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. 2020 Mar 16;10(1):4830.
doi: 10.1038/s41598-020-61634-7.

Self-regulation Versus Social Influence for Promoting Cooperation on Networks

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Free PMC article

Self-regulation Versus Social Influence for Promoting Cooperation on Networks

Dario Madeo et al. Sci Rep. .
Free PMC article

Abstract

Cooperation is a relevant and controversial phenomenon in human societies. Indeed, although it is widely recognized essential for tackling social dilemmas, finding suitable policies for promoting cooperation can be arduous and expensive. More often, it is driven by pre-established schemas based on norms and punishments. To overcome this paradigm, we highlight the interplay between the influence of social interactions on networks and spontaneous self-regulating mechanisms on individuals behavior. We show that the presence of these mechanisms in a prisoner's dilemma game, may oppose the willingness of individuals to defect, thus allowing them to behave cooperatively, while interacting with others and taking conflicting decisions over time. These results are obtained by extending the Evolutionary Game Equations over Networks to account for self-regulating mechanisms. Specifically, we prove that players may partially or fully cooperate whether self-regulating mechanisms are sufficiently stronger than social pressure. The proposed model can explain unconditional cooperation (strong self-regulation) and unconditional defection (weak self-regulation). For intermediate self-regulation values, more complex behaviors are observed, such as mutual defection, recruiting (cooperate if others cooperate), exploitation of cooperators (defect if others cooperate) and altruism (cooperate if others defect). These phenomena result from dynamical transitions among different game structures, according to changes of system parameters and cooperation of neighboring players. Interestingly, we show that the topology of the network of connections among players is crucial when self-regulation, and the associated costs, are reasonably low. In particular, a population organized on a random network with a Scale-Free distribution of connections is more cooperative than on a network with an Erdös-Rényi distribution, and, in turn, with a regular one. These results highlight that social diversity, encoded within heterogeneous networks, is more effective for promoting cooperation.

Conflict of interest statement

The authors declare no competing interests.

Figures

Figure 1
Figure 1
SR-EGN equation. Strategy dynamics of player v (green node) is ruled by the SR-EGN equation (green box). It includes two terms: the social influence term ϕv/xv (blue box) and the self-regulation term βvfv (orange box). The arrows represent the interactions of player v with neighbors (blue) and with himself (orange).
Figure 2
Figure 2
Flow and dynamics. The value of the derivative x˙v is plotted as a function of xv and βv, with kv=10, together with attractive (black) and repulsive (white) steady states. For a T-driven game (T=3, S=1 and ρ=2), the time derivatives of xv for a generic player connected only to full defectors (x¯v=0) and only to full cooperators (x¯v=1) are shown in (A.1,B.1), respectively. Similarly, (C.1,D.1) show the time derivatives of xv for a S-driven game (T=2, S=2 and ρ=2), assuming a neighborhood of full defectors and full cooperators, respectively. Vertical dashed lines are drawn for βv=kv/ρ and βv=kvρ, thus separating the regions D, U and C. Some examples of the time courses of xv(t) (red) and of x¯v(t) (blue) for a player vU are depicted in A.2, B.2, C.2 and D.2 for βv<kv, and in A.3, B.3, C.3 and D.3 for βv>kv.
Figure 3
Figure 3
Average distribution of strategies and convergence speed. Four different setups are considered: Erdös-Rényi for T-driven (A) and S-driven (B) games, and Scale-Free for T-driven (C) and S-driven (D) cases. 500 graphs with N=1000 nodes and average degree k¯=10 have been generated for each topology. For different values of the parameter βv=β{0,,20} and using random initial conditions in the set (0,1), the SR-EGN equation is simulated until a steady state is reached. The values of T and S are the same as in Fig. 2. The average distribution of strategies of the whole population is shown for defectors (red), partial cooperators (orange), and cooperators (yellow). The blue lines represent the convergence speed, experimentally estimated as the inverse of the time required to reach the steady state.
Figure 4
Figure 4
SR-EGN equation: schematic representation of game transitions.
Figure 5
Figure 5
Average cooperation vs. average degree. The average cooperation level x^ of the whole population at steady state is reported for T-driven (A) and S-driven (B) games as a function of the average degree k¯{2,4,,12}. The population is composed by N=1000 players and it is organized over regular (magenta), Erdös-Rényi (green) and Scale-Free (blue) random networks. Similarly, in (C,D) the average cooperation x^U of the subpopulation of players vU, is depicted. The values have been averaged over 500 simulations for each network topology and for each game. In all cases, βv=β=5. The values of T and S are the same as in Fig. 2.
Figure 6
Figure 6
Selfishness and altruism within heterogeneous populations. Using the same experimental setup developed for Fig. 3, for each player, we report a dot representing the value cv (x-axis) and his degree kv (y-axis). The color of each dot indicates the degree kv of player v, thus allowing to distinguish among classes C (green dots), U (blue dots) and D (magenta dots). The self-regulation parameter β is set to 10 for all players. The black lines represent the distribution (%) of the indicator cv over the whole population.

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