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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.

The theoretical literature from 1985 to the present on the evolution of learning strategies in variable environments is reviewed, with the focus on deterministic dynamical models that are amenable to local stability analysis, and on deterministic models yielding evolutionarily stable strategies. Individual learning, unbiased and biased social learning, mixed learning, and learning schedules are considered. A rapidly changing environment or frequent migration in a spatially heterogeneous environment favors individual learning over unbiased social learning. However, results are not so straightforward in the context of learning schedules or when biases in social learning are introduced. The three major methods of modeling temporal environmental change – coevolutionary, two-timescale, and information decay – are compared and shown to sometimes yield contradictory results. The so-called Rogers’ paradox is inherent in the two-timescale method as originally applied to the evolution of pure strategies, but is often eliminated when the other methods are used. Moreover, Rogers’ paradox is not observed for the mixed learning strategies and learning schedules that we review. We believe that further theoretical work is necessary on learning schedules and biased social learning, based on models that are logically consistent and empirically pertinent.

Human evolution depends on gene-culture co-evolution: genetically determined behavior rules co-evolve with socially learned information. Social transmission of information can occur vertically from parent to child, horizontally from peer to peer, or obliquely from unrelated individuals of different generations. Models for the emergence of cumulative culture have in their majority studied gene-culture co-evolution under random interactions between individuals, which rules out vertical transmission. Yet, vertical transmission is thought to be a primary form of transmission in nature. In order to study gene-culture co-evolution within the family, we derive here the invasion fitness of a mutant allele that influence a deterministic environmental state variable (e.g., amount of knowledge or skill, stock of culture, or any other non-innate environmental feature) to which diploid carriers of the mutant are exposed in subsequent generations. Under the assumption that this environment has a single attractor point, we show that singular strategies and the concomitant environmental state they generate can be evaluated analytically, thereby making gene-culture co-evolution in the family with multigenerational effects mathematically tractable. We then apply our results to model the co-evolution between individual and social learning rules on one hand, and the amount of adaptive information these rules generate on the other. We find that vertical transmission generally favours the accumulation of adaptive information due to kin selection effects, but that when adaptive information is learned as efficiently between family members as between unrelated individuals, this effect is moderate in diploids. We also show that vertical transmission prevents evolutionary branching and therefore can have a qualitative impact on gene-culture co-evolutionary dynamics. More generally, our model is of relevance to study scenarios of niche construction and gene-culture co-evolution within the family, where modifications of the environment by individuals have long-lasting effects on genetic lines of descent.

Humans have created highly developed cultures, brought about by iterative improvements in technology. Using a mathematical model, I investigated the conditions under which cultural traits tend to be improved for a higher level of culture to evolve. In the model, I consider three ways of learning: individual learning, simple social learning, and improvements of socially learned cultural traits (social improvement). I obtain the evolutionarily stable number of cultural traits acquired through each way of learning. I show that organisms improve many socially learned cultural traits under the following conditions: (1) environmental stability is intermediate; (2) the environment is severe; (3) the success rate of individual learning is high; (4) the utility of cultural traits acquired by individual learning is large; (5) the accuracy of social learning is high; and (6) the increase in the utility of beneficial cultural traits attained by social improvement is large. I also show that when organisms have greater ability for social improvement, the average utility of the beneficial cultural trait increases, the proportion of beneficial cultural traits among all cultural traits decreases, and the total number of cultural traits acquired by the three ways of learning is constant. These results shed light on the origins of human cumulative culture.

Humans strongly depend on individual and social learning, both of which are highly effective and accurate. I study the effects of environmental change on the evolution of the effectiveness and accuracy of individual and social learning (individual and social learning levels) and the number of pieces of information learned individually and socially (individual and social learning capacities) by analyzing a mathematical model. I show that individual learning capacity decreases and social learning capacity increases when the environment becomes more stable; both decrease when the environment becomes milder. I also show that individual learning capacity increases when individual learning level increases or social learning level decreases, while social learning capacity increases when individual or social learning level increases. The evolution of high learning levels can be triggered when the environment becomes severe, but a high social learning level can evolve only when a high individual learning level can simultaneously evolve with it.