Spatial evolutionary public goods game on complete graph and dense complex networks

We study the spatial evolutionary public goods game (SEPGG) with voluntary or optional participation on a complete graph (CG) and on dense networks. Based on analyses of the SEPGG rate equation on finite CG, we find that SEPGG has two stable states depending on the value of multiplication factor r, illustrating how the “tragedy of the commons” and “an anomalous state without any active participants” occurs in real-life situations. When r is low (), the state with only loners is stable, and the state with only defectors is stable when r is high (). We also derive the exact scaling relation for r*. All of the results are confirmed by numerical simulation. Furthermore, we find that a cooperator-dominant state emerges when the number of participants or the mean degree, 〈k〉, decreases. We also investigate the scaling dependence of the emergence of cooperation on r and 〈k〉. These results show how “tragedy of the commons” disappears when cooperation between egoistic individuals without any additional socioeconomic punishment increases.

T he emergence and evolution of cooperation is central to understanding the evolution and human activityassociated dynamics. One of the most popular theoretical frameworks that is used to shed light on such issues is evolutionary game theory. Game theory has also been successfully applied in diverse fields such as evolutionary biology and psychology 1 , computer science and operations research 2,3 , political science and military strategy 4,5 , cultural anthropology 6 , ethics and moral philosophy 7 , economics 8,9 , traffic flow research 10,11 and public health 12 . When preferences and goals of participating agents are in conflict, game theory can explain and predict interactive decisions 13 . The central aim of game theory research is to determine conditions needed for cooperation to emerge between egoistic individuals [14][15][16] . Two of the most famous models for game theory include the prisoner's dilemma (PD) and public goods game (PGG) 17 . While the PD for a pairwise interaction attracted the attention of biologists and social scientists, PGG for group interactions was the focus of studies in experimental economics 18 . The PGG was often studied to identify effects of collective action arising from joint group decisions. Although sometimes the group interactions can be modeled as repeated simple pair interactions as with the PD, the most fundamental unit of the game is irreducible multi-agent nature 13,19,20 . The PGG offers valuable insight into prevailing socioeconomic problems such as pollution, deforestation, mining, fishing, climate control and environmental protection 13 . In identifying potential solutions to these issues, PGGs with various strategies 13,17, have been suggested and studied. Economists have mainly studied PGG with two strategies, C and D, in which all agents participate and share a single common pool [21][22][23][24] .
In this report, we focus on a PGG with voluntary participation 25 in which three strategic players (cooperators (C), defectors (D) and loners (L)) are considered. Each C contributes c to the common pool, whereas D attempts to exploit the resource at no cost. Then, each C gets the payoff P C as P C 5 rcn C /(n C 1 n D ) 2 c, whereas each D obtains P D as P D 5 rcn C /(n C 1 n D ). Here, n C (n D ) denotes the number of C's (D's) participating in the game, and r(.1) is the multiplication factor, which describes synergistic effects of cooperation. In contrast, L refuses to participate in the game and relies only on private payoff s. In this report the condition, 0 , s , c(r 2 1), is imposed 25 .
Recently, the spatial evolutionary PGG (SEPGG) has been intensively studied to understand how steadystate strategies emerge on various structures and to identify characteristic features of such steady-state strategies 17,25,26,[28][29][30][31][32][33] . In the SEPGG, each agent is assigned to a node on a lattice or network. In a unit game of the SEPGG, only a randomly selected agent and its linked neighbors participate 26 . Then, in each update of the SEPGG, a randomly selected agent i adopts the strategy of a randomly selected neighbor j of i with a transition probability f ij that depends on payoffs P i and P j 17 . The SEPGG studies have revealed interesting results such as cyclic dominance 25,27 , transition nature 26 , and payoff distribution 28 . The effects of underlying topology on the SEPGG properties 17,28-33 have also been found, such as the spatial reciprocity on diluted networks 34 and multiplex networks [35][36][37][38][39][40] .
Since the SEPGG on regular lattices and sparse networks has considered only local interactions, the number of participants in a unit game centered at a node i cannot exceed k i 1 1, where k i is the degree (or coordination number) of i. Thus, the SEPGG on sparse networks is hardly a theoretical model of real-life examples with very large participants such as taxes, provision levels, tolls, user fees, etc. 48 . Such cases involving public resources which anyone can overuse can be mapped into ''tragedy of the commons'' problem 49,50 . However, SEPGG in which all agents participate in a unit game has been rarely studied. Thus, we focus on SEPGG with very large participants.
