bebefit-to-cost ratio b/cBebefit-to-cost ratio b/c =1.1=1.0=0.9 d Bebefit-to-cost ratio b/c e f Figure 2: (Color online) The rule is in good agreement with numerical simulations. The upperand lower rows show the simulation results of the fixation probability for the random regular graphwith degree k (a , b , and c) and the random graph with average degree k (d , e , and f ) . Both areperformed for population size N = 400 with weak selection w = 0 .
01. Fixation probability ρ C isapproximated by 10 independent runs. The horizontal dotted line in each figure represents thefixation probability of a neutral mutant, 1 /N . And the arrows point to the theoretical critical valueof b/c favoring cooperation, i.e., ρ C > /N . The first to third columns correspond to the publicgoods game with discounting δ = 0 .
9, linearity δ = 1 .
0, and synergy δ = 1 .
1. We find that the ruleapplies to both random regular graphs and random graphs. Note that, as δ is too big or small,accumulation of the payoff in the common pool is changing rapidly. Any extremely big or smallcontribution bδ j − of any later j th cooperator are fabricated, thus only 0 . ≤ δ ≤ . (f ) with high average degree ( k = 10) sincethe derivation of the rule is at large N and pair approximation is formulated for the regular graphwithout any loops.We study the emergence of cooperation by comparing the fixation probability [5, 12, 38] of asingle cooperator ( ρ C ) invading a wild population of defective type under weak selection with thatunder neutrality 1 /N [4]. If ρ C > /N then natural selection favors cooperator replacing defector[4], so we see that natural selection favors the emergence of cooperation. We see that naturalselection favors the stabilization of cooperation if ρ D < /N , that is, natural selection opposes thefixation of defectors. And if ρ C > ρ D , we see that natural selection favors cooperator over defector[4]. We obtain the fixation probability of both cooperation and defection by the pair approximation(see Appendix A and B). For large population size and weak selection, we have a rule: ρ C > N if4
30 60 90 120 150 180 210 240 270 30005000100001500020000250003000035000 2 30 60 90 120 150 180 210 240 270 300050100150200250300 =0.95 C <1/N C >1/N D >1/N D <1/N a C r i t i c a l v a l ue o f b / c C = D =1/N for =1 C =1/N for =0.95 D =1/N for =0.95 C r i t i c a l v a l ue o f b / c k =1.05 C < / N D >1/N D <1/N C >1/N b C = D =1/N for =1 C =1/N for =1.05 D =1/N for =1.05 k Figure 3: (Color online) The critical value of the benefit-to-cost ratio for natural selection favoringemergence and stabilization of cooperation (defection). We set δ = 0 . (a) and 1 . (b) torepresent respectively the weak discounting and synergy in structured populations with N = 5 × numerically. For discounting effect δ <
1, both the critical values for ρ C = 1 /N and ρ D = 1 /N increase rapidly with average degree k . Yet the critical value for ρ D = 1 /N is greater than thatof ρ C = 1 /N , and it also increases much faster. This shows that in the discounting public goodsgame, with the increase in the number of neighbors, it is easier for a cooperator to be invasive( ρ C > /N ) than to be stabilized ( ρ D < /N ). For synergy effect δ >
1, the critical value for ρ C = 1 /N is greater than that of ρ D = 1 /N for any neighbor size. This shows that it is easier forcooperation to be stabilized ( ρ D < /N ) than to be invasive ( ρ C > /N ). Interestingly, both thecritical values for ρ C = 1 /N and ρ D = 1 /N are non-monotonic and is a one-hump function of k .This shows for both the emergence and the stabilization of cooperation in the public goods gamewith weak synergy, a moderate number of neighbors is the worst.and only if b/c > ( k +1) k +3 for δ = 1 δ +4 δ +3 for k = 2 ( N − k +1) Q Nf ( δ ) otherwise , (2)and ρ D < N if and only if b/c > ( k +1) k +3 for δ = 1 δ +4 δ +3 for k = 2 ( N − k +1) Q Ng ( δ ) otherwise (3)where the notations Q , f ( δ ), and g ( δ ) can be found in Appendix B.4.For the linear public goods game ( δ = 1), natural selection favors not only the emergence ofcooperation but also its stabilization if and only if b/c > ( k + 1) / ( k + 3) . (4)5n this occasion, the rule is equivalent to r > n / ( n + 2) where r is the multiplication factor ofthe common pool. Since n / ( n + 2) < n , it implies theoretically that cooperative dilemma can berelaxed in structured populations compared with that in the well mixed case, without invoking anyother additional mechanisms.We get ρ C > ρ D if and only if b/c > / k = 2in our model). And the same result is also found in [39], where, using a direct approach withoutapproximations, van Veelen et al. pointed that ρ C > ρ D if and only if b/c > n/ (4 − /n ). Asgroup size n = 3, the linear public goods game on cycle coincides with that in the structuredpopulation with k = 2, and both deduce b/c > /
5. What’s more is that the general rule todetermine the emergence of cooperation is found to be in good agreement with computer numericalsimulations (see the first row of Fig. 2). And it also approximately applies to heterogeneousstructured populations (see the second row of Fig. 2).Based on the equation (4), we find that ρ C > ρ D is also equivalent with the emergence ρ C > /N and stabilization ρ D < /N of cooperation. Furthermore, the following equivalence holds ρ C > N ⇔ ρ C > ρ D ⇔ ρ D < N (5)for the public goods game which is either linear δ = 1 or on a cycle k = 2. That is to say, for largestructured population under linear public goods game or public goods game with nonlinearity inindividual payoff on cycle, we have that natural selection favors emergence of cooperation if andonly if it favors stabilization of cooperation. Further, we show that the critical value, both for theemergence and the stabilization of cooperation, is continuous with the discounting or synergy factor δ (see Appendix C.1). Hence the equivalent proposition (5) applies for infinitesimal nonlinearity(see Appendix C.2).From the rule (see inequalities (2) and (3)), we theoretically get the critical benefit-to-cost ratio b/c for the emergence and stabilization of cooperation (defection) with the two factors combined,saying the spatial reciprocity and nonlinearity in payoff induced by multiple interactions. Fig. 3shows that weak discounting significantly inhibits both the emergence and stabilization of coopera-tion, whereas the weak synergy favors them greatly. In contrast with the linear public goods game,the critical ratios b/c for ρ C = 1 /N and ρ D = 1 /N are no longer overlapping for nonlinear payoffeffects (see Fig. 3). In other words, taking into account either discounting or synergy, a particularform of nonlinearity in payoff, the emergence and the stabilization of cooperation are no longerequivalent as in the linear public goods case (see equivalent proposition (5)).For multiple interactions such as linear public goods, similar to pairwise interactions [12], coop-eration will also be impeded with an increase of the number of neighbors (see light gray up trianglein Fig. 3). Discounting in payoff significantly inhibits cooperation (see Fig. 3a). In particular,in this case, with the increase in the size of neighbourhood, it will become even harder for theemergence and stabilization of cooperation. The critical benefit-to-cost ratio is increasing muchrapidly than its linear public goods game counterpart.For weak synergy, the critical benefit-to-cost ratio b/c for emergence or stabilization of cooper-ation still first increases as the growth of every player’s number of neighbors (see Fig. 3b). But itdecreases as neighbor size is big enough and tends towards zero. This illustrates that for small sizeof neighbor, increasing interaction range, i.e. k , is detrimental for cooperation, which is consistentwith linear public goods game; yet the interesting story comes along when the interaction range isrelatively large, in this case, increasing the interaction range is beneficial for cooperation, which isseldom observed in cooperative dilemmas. An intuitive understanding can be: for small neighborsize, the local competition plays a key role. Even though the public goods are exponentially in-creasing with additional cooperator in a group, the group is small in size generally, thus the defector6
