Diversity-induced resonance in the response to social norms

In this paper we focus on diversity-induced resonance, which was recently found in bistable, excitable, and other physical systems. We study the appearance of this phenomenon in a purely economic model of cooperating and defecting agents. An agent's contribution to a public good is seen as a social norm, so defecting agents face a social pressure, which decreases if free riding becomes widespread. In this model, diversity among agents naturally appears because of the different sensitivities towards the social norm. We study the evolution of cooperation as a response to the social norm (i) for the replicator dynamics and (ii) for the logit dynamics by means of numerical simulations. Diversity-induced resonance is observed as a maximum in the response of agents to changes in the social norm as a function of the degree of heterogeneity in the population. We provide an analytical, mean-field approach for the logit dynamics and find very good agreement with the simulations. From a socioeconomic perspective, our results show that, counterintuitively, diversity in the individual sensitivity to social norms may result in a society that better follows such norms as a whole, even if part of the population is less prone to follow them.

Figure 1

(top) Asymptotic fraction of cooperators $n_c$ dependent on the sensitivity to the social norm, $\theta \equiv \Theta$, which is equal for all agents in the model without diversity. (bottom) Fluctuations $\xi$ around the expected action, which is free riding ($\sigma =0$) for $\Theta <2$ and cooperation for $\Theta >2$. (left) Replicator dynamics. Different curves correspond to different values of $\epsilon$: circles, 0.01; squares, 0.02; diamonds, 0.05; triangles, 0.10. (right) Logit dynamics. Different curves correspond to different values of $\beta$: circles, 0.1; squares, 1.0; diamonds, 2.5; triangles, 10. The other parameters are described in the main text. In the top right panel, the lines correspond to the analytical treatment, developed in Sec. 4.

Figure 2

(top) Asymptotic fraction of cooperators $n_c$ and (bottom) susceptibility ${\xi }^2$ dependent on the sensitivity to the social norm $\theta$, which is different for all agents in the model with diversity. $\Delta \theta$ is the variance of the distribution $g\left(\theta \right)$ with mean value $\Theta =2$. (left) Replicator dynamics. Different curves correspond to different values of $\epsilon$: circles, 0.02; squares, 0.05; diamonds, 0.07; triangles, 0.10. (right) Logit dynamics. Different curves correspond to different values of $\beta$: circles, 2.0; squares, 2.25; diamonds, 2.50; triangles, 2.75. In the top panels, we have selected two different initial conditions [$n_c\left(0\right)=0.1$ and $0.9$] for both kinds of dynamics. The other parameters are described in the main text. In the top right panel, the lines correspond to the analytical treatment, developed in Sec. 4.

Figure 3

Response of the system in the presence of a periodic square-wave forcing with logit dynamics. In all the plots, $\beta =2.5$. The left column shows (top) spectral amplification factor $R$, Eq. (11), (middle) maximum and minimum levels of cooperation attained during the evolution of $n_c$ for the system, and (bottom) the susceptibility. Each symbol corresponds to a different signal amplitude: $\Delta \alpha =0.05,0.1,0.2,0.5$ are shown as circles, squares, diamonds, and triangle, respectively. Analytical results (see main text) are represented with solid lines. In the right column, we depict the time dependency of the macroscopic state $n_c$ (solid black lines) for three different values of the parameter $\Delta \theta$. The values are $\Delta \theta =0.7,1.2,1.7$ in the top, middle, and bottom plots, respectively. The dotted line represents the social pressure ($\Delta \alpha =0.1$), while the thin green dashed line (only in the middle plot) shows the signal applied (not on the same scale, for clarity). Other parameters are $T={10}^3$, $N={10}^4$, $r=5$, $\Theta =2$, $\alpha =1$.

Figure 4

Response of the system in the presence of a periodic forcing with logit dynamics. In all the plots, $\Delta \alpha =0.05$. (top) Spectral amplification factor $R$, Eq. (11). (middle) Maximum and minimum levels of cooperation attained during the evolution of $n_c$ for the system. (bottom) The system susceptibility. Each symbol (and color) corresponds to a different value of the inverse randomness: $\beta =2,2.32,2.5,2.75$ are shown by circles, squares, diamonds, and triangles, respectively. With symbols we represent the results obtained by means of computer simulations, while the analytical results are presented as solid lines. Other parameters are $T={10}^3$, $N={10}^4$, $r=5$, $\Theta =2$, $\alpha =1$.

Figure 5

Effect of a periodic signal applied to the social norm as a function of the population diversity $\Delta \theta$ for the replicator dynamics. In all plots, $\Delta \alpha =0.05$. (top) Spectral amplification factor $R$, Eq. (11). (middle) Maximum and minimum levels of cooperation attained during the evolution of $n_c$ for the system. (bottom) The system's susceptibility. Each curve corresponds to a different value randomness: $\epsilon =0.04,0.045,0.05,0.055$ are shown as circles, squares, diamonds, and triangles, respectively. Other parameters are $T={10}^3$, $N={10}^4$, $r=5$, $\theta =2$, $\alpha =1$.