Coupling with opinion dynamics promotes prosocial behavior in multilayer networks

The gist: Coupling a social dilemma layer to a non-dilemmatic constant-selection layer robustly enhances cooperation in many cases, across different multilayer networks, updating rules, and payoff schemes.

Model Setup

The model develops an evolutionary game on two-layer networks where one layer supports the prisoner’s dilemma game and the other follows constant-selection dynamics. In layer 1, individuals play a donation game, which is a special case of the prisoner’s dilemma game, with each of their neighbors in layer 1 and that the evolutionary dynamics are governed by constant selection in layer 2. The total payoff for each individual is assumed to be the sum of payoffs across both layers, i.e., ui = u[1]i + u[2]i.

Evolutionary Dynamics and Rules

The evolutionary dynamics are governed by the death-Birth (dB) updating rule in layer 1 and constant selection dynamics in layer 2. The dB-dB rule is used for the combined updating rule, while the dB-Bd rule is also investigated for layer 2. The payoff of an individual in layer 1 is given by u[1]i = -cx[1]i + X N j=1 bpijx[1]j. In layer 2, the fitness of the mutant and resident type is denoted by r and 1, respectively, with the payoff given by u[2]i = x[2]i(r − 1) + 1.

Conditions for Cooperation

The theoretical analysis reveals conditions under which cooperation is favored in layer 1 and mutant is favored in layer 2. Under the dB-dB rule, the condition under which cooperation is favored in layer 1 is given by cθξ[1]2 + b/θξ[1]1 − θξ[1]3− (r − 1)ϕξ[1,2]2,0 > 0. Similarly, the condition under which the mutant is favored in layer 2 is given by -(r − 1)θξ[1]2 + cϕξ[1,2]2,0 + b/ϕξ[1,2]0,1− ϕξ[1,2]2,1 > 0.

Network Structure Effects

The effect of the constant-selection layer to modulate (b/c)∗ changes depending on the case: if ϕξ[1,2]2,0 > 0 and 0 ≤ r < 1, then (b/c)∗ is lower in the two-layer than the one-layer network if 0 ≤ r < 1, with the two-layer network easing cooperation. In case 3, if ϕξ[1,2]2,0 = 0, then (b/c)∗ does not depend on r such that (b/c)∗ remains the same as the case of r = 1.

Empirical Observations and Generalizations

Numerical simulations on larger networks and empirical two-layer networks show that coupling with the constant-selection dynamics can facilitate cooperation on networks. For example, in the coupled ring network, (b/c)∗ decreases from 8/3 in the case of the one-layer ring network to 1.74 when the donation game with b/c = 10 is played on the other ring layer. The results reinforce our main finding that coupling with the constant-selection dynamics can facilitate cooperation on networks.

Future Directions and Limitations

The present work has the following major limitations: first, as in previous studies relying on mathematical frameworks of fixation in multilayer networks, it is computationally demanding to check conditions for favoring cooperators and mutants. Second, as for a plethora of work in mathematically founded studies of evolutionary game and graph theory, we assumed weak selection. An important direction for future research is to test this prediction by controlled experiments and data-driven multilayer network modeling. The present result implies that deploying a new opinion or fad in the unanimity of a resident opinion may enhance cooperation in the opposite network layer.

Summary of Findings

The study provides a principled approach to engineering cooperation by embedding individuals within diverse networked settings, even those unrelated to direct social dilemmas. The coupling with the constant-selection dynamics can enhance prosociality in the donation game layer, much as coupling between two donation-game layers can enhance prosociality relative to the one-layer case. The condition for favoring cooperation (i.e., (b/c)∗) is reasonably responsive to the bias in the strength of the two opinions competing in the other layer, and conversely the donation game parameters (i.e., b and c) similarly spill over to affect the condition for favoring the mutant type. The results are verified across different networks, updating rules, and goods schemes.

Methods

The generation of two-layer BA networks involved growing the network by sequentially adding nodes with mL edges according to the preferential attachment rule and shuffling the node label of layer 2 to avoid the tendency that individuals with small indices (i.e., joining the network early in the growth process) are hubs in both layers such that the node’s degree is positively correlated between the two layers. Empirical two-layer networks used include the Vickers–Chan 7th Graders (VC7) network and the Lazega Law Firm (LLF) network.