25 50 75 100 125 1500246810 2 10 20 30 40 500.00.51.01.52.02.53.03.5 2 10 20 30 40 500.00.51.01.52.02.53.03.52 10 20 30 40 500.00.51.01.52.02.53.03.5 2 10 20 30 40 500.00.51.01.52.02.53.03.5 2 10 20 30 40 500.00.51.01.52.02.53.03.5 =2.3=1.9a=1.1 C =1/N D =1/N C r i t i c a l v a l ue o f b / c b=1.3 c=1.5 C r i t i c a l v a l ue o f b / c d e k k f=3.3 k Figure 4: (Color online) Critical value of benefit-to-cost ratio degenerates to a monotonic functionof average degree k from one-hump function with the enhancement of synergy. We set differentsynergy factors to investigate its effects on the critical value of b/c with population size N = 5 × .Specific values of δ are marked on each panel. Orange star and cyan square are used to indicate ρ C = 1 /N and ρ D = 1 /N respectively. The critical value decreases with the increase of δ . Andit is a parabolic function of k when synergy is weak ( a , b , c , d , and e ), whereas monotonic whensynergy is strong ( f ). Under the synergistic effect, big k or δ can drive the critical value to approach0. The intuitive understanding behind this is the competition between the two factors. For large k ,the population is approximately well mixed, thus the local interaction diminishes by the replicatorequation [41], and cooperation with synergy thrives, leading to a decrease of the critical value withincreasing k . For small k , the local interaction plays an important role if the synergy effect is notstrong. That is to say for δ slightly more than the unit, increasing k does inhibit the fixation forboth strategies as in the pairwise Prisoner’s Dilemma [12]. To sum up, for small δ >
1, there is ahump for the critical value with the increase of neighbors. For large δ , however, even for small k ,the synergy effect is strong enough to outperform the locality of the population structure. By thesame argument, the critical value is monotonically decreasing with the neighbors.7ould reproduce more efficiently with the increase of size. This leads to the increase of the criticalbenefit-to-cost ratio. For large size of neighbors, however, the exponential increase in the accu-mulation of public goods with an additional cooperator outperforms the reproduction of defectors.For example, with the j th (0 ≤ j ≤ k + 1) cooperator’s benefit bδ j − to common pool, big j induceslarge payoff bδ j − / ( k + 1) to every player in the same group with weak synergistically enhancedeffect δ as well as the baseline benefit b . This rather large payoff paves the way for emergenceand stabilization of cooperation for large neighbor size. Therefore, it is the worst for emergence ofcooperation when the number of neighbors is moderate. Another intuitive explanation is that forweak synergy effect, the replicator dynamics allows the coexistence of cooperators and defectors[41], which is quite similar to the snowdrift game. For the snowdrift game, however, local interac-tion can inhibit cooperation [48] in contrast with the prisoner’s dilemma [49]. Here, we explicitlyshow up to how many numbers would be the worst for cooperation in such scenario. It shows thatsynergistic interactions work with strangers in well mixed populations [41]. Furthermore, we alsofind that the synergistic interactions do not work with neighbors in structured populations.For stronger synergy, the critical benefit-to-cost ratio always decreases with the neighbor size(see Fig. 4). Thus in this case cooperation could be promoted significantly with the increase of everyplayer’s number of neighbors under strong synergy, which intrinsically differs from effect of linearity,discounting, or weak synergy. This is because the synergy effect is so great that it effectively is acoexistence game where best relies should be of the population minority. In this case, enlargingthe interaction range paves the way for cooperator mutants to be more like an minority yieldingan enhancement of cooperation level. Our results strongly suggest that the intrinsic multipleinteractions, where payoffs are nonlinear in general, can cause the so called sea-sickness [50]. In conclusion, we find a rule theoretically elucidating the critical value of benefit-to-cost ratio b/c up to which cooperation emerges and is stabilized. In addition numerical simulations verify thevalidity of the rule as well as its feasibility for random graph. For linear public goods game onany regular graph and any public goods game with synergy and discounting on a cycle, we find anequivalent proposition that the rule determines not only the emergence, but also the stabilizationof cooperation. What’s more, in public goods with synergy, we present that it can be the worst forthe emergence of cooperation, as the number of neighbors is moderate. We find that synergisticinteractions work with strangers but not with neighbors, and cooperation with both synergistic andlocal interactions can be worse than that with each alone. Our work suggests that there can be abig shadow in the effects of combinational mechanisms on the evolution of cooperation.Over the past two decades, structured populations depicted by networks have been taken intoconsideration to study the evolution of cooperation by virtue of evolutionary graph theory[11, 20,24, 51]. It has been shown that cooperation can flourish in both static network and dynamicnetwork [8, 12, 20, 25, 51, 52, 53] (for an exception, please see [48]). The main reason is that thesepopulation structures can lead to a clustering of cooperative individuals [8, 12, 21, 24, 54], withinwhich cooperators can survive by enjoying the benefits from mutual cooperation even though somecooperators are exploited by defectors along cluster boundaries. We indicate that in a non-additivepublic goods game, where nonlinearity in payoff arises, this clustering (see equations (25) and (26)in Appendix A.2) is not always beneficial when the neighbors are few in number: in a synergy publicgoods scenario, where the latter cooperators in the group contribute significantly much more thanthe previous cooperators, the worst case for cooperation emerges when the number of neighbors ismoderate, not too big nor too small. This means that synergistic interactions work with strangers8ut not with neighbors. Our results show that the interaction between different mechanisms [55]might trigger novel unexpected results. The combination of different factors with each promotingcooperation alone can be worse than every factor alone in promoting cooperation. Thus, it may bepromising to investigate the combination of previous mechanisms promoting cooperation.We find that the rules governing the emergence and stabilization of cooperation are equivalentfor linear public goods games, which is validated by numerical simulations on homogenous as wellas heterogenous structured populations. The rule simply asks the benefit to cost ratio b/c to exceeda critical value ( k + 1) / ( k + 3), where k is the average number of neighbors in the population. Infact, for any number of neighbors k , the numerator of the critical value is ( k + 1) , which suggeststhe number of individuals relevant to the payoff of the focal individual with recounting (see Fig. 1),i.e., the product of the group size k + 1 and the average number of the public goods game everyplayer involved in, k +1. Therefore, in this case, as in a well mixed population, multiple interactionssignificantly inhibit the cooperation than its pairwise counterpart. For public goods games on acycle, with either linearity, discounting, or synergy. The equivalence still holds between the rulesfacilitating the emergence of cooperation and that governing the stabilization. Therefore the samecriterion applies to determine under what condition the average abundance of cooperation exceedsthat of defection in the mutation-selection equilibrium under small mutation [56, 57].The equivalence falls down for general population structures and nonlinear public goods game.For the synergy effect, the emergence and the stabilization of cooperation are facilitated significantlyfor any number of neighbors compared with the linear public goods game. Being stabilized, in thiscase, is much easier than the emergence. For the discounting effect, both the emergence and thestabilization of cooperation are inhibited significantly for any number of neighbors compared withthe linear public goods game. Being stabilized, in this case, is much harder than the emergence.Therefore, both synergy and discounting has a more significant role in the stabilization comparedwith the emergence.As applications, for microbes with either synergy and discounting public goods, in particular,if the public goods are diffusive [58], the average degree of the network suggests the diffusion rateof the public goods. Our result suggests that for the discounting public goods game, only a lowdiffusion rate of public goods can make the cooperator cells thrive; For the synergy public goodsgame, however, it is always better than the discounting case. The interesting result lies in the factthat cooperation is better off for both very low and very high diffusion, whereas is worst off formoderate diffusion rate. Experimental validation along this line might be interesting. Author Contributions
A.L., B.W. and L.W. devised and analysed the model. A.L. performed theoretical derivation andnumerical simulations. A.L., B.W. and L.W. analysed the results and wrote the paper.
