Conformism

Conformists choose the most common option in their sample. An interesting consequence of this behavior is that it leads to uniformity of choice. If slightly more than half of a conformist population chooses A, eventually, all individuals will choose A (see top row of figure 1.14). Higher sample sizes make it easier to detect which option is actually the more frequent one and are thus associated with higher conformity, in the sense of a more aligned behavior of the conformists.

If transmission were unbiased (like for random copying), the frequency of A choices would not change (except by drift), and if transmission were anti-conformist, it would tend to return to 50% a choice. A conformist population would, however, tend to exaggerate even small deviations. This exaggeration is even stronger for higher sample sizes. An illustration of this is found in the bottom left panel of figure 1.14, were the behavior of conformists of different sample sizes is shown. We simulated their behavior in the same environment when being paired with an equal number of individual learners, who are not shown. This example shows that instead of matching the environment, as individual learners would, conformists overmatch. This is validated by a linear regression (bottom right panel of figure 1.14), which results in a slope of 1.903 (1.853-1.953, 95% confidence bounds, R² = 0.525).

When a conformist has a higher sample size, she is more likely to actually choose the option that is most common in the population. As long as con-formists are sufficiently rare, this should lead to a better performance and hence fitness. This can be seen in the left panel of figure 1.15. Conformists with higher sample size are, however, also more likely to collectively choose the incorrect option. Maladaptive behavior is possible earlier for higher sample sizes, and the incorrect option is more likely to be chosen once that becomes a possibility. Higher sample sizes thus have advantages and disad-vantages.

Next we performed numerical evolutionary simulations starting with a population consisting predominantly of individual learners and a minority of conformists. We simulated conformists with sample size 3, 5, and 7.

Figure 1.14:Top row: recurrence relation for conformists. Top left: compari-son between different sample sizes (as indicated by the numbers). There are three equilibria, nobody chooses A, everybody chooses A (“conformist” behaviors, both stable), and 50% choose A (unstable). Higher sample sizes result in stronger con-formity in the sense of faster convergence towards total concon-formity. Top right: The recurrence relation can be used to trace the convergence to conformity, as indicated by the arrows. Bottom left: Behavior of conformists over time. Conformists with sample size of either 3 (×), 5 (•), or 7 () were paired with an equal number of individual learners (not shown). The proportion of A choices by conformists is shown on the left y-axis, the environment in the form ofpA−pB (solid line) on the right y-axis. Bottom right: Conformists (here with sample size 3) overmatch, since small changes in the differences betweenpAandpB may lead to large swings in the proportion of A choices.

Figure 1.15: Fitness (left) and evolution (right) of conformists, depending on their sample size. Left: analytically derived fitness (stable: solid line, unstable: dashed line). Performance of individual learners (dotted line) was assumed to be 59%, as measured previously, and is independent of the frequency of conformists. Con-formists with higher sample size are initially more likely to choose correctly than those with small sample size, and thus have a higher fitness at low frequencies.

They are, however, also more likely to collectively choose the wrong option, as is evidenced by the earlier appearance of the bifurcation that marks the emergence of the maladaptive equilibrium. Right: Comparison of evolutionary simulations of the frequency of conformists (solid line) with different sample sizes as indicated when competing with individual learners (not shown). Conformists with lower sample size have higher equilibrium frequencies (as measured by the mean frequency during the last 500 generations, dash-dotted line) but take more time to reach it.

Our analytical findings suggest that 1) conformists with higher sample size invade faster because they initially have a higher fitness and 2) conformists with lower sample size have a higher equilibrium frequency because it takes more of them before maladaptive behavior occurs. In the right panel of figure 1.15, we show the simulation results. Although the invasion speed of conformists with sample size 5 and 7 is indistinguishable, both invade faster than conformists with sample size 3. Furthermore, the latter have the highest equilibrium frequency, calculated as the arithmetic mean of the last 500 generations, 79.6% to be precise. Sample size of 5 leads to an equilibrium frequency of 72.9% and 7 of 67.4%. All differences are significant (all p <

10⁻¹¹, Wilcoxon rank sum test). The numerical simulations therefore reflect our predictions from the analytical findings.

Instead of directly simulating the evolutionary path of conformists, we can use a different method to gauge their evolution. For this, we fix the frequency of the involved strategies and simulate their performance a large number of times. This will let us estimate precisely the performance of the strategies at different frequencies, and since we know that a strategy with a higher performance than its competitor also has higher fitness, we can infer the evolutionary path of the strategy from this. When performance is equal, we should expect that the corresponding frequency is an equilibrium. From

58.96±0.16% for individual learners, ±S.E.M.); it thus correctly predicts the equilibrium frequency. Additionally, as conformists perform better when less frequent and worse when more frequent, the performance measure allows us to correctly infer that the equilibrium is stable. All in all, the results from the performance approach corroborate our earlier findings and thus validate the approach.

For sample sizes of 5 and 7, we simulated the performance of conformists at the equilibrium frequency suggested by the evolutionary simulations. We found performance to be 58.11±0.55% and 59.20±0.55%, respectively, very close to the performance of individual learners (58.96±0.16%). Therefore, the performance approach would yield the same equilibrium frequency as the evolutionary simulations for the given sample size, confirming this approach to be a good substitute for evolutionary simulations.