Conclusion

The sole overarching message derived from our analysis is that coupling with the constant-selection dynamics, or the biased voter model dynamics, can enhance prosociality in social dilemma games played in the opposite network layer, similar to the coupling of two network layers in both of which social dilemma games are played. The present work has the following major limitations: first, as in previous studies relying on mathematical frameworks of fixation in multilayer networks, it is computationally demanding to check conditions for favoring cooperators and mutants. Second, as for a plethora of work in mathematically founded studies of evolutionary game and graph theory, we assumed weak selection. The sole overarching message derived from our analysis is that coupling with the constant-selection dynamics, or the biased voter model dynamics, can enhance prosociality in social dilemma games played in the opposite network layer, similar to the coupling of two network layers in both of which social dilemma games are played. The present work has the following major limitations: first, as in previous studies relying on mathematical frameworks of fixation in multilayer networks, it is computationally demanding to check conditions for favoring cooperators and mutants. Second, as for a plethora of work in mathematically founded studies of evolutionary game and graph theory, we assumed weak selection. The sole overarching message derived from our analysis is that coupling with the constant-selection dynamics, or the biased voter model dynamics, can enhance prosociality in social dilemma games played in the opposite network layer, similar to the coupling of two network layers in both of which social dilemma games are played. The present work has the following major limitations: first, as in previous studies relying on mathematical frameworks of fixation in multilayer networks, it is computationally demanding to check conditions for favoring cooperators and mutants. Second, as for a plethora of work in mathematically founded studies of evolutionary game and graph theory, we assumed weak selection.

1. Bold header: Incorporate Non-dilemmatic Social Dynamics into Agent Modeling. This allows AI systems to model agents participating in a diversity of interactions, not just Prisoner’s Dilemma games, by coupling a layer supporting the dilemma game with one following constant-selection dynamics, representing biased but non-dilemmatic competition.

2. Bold header: Engineering Cooperation via Network Layer Coupling. The AI can be engineered to enhance cooperation in a social dilemma layer by embedding agents within diverse networked settings, as the paper suggests, because embedding individuals within diverse networked settings—even those unrelated to direct social dilemmas—can be a principled approach to engineering cooperation.

3. Bold header: Dynamic Threshold Modulation for Cooperation. The system can dynamically adjust the threshold for cooperation based on the parameters of non-dilemmatic dynamics; specifically, it can modulate (b/c)∗ by the bias in the strength of the two opinions competing in the other layer and how the donation game parameters (i.e., b and c) similarly spill over to affect the condition for favoring the mutant type.

4. Bold header: Adaptability Across Network Topologies. The AI can exhibit robust performance across different network structures, as results are verified across different multilayer networks, including coupled ring networks, complete graphs, and complete bipartite graphs, demonstrating that coupling with the constant-selection dynamics can facilitate cooperation on networks.

5. Bold header: Rule-Dependent Dynamics for Selection Control. The system's evolutionary trajectory can be controlled by switching between updating rules; for instance, comparing the effects of the dB-dB rule versus the dB-Bd rule reveals that (b/c)∗ under the dB-Bd rule is 10.1 times more sensitive to variation in r than under the dB-dB rule in the two-layer network shown in Fig. 4(b).

Various theoretical and empirical studies have accounted for why humans cooperate in competitive environments. Although prior work has revealed that network structure and multiplex interactions can promote cooperation, most theory assumes that individuals play dilemma games in all social contexts. However, real-world agents may participate in a diversity of interactions, not all of which present dilemmas. We develop an evolutionary game model on multilayer networks in which one layer supports the prisoner's dilemma game, while the other follows constant-selection dynamics, representing biased but non-dilemmatic competition, akin to opinion or fad spreading. Our theoretical analysis reveals that coupling a social dilemma layer to a non-dilemmatic constant-selection layer robustly enhances cooperation in many cases, across different multilayer networks, updating rules, and payoff schemes. These findings suggest that embedding individuals within diverse networked settings---even those unrelated to direct social dilemmas---can be a principled approach to engineering cooperation in socio-ecological and organizational systems.