Acknowledgements
A. Li and L. Wang are supported by NSFC (Grants No. 61375120 and No. 61020106005). B.Wu greatly acknowledges the sponsorship by the Max-Planck Society. We are indebted to YongWang, Maozheng Guo, Guangming Xie, James Price and Arne Traulsen for helpful discussions.Discussions with Xi Weng and Te Wu are also acknowledged.9
The calculation of fixation probability under the framework ofpair approximation
We adopt the fixation probability to investigate the emergence and stabilization of cooperation,which has been used by many researchers for finite population [59, 5, 12, 60]. The updating processand dynamics can be got by pair approximation [12, 61]. And the method to derive fixationprobability of cooperation or defection based on multiple interactions is similar to that based onpairwise interactions [12].
A.1 Updating process
For the structured population, let X and Y denote the strategy of cooperation or defection. And p X and p XY are assigned respectively to the frequency of strategy X and XY pairs. q X | Y indicatesthe conditional probability of finding a player of strategy X for a player of strategy Y . From thenotations interpreted above, we have p C + p D = 1 ,q C | X + q D | X = 1 ,p XY = p Y · q X | Y ,p CD = p DC . Furthermore, under the framework of pair approximation, we find p C and q C | C are sufficient todescribe the system since p D = 1 − p C ,q D | C = 1 − q C | C , (6) p CD = p DC = p C · q D | C = p C (1 − q C | C ) , (7) q C | D = p CD p D = p C (1 − q C | C )1 − p C , (8) q D | D = 1 − q C | D = 1 − p C + p C q C | C − p C , (9) p DD = p D q D | D = 1 − p C + p C q C | C . (10)We assume that a selected individual, cooperator C , who has k neighbors which consist of k C cooperators (C ) and k D defectors (D ) (see Fig. 1 in main text), is replaced by a defector withprobability Λ under the death-birth (DB) process, thus we haveΛ = k D (1 − w + wπ CD ) k C (1 − w + wπ CC ) + k D (1 − w + wπ CD )= k D k + k C k D ( π CD − π CC ) k w + O ( w )with k C + k D = k , where π DC , π DD mean the payoff of C , D after playing public goods gamerespectively, and O ( w i ) means that the error is of order O ( w i ), where i is a positive integer.In order to compute the payoff of an individual after playing public goods games, we shouldconsider the population structure around the individual. Here, we list the the neighbor and the10umber of cooperators as well as defectors among k neighbors of X ab and her neighbors asX ab X a : 1 (cid:26) C a1 : k C D a2 and D a3 : k D C ab1 : ( k − q C | X X ab : 1C ab11 : ( k − q C | C D ab12 : ( k − q D | C D ab2 : ( k − q D | X X ab : 1C ab21 : ( k − q C | D D ab22 : ( k − q D | D where X can be C or D, a = 1 or 2, and b = 1, 2, or 3.Therefore, we know that there are ( k − q C | C cooperators (we call them C ), except C , and( k − q D | C defectors (we call them D ) among the k neighbors of C (see Fig. 1 in main text).Hence, we have π CC = P C (cid:0) ( k − q C | C + 2 (cid:1) + ( k − q C | C · P C (cid:0) ( k − q C | C + 2 (cid:1) +( k − q D | C · P C (cid:0) ( k − q C | D + 1 (cid:1) + P C ( k C + 1)considering population structure around C . The first term in the right-hand side of the aboveequation is the payoff got by C from the public goods game centered by itself, and the last threeterms are the sum of the payoffs from the games organized by its neighbors. Similarly, among the k neighbors of D (see Fig. 1 in main text), there are ( k − q C | D cooperators (we call them C ),except C , and ( k − q D | D defectors (we call them D ). Thus we have π CD = P D (cid:0) ( k − q C | D + 1 (cid:1) + ( k − q C | D · P D (cid:0) ( k − q C | C + 1 (cid:1) +( k − q D | D · P D (cid:0) ( k − q C | D (cid:1) + P D ( k C + 1)considering population structure around D . From the point of pair approximation, we know thatD can obtain the same payoff as D .If the selected cooperator is replaced by a defector, p C will decreases by N with probabilityPr (cid:16) ∆ p C = − N (cid:17) = p C k X k C =0 (cid:18) kk C (cid:19) q k C C | C q k D D | C Λ . Simultaneously, as the decrease of p C , the number of CC -pairs is changing. And CC -pairswill decrease by k C after a defector replacing a cooperator. Hence p CC decreases by k C kN/ withprobability Pr (cid:16) ∆ p CC = − k C kN (cid:17) = p C (cid:18) kk C (cid:19) q k C C | C q k D D | C Λ . If the selected individual is a defector D , we obtain π DC = P C (cid:0) ( k − q C | C + 1 (cid:1) + P C ( k C )+( k − q C | C · P C (cid:0) ( k − q C | C + 2 (cid:1) +( k − q D | C · P C (cid:0) ( k − q C | D + 1 (cid:1) π DD = P D (cid:0) ( k − q C | D (cid:1) + P D ( k C )+( k − q C | D · P D (cid:0) ( k − q C | C + 1 (cid:1) +( k − q D | D · P D (cid:0) ( k − q C | D (cid:1) π DC ( π DD ) indicates the payoff of D ’s neighbor C (D or D ) who adopts the strategy ofcooperation (defection) after playing public goods games. Thus, D is replaced by a cooperatorduring the updating process with probabilityΓ = k C (1 − w + wπ DC ) k C (1 − w + wπ DC ) + k D (1 − w + wπ DD ) . And, similarly we have Pr (cid:16) ∆ p C = 1 N (cid:17) = p D k X k C =0 (cid:18) kk C (cid:19) q k C C | D q k D D | D Γ , and Pr (cid:16) ∆ p CC = 2 k C kN (cid:17) = p D (cid:18) kk C (cid:19) q k C C | D q k D D | D Γ . A.2 Updating dynamics
