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 , etc.

To obtain stationary states from general initial configurations with , and , early time behaviors of ρ C , ρ D and ρ L must be considered. Early time behaviors of ρ C , ρ D and ρ L are determined based on competition between two terms of Eqs. (1)– (3), respectively. As ρ C ρ D tanh(−βc/2) ≤ 0 in Eq. (1) and ρ C ρ D tanh(βc/2) ≥ 0 in Eq. (2) for any non-negative ρ C , ρ D , β and c, two distinctive steady states are achievable depending on the value of ρ C . When , in Eq. (1), in Eq. (2) and in Eq. (3). Thus, and , which make in Eq. (1) and after some time. From these relations we find that the state of { , , } appears when . Similarly, when , and , which also make in Eq. (3) and after some time. As a result, when , the state of { , , } appears. We call this state the D-state. In contrast, when , and , which make in Eq. (2) and after some time. Thus, the state of { , , } appears. We call this state the L-state. As the D-state or the L-state appears depending on the condition , 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 { , , }, the rate equation (1) becomes

because . By solving Eq. (4) for time t, we obtain

Similarly, the rate equation (3) also becomes

When , and

As ρ C decreases with t, the condition for the D-state breaks down for t > t*. From the Eq. (5) and the condition with , . Therefore, on the CG with N → ∞, the L-state is the only stationary state. However, on the CG with finite N, the nonzero-minimum of ρ L is 1/N and thus ρ L = 0 if ρ L (t) < 1/N. Therefore, if , then ρ L (t > t*) = 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 , the D-state becomes stable, where

or

More specifically, this D-state for high r or 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 , the L-state becomes stable. This L-state for 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 . No C-dominant stationary state is found on the CG even for high r. Compared to the C-dominant stationary states on a square lattice17,26 and on sparse networks28,30,31,32,33 for high r, the stationary states on the CG are unique and anomalous.

In Fig. 1, ρ C (t), ρ D (t) and ρ L (t) from a single run of simulation on the CG with N = 105 are plotted. ρ C (t) and ρ L (t) decay exponentially in the early time regardless of the r value. For , the time dependences of ρ C (t) and ρ L (t) are sustained throughout and the stationary D-state eventually appears as shown in Fig. 1(a) (r = 2000). In contrast, when , ρ L (t) increases after some time or for t > t* and the L-state eventually appears as shown in Fig. 1(b) (r = 60). Hence, simulation data presented in Fig. 1 exactly reproduce the analytical results of rate equations (1)– (3). More specifically, early time behaviors of ρ L ~ exp(−t) and are confirmed by fittings to simulation data as shown in Figs. 1(a) and 1(b). Furthermore, the crossover time t* for r = 60 is t* = 8.86 in Fig. 1(b), which is nearly identical to t* obtained from .

Figure 1 Simulation results of the SEPGG on the CG. Plots of ρ C (t), ρ D (t) and ρ L (t) of the SEPGG with c = 1, σ = 1 and β = 1 from a single simulation run with N = 105. The dotted horizontal line denotes the value of 1/N. (a) When r = 2000, the stationary D-state appears. By fitting the data to Eqs. (5) and (7), ρ C ~ exp(~α C t) with (solid line) and ρ L ~ exp(−α L t) with α L = 1.00(2)(~1.0) (dash-dotted line) are obtained. (b) When r = 60, the stationary L-state eventually appears. The vertical dashed line denotes the value of . By the fitting, ρ C ~ exp(−α C t) with (solid line) and ρ L ~ exp(−α L t) with α L = 0.97(4)(~1.0) (dash-dotted line) are obtained for t < t*. Full size image

When and in the limit of N → ∞, the time dependences of ρ C , ρ D and ρ 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 ρ C (t) decreases. Thus, D cannot receive enough payoff50, causing ρ D (t) to decrease and ρ L (t) to increase. Finally, most agents become L, as no one remains in the commons. Consequently, the stationary L-state eventually appears for .

