Non-dilemmatic social dynamics promote cooperation in multilayer networks

TL;DR

The 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.

Abstract

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 similar 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.

Figures (14)

• Figure 1: Schematic of the two-layer game. The donation game is played in layer 1. The constant-selection dynamics occurs in layer 2. The total payoff for each individual, which drives evolutionary dynamics in both layers, is the sum of the payoffs that the individual obtains from both layers. The figure shows the total payoffs for individuals 5 and 6 as examples. In both layers, replica nodes (shown by circles) that have a high total payoff tend to spread its type to its neighbors.
• Figure 2: $(b/c)^*$ for various values of $r$. (a) $\theta_{1}-\theta_{3}>0$. (b) $\theta_{1}-\theta_{3}<0$. In (a), cooperation is favored when $b/c$ is larger than the value specified by the line, whose slope depends on $\phi_{2,0}$. In (b), spite is favored when $b/c$ is smaller than the value specified by the line depending on the $\phi_{2,0}$ value.
• Figure 3: Distribution of $dr^*/db$ and $dr^*/dc$ over the 2,763,739 unique pairs of the two-layer network with six individuals and initial condition. This is a non-smoothed two-dimensional histogram.
• Figure 4: Selection of cooperators and mutants in four two-layer networks. Each row corresponds to the two-layer network and its initial condition visualized in the left panel. The colors of the replica nodes are the same as those used in Fig. \ref{['fig:general_rule_multilayer']}; orange, black, blue, and red represent cooperator, defector, resident, and mutant, respectively. The second column of each row of the figure shows the results under the dB-dB updating rule. The third column shows the results under the dB-Bd updating rule. Each colored region shows the $(b/c, r)$ region in which selection favors or disfavors cooperators in layer 1 or mutants in layer 2. (a) Coupled ring networks with $N=10$ individuals. We obtain $(b/c)^* = 8/3$ for the uncoupled one-layer ring, shown by the dotted lines. (b) Coupled heterogenous networks with $N=6$. We obtain $(b/c)^* = -15.89$ for the uncoupled layer-1 network. (c) Coupled complete graphs with $N=10$. The one-layer counterpart yields $(b/c)^*= -9$. (d) Coupled complete bipartite graphs with $N=10$. The one-layer counterpart yields $(b/c)^* = \infty$.
• Figure 5: Evolution of cooperation in coupled ER and BA networks. (a) ER, dB-dB rule. (b) BA, dB-dB rule. (c) ER, dB-Bd rule. (d) BA, dB-Bd rule. The upper part of each panel shows the fraction of pairs of one-layer network and its initial condition that yield cooperation when $b/c > (b/c)^*$ with a threshold value $(b/c)^* > 0$. The fraction values are the same between (a) and (c) and between (b) and (d) because this fraction only depends on the layer-1 network. The lower part of each panel shows the median along with the 5th and 95th percentiles (in square brackets) of $\left| \text{d}(b/c)^*/\text{d}r \right|$ for pairs of two-layer network and initial condition. Each two-layer network is composed of $N=15$ individuals.

Theorems & Definitions (1)

• Lemma 1