Intergroup information exchange drives cooperation in the public goods game

In this paper we explore the onset of cooperative traits in the public goods game. This well-known game involves $N$-agent interactions and thus reproduces a large number of social scenarios in which cooperation appears to be essential. Many studies have recently addressed how the structure of the interaction patterns influences the emergence of cooperation. Here we study how information about the payoffs collected by each individual in the different groups it participates in influences the decisions made by its group partners. Our results point out that cross-information plays a fundamental and positive role in the evolution of cooperation for different versions of the public goods game and different interaction structures.

Figure 1

Average fraction of cooperators $\langle c\rangle$, as a function of the normalized enhancement factor $r/\langle G\rangle$, for a SF distribution of the membership frequency $k_i$ (i.e., number of groups to which an agent belongs) and a fixed group size $G_g=10$. Different panels correspond to different values of the parameter $\alpha$ that models the intergroups' information exchange: $\alpha =1$ corresponds to the case in which agents have full information about their partners' payoffs, while for $\alpha =0$ agents only have knowledge of payments from their games. Solid red lines correspond to the FCG formulation of the PGG, while dashed blue lines correspond to the FCI version. Results are averaged over ${10}^2$ realizations (and different membership matrices) for each value of $r/\langle G\rangle$. Other parameter values are $N=N_G={10}^3,\beta =0.1,\zeta =1$.

Figure 2

Average fraction of cooperators $\langle c\rangle$, as a function of the normalized enhancement factor $r/\langle G\rangle$, when the sizes of the groups $G_g$ are distributed according to a power law and all the agents belong to the same number of groups $k_i=10$. Different panels correspond to different values of the inter-groups' information exchange ratio, from $\alpha =1$ (maximum) to $\alpha =0$ (minimum). Solid lines red are for the FCG formulation, while dashed blue lines are for the FCI version. Results are averaged over ${10}^2$ realizations performed in different networks. Other parameters are equal to those used in Fig. 1.

Figure 3

Critical value $r_c/\langle G\rangle$ of the normalized enhancement factor as a function of the inter-groups information exchange ratio $\alpha$ for the FCG and FCI formulations of the PGG. Left panel corresponds to a SF distribution of the number of groups to which an agent belongs, $k_i$, and a fixed group size $G_g=10$. Right panels corresponds to the case in which the group size distribution $\left(P\left(G_g\right)\right)$ follows a power law while all the agents belong to the same number of groups $k_i=10$. Other parameters are equal to those used in Fig. 1.

Figure 4

Average fraction of cooperators $\langle c\rangle$ versus the normalized enhancement factor $r/\langle G\rangle$ for different values of the parameter $\beta$ corresponding to the inverse temperature of the system (i.e., the lower $\beta$, the more random is imitation), when the group size distribution $P\left(G_g\right)$ follows a power law and all the agents belong to the same number of groups $k_i=10$. Left panel corresponds to the FCG formulation of the PGG, while right panel corresponds to the FCI one. Results are averaged over ${10}^2$ realizations performed in different networks. Other parameters are equal to those used in Fig. 1.