Cooperation driven by success-driven group formation

In the traditional setup of the public goods game all players are involved in every available group and the mutual benefit is shared among competing cooperator and defector strategies. However, in real life situations the group formation of players could be more sophisticated because not all players are attractive enough for others to participate in a joint venture. What if only those players who are successful enough to the neighbors can initiate a group formation and establish a game? To elaborate this idea we employ a modified protocol and demonstrate that a carefully chosen threshold to establish a joint venture could efficiently improve the cooperation level even if the synergy factor would suggest a full defector state otherwise. The microscopic mechanism that is responsible for this effect is based on the asymmetric consequences of competing strategies: while the success of a cooperator provides a long-time well-being for the neighborhood, the temporary advantage of defection cannot be maintained if the protocol is based on the success of leaders.

Figure 1

Stationary fractions of the four possible states of players in dependence on the threshold value $H$ as obtained for $r=3.5$ [in Fig. 1] and for $r=4.5$ [in Fig. 1]. Cooperators (defectors) with high and low merit are denoted by $C_h$ and $C_l$ ($D_h$ and $D_l$) respectively. The application of low-$H$ values has no particular impact on the strategy evolution: only defectors survive at low-$r$ value, while cooperators and defectors coexist at high-$r$ value. If we increase the threshold value then cooperators become dominant gradually and the system reaches the full-cooperator state. Increasing the $H$ value further the system is trapped into a frozen state where both high-merit strategies extinct at an early stage of evolution. The border of this critical $H$ value is marked by thin dashed line in both panels. In the bottom panel arrow marks the critical $H$ value where success-driven group formation already improves the cooperation level. The error bars are comparable to the width of curves.

Figure 2

Stationary fractions of the four possible states of players in dependence on synergy factor $r$ for low [$H=1.5$ in Fig. 2] and high threshold value [$H=3.8$ in Fig. 2]. For low-$H$ value the system behavior is similar to the classical model where higher $r$ always results in higher fraction of cooperators. Interestingly, increasing $r$ does not necessarily elevate the cooperation level for high $H$.

Figure 3

Time evolution of random initial state as obtained for $r=3.5$ and $H=5$ on a square lattice using $L=100$ system size. Cooperators with high merit are marked by dark blue (dark gray), whereas the same strategy with low merit is denoted by light blue (light gray). Defectors with low merit are marked by red (middle gray) and defectors with high merit are denoted by a green color (dotted lighted gray), as indicated by the legend on the top. Snapshots were taken at 10, 200, 400, and 800 MCSs. Because of the high threshold value almost all players become in a low-merit state, shown in (a), and only a tiny seed of cooperator domains is capable to collect the necessary high payoff to enjoy high-merit state. Due to a neutral drift between low-merit strategies a slow coarsening starts, as illustrated in (b). Importantly, high-merit cooperator domains grow undisturbed way deep in low-merit domains, which offers a protecting shield against defectors. Even if a defector becomes a neighbor of a high-merit cooperator domain, as shown in (c), the resulting high payoff and the related high merit offer just a temporary advantage to $D_h$ player: when a neighboring cooperator follows the victor then the latter's payoff, hence merit, falls down and the mentioned defector will be unable to utilize cooperator neighbors efficiently anymore. In the stationary state, which is shown in (d), the majority of players remain in a cooperator status and just a small portion of players could enjoy temporarily the benefit of defector strategy. Note that the system would be in full-defector state in the absence of success-driven group formation protocol for this low value of synergy factor. Further details are given in the main text.

Figure 4

Full $r\text{−}H$ phase diagram for public goods game when success-driven group formation is applied. Solid red line denotes continuous phase transition, while dashed blue line marks discontinuous phase transition. $D$ ($C$) denotes full defector (cooperator) phase while $D+C$ marks the phase where competing strategies coexist. ${(D+C)}_i$ denotes a phase where the application of success-driven group organization results in a higher cooperation level comparing to the traditional model. Gray dotted line denotes the border of fixed $F$ state where high-ranking players extinct at an early stage of evolution.

Figure 5

Stationary fractions of the four possible states of players in dependence on threshold value $H$ as obtained for $r=3.5$ on random graph where $\langle k\rangle =4$ uniform degree distribution was applied. If threshold $H$ is increased above a critical value, marked by an arrow, then success-driven group formation mechanism is capable to elevate the cooperation level significantly. Above a critical value of $H$, denoted by a dashed line, the demand towards group organizers becomes so high that nobody can fulfill it and the system is trapped into a state of low-merit players.

Figure 6

Success-driven group formation in well-mixed populations. Top panels show how the fractions of players' state change when the system evolves from a random initial state for different threshold values of $H$. If $H$ is low, shown in (a), then all cooperators die out and the system terminates into a full-defector state. If $H$ is high enough, shown in (b), then all high-merit players extinct independently of their strategies and only low-merit players remain who are unable to organize groups and play public goods game. Threshold values are $H=2.95$ and 3.0 respectively, while $r=4.0$ for both cases. (c) depicts the critical value of $H$ in dependence on synergy factor, which is the border line of the above mentioned two possible trajectories.