Cultural and evolutionary dynamics with best-of-k learning when payoffs are uncertain

Social learning not only takes the form of random copying of other individuals, but also involves learners' choice of what to learn or from whom to learn. Best-of-k learning refers to a kind of success-biased social learning strategy in which a learner randomly samples k exemplars from the population and imitates the most "successful" one, or the one gaining the highest payoff. While it is intuitive that best-of-k learning can promote the spread of superior variants and thereby enable cumulative cultural evolution, a previous mathematical analysis has shown that it may sometimes result in maladaptive cultural evolution when the payoffs associated with cultural variants vary stochastically. If so, best-of-k learners may be selectively disfavored and in the long run replaced by unbiased learners, who simply copy someone chosen at random. Here we develop new mathematical models that are more simplified and mathematically tractable than the previous model to achieve a fuller analysis of cultural and evolutionary dynamics involving best-of-k learning and stochastic payoffs. We find that best-of-k learning, unlike unbiased learning, can facilitate the invasion of an on average inferior variant that sometimes gives a very high payoff, destabilize a population fixed with a variant that is on average superior but occasionally results in a very low payoff, and maintain cultural polymorphism at equilibrium. Considering gene-culture coevolution of learning rules and cultural variants, under the assumption that social learning is always faithful, it is shown that a population of best-of-k learners at the culturally polymorphic state can always be invaded by unbiased learners and eventually converges to a culturally monomorphic state. Nonetheless, we show that best-of-k learning can be stable against invasion by unbiased learning if social learning is sometimes combined with individual learning.