In this report, the SEPGG with three strategies on a complete graph (CG) and dense complex networks is considered to understand the SEPGG with large participants. The CG is a simple undirected graph in which any node on the graph is linked to all other nodes. Thus, the number of links on the CG is N(N 2 1)/2, where N is the number of nodes. In the SEPGG on the CG, all agents participate in a unit game. From analytically exact rate equations of the SEPGG on the CG, two stationary states depending on r and N are found. The state with only L agents (or L-state) is stable for low r =r Ã ð Þ. The state with only D agents (or D-state) is stable for high r ?r Ã ð Þ. r* at which the crossover from the L-state to the D-state occurs is analytically obtained and also confirmed by numerical simulation. In the SEPGG on the CG, a C-dominant state cannot be stable even for very high r. These stationary states on the CG are very peculiar compared to the C-dominant state (or C-state) on regular lattices and sparse networks for very high r 28,30-33 . The L-state on the CG is also very peculiar in the sense that the L-state occurs only for s . c(r 2 1) in the PGG game with the well-mixed population 26 , whereas the L-state on the CG occurs even when 0 , s , c(r 2 1) or r is quite high.
More specifically, the time evolution of the SEPGG on the CG for high r is shown to have the following stages. In early time, the numbers of both C and L agents decrease, whereas the number of D agents hardly varies. Eventually, the D-state becomes stable. Hence, the time evolution of the SEPGG for high r describes key processes to the ''tragedy of the commons'' very well 49,50 , because the key processes are the following processes: First, the most of agents overuse the public resource in the commons as defector. Then, the overuse of the public resource will ruin it.
Ref. 26 revealed that the dominant state on sparse networks for high r is the C-state. Hence, we investigate crossover behaviors of the L-state or the D-state on dense networks such as the CG to a C-state on sparse networks by numerical simulation. For low r, first the crossover from the L-state to a D-state occurs, and the D-state successively crosses over to a C-state as mean-degree AEkae decreases. Furthermore, the D-state for moderate AEkae remains even in the limit N R '. We also quantitatively find that cooperation gradually increases as the number of participants or AEkae decreases, which is the origin of two crossovers. Hence, the crossovers for low r describe how the enhanced cooperation on sparse networks with low AEkae overcomes ''tragedy of the commons'', resulting in the C-state. For high r the direct crossover from the D-state to the C-state occurs. This direct crossover is nearly the same as that from the D-state to the C-state for low r.

Results
SEPGG on the complete graph. From f ij in Eq. (11) using {P i } on the CG, exact rate equations of densities on the CG are written as and where _ r C~d r C =dt, etc. To obtain stationary states from general initial configurations with r I C~rC t~0 ð Þw0, r I D w0 and r I L w0, early time behaviors of r C , r D , and r L must be considered. Early time behaviors of r C , r D , and r L are determined based on competition between two terms of Eqs. (1)-(3), respectively. As r C r D tanh(2bc/2) # 0 in Eq. (1) and r C r D tanh(bc/ 2) $ 0 in Eq. (2) for any non-negative r C , r D , b and c, two distinctive steady states are achievable depending on the value of r C . When (1) and _ r C v0 after some time. From these relations we find that the state of {r D^1 , (3) and _ r L v0 after some time. As a result, when r C ws 1{r L cr , the state of {r D^1 , r C =1, r L =1} appears. We call this state the D-state. In contrast, when r C vs 1{r L cr , _ r L w0 and _ r C v0, which make r C r D tanh bc 2 =1 in Eq. (2) and _ r D v0 after some time. Thus, the state of {r L^1 , r C =1, r D =1} appears. We call this state the L-state. As the D-state or the L-state appears depending on the condition r C vs 1{r L cr , we now examine the stability of the D-state based on rate equations (1)-(3). If the D-state is unstable, the L-state should be stable.
