The level of tolerance of individuals, individual thinking, and the formation of social norms

Abstract

In this work, we study the impact of considering agents with distinct norm heterogeneity tolerance levels, and with distinct initial levels of thinking, in the evolution of social norms and levels of thinking in an artificial society. To this end, we generalize an agent-based model first proposed by Joshua M. Epstein to consider agents with distinct tolerance levels to norm heterogeneity in their neighborhoods. In this model, agents that are located in a cyclic network can choose between two norms, want to conform to their neighborhoods, and decide how much to think—i.e., how many neighbors to consult in deciding which norm to follow—analyzing the norm heterogeneity in their surroundings. Through computer simulations of the model, we obtain the following results: (i) when agents have distinct levels of tolerance, the society converges, for a wide range of initial levels of thinking, to a steady state showing higher global diversity (measured by the number of stable local groups formed at the steady state, where contiguous agents within each group conform to the same norm) associated with higher levels of thinking than in scenarios where all agents have the same tolerance level; (ii) for lower initial levels of thinking, more initial thinking implies in faster convergence to the steady state in the cases of heterogeneous tolerance levels, and when agents present homogeneous maximum intolerance levels; (iii) in all scenarios, more thinking is required in the process of reaching the steady state (the equilibrium of the model), than to maintain this equilibrium afterward; (iv) our model was able to generate distinct average levels of thinking across groups; and (v), higher levels of initial thinking imply in a society with less global diversity at the steady state, with this inverse relationship following a broken power law. Finally, we show that some of our results are similar to results presented in the literature.

Appendices

Appendix

Simulations of the model with noise

We consider the following stopping criterion for the steady state in a world with noise: if the moving average (MA) of the average radius of all agents was kept unchanged in the last 10 iterations, then the steady state was reached. The moving average is calculated considering the last 300 cycles and is the same between two iterations when $| \text {MA}_i - \text {MA}_ {i + 1} | < 0.001$.

Since in this case, it is more difficult to quantitatively measure the number of groups in society, in Fig. 13, we present the average population radius in the steady state as a function of the maximum initial radius for different tolerance levels, considering the following levels of noise: $p=0.01,\;0.05,\;0.10,\;0.25,\;0.50,$ 1.00. For lower level of noise ($p=0.01,\;0.05,\;0.10$), these results are qualitatively similar to the previous case without noise. However, the dependence of the steady state ($r_\mathrm{AVG}^\mathrm{SS}$) on the initial maximum level of thinking (R) becomes less intense as the noise level increases, such that for high levels of noise—$p=0.25,\;0.50,\;1.00$—this dependence completely disappears, remaining only the dependence on the tolerance level. In general, a higher level of noise implies in more thinking at the steady state for all homogeneous tolerance levels. In a very intolerant society ($T=0$), noises greater or equal to 0.1 make individuals to show maximum thinking in the steady state, $R=95$, while a society with heterogeneous tolerances converges to an intermediate value of thinking, $r_\mathrm{AVG}^\mathrm{SS}\approx 10$, which becomes independent of the level of noise. In the presence of noise, the convergence to the steady state is slower than all the cases without noise discussed before.

In “Homogeneous tolerance scenario with noise, variable tolerances, and R = 10”, we present, in Figs. 14, 15, 16 and 17, simulations with the evolution of the society considering illustrative noise levels $p=0.10,\;0.25,\;0.50,\;1.00$. For each of these noisy scenarios, we consider a constant maximum initial radius of $R=10$, and the following tolerance levels: $T=0.0000,0.0125,0.0250,0.0500,$ 0.0750, 0.1000, 0.2500, 0.5000. The main differences of these results to those without noise—Figs. 1, 2—are the fact that in a noisy environment not all the agents in a local group follow the same norm, but only a majority of them, and the steady state is not static, i.e., the composition and size of each local group change with time. Indeed, in these results, we identify not only local conformity, and global diversity as in the model without noise, but also the presence of punctuated equilibria of norms (the fact that a current norm can eventually be replaced by a new one, as a consequence of chance events), what also characterizes a more complex spatio-temporal dynamics [10, 24].