If we assume every DB updating incident takes place in one unit of time, the derivatives of ˙ p C and˙ p CC can be written as˙ p C = 1 N Pr (cid:18) ∆ p C = 1 N (cid:19) + (cid:18) − N (cid:19) Pr (cid:18) ∆ p C = − N (cid:19) , (11)and ˙ p CC = k X k C =0 k C kN Pr (cid:18) ∆ p CC = 2 k C kN (cid:19) + k X k C =0 (cid:18) − k C kN (cid:19) Pr (cid:18) ∆ p CC = − k C kN (cid:19) . (12)We havePr (cid:16) ∆ p C = 1 N (cid:17) = p D k X k C =0 (cid:18) kk C (cid:19) q k C C | D q k D D | D k C k + p D k X k C =0 (cid:18) kk C (cid:19) q k C C | D q k D D | D k C k D ( π DC − π DD ) k w + O ( w )= p D k X k C =1 k ! k C !( k − k C )! k C k q k C C | D q k − k C D | D + p D ( π DC − π DD ) w k X k C =0 k ! k C !( k − k C )! k C ( k − k C ) k q k C C | D q k − k C D | D + O ( w )= p D q C | D k X k C =1 ( k − k C − k − k C )! q k C − C | D q k − k C D | D + O ( w )+ p D q C | D q D | D k − k ( π DC − π DD ) w k − X k C =1 ( k − k C − k − k C − q k C − C | D q k − k C − D | D = p D q C | D + p D q C | D q D | D k − k ( π DC − π DD ) w + O ( w )= p CD + k − k ( π DC − π DD ) p CD q D | D w + O ( w ) , (13)12ndPr (cid:16) ∆ p C = − N (cid:17) = p C k X k C =0 (cid:18) kk C (cid:19) q k C C | C q k D D | C (1 − k C k )+ p C k X k C =0 (cid:18) kk C (cid:19) q k C C | C q k D D | C k C k D ( π CD − π CC ) k w + O ( w )= p C − p C k X k C =0 k ! k C !( k − k C )! k C k q k C C | C q k − k C D | C + p C k X k C =0 k ! k C !( k − k C )! k C ( k − k C ) k q k C C | C q k − k C D | C ( π CD − π CC ) w + O ( w )= p C − p C q C | C k X k C =1 ( k − k C − k − k C )! q k C − C | C q k − k C D | C + O ( w )+ p C q C | C q D | C k − k ( π CD − π CC ) w k − X k C =1 ( k − k C − k − k C − q k C − C | C q k − k C − D | C = p C − p C q C | C + p C q C | C q D | C k − k ( π CD − π CC ) w + O ( w )= p CD + k − k ( π CD − π CC ) p CD q C | C w + O ( w ) . (14)Substituting equations (13) and (14) into equation (11), we obtain˙ p C = 1 N k − k p CD (cid:2) q D | D ( π DC − π DD ) + q C | C ( π CC − π CD ) (cid:3) w + O ( w ) . (15)13he first term in the right-hand side of the equation (12) is k X k C =0 k C kN Pr (cid:18) ∆ p CC = 2 k C kN (cid:19) = k X k C =0 k C kN p D (cid:18) kk C (cid:19) q k C C | D q k D D | D (cid:18) k C k + O ( w ) (cid:19) = 2 p D N k k X k C =0 (cid:18) kk C (cid:19) q k C C | D q k D D | D k C + O ( w )= 2 p D N k k X k C =1 k C k ! k C !( k − k C )! q k C C | D q k − k C D | D + O ( w )= 2 p D N k k X k C =1 q C | D ( k C − k !( k C − k − k C )! q k C − C | D q k − k C D | D + O ( w )= 2 p CD N k (cid:20) k X k C =1 ( k C − k !( k C − k − k C )! q k C − C | D q k − k C D | D + k X k C =1 k ( k − k C − k − k C )! q k C − C | D q k − k C D | D (cid:21) + O ( w )= 2 p CD N k k X k C =2 q C | D k ( k − k − k C − k − k C )! q k C − C | D q k − k C D | D + k + O ( w )= 2 p CD N k (cid:2) q C | D k ( k −
1) + k (cid:3) + O ( w )= 2 p CD N k (cid:2) k − q C | D (cid:3) + O ( w ) , and the second term is k X k C =0 (cid:18) − k C kN (cid:19) Pr (cid:18) ∆ p CC = − k C kN (cid:19) = − k X k C =0 k C kN p C (cid:18) kk C (cid:19) q k C C | C q k D D | C (cid:18) k D k + O ( w ) (cid:19) = − p C N k k X k C =0 (cid:18) kk C (cid:19) q k C C | C q k D D | C k C k D + O ( w )= − p C N k k − X k C =1 k ! k C !( k − k C )! q k C C | C q k − k C D | C k C ( k − k C ) + O ( w )= − p C N k k − X k C =1 q C | C q D | C k ( k − k − k C − k − k C − q k C − C | C q k − k C − D | C + O ( w )= − k − N k p CD q C | C + O ( w ) . Hence, we obtain ˙ p CC = 2 N k p CD (cid:2) k − q C | D − q C | C ) (cid:3) + O ( w ) . (16)14rom equations (15) and (16), we have˙ q C | C = dd t (cid:18) p CC p C (cid:19) = 2 N k p CD p C (cid:2) k − q C | D − q C | C ) (cid:3) + O ( w ) . (17)The system is described only by p C and q C | C . Rewriting the r.h.s’s of equations (15) and (17)as functions of p C and q C | C yields the closed dynamic system (cid:26) ˙ p C = wF (cid:0) p C , q C | C (cid:1) + O ( w )˙ q C | C = F (cid:0) p C , q C | C (cid:1) + O ( w ) (18)where F (cid:0) p C , q C | C (cid:1) = 1 N k − k p C (1 − q C | C ) h q C | C ( π CC − π CD )+ 1 − p C + p C q C | C − p C ( π DC − π DD ) i ,F (cid:0) p C , q C | C (cid:1) = 2 N k (1 − q C | C ) (cid:20) k − p C − q C | C − p C (cid:21) . With 0 < w ≪
1, above system can be reduced as (cid:26) ˙ p C = wF (cid:0) p C , q C | C (cid:1) w ˙ q C | C = wF (cid:0) p C , q C | C (cid:1) . For q C | C , whose velocity can be large when w is small and wF ( p C , q C | C ) = 0, may rapidly convergeto the root defined by F ( p C , q C | C ) = 0 as time t → + ∞ . Thus we get q C | C = k − k − p C + 1 k − p C = wF (cid:18) p C , k − k − p C + 1 k − (cid:19) . From equation (19), we know that all variables in the dynamically evolutionary system (18) canbe described only by p C when it is stable. And from equations (6) ∼ (10), we have q D | C = k − k − − p C ) , (20) p CD = p DC = k − k − p C (1 − p C ) , (21) q C | D = k − k − p C , (22) q D | D = 1 − k − k − p C , (23) p DD = (1 − p C ) (cid:18) − k − k − p C (cid:19) . (24)15ere we obtain q C | C − q C | D = 1 k − q D | D − q D | C = 1 k − k neighbors, there is a high average probability for cooperators or defectors to form clusters whereevery individual with the same strategy. We know that the above equation is irrelevant to the payoffforms and the existence of nonlinearity in individual fitness. The relation is also proposed