Social learning may lead to population level conformity without individual level frequency bias

A requirement of culture, whether animal or human, is some degree of conformity of behavior within populations. Researchers of gene-culture coevolution have suggested that population level conformity may result from frequency-biased social learning: individuals sampling multiple role models and preferentially adopting the majority behavior in the sample. When learning from a single role model, frequency-bias is not possible. We show why a population-level trend, either conformist or anticonformist, may nonetheless be almost inevitable in a population of individuals that learn through social enhancement, that is, using observations of others’ behavior to update their own probability of using a behavior in the future. The exact specification of individuals’ updating rule determines the direction of the trend. These results offer a new interpretation of previous findings from simulations of social enhancement in combination with reinforcement learning, and demonstrate how results of dynamical models may strongly depend on seemingly innocuous choices of model specifications, and how important it is to obtain empirical data on which to base such choices.

Proof for incr C (x) and Bs last Define f * by (1) It is straightforward to verify that the definition of f * implies We analyze the effect of a series of observations with Bs last by treating the bs and the Bs separately. For the first (1−f )·τ time steps the observed behavior is always b, yielding x t+1 = (1 − γ C ) · x t + γ C · x 2 t , which quickly tends to zero. For any > 0 and for sufficiently large t we therefore have x t+1 < (1 + − γ C ) · x t , and for sufficiently large τ we consequently have The second stage consists of the remaining f ·τ time steps in which the observed behavior is always B, yielding x t+1 = (1 + γ C ) · x t − γ C · x 2 t < (1 + γ C ) · x t for all time steps in the second stage. Combining the two stages, we have for all sufficiently large τ that The right-hand side expression is increasing in f and decreasing in . By continuity it then follows that for all f < f * we can choose an > 0 such that for all sufficiently large τ . By Eq. 2 the right-hand expression equals x 0 · 1 τ = x 0 = 1/2 < f . Hence, x τ < 1/2 < f for all f < f * and all sufficiently large τ .
Proof for incr C (x) and Bs first To study the case of Bs first, we can mimic the proof for the previous case if we study 1 − x t instead of x t . Corresponding to Eq. 4 we then obtain 1 for all sufficiently large τ . By Eq. 2 the right-hand expression equals which tends to zero when τ tends to infinity, as f > 1/2. Hence, Eq. 5 implies that x τ tends to 1, and hence that x τ > f for all sufficiently large τ .

Proof for incr N (x) and Bs first
After the initial f · τ observations of B we necessarily have x f τ ≤ 1. Each of the (1 − f ) · τ subsequent observations of b will then, according to specification incr N , result in a decrease by a factor of (1 − γ N ). The final value therefore satisfies It follows that x τ tends to zero when τ tends to infinity. Hence x τ < f for all sufficiently large τ .

Proof for incr N (x) and Bs last
To study the case of Bs last, we can mimic the proof for the previous case if we study 1 − x t instead of x t . It follows that 1 − x τ tends to zero when τ tends to infinity. Hence x τ > f for all sufficiently large τ .

Proof for incr A (x)
The above proofs for the case of incr = incr N apply also to the case of incr = incr A after replacing (1 − γ N ) with (1 − γ A ) 2 .

Supplementary material for "Social learning may lead to population level conformity without individual level frequency bias" by Kimmo Eriksson, Daniel Cownden, and Pontus Strimling
Python code for the simulations import numpy as np import pdb import matplotlib.pylab as pl from matplotlib import cm