The Evolution of Facultative Conformity Based on Similarity

Conformist social learning can have a pronounced impact on the cultural evolution of human societies, and it can shape both the genetic and cultural evolution of human social behavior more broadly. Conformist social learning is beneficial when the social learner and the demonstrators from whom she learns are similar in the sense that the same behavior is optimal for both. Otherwise, the social learner’s optimum is likely to be rare among demonstrators, and conformity is costly. The trade-off between these two situations has figured prominently in the longstanding debate about the evolution of conformity, but the importance of the trade-off can depend critically on the flexibility of one’s social learning strategy. We developed a gene-culture coevolutionary model that allows cognition to encode and process information about the similarity between naive learners and experienced demonstrators. Facultative social learning strategies that condition on perceived similarity evolve under certain circumstances. When this happens, facultative adjustments are often asymmetric. Asymmetric adjustments mean that the tendency to follow the majority when learners perceive demonstrators as similar is stronger than the tendency to follow the minority when learners perceive demonstrators as different. In an associated incentivized experiment, we found that social learners adjusted how they used social information based on perceived similarity, but adjustments were symmetric. The symmetry of adjustments completely eliminated the commonly assumed trade-off between cases in which learners and demonstrators share an optimum versus cases in which they do not. In a second experiment that maximized the potential for social learners to follow their preferred strategies, a few social learners exhibited an inclination to follow the majority. Most, however, did not respond systematically to social information. Additionally, in the complete absence of information about their similarity to demonstrators, social learners were unwilling to make assumptions about whether they shared an optimum with demonstrators. Instead, social learners simply ignored social information even though this was the only information available. Our results suggest that social cognition equips people to use conformity in a discriminating fashion that moderates the evolutionary trade-offs that would occur if conformist social learning was rigidly applied.

The evolution of facultative conformity based on similarity Choices are also in {0, 1}. In any generation, if an individual chooses 0 in state 0 or chooses 1 in state 1, she receives a relatively high payoff, v H . If she chooses 0 in state 1 or chooses 1 in state 0, she receives a relatively low payoff, v L , where 0 < v L < v H . Let C t+1 be a random variable designating the choice of a randomly selected learner in t + 1.
Before making a choice, a learner receives three pieces of information. First, the learner receives a private signal about her environmental state, which is a random variable,S. The support is some subset of R, and realizations are denoteds. We specify howS is distributed below. For the moment, however, we simply note that, because private signals are noisy but informative,S is distributed in a way that depends on the learner's environmental state, and we denote conditional distributions generically as P (S =s | Z = z). These distributions depend on the state in t + 1 but not the state in t, i.e. P (S =s | Y = y, Z = z) = P (S =s | Z = z).
Second, the learner in t + 1 randomly samples N demonstrators from t and observes how many exhibit behavior 1. This is a random variable, I, with support {0, . . . , N } and realizations i. Let p t designate the frequency of choice 1 among demonstrators. By extension, I ∼ binomial(N, p t ). Note that, distributions for I depend on the state in t but not the state in t + 1, i.e. P (I = i | Y = y, Z = z) = P (I = i | Y = y).
Finally, the learner receives a private signal indicating if she is learning in the same environmental state as that of the demonstrators. This signal is a random variable, A, with support {0, 1} and realizations a. A = 1 indicates the same state. Distributions for A depend on both the state faced by demonstrators and the learner's state. We assume that signals are accurate with probability φ, which means Now we turn to the cognitive representation of the decision-making task.
The cognition of learners may or may not accurately represent the actual structure of the task. The point is simply that cognition, whether accurate or not, encodes a representation of the task, and this representation allows learners to choose. We use a hat to indicate quantities that are cognitive representations. This approach will lead to many hats, for which we apologize, but we hope this notation helps to distinguish clearly between the actual structure of the decision-making task and the cognitive representation of this structure.

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2/14 S1 Appendix The evolution of facultative conformity based on similaritŷ q 0 represents the probability that demonstrators choose 1 in state 0, andq 1 represents the probability that demonstrators choose 1 in state 1. Additionally, learner cognition encodes a prior probability distribution over the four possible (y, z) combinations, whereŵ yz represents P (y, z). Cognition also represents the signal quality pertaining to A asφ. Given these encoded values, after observing a, i, ands, the subjective posterior follows, LetÊ[V (c t+1 )] be the expected value of choosing c t+1 from the learner's perspective.
With an exogenous payoff normalized to 1, (2)  Rather, given a cognitive representation of the decision-making task, the learner simply observes a, i, ands, checks condition (3), and chooses accordingly.
To clarify that cognition encodes information about the relationship between demonstrators and learners, note thatr =ŵ 10 +ŵ 11 encodes the prior probability that PLOS 3/14 S1 Appendix The evolution of facultative conformity based on similarity Y = 1, andû =ŵ 01 +ŵ 11 encodes the prior probability that Z = 1. Finally, D =ŵ 00ŵ11 −ŵ 01ŵ10 is the encoded value for the covariance between Y and Z. In this sense,D captures the encoded prior about the relationship between demonstrators and learners. This representation is modified after observing a, wherê Condition (3) comes in two versions, one if A = 0 and the other if A = 1.
To proceed, we return to the actual structure of the decision-making task.
We introduce simplifying assumptions similar to Perreault and colleagues [1]. Private signals are normally distributed conditional on state. If Z = 0, the distribution has mean µ 0 < 0 and variance σ 2 . If Z = 1, the distribution has mean µ 1 = −µ 0 and variance σ 2 . To normalize private signals, lets = bs, where b = µ 1 , and thus s =s/b is unit-free. By normalizing in this way, we only need a single quantity to fully specify the stochastic properties of the private signals upon which individual learning depends.
Define this single quantity as α = σ 2 /(2µ 2 1 ). To avoid having the clutter of always writing out conditional distributions for normally distributed signals, denote the density for s, given z, as f z (s; α) and the associated cumulative probability as F z (s; α).