To analyze the dependence of stationary states on the multiplication factor r, , and are obtained from simulations for various N and r by averaging over 1,000 realizations. Simulation results of and 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, and in Fig. 2 exactly depend on the single scaling parameter r 0 defined as . The scaling behaviors confirm that the L-state crosses over to the D-state at as Eq. (9).

Figure 2 Simulation results of the SEPGG on the CG for various r and N. Plots of (a) and (b) against for N = 103, 104, 105 and 106. c = 1, σ = 1 and β = 1 are used. Inset of (a): Plots of against r. Inset of (b): Plots of against r. Full size image

Crossover from the behavior on dense networks to that on sparse networks

A dense network is a network in which the mean-degree 〈k〉 satisfies 〈k〉 ∝ N51. For example, the CG is a typical dense network, as 〈k〉 = N − 1 in the CG. In a sparse network, 〈k〉 = finite51. 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 networks30,33 and two dimensional square lattices17,26. 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 〈k〉. For a given N and 〈k〉, , and are obtained by averaging over 2,000 realizations. Typical crossover behaviors for r 0 = 0.3 are shown in Fig. 3. As shown in Fig. 3(a), two crossovers occur successively as 〈k〉 decreases. The L-state is stable when 〈k〉 is quite high. The C-state of { , , } is stable when 〈k〉 is low enough. For moderate 〈k〉 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 〈k〉 decreases.

Figure 3 Simulation results of the SEPGG on random networks for . (a) Plots of , and against 〈k〉 for N = 16000. c = 1, σ = 1 and β = 1 are used. (b) Plots of against 〈k〉 for N = 4000, 8000, 16000 and 32000. Here, and are not shown, because for high 〈k〉 and for low 〈k〉. (c) Plots of 〈k〉 1 and 〈k〉 2 against N. The straight lines denotes fittings of with and with to corresponding data. (d) Plot of Δ 〈k〉 (≡〈k〉 1 − 〈k〉 2 ) against N. (e) Plot of against with ν 1 in (c). (f) Plot of against with ν 2 in (c). Full size image

The stability of the D-state for moderate 〈k〉 in the limit N → ∞ is studied using the following methods. From simulation data of , and as in Figs. 3(a) and 3(b), we first obtain 〈k〉 1 at which relations and hold simultaneously. We also obtain 〈k〉 2 at which and hold. For example, dependences of 〈k〉 1 and 〈k〉 2 on N for r 0 = 0.3 are shown in Fig. 3(c). The dependence of Δ 〈k〉 (≡〈k〉 1 − 〈k〉 2 ) is also shown in Fig. 3(d). As shown in Fig. 3(d), Δ 〈k〉 increases monotonically with N, guaranteeing the stability of the D-state for moderate 〈k〉 in the limit N → ∞. Furthermore, as shown in Fig. 3(c), 〈k〉 1 and 〈k〉 2 satisfy power laws and . By fitting these power laws to data presented in Fig. 3(c), crossover exponents are obtained as ν 1 = 0.898(2), ν 2 = 0.520(2). The result ν 1 > ν 2 also guarantees the stability of the D-state for moderate 〈k〉. The crossover property from the L-state to the D-state presented in Fig. 3(b) is adequately described by the single exponent ν 1 obtained in Fig. 3(c). for higher 〈k〉 and various N are plotted against the scaling variable with the obtained ν 1 as in Fig. 3(e), which shows that for higher 〈k〉 is a function of the single scaling variable . As shown in Fig. 3(f), crossover from the D-state to the C-state also satisfies the scaling property that for lower 〈k〉 is a function of the single scaling variable with the obtained exponent ν 2 . Using the same method ν 1 's and ν 2 's for various low r 0 (<1) are obtained as shown in Fig. 4. Because ν 1 > ν 2 in Fig. 4, the D-state for moderate 〈k〉 and low r 0 (<1) is stable in the limit N → ∞.