in [12]for pairwise interactions. Hence, it explicitly illustrates the fact of clustering by local interactionfor multiple and pairwise games in finite structured population.Then, as δ = 1, we know π DC − π DD = P C (cid:0) ( k − q C | C + 1 (cid:1) + ( k − q C | C · P C (cid:0) ( k − q C | C + 2 (cid:1) +( k − q D | C · P C (cid:0) ( k − q C | D + 1 (cid:1) − P D (cid:0) ( k − q C | D (cid:1) − ( k − q C | D · P D (cid:0) ( k − q C | C + 1 (cid:1) − ( k − q D | D · P D (cid:0) ( k − q C | D (cid:1) − c = b ( k + 1)(1 − δ ) (cid:20) − δ ( k − q C | C +1 + ( k − q C | C − ( k − q C | C δ ( k − q C | C +2 +( k − q D | C − ( k − q D | C δ ( k − q C | D +1 − δ ( k − q C | D − ( k − q C | D +( k − q C | D δ ( k − q C | C +1 − ( k − q D | D + ( k − q D | D δ ( k − q C | D (cid:21) − ( k + 1) c = b ( k + 1)(1 − δ ) (cid:26) − ( k − q C | C δ ( k − q C | C +2 + (cid:2) ( k − q C | D − (cid:3) δ ( k − q C | C +1 − ( k − q D | C δ ( k − q C | D +1 + (cid:2) ( k − q D | D + 1 (cid:3) δ ( k − q C | D (cid:27) − ( k + 1) c = b ( k + 1)(1 − δ ) (cid:26)(cid:2) − ( k − p C − (cid:3) δ ( k − p C +3 + (cid:2) ( k − p C − (cid:3) δ ( k − p C +2 + (cid:2) ( k − p C − k + 2 (cid:3) δ ( k − p C +1 + (cid:2) − ( k − p C + k (cid:3) δ ( k − p C (cid:27) − ( k + 1) c = b ( k + 1)(1 − δ ) (cid:20) − ( k − p C δ ( k − p C +3 + ( k − p C δ ( k − p C +2 + ( k − p C δ ( k − p C +1 − ( k − p C δ ( k − p C − δ ( k − p C +3 − δ ( k − p C +2 + 2 δ ( k − p C +1 − kδ ( k − p C +1 + kδ ( k − p C (cid:21) − ( k + 1) c = b ( k − δ − k + 1 δ ( k − p C p C + b ( δ + 2 δ + k ) k + 1 δ ( k − p C − ( k + 1) c. (27)Using π DC − π DD − (cid:0) π CC − π CD (cid:1) = P C (cid:0) ( k − q C | C + 1 (cid:1) − P C (cid:0) ( k − q C | C + 2 (cid:1) + P D (cid:0) ( k − q C | D + 1 (cid:1) − P D (cid:0) ( k − q C | D (cid:1) = b (1 − δ ) k + 1 δ ( k − p C ,
16e get π CC − π CD = b ( k − δ − k + 1 δ ( k − p C p C + b (cid:2) δ + 2 δ + k − (cid:3) k + 1 δ ( k − p C − ( k + 1) c. (28)As δ = 1, we obtain π DC − π DD = π CC − π CD = b ( k + 3) k + 1 − ( k + 1) c. From above equation, we find both equation (27) and (28) are available as δ = 1. A.3 Fixation probability
Using Kolmogorov backward equation [59, 60, 62], we get the fixation probability, φ C ( p ) of thestrategy cooperation with initial frequency p , satisfies the ordinary differential equation as M ( p C ) d φ C ( p )d p + V ( p C )2 d φ C ( p )d p = 0 , with boundary conditions φ C (0) = 0 φ C (1) = 1where M ( p C ) and V ( p C ) are respectively the mean and the variance of p C , the amount of changein strategy frequency per generation. The solution of the above differential equation with theboundary conditions can be written φ C ( p ) = R p G ( x ) d x R G ( x ) d x where G ( x ) = e − R M ( x ) V ( x ) d x . Now, let’s calculate − M ( x ) /V ( x ). In a short time interval, ∆ t , we have M ( p C ) = E[∆ p C ]∆ t = ∆ tN (cid:20) Pr (cid:18) ∆ p C = 1 N (cid:19) − Pr (cid:18) ∆ p C = − N (cid:19)(cid:21) t ≈ N k − k p CD h q D | D ( π DC − π DD ) + q C | C ( π CC − π CD ) i w = wN k − k k − k − p C (1 − p C ) (cid:20)(cid:18) kk − − q C | C (cid:19) ( π DC − π DD ) + q C | C ( π CC − π CD ) (cid:21) = w ( k − p C (1 − p C ) N k ( k − (cid:26) k ( π DC − π DD ) + [( k − p C + 1] (cid:2) ( π CC − π CD ) − ( π DC − π DD ) (cid:3) (cid:27) = w ( k − p C (1 − p C ) N k ( k − ( k (cid:20) b ( k − δ − k + 1 δ ( k − p C p C + b ( δ + 2 δ + k ) k + 1 δ ( k − p C − ( k + 1) c (cid:21) + [( k − p C + 1] b ( δ − δ ( k − p C k + 1 ) = w ( k − p C (1 − p C ) N k ( k − ( b ( δ − k − δ ( k − p C p C + b (cid:2) ( k + 1) δ + 2 kδ + k − (cid:3) k + 1 δ ( k − p C − k ( k + 1) c ) , (29)17nd V ( p C ) = Var[∆ p C ]∆ t = 1∆ t E "(cid:18) ∆ p C − E[∆ p C ]∆ t (cid:19) ∆ t = " N − p C ] N ∆ t + (cid:18) E[∆ p C ]∆ t (cid:19) Pr (cid:18) ∆ p C = 1 N (cid:19) + " N + 2 E[∆ p C ] N ∆ t + (cid:18) E[∆ p C ]∆ t (cid:19) Pr (cid:18) ∆ p C = − N (cid:19) ≈ N (cid:20) Pr (cid:18) ∆ p C = 1 N (cid:19) + Pr (cid:18) ∆ p C = − N (cid:19)(cid:21) ≈ p CD N = 2 p C ( k − − p C ) N ( k − . (30)Then we obtain − M ( x ) V ( x ) = − N wk ( b ( δ − k − δ ( k − x x + b (cid:2) ( k + 1) δ + 2 kδ + k − (cid:3) k + 1 δ ( k − x − k ( k + 1) c ) = w ( − N b ( δ − k − k + 1) k ( k + 1) δ ( k − x x − N b (cid:2) ( k + 1) δ + 2 kδ + k − (cid:3) k ( k + 1) δ ( k − x + N ( k + 1) c ) . Therefore, we have − M ( x ) V ( x ) = w h Rδ ( k − x x + Sδ ( k − x + N ( k + 1) c i where R = − N b ( δ − k − k S = − N b (cid:2) ( k + 1) δ + 2 kδ + k − (cid:3) k ( k + 1) . A.3.1 Fixation probability for δ = 1As δ = 1, k ≥
2, we have R = 0 , S = − N b ( k + 3) k + 1 , and − M ( x ) V ( x ) = [ S + c ( k + 1) N ] w. Then we get G ( x ) = e R [ S + c ( k +1) N ] w d x ≈ n S + c ( k + 1) N ] wx o C C is a constant. Thus, φ C ( p ) = p + [ S + c ( k + 1) N ] p w S + c ( k + 1) N ] w ≈ p + h S + c ( k + 1) N i p w − h S + c ( k + 1) N i p w = p + h S + c ( k + 1) N i w p − p )= p + N (cid:20) b ( k + 3) k + 1 − c ( k + 1) (cid:21) p (1 − p ) w. And as δ = 1, k ≥
2, we also have φ D ( p ) = p − N (cid:20) b ( k + 3) k + 1 − c ( k + 1) (cid:21) p (1 − p ) w. A.3.2 Fixation probability for δ = 1 and k = 2As δ = 1, and k = 2, we have Z − M ( x ) V ( x ) d x = Z h ( Rx + S ) δ ( k − x + c ( k + 1) N i w d x = w " ( Rx + S ) δ ( k − x ( k −
2) ln δ − Z R δ ( k − x ( k −
2) ln δ d x + c ( k + 1) N x + C = c ( k + 1) N xw + δ ( k − x ( k −
