On random conformity bias in cultural transmission of polychotomous traits

Mathematical models of conformity and anti-conformity have commonly included a set of simplifying assumptions. For example, (1) there are $m=2$ cultural variants in the population, (2) naive individuals observe the cultural variants of $n=3$ adult “role models,” and (3) individuals’ levels of conformity or anti-conformity do not change over time. Three recent theoretical papers have shown that departures from each of these assumptions can produce new population dynamics. Here, we explore cases in which multiple, or all, of these assumptions are violated simultaneously: namely, in a population with $m$ variants of a trait where conformity (or anti-conformity) occurs with respect to $n$ role models, we study a model in which the conformity rates at each generation are random variables that are independent of the variant frequencies at that generation. For this model a class of symmetric constant equilibria exist, and it is possible that all of these equilibria are simultaneously stochastically locally stable. In such cases, the effect of initial conditions on subsequent evolutionary trajectories becomes very complicated.