Figure 4 Plots of exponents ν 1 and ν 2 against r 0 . c = 1, σ = 1 and β = 1 are used. In the limit N → ∞, the D-state for moderate 〈k〉 is stable, because ν 1 > ν 2 . Full size image

Furthermore, the dependences of , and on 〈k〉 for low r 0 in Fig. 3(a) are quite similar to the time dependences of ρ C (t), ρ D (t) and ρ 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 〈k〉 there are also enough Cs. For moderate 〈k〉 D governs the system. When 〈k〉 becomes high enough, L dominates. Hence, it is very interesting to compare dynamical behaviors on the CG to static crossover behaviors depending on 〈k〉.

We thus now focus on the time dependence of ρ C (t), ρ D (t) and ρ L (t) for various 〈k〉 to understand crossover behaviors for low r 0 in Fig. 3(a). The time dependences of ρ C , ρ D and ρ L for moderate 〈k〉 are shown in Fig. 5(a) and those for low 〈k〉 are shown in Fig. 5(b). For high 〈k〉, the time dependence is nearly identical to that on the CG shown in Fig. 1(b). For moderate 〈k〉 and high 〈k〉, ρ C and ρ L decrease, but ρ D increases in early time. However, the stationary state is strongly affected by the subsequent time dependence of ρ C . If 〈k〉 is quite high or if , ρ C decays quickly and ρ D cannot receive enough payoff. As a result, ρ L increases for t > t* and the stationary L-state appears as explained in Fig. 1(b). In contrast, for moderate 〈k〉 or , ρ C (t) decreases relatively slowly and ρ L (t) never have a chance to increase reversely before the time at which ρ L (t) ≤ 1/N [see Fig. 5(a)]. This means that the cooperation is effectively enhanced for moderate 〈k〉 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 〈k〉 or , ρ C (t) never decreases as on sparse networks28,30,31,32,33 [see Figs. 5(b)] and . Hence, the crossover from the D-state to the C-state (or C-dominant state) occurs for 〈k〉 ~ 〈k〉 2 as 〈k〉 decreases.

Figure 5 Time dependence of ρ C (t), ρ D (t) and ρ L (t) on random networks with N = 16000 for r 0 = 0.3. Plots of ρ C (t), ρ D (t) and ρ L (t) (a) for moderate 〈k〉 ( = 30) and (b) for low 〈k〉 ( = 10). (a) For moderate 〈k〉 ( = 30), ρ D increases with t, whereas ρ C and ρ L decreases. Finally, the stationary D-state emerges. (b) For low 〈k〉 ( = 10), ρ C increases with t, whereas ρ D and ρ L decreases. Finally, the stationary C-state appears. The time dependences for high 〈k〉 are not shown, because they are nearly the same as those shown in Fig. 1(b). Full size image

The two crossovers for low r 0 thus derive from a gradual increase of cooperation as the number of participants (or 〈k〉) 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 group45,46. However, this dependence on the group size is not necessarily accurate, because a recent study on PGG44 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 ( = 10), the D-state is stable when 〈k〉 is quite high. The C-state is stable when 〈k〉 is low enough. Therefore, for high r 0 , the direct crossover from the D-state to the C-state occurs as 〈k〉 decreases. To analyze the dependence of this direct crossover on N, 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 , where at 〈k〉 3 both and hold. From the dependence of 〈k〉 3 on N, is obtained for r 0 = 10. This direct crossover satisfies the scaling property that is a function of the single scaling variable with ν 3 = 0.51. As shown in Fig. 6(d), ν 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 ν 3 increases as r 0 increases. As the D-state is always stable on the CG or dense networks with 〈k〉 ∝ N, the upper bound of ν 3 should be equal to 1. We also confirm that the time dependences of ρ C (t), ρ D (t) and ρ L (t) for high r 0 are nearly the same as those in Fig. 1(a) for high 〈k〉 and as those in Fig. 5(b) for low 〈k〉, respectively. Hence, this direct crossover is nearly identical to the second crossover from the D-state to the C-state for low r 0 .