2) ln δ (cid:20) Rx + S − R ( k −
2) ln δ (cid:21) w + C w where C and C are constants, and then we obtain G ( x ) = e (cid:26) c ( k +1) Nx + δ ( k − x ( k −
2) ln δ h Rx + S − R ( k −
2) ln δ i + C (cid:27) w ≈ ( c ( k + 1) N x + δ ( k − x ( k −
2) ln δ (cid:20) Rx + S − R ( k −
2) ln δ (cid:21) + C ) w. Let A = ( k −
2) ln δ , then Z p G ( x ) d x ≈ x (cid:12)(cid:12)(cid:12) p + " c ( k + 1) N x (cid:12)(cid:12)(cid:12)(cid:12) p + δ ( k − x (cid:0) Rx + S − RA (cid:1) A (cid:12)(cid:12)(cid:12)(cid:12) p − Rδ ( k − x A (cid:12)(cid:12)(cid:12)(cid:12) p + C (cid:12)(cid:12)(cid:12)(cid:12) p w = p + " c ( k + 1) N p A δ ( k − p (cid:18) Rp + S − RA (cid:19) − A (cid:18) S − RA (cid:19) − Rδ ( k − p A + RA + C p w = p + (cid:20) c ( k + 1) N p (cid:18) RA p + SA − RA (cid:19) δ ( k − p − (cid:18) SA − RA (cid:19) + C p (cid:21) w and Z G ( x ) d x ≈ (cid:20) c ( k + 1) N (cid:18) RA + SA − RA (cid:19) δ ( k − − (cid:18) SA − RA (cid:19) + C (cid:21) w. φ C ( p ), with initial frequency p C ( t =0) = p can be written φ C ( p ) = A + B wC + D w where A = p B = c ( k + 1) N p (cid:18) RA p + SA − RA (cid:19) δ ( k − p − (cid:18) SA − RA (cid:19) + C p,C = 1 D = c ( k + 1) N (cid:18) RA + SA − RA (cid:19) δ ( k − − (cid:18) SA − RA (cid:19) + C . Expanding φ C ( p ) in a Taylor series at w = 0, we obtain φ C ( p ) ≈ p + ( B − D p ) w = p + (cid:20)(cid:18) c ( k + 1) N p + Q (cid:19) ( p −
1) + (cid:18) RA p + Q (cid:19) δ ( k − p − (cid:18) RA + Q (cid:19) pδ k − (cid:21) w where Q = SA − RA = − AN b (cid:2) ( k + 1) δ + 2 kδ + k − (cid:3) + 2 N b ( δ − k − k + 1) A k ( k + 1)= N b δ − k − k + 1) − A (cid:2) ( k + 1) δ + 2 kδ + k − (cid:3) A k ( k + 1) = N b Q Q ,RA p + Q = − pN b ( δ − k − k + 1) A k ( k + 1) + N bQ A k ( k + 1) = ApN b (1 − δ )( k − k + 1) + N bQ A k ( k + 1)= N b pQ + Q Q ,RA + Q = N b Q + Q Q , and the notations are used as Q = A k ( k + 1) = k ( k + 1)( k − ln δ,Q = 2( δ − k − k + 1) − A (cid:2) ( k + 1) δ + 2 kδ + k − (cid:3) = 2( δ − k − k + 1) − (cid:2) ( k − k + 1) δ + 2 k ( k − δ + ( k − k − (cid:3) ln δ,Q = A (1 − δ )( k − k + 1) = (1 − δ )( k + 1)( k − ln δ. Furthermore we have φ C ( p ) ≈ p + (cid:26)(cid:20) c ( k + 1) N p + N b Q Q (cid:21) ( p −
1) +
N b pQ + Q Q δ ( k − p − N b Q + Q Q pδ k − (cid:27) w with δ = 1, and k = 2.And we also have φ D ( p ) ≈ p − (cid:26)(cid:20) c ( k + 1) N p + N b Q + Q Q δ k − (cid:21) ( p − − N b ( p − Q − Q Q δ ( k − − p ) − N bp Q Q (cid:27) w with δ = 1, and k = 2. 20 .3.3 Fixation probability for k = 2Using the same method as above, we have φ C ( p ) ≈ p + N (cid:20) (3 δ + 4 δ + 3) b − c (cid:21) p (1 − p ) w with k = 2, and φ D ( p ) ≈ p − N (cid:20) (3 δ + 4 δ + 3) b − c (cid:21) p (1 − p ) w with k = 2. A.3.4 Conclusion of the fixation probability for cooperation and defection
By the above tedious calculation, we obtain the fixation probability of strategy cooperation withinitial frequency p as φ C ( p ) = p + N h b ( k +3) k +1 − c ( k + 1) i p (1 − p ) w for δ = 1 p + N h (3 δ +4 δ +3) b − c i p (1 − p ) w for k = 2 p + nh c ( k +1) N p + N b Q Q i ( p −
1) +
N b pQ + Q Q δ ( k − p − N b Q + Q Q pδ k − o w otherwise . (31)And through similarly calculative process, we have the fixation probability of strategy defectionwith initial frequency p as φ D ( p ) = p − N h b ( k +3) k +1 − c ( k + 1) i p (1 − p ) w for δ = 1 p − N h (3 δ +4 δ +3) b − c i p (1 − p ) w for k = 2 p − nh c ( k +1) N p + N b Q + Q Q δ k − i ( p − − N b ( p − Q − Q Q δ ( k − − p ) − N bp Q Q o w otherwise . (32) B Criteria for emergence and stabilization of cooperation
As to the criteria for emergence and stabilization of cooperation, we compare (the fixation probabil-ity with which a cooperator will invade and take over a population of N − /N (thefixation probability of a neutral mutant [4]). If ρ C > /N then natural selection favors cooperatorreplacing defector [4] , we call that natural selection favors the emergence of cooperation. We callthat natural selection favors the stabilization of cooperation if ρ D < /N , that is, natural selectionopposes the fixation of defectors. And if ρ C > ρ D , we call that natural selection favors cooperatorover defector [4] . For mathematical feasibility, we obtain the criteria according to the followingthree occasions. B.1 As δ = 1 For large N and δ = 1, the fixation probability of a single cooperator (defector) in a population of N − ρ C = φ C (cid:0) N (cid:1) ( ρ D = φ D (cid:0) N (cid:1) ). That is, ρ C = φ C (cid:18) N (cid:19) ≈ N + 12 (cid:20) b ( k + 3) k + 1 − c ( k + 1) (cid:21) (cid:18) − N (cid:19) w, (33) ρ D = φ D (cid:18) N (cid:19) ≈ N − (cid:20) b ( k + 3) k + 1 − c ( k + 1) (cid:21) (cid:18) − N (cid:19) w. (34)21ence, we have ρ C > N if and only if b/c > ( k + 1) k + 3 , (35)and ρ D < N if and only if b/c > ( k + 1) k + 3 . (36) B.2 As δ = 1 and k = 2 For large N , δ = 1, and k = 2, we have ρ C = φ C (cid:18) N (cid:19) ≈ N + wQ (cid:20) − NN c ( k + 1)2 Q + (1 − N ) bQ + ( Q + N Q ) bδ k − N − ( Q + Q ) bδ k − (cid:21) = 1 N + wQ (cid:20) − NN c ( k + 1)2 Q + bf ( δ ) (cid:21) where f ( δ ) = ( N Q + Q ) δ k − N − ( Q + Q ) δ k − − ( N − Q .As 0 < δ <
1, we know that Q <
0, then ρ C > N ⇔ wQ (cid:20) − NN c ( k + 1)2 Q + bf ( δ ) (cid:21) > ⇔ − NN c ( k + 1)2 Q + bf ( δ ) < ⇔ bf ( δ ) c < ( N − k + 1) Q N .