In the D-state with {r D^1 , r C =1, r L =1}, the rate equation (1) because r L =r D . By solving Eq. (4) for time t, we obtain Similarly, the rate equation (3) also becomes When r C ?s 1{r L cr , tanh b 2 rcr C 1{r L {s ! 1 and As r C decreases with t, the condition r C ws 1{r L cr for the D-state breaks down for t . t*. From the Eq. (5) and the condition the CG with N R ', the L-state is the only stationary state. However, on the CG with finite N, the nonzero-minimum of r L is 1/N and thus v1=N, then r L (t . t*) 5 0 and the D-state is still the stationary state. These results mean that the SEPGG on the CG with finite N has the following stationary state. For r?r Ã , the D-state becomes stable, where or More specifically, this D-state for high r or r?r Ã has never been found on regular lattices and sparse networks. As emphasized in our introductory remarks, this state also describes ''tragedy of the commons'' very well. In contrast, for r=r Ã , the L-state becomes stable. This L-state for r=r Ã has never been found on regular lattices and sparse networks either. The L-state is also anomalous and surprising, because no body remains as an active participant in the PGG for r=r Ã . No C-dominant stationary state is found on the CG even for high r.  1(b). Furthermore, the crossover time t* for r 5 60 is t* 5 8.86 in Fig. 1(b), which is nearly identical to t* When r= c s N tanh bc 2 ð Þ and in the limit of N R ', the time dependences of r C , r D and r L on the CG shown in Fig. 1(b) effectively present the process to the anomalous L-state with no active participants. The process means the following three steps. First, most agents defect one another. C then changes his strategy to D, and r C (t) decreases. Thus, D cannot receive enough payoff 50 , causing r D (t) to decrease and r L (t) to increase. Finally, most agents become L, as no one remains in the commons. Consequently, the stationary L- To analyze the dependence of stationary states on the multiplication factor r, r S C~rC t?? ð Þ, r S D~rD t?? ð Þ, and r S L~rL t?? ð Þ are obtained from simulations for various N and r by averaging over 1,000 realizations. Simulation results of r S D and r S L for various N and r are shown in the insets of Fig. 2. As shown in insets of Fig. 2, the crossover value of r, i.e., r*, from the stationary L-state to the stationary D-state increases with N as expected from Eq. (9). More specifically, r S D N,r ð Þ and r S L N,r ð Þ in Fig. 2  Crossover from the behavior on dense networks to that on sparse networks. A dense network is a network in which the mean-degree AEkae satisfies AEkae / N 51 . For example, the CG is a typical dense network, as AEkae 5 N 2 1 in the CG. In a sparse network, AEkae 5 finite 51 . In the SEPGG on the CG, either the L-state or the D-state is stable depending on r and N and the C-dominant state cannot be stable. In contrast, the C-dominant state is stable for relatively high r in the SEPGG on sparse networks such as random networks 30,33 and two dimensional square lattices 17,26 . Therefore, it is interesting to study how crossover from the L-state and the D-state on dense networks to the C-dominant state on sparse networks occurs for given values of r and N. We first investigate how the L-state on dense networks crosses over to the C-dominant state on sparse networks. Since the L-state is stable for low r 0 on the CG as shown in Fig. 2, the crossover behaviors for low r 0 are studied by simulations on random networks with AEkae. For a given N and AEkae, r S C , r S D , and r S L are obtained by averaging over 2,000 realizations. Typical crossover behaviors for   r 0 5 0.3 are shown in Fig. 3. As shown in Fig. 3(a), two crossovers occur successively as AEkae decreases. The L-state is stable when AEkae is quite high. The C-state of {r S C~1 , r S D~0 , r S L~0 } is stable when AEkae is low enough. For moderate AEkae the D-state is stable. Therefore, for low r 0 , the stationary state is first changed from the L-state to a D-state and crossover from the D-state to a C-state occurs as AEkae decreases.
The stability of the D-state for moderate AEkae in the limit N R ' is studied using the following methods. From simulation data of Þ as in Figs. 3(a) and 3(b), we first obtain AEkae 1 at which relations r S L k h i 1 ,N À Á~1 =2 and r S D k h i 1 ,N À Á~1 =2 hold simultaneously. We also obtain AEkae 2 at which r S D k h i 2 ,N À Á~1 =2 and r S C k h i 2 ,N À Á~1 =2 hold. For example, dependences of AEkae 1 and AEkae 2 on N for r 0 5 0.3 are shown in Fig. 3(c). The dependence of D AEkae (;AEkae 1 2 AEkae 2 ) is also shown in Fig. 3(d). As shown in Fig. 3(d), D AEkae increases monotonically with N, guaranteeing the stability of the D-state for moderate AEkae in the limit N R '. Furthermore, as shown in Fig. 3(c), AEkae 1 and AEkae 2 satisfy power laws k h i 1^N n1 and k h i 2^N n2 . By fitting these power laws to data presented in Fig. 3(c), crossover exponents are obtained as n 1 5 0.898(2), n 2 5 0.520 (2). The result n 1 . n 2 also guarantees the stability of the D-state for moderate AEkae. The crossover property from the L-state to the D-state presented in Fig. 3(b) is adequately described by the single exponent n 1 obtained in Fig. 3(c). r S D k h i,N ð Þ 0 s for higher AEkae and various N are plotted against the scaling variable k h i=N n 1 with the obtained n 1 as in Fig. 3(e), which shows that r S D k h i,N ð Þfor higher AEkae is a function of the single scaling variable k h i=N n1 . As shown in Fig. 3(f), crossover from the D-state to the C-state also satisfies the scaling property that r S D k h i,N ð Þfor lower AEkae is a function of the single scaling variable k h i=N n2 with the obtained exponent n 2 . Using the same method n 1 's and n 2 's for various low r 0 (,1) are obtained as shown in Fig. 4. Because n 1 . n 2 in Fig. 4, the D-state for moderate AEkae and low r 0 (,1) is stable in the limit N R '.
