Models of Cultural Evolution
Mathematical models of cultural transmission, biased learning, and gene-culture coevolution
A Brief History
Culture as an Evolutionary System
Culture evolves. Ideas, beliefs, norms, and practices spread through populations in ways that bear striking parallels to genetic evolution, yet with fundamental differences. Where genes are copied with high fidelity from parent to offspring, cultural traits can be transmitted vertically (parent to child), obliquely (any elder to younger), or horizontally (peer to peer), and the fidelity of transmission can vary dramatically. Most remarkably, cultural evolution can proceed orders of magnitude faster than genetic evolution.
The field of dual inheritance theory, pioneered by Luigi Luca Cavalli-Sforza and Marcus Feldman (1981) and independently by Robert Boyd and Peter Richerson (1985), applies the mathematical machinery of population genetics to cultural traits. In this framework, cultural traits are units of information that spread through populations subject to transmission, mutation, and selection pressures.
Vertical transmission: Knowledge, skills, and beliefs pass from parent to offspring, resembling genetic inheritance. This is the mode captured by much family tradition and early education.
Oblique transmission: Younger individuals acquire cultural traits from unrelated elders or authority figures — teachers, mentors, priests. This can create rapid cultural change as each generation samples ideas from a broad pool of predecessors.
Horizontal transmission: Cultural traits spread among peers of similar age. Fashion, slang, and new technologies often spread horizontally. This mode can create avalanche-like adoption dynamics.
The consequence is that a single beneficial cultural innovation can sweep through a population in a few generations, whereas a similarly advantageous genetic mutation might take thousands. Cultural evolution is thus a dominant force shaping human societies, and understanding it requires rigorous dynamical models.
Biased Transmission
Not all cultural traits spread with equal ease. Cultural transmission is inherently biased: some ideas are intrinsically more attractive (content bias), some individuals are more influential (prestige bias), and most populations show a tendency to copy the majority (conformist bias).
Consider the simplest model: a population with two cultural variants, present in frequencies $p$ and $(1-p)$. Transmission is biased toward one