We have f ( δ ) = ( N Q + Q ) δ k − N − ( Q + Q ) δ k − − ( N − Q < < δ < ρ C > N ⇔ b/c > ( N − k + 1) Q N f ( δ )for 0 < δ < δ >
1, we know that Q >
0, then ρ C > N ⇔ wQ (cid:20) − NN c ( k + 1)2 Q + bf ( δ ) (cid:21) > ⇔ − NN c ( k + 1)2 Q + bf ( δ ) > ⇔ bf ( δ ) c > ( N − k + 1) Q N .
We have f ( δ ) > δ >
1. Thus, we get ρ C > N ⇔ b/c > ( N − k + 1) Q N f ( δ )for δ >
1. 22o, for large N , δ = 1, and k = 2, we have ρ C > N if and only if b/c > ( N − k + 1) Q N f ( δ ) (37)where f ( δ ) = ( N Q + Q ) δ k − N − ( Q + Q ) δ k − − ( N − Q .For ρ D , we have ρ D = φ D (cid:18) N (cid:19) ≈ N − wQ (cid:26) − NN c ( k + 1)2 Q − b ( N − Q + Q ) δ k − + b h N Q + ( N − Q i δ ( k − N − N − bQ (cid:27) = 1 N − wQ (cid:20) − NN c ( k + 1)2 Q + bg ( δ ) (cid:21) where g ( δ ) = [ N Q + ( N − Q ] δ ( k − N − N − ( N − Q + Q ) δ k − − Q . We have Q < g ( δ ) < < δ <
1, while Q > g ( δ ) > δ >
1. Thus, using the same method, we alsoobtain ρ D < N if and only if b/c > ( N − k + 1) Q N g ( δ ) (38)for large N , δ = 1, and k = 2. B.3 As k = 2 Using the same method as above, we have ρ C > N , ρ D < N if and only if b/c > δ + 4 δ + 3 , (39)and ρ D < N if and only if b/c > δ + 4 δ + 3 (40)for large N and k = 2. B.4 A rule for the evolution of cooperation
A rule is obtained theoretically for the evolution of cooperation under combination of structuredpopulation and multiple interactions, that is, we have proposition that ρ C > N if and only if b/c > ( k +1) k +3 for δ = 1 δ +4 δ +3 for k = 2 ( N − k +1) Q Nf ( δ ) otherwise , (41)and ρ D < N if and only if b/c > ( k +1) k +3 for δ = 1 δ +4 δ +3 for k = 2 ( N − k +1) Q Ng ( δ ) otherwise (42)23rom inequalities (35), (37), (39), and (36), (38), (40) for large N , k >
1, and δ >
0, where f ( δ ) = ( N Q + Q ) δ k − N − ( Q + Q ) δ k − − ( N − Q , (43) g ( δ ) = h N Q + ( N − Q i δ ( k − N − N − ( N − Q + Q ) δ k − − Q , (44)and Q = k ( k + 1)( k − ln δ,Q = 2( δ − k − k + 1) − (cid:2) ( k − k + 1) δ + 2 k ( k − δ + ( k − k − (cid:3) ln δ,Q = (1 − δ )( k + 1)( k − ln δ. For the linear public goods game on the cycle ( δ = 1 and k = 2), ρ C > ρ D is equivalent to b/c > /
5, which coincides with [39] theoretically. Through numerical simulations for k > ρ C > /N , is in good agreement with computernumerical simulations (see Fig. 2).Furthermore, from the inequalities (41) and (42), we know that as δ = 1 and k = 2 the conditionsfor ρ C > N and ρ D < N are mainly different in f ( δ ) and g ( δ ). Here we expend f ( δ ) and g ( δ ) inTaylor series around δ = 1 to investigate the difference. For each δ in interval (0 . , . f ( δ ) = k ( k + 3)( k − ( N − N ( δ − + (4 − k )( N − N + 2) − k (19 N − N −
16) + 2 k ( N − N ( k − ( δ − + o (( δ − ) (45) g ( δ ) = k ( k + 3)( k − ( N − N ( δ − + (3 k − N − N + 2) − k (35 N − N + 16) + k (4 N − N + 2)12 N ( k − ( δ − + o (( δ − ) (46)where o (( δ − ) captures the error is the higher order infinitesimal of ( δ − . Therefore we obtain f ( δ ) and g ( δ ) display the difference in forth order of δ −
1, and it is infinitesimal as δ around 1. C Theoretical analysis for the critical value of b/c favoring evolu-tion of cooperation
C.1 Evaluation on continuity for critical value of b/c
We first calculate the limitation of ( N − k +1) Q Nf ( δ ) and ( N − k +1) Q Ng ( δ ) at δ = 1 and k = 2. Letlim δ → ( N − k + 1) Q N f ( δ ) = ( N − k + 1)2 N ( L + L )24here L = lim δ → h Q (cid:16) δ k − N − δ k − (cid:17) . Q i = lim δ → (1 − δ )( δ k − N − δ k − ) k ( k −
2) ln δ = lim δ → − δ k − N +2 − δ k ) + (1 − δ ) h k − N δ k − N − ( k − δ k − i k ( k −
2) ln δ = lim δ → δ k ( k − ( − (cid:20)(cid:18) k − N + 2 (cid:19) δ k − N +1 − kδ k − (cid:21) − δ (cid:20) k − N δ k − N − ( k − δ k − (cid:21) +(1 − δ ) "(cid:18) k − N (cid:19) δ k − N − − ( k − δ k − = 2 k (cid:18) − N (cid:19) (47)and L = lim δ → n Q h(cid:16) δ k − N − δ k − (cid:17) + ( N − (cid:16) δ k − N − (cid:17)i . Q o = lim δ → L z }| {(cid:8) δ − k + 1) − (cid:2) ( k + 1) δ + 2 kδ + ( k − (cid:3) ln δ (cid:9) L z }| {h ( δ k − N − δ k − ) + ( N − δ k − N − i k ( k + 1)( k − ln δ | {z } L = lim δ → L ′ L + L L ′ L ′ (cid:0) L (1) = L (1) = L (1) = L ′ (1) = 0 (cid:1) = lim δ → L ′′ L + 2 L ′ L ′ + L L ′′ L ′′ (cid:0) L ′ (1) = L ′′ (1) = 0 (cid:1) = lim δ → L ′′′ L + 3 L ′′ L ′ + 3 L ′ L ′′ + L L ′′′ L ′′′ (cid:0) L ′ (1) = 0 , L ′′ (1) = 0 , L ′′′ = 0 (cid:1) = lim δ → L ′ L ′′ L ′′′ = 3( − k + k + 4)( k − k ( k + 1)( k − (cid:18) N − (cid:19) = − k + k + 42 k ( k + 1) (cid:18) N − (cid:19) (48)25ith L ′ = 4( k + 1) δ − [2( k + 1) δ + 2 k ] ln δ − ( k + 1) δ + 2 kδ + k − δ ,L ′ = ( k − δ k − N − − δ k − ) ,L ′ = 3 k ( k + 1)( k − ln δδ ,L ′′ = ( k − (cid:20)(cid:18) k − N − (cid:19) δ k − N − − ( k − δ k − (cid:21) ,L ′′ = 3 k ( k + 1)( k − δ (cid:0) δ − ln δ (cid:1) ,L ′′′ (1) = 6 k ( k + 1)( k − δ . Therefore we get lim δ → ( N − k + 1) Q N f ( δ ) = ( k + 1) k + 3 . (49)Using the same method, we also obtainlim δ → ( N − k + 1) Q N g ( δ ) = ( k + 1) k + 3 . (50)Thus, we find the greatest lower bounds which support ρ C > /N and ρ D < /N is continuousat δ = 1 respectively. Also, we could obtain that the critical value is also continuous at k = 2,or, k = 2 and δ = 1. Thus, the critical values in equalities (41) and (42) can be proven as ancontinuous function of k and δ for k > δ > C.2 The equivalent proposition with infinitesimal nonlinearity