Furthermore, the dependences of r S C , r S D , and r S L on AEkae for low r 0 in Fig. 3(a) are quite similar to the time dependences of r C (t), r D (t), and r L (t) on the CG for low r 0 shown in Fig. 1(b). In Fig. 1(b), initially there are enough Cs. As t increases, D governs the system. Finally L dominates, because D cannot receive enough payoff. Likewise, in Fig. 3(a), for low AEkae there are also enough Cs. For moderate AEkae D governs the system. When AEkae becomes high enough, L dominates. Hence, it is very interesting to compare dynamical behaviors on the CG to static crossover behaviors depending on AEkae.
We thus now focus on the time dependence of r C (t), r D (t), and r L (t) for various AEkae to understand crossover behaviors for low r 0 in Fig. 3(a). The time dependences of r C , r D , and r L for moderate AEkae are shown in Fig. 5(a), and those for low AEkae are shown in Fig. 5(b). For high AEkae, the time dependence is nearly identical to that on the CG shown in Fig. 1(b). For moderate AEkae and high AEkae, r C and r L decrease, but r D increases in early time. However, the stationary state is strongly affected by the subsequent time dependence of r C . If AEkae is quite high or if k h i? k h i 1 , r C decays quickly and r D cannot receive enough payoff. As a result, r L increases for t . t* and the stationary L-state appears as explained in Fig. 1(b). In contrast, for moderate AEkae or k h i 2 = k h i= k h i 1 , r C (t) decreases relatively slowly, and r L (t) never have a chance to increase reversely before the time at which r L (t) # 1/N [see Fig. 5(a)]. This means that the cooperation is effectively enhanced for moderate AEkae and D receives enough payoff until L disappears due to the enhanced cooperation. This first crossover is quite similar to the crossover from the L-state in Fig. 1(b) to D-state in Fig. 1(a) on the CG. For low AEkae or k h i= k h i 2 , r C (t) never decreases as on sparse networks 28,[30][31][32][33]], and r S C wr S D . Hence, the crossover from the D-state to the C-state (or C-dominant state) occurs for AEkae , AEkae 2 as AEkae decreases.
The two crossovers for low r 0 thus derive from a gradual increase of cooperation as the number of participants (or AEkae) decreases. Therefore, the crossovers that describe the disappearance of both the anomalous state with no active participants and ''tragedy of the   commons'' quantitatively show that agents in the larger group hardly cooperate relative to those in the smaller group 45,46 . However, this dependence on the group size is not necessarily accurate, because a recent study on PGG 44 reported that increasing the group size does not necessarily lead to mean-field behaviors. Finally, we study the crossover from the D-state to a C-state for high r 0 (.1). Typical crossover behaviors for high r 0 are shown in Fig. 6(a). As shown in Fig. 6(a), for high r 0 (510), the D-state is stable when AEkae is quite high. The C-state is stable when AEkae is low enough. Therefore, for high r 0 , the direct crossover from the D-state to the Cstate occurs as AEkae decreases. To analyze the dependence of this direct crossover on N, r S D k h i,N ð Þ 0 s for various N are obtained by simulation as shown in Fig. 6(b). The dependence of the direct crossover on N can be obtained by the ansatz k h i 3 *N n3 , where at AEkae 3 both r S D k h i 3 ,N À Á~1 =2 and r S C k h i 3 ,N À Á~1 =2 hold. From the dependence of AEkae 3 on N, n 3^0 :51 1 ð Þ is obtained for r 0 5 10. This direct crossover satisfies the scaling property that r S D k h i ð Þ is a function of the single scaling variable k h i=N n3 with n 3 5 0.51. As shown in Fig. 6(d), n 3 's for various high r 0 (.1) are obtained using the same method. The data in Fig. 6(d) show that the value of n 3 increases as r 0 increases. As the D-state is always stable on the CG or dense networks with AEkae / N, the upper bound of n 3 should be equal to 1. We also confirm that the time dependences of r C (t), r D (t), and r L (t) for high r 0 are nearly the same as those in Fig. 1(a) for high AEkae and as those in Fig. 5(b) for low AEkae, respectively. Hence, this direct crossover is nearly identical to the second crossover from the D-state to the C-state for low r 0 .

