Polychotomous traits and evolution under conformity

Conformist and anticonformist transmission of dichotomous cultural traits (i.e., traits with two variants) have been studied both experimentally, in many species, and theoretically, with mathematical models. Signatures of types of conformity to polychotomous traits (with more than two variants; e.g., baby names and syllables in bird song) have been inferred from population-level data, but there are few models that include individual-level biases among more than two discrete variants. We generalize the standard dichotomous trait conformity model by Boyd and Richerson to incorporate [Formula: see text] role models and [Formula: see text] variants. Our analysis shows that in the case of [Formula: see text] role models, under anticonformity, the central polymorphic equilibrium [Formula: see text] is globally stable, whereas under conformity, if initially the frequencies of [Formula: see text] variants are all equal to the maximum variant frequency in the population, there is global convergence to an equilibrium in which the frequencies of these variants are all [Formula: see text] and all other variants are absent. With a general number n of role models, the same result holds with conformity, whereas under anticonformity, global convergence is not guaranteed, and there may be stable frequency cycles or chaos. If both conformity and anticonformity occur for different configurations of variants among the n role models, a variety of novel polymorphic equilibria may exist and be stable. Future empirical studies may use this formulation to directly quantify an individual's level of (anti)conformist bias to a polychotomous trait.