For large structured population under linear public goods game or public goods game with nonlin-earity in individual fitness on cycle, we have that natural selection favors emergence of cooperationif and only if it favors stabilization of cooperation (see equivalent proposition (5) in main text).Here, we give the validation of the equivalent proposition for infinitesimal nonlinearity.Expending the critical values in equalities (41) (indicated by
Critical V alue C ) and (42) (in-dicated by
Critical V alue D ) in Taylor series at δ = 1, we have Critical V alue C ≈ ( k + 1) k + 3 − (1 + k ) (cid:2) ( N − k ) + ( N + 1)(3 k + k ) (cid:3) N k (3 + k ) ( δ − k + 1) N k (3 + k ) h N − + 4(7 N − N + 28) k + 4(7 N − N + 1) k + 2(53 N − N − k + (65 N + 37 N + 5) k +6(2 N + 6 N + 1) k + ( N + 5 N + 1) k i ( δ − Critical V alue D ≈ ( k + 1) k + 3 − (1 + k ) (cid:2) ( N − − − k ) + (2 N − k + k ) (cid:3) N k (3 + k ) ( δ − k + 1) N k (3 + k ) h N − + 4(11 N + 10 N − k + 4( − N +59 N + 1) k + 2( − N + 121 N − k + (107 N − N + 5) k +6(9 N − N + 1) k + (7 N − N + 1) k i ( δ − . N comparing linear public goods game.Besides, for large N , δ →
1, and k ≥
2, we have ρ C > N ( ρ D < N ) if and only if b/c > ( k + 1) k + 3 . (51)And for large structured population under public goods game with infinitesimal nonlinearity inindividual fitness, we have ρ C > N ⇔ ρ C > ρ D ⇔ ρ D < N . (52)
D Proof of the relation between f ( δ ) and Let f ( δ ) = ( N Q + Q ) δ k − N − ( Q + Q ) δ k − − ( N − Q = Q f ( δ ) + f ( δ )where f ( δ ) = δ k − N − δ k − + ( N − (cid:16) δ k − N − (cid:17) , and f ( δ ) = Q (cid:16) δ k − N − δ k − (cid:17) .For Q , we have Q = 2( δ − k − k + 1) − (cid:2) ( k − k + 1) δ + 2 k ( k − δ + ( k − k − (cid:3) ln δQ ′ = f ( δ ) δ ( k − Q ′′ = (cid:20) − δ + k − kδ − k + 1) δ (cid:21) ( k + 1)( k − Q ′′′ = 2 (cid:2) − k + kδ (1 − δ ) + 1 − δ (cid:3) δ ( k − f ( δ ) = ( k + 1) (cid:2) (3 − δ ) δ − k + 1 (cid:3) − kδ (ln δ + 1). Besides, Q , Q , Q , and the first tothird derivative of Q are all functions of δ with the parameter k . D.1 The proof of f ( δ ) < when < δ < For large N , k = 2, and 0 < δ <
1, we derive f ( δ ) < < δ < f ( δ ) < Q < δ k − N − δ k − > f ( δ ) = N ( δ k − N −
1) + (1 − δ k − ). 0 < − δ k − < δ k − N − < < δ <
1. Hence,we have f ( δ ) < N > − δ k − N .(iii). As 0 < δ < Q ′′′ < Q ′′ (1) = k ( k − k − >
0, thus we have Q ′′ > < δ < Q ′ (1) = ( − k + k + 4)( k − <
0, we obtain that Q ′ < < δ <
1. And with Q (1) = 0,we eventually get Q > < δ < f ( δ ) < < δ < .2 The proof of f ( δ ) > when δ > For large N , k = 2, and δ >
1, we derive f ( δ ) > δ > f ( δ ) > Q < δ k − N − δ k − < f ′ ( δ ) = ( k − (cid:16) δ k − N − − δ k − (cid:17) < δ >
1. With f (1) = 0, we know that f ( δ ) < δ > ≤ δ , i.e. , δ ≥ e we have Q < Q ′ < Q (1) = 0.(iv). Before proving Q < < δ < e , let us consider the function h ( x ) = 3(1 + 2 x ) − (3 − x )e x first. We want to point out as follows that h ( x ) > < x < / h ( x ), we have h ′ ( x ) = 4e x ( x −
1) + 6 h ′′ ( x ) = 4e x (2 x − h ′′ ( x ) < < x < /
2, and h ′′ ( x ) > x > /
2. we know that h ′ ( x ) reachesits minimum at x = 1 / < x < /
2. Thus, we obtain h ′ ( x ) > < x < / h ′ (1 /
2) = 6 − >
0. And then we find h ( x ) > < x < / h (0) = 0.Let δ = e x , and then 1 < δ < e is equivalent to 0 < x < . Hence, for 0 < x < , we have f ( δ ) = ( k + 1) (cid:2) (3 − δ ) δ − k + 1 (cid:3) − kδ (ln δ + 1)= ( k + 1) (cid:2) (3 − x )e x − k + 1 (cid:3) − k e x ( x + 1) < ( k + 1) [3(1 + 2 x ) − k + 1] − k (1 + x )( x + 1)= − kx + (2 k + 6) x − k + 4 + k with the inequalities 3(1 + 2 x ) > (3 − x )e x and e x > x . The maximum value of f ( δ ) is f ( k ) = − k + k + 7 + k for 0 < x < . And f ( k ) < k >
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0. Because f ′ (1) = 4 > δ → + ∞ f ′ = −∞ , we derive that f ( δ ) reaches itsmaximum value at δ where f ′ ( δ ) = 0. Through Matlab, we can get f ( δ ) < − .
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