In “Homogeneous tolerance scenario with noise, variable maximum initial radius, and T = 0.05”, we present, in Figs. 18, 19, 20 and 21, the evolution of the society considering the fixed level of tolerance $T=0.05$, and maxima initials radii $R=1,10,28,42,56,70,84,95$, for the same levels of noise as before.

In “Non-homogeneous tolerance scenario with noise”, we present, in Figs. 22, 23, 24 and 25, the evolution of the society considering non-homogeneous tolerance levels, and maxima initials radii $R=1,10,28,42,56,70,84,95$, also for the same noisy scenarios. A distinct behavior of these scenarios is that intense thinking permeates society, for all maximum initial radius considered. Besides, when $p=0.5$ (Fig. 24), we have an interesting result that society alternates periods when one norm dominates with other when the opposite norm becomes a majority, with the formation of local groups playing a minor role.

As we can see in all these figures, a higher level of noise implies, in general, in the formation of smaller local groups. In the limiting case—when all agents decide what norm to follow randomly, $p=1$—the society is completely mixed, showing the highest level of global diversity in the sense considered in this work (Figs. 17, 21, 25).

Finally, note that for a very intolerant society , $T=0$, in all noisy scenarios, the level of thinking always increases to its maximum possible value, $r_\mathrm{AVG}^\mathrm{SS}=95$. On the contrary, for more tolerant societies, $T=0.25,0.5$, the level of thinking always decreases to its minimum possible value, $r_\mathrm{AVG}^\mathrm{SS}=1$.

Fig. 13

Average population radius in the steady state as a function of the maximum initial radius for different levels tolerance and noise

Homogeneous tolerance scenario with noise, variable tolerances, and $R=10$

Fig. 14

Some examples of the evolution of society considering the noise level $p=0.1$, the maximum initial radius $R=10$, and levels of tolerance $T=0.0000,0.0125,0.0250,0.0500,0.0750,0.1000,0.2500,0.5000$

Fig. 15

Some examples of the evolution of society considering the noise level $p=0.25$, the maximum initial radius $R=10$, and levels of tolerance $T=0.0000,0.0125,0.0250,0.0500,0.0750,0.1000,0.2500,0.5000$

Fig. 16

Some examples of the evolution of society considering the noise level $p=0.5$, the maximum initial radius $R=10$, and levels of tolerance $T=0.0000,0.0125,0.0250,0.0500,0.0750,0.1000,0.2500,0.5000$

Fig. 17

Some examples of the evolution of society considering the noise level $p=1$, the maximum initial radius $R=10$, and levels of tolerance $T=0.0000,0.0125,0.0250,0.0500,0.0750,0.1000,0.2500,0.5000$

Homogeneous tolerance scenario with noise, variable maximum initial radius, and $T=0.05$

Fig. 18

Some examples of the evolution of society considering the noise level $p=0.1$, the level of tolerance $T=0.05$, and maximum initial radius $R=1,10,28,42,56,70,84,95$

Fig. 19

Some examples of the evolution of society considering the noise level $p=0.25$, the level of tolerance $T=0.05$, and maximum initial radius $R=1,10,28,42,56,70,84,95$

Fig. 20

Some examples of the evolution of society considering the noise level $p=0.5$, the level of tolerance $T=0.05$, and maximum initial radius $R=1,10,28,42,56,70,84,95$

Fig. 21

Some examples of the evolution of society considering the noise level $p=1$, the level of tolerance $T=0.05$, and maximum initial radius $R=1,10,28,42,56,70,84,95$

Non-homogeneous tolerance scenario with noise

Fig. 22

Some examples of the evolution of society considering non-homogeneous levels of tolerance, the noise level $p=0.1$, and maximum initial radius $R=1,10,28,42,56,70,84,95$

Fig. 23

Some examples of the evolution of society considering non-homogeneous levels of tolerance, the noise level $p=0.25$, and maximum initial radius $R=1,10,28,42,56,70,84,95$

Fig. 24

Some examples of the evolution of society considering non-homogeneous levels of tolerance, the noise level $p=0.5$, and maximum initial radius $R=1,10,28,42,56,70,84,95$

Fig. 25

Some examples of the evolution of society considering non-homogeneous levels of tolerance, the noise level $p=1$, and maximum initial radius $R=1,10,28,42,56,70,84,95$