Different perceptions of social dilemmas: Evolutionary multigames in structured populations

Motivated by the fact that the same social dilemma can be perceived differently by different players, we here study evolutionary multigames in structured populations. While the core game is the weak prisoner's dilemma, a fraction of the population adopts either a positive or a negative value of the sucker's payoff, thus playing either the traditional prisoner's dilemma or the snowdrift game. We show that the higher the fraction of the population adopting a different payoff matrix the more the evolution of cooperation is promoted. The microscopic mechanism responsible for this outcome is unique to structured populations, and it is due to the payoff heterogeneity, which spontaneously introduces strong cooperative leaders that give rise to an asymmetric strategy imitation flow in favor of cooperation. We demonstrate that the reported evolutionary outcomes are robust against variations of the interaction network, and they also remain valid if players are allowed to vary which game they play over time. These results corroborate existing evidence in favor of heterogeneity-enhanced network reciprocity, and they reveal how different perceptions of social dilemmas may contribute to their resolution.

Figure 1

Evolution of cooperation (top panel) and the average payoff of the population (bottom panel) in the multigame environment on the square lattice. Depicted results were obtained for $\rho =1$ and different values of $\Delta$, as indicated in the legend. Here $\rho =1$ means that all players use either $S=+\Delta$ or $-\Delta$ (none use $S=0$). Larger values of $\Delta$ allow cooperators to survive at larger values of $T$. Importantly, this improvement in $f_C$ is also accompanied by a suitable increase in the average payoff of the population, as shown in the bottom panel.

Figure 2

Evolution of cooperation in the multigame environment on the square lattice, as obtained in dependence on $\rho$ and $\Delta$. The color map encodes the stationary fraction of cooperators $f_C$. It can be observed that the dependence of $f_C$ on both $\rho$ and $\Delta$ is monotonous and that it is thus beneficial for the population to be in the most heterogeneous state possible. Depicted results were obtained for $T=1.1$, but qualitatively equal evolutionary outcomes can be observers also for other values of $T$.

Figure 3

The top panel depicts the level of cooperation in the two subpopulations on the square lattice, where $f_{C_{+}}$ denotes the fraction of cooperators among those who use $S=+\Delta$, while $f_{C_{-}}$ denotes the fraction of cooperators in the group where $S=-\Delta$ is used. For reference, we also plot the cooperation level in the corresponding homogeneous population, where every player uses $S=0$. Expectedly, the level of cooperation is largest in the subpopulation where players use $S=+\Delta$. Much more surprisingly, however, the level of cooperation in the subpopulation where players use $S=-\Delta$ still significantly exceeds the baseline outcome of the homogeneous weak prisoner's dilemma game. The bottom panel depicts the difference between the level of cooperation in the homogeneous and the heterogeneous multigame environment $\Delta f_C$, along with the difference in the strategy invasion flow $\Delta \gamma$ between the two “$+$” and “$-$” subpopulations (see main text for details). These results were obtained with $\rho =1$ and $\Delta =0.2$, but remain qualitatively identical also for other parameter values.

Figure 4

Evolution of cooperation in the multigame environment on the scale-free network with degree-normalized payoffs. Depicted results were obtained when only 2% of the hubs (high-degree players) used $S=+\Delta$, while the rest of the population used a moderately negative $S$ value (see main text for details). As on the square lattice (see top panel of Fig. 1), larger values of $\Delta$ (see legend) allow cooperators to survive at larger values of $T$.

Figure 5

Evolution of cooperation in the time-varying multigame environment on the square lattice. Depicted results were obtained when players could choose between $S=+\Delta$ and $-\Delta$ with equal probability at each instance of the game (see legend for the applied $\Delta$ values). As on the square lattice with time invariable subpopulations (see top panel of Fig. 1), larger values of $\Delta$ allow cooperators to survive at larger values of $T$, although in this case the positive impact on the evolution of cooperation is less strong.