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4/14 S1 Appendix The evolution of facultative conformity based on similarity Turning once again to the cognitive representation of the decision-making task, assume that cognition reflects simplifications analogous to those outlined immediately above. Specifically, letα =σ 2 /(2μ 2 1 ). This means that a learner, in a fashion parallel to reality, views private signals about the environmental state as conditionally normally distributed accordingα. In addition,r =û = 1/2 and D = (1 − 2γ)/4. Finally,q 1 = 1 −q 0 =q, which simply means that a learner's cognition encodes the idea that demonstrators make choices that are optimal, from the perspective of the demonstrators, with probabilityq. Altogether,α summarizes the cognition of individual learning by encoding the information necessary for a learner to interpret a private signal about the environmental state she faces.γ,q, andφ specify the cognition of social learning in the sense thatq summarizes the behavior of demonstrators,γ summarizes the learner's prior regarding the relationship between demonstrators and learners, andφ summarizes how a learner updates her view of this relationship after observing a.
If we denote the cognitive representation of density functions for s given z asf z (s), then the left side of (3) simply becomesf 1 (s)/f 0 (s) = exp{s/α}. Further note that the learner's cognitive representations of observing i take the form, By extension, the condition for choosing behavior 1 after observing A = 0, i, and s is where 1 1 We introduce N (i;q,γ,φ, A = 0) and B(i;q,γ,φ, A = 0) because expressions below would otherwise become unruly. To derive, substituter =û = 1/2,D = (1 − 2γ)/4, and expression (6) in the right side of (4) and simplify.
Given A = 0 and i, which are observed, and z, which is not observed, the probability that the learner chooses behavior 1 is thus Notice that F z (·) is the actual distribution function for s given z, which is what ultimately matters when specifying the probability that a learner observes a value of s that satisfies condition (8).
Given A = 1 and i, which are observed, and z, which is not observed, the probability that the learner chooses behavior 1 is To specify how a learning system based on (8) and (11) evolves, we treat cognitive encodings as genotypes and derive the resulting gene-culture coevolutionary system. To see how this works, our task is to derive expressions for both the cultural evolutionary process and the linked genetic evolutionary process. For the cultural evolutionary process, we need expressions for learner choice that depend on neither a, i, or s, all of which vary across learners within a generation 3 . To specify these expressions, recall that p t designates the actual proportion of demonstrators in t choosing behavior 1.
We would like to emphasize four results. First, when the signal of similarity is not reliable (φ = 0.5), facultative strategies that condition on a do not evolve (Figs A and   B). Second, when the signal of similarity is informative (e.g. φ = 0.9), facultative strategies evolve when individual learning is relatively effective. In this case, individual learning has a relatively strong influence when the signal of similarity indicates discordance (Figs A and B), and the learning system exhibits strongly positive social influence with the "S" shape of conformity when the signal of similarity indicates concordance (Figs A and B). Third, when the signal of similarity is informative (e.g. φ = 0.9), facultative strategies may or may not evolve when individual learning is relatively ineffective. If the probability of discordance is low (e.g. γ = 0.01), strategies are not meaningfully facultative, and they exhibit positive social influence with the "S" shape of conformity (Fig A). If the probability of discordance is higher (e.g. γ = 0.1), strategies are strongly facultative. They exhibit negative social influence when the signal of similarity indicates discordance (Fig B) and positive social influence when the signal indicates concordance (Fig B). Adjustments in this case are asymmetric, but relatively reliable signals and relatively high probabilities of discordance reduce the asymmetry (S2 Appendix).
Finally, steady-state learning is polymorphic in terms of cognition but not in terms of phenotype. Specifically, for all parameter combinations, two or more cognitive representations are present in equilibrium. They are essentially indistinguishable phenotypically, but they represent the structure of the decision-making task in very different ways. To see the intuition, imagine that copying the majority behavior among demonstrators is advantageous. Cognition can produce conformity as a behavioral response in at least two different ways. It can encode the idea that demonstrators are biased toward the demonstrator optimum, and demonstrators and learners have the same optimum. Alternatively, it can encode the idea that demonstrators are biased PLOS 9/14 S1 Appendix The evolution of facultative conformity based on similarity toward the demonstrator sub-optimum, and demonstrators and learners have different optima. Importantly, within the context of our model, these two encodings can be behaviorally equivalent. They cannot, however, both be accurate representations of reality. If selection favors accurate representations in other decision-making domains, this might eliminate one or more representations in the domain we consider. In any case, selection ultimately responds to phenotype, and phenotypes are equivalent.  Rows vary according to whether the signal indicating similarity is uninformative (φ = 0.5) or informative (φ = 0.9) and whether individual learning is relatively accurate (µ1/σ = 1) or inaccurate (µ1/σ = 1/9). The horizontal dashed lines show a learning system that ignores demonstrator behavior and relies only on individual learning. The diagonal dashed lines show an unbiased learning system that does not generate cultural evolution.