Discussion
In summary, we have studied the SEPGG on the CG and complex dense networks to understand behaviors of the SEPGG with very large participants. By analyses of the rate equations, we have shown that the L-state of {r C =1, r D =1, r L^1 } is stable on the CG for r , r* with r Ã * s c N tanh bc 2 ð Þ . In contrast, the D-state of {r C =1, r D^1 , r L =1}, representing ''tragedy of the commons'', is stable for r . r*.
These analytic results on the CG have been confirmed by simulation.
We have also studied crossover behaviors from the L-state or the D-state on dense networks to the C-dominate state on sparse networks by numerical simulation on random networks with a mean degree AEkae. For r , r*, the L-state first crosses over to a D-state, and successively this D-state crosses over to a C-state as AEkae decreases. We have investigated the dependence of the crossovers on N for low r 0 using the ansatz k From the numerical simulations, n 1 and n 2 have been obtained. Since n 1 . n 2 for r , r*, we have found that the D-state for moderate AEkae is stable even in the limit N R '. We have also studied the time dependences of r C , r D , and r L on random networks with AEkae to understand the crossover behaviors for r , r*. For moderate AEkae, the D-state is stable, because r C decreases relatively slowly. For low AEkae, cooperation is enhanced and the C-state is stable. The two crossovers for r , r* derive from a gradual increase of cooperation as the number of participants (or AEkae) decreases. The crossovers thus show how the enhanced cooperation on sparse networks with low AEkae produces the C-state, overcoming both the anomalous state with no active participants and ''tragedy of the commons'' for low r 0 .
For high r 0 , the D-state is stable when AEkae is high. The C-state is stable when AEkae is low. Therefore, for high r 0 , the direct crossover from the D-state to the C-state occurs as AEkae decreases. The dependence of the direct crossover on N has been also analyzed by the ansatz k h i 3 *N n3 , where the D-state appears for k h i? k h i 3 and the C-state appears for k h i= k h i 3 . From the numerical simulations, n 3 has been obtained. The value of n 3 increases to 1 as r 0 increases, because the Dstate always appears on the CG or dense networks with AEkae / N. The crossovers thus describe how the enhanced cooperation on sparse networks with low AEkae overcomes ''tragedy of the commons'' and makes the C-state for high r 0 .
Finally, the cyclic dominance in Ref. 25 can also be found for very low r and AEkae. For example, for r 0 5 0.1, the crossover from the Cstate to the cyclic dominance occurs at k h i^10 on the network with the size N 5 10 4 . This crossover behavior is not explained quantitatively here, because the crossover occurs only on sparse networks.

Methods
Let us define the SEPGG model on a given graph or network in detail. Each agent is assigned to a node on the network. Variable s i of the agent on node i represents the  (510), r C increases with t, whereas r D and r L decreases. Finally, the stationary C-state appears. The time dependences for high AEkae are not shown, because they are nearly the same as those shown in Fig. 1(b). In each update of SEPGG on the network, an agent i is randomly selected. Then, the payoff P i of i depends on the strategies of k i 1 1 participants, where k i is the degree of i. If n i,C is the number of agents with C, n i,D is the number of agents with D, and n i,L is the number of agents with L among the k i 1 1 participants, n i,C 1 n i,D 1 n i,L 5 k i 1 1. P i is thus given by Here, c is the cost contributed by a C to the common pool, r(.1) is the multiplication factor and s is the fixed payoff of a L 26 . We impose the condition 0 , s , c(r 2 1) as in Ref. 25. Even if only one active participant remains, the payoff of the agent still follows Eq. (10). Then, the strategy of i is updated through the comparison of P i with P j of a randomly selected neighbor j among k i neighbors in order to select a better strategy. If s i ? s j , the agent i stochastically adopts the strategy s j of the neighbor j with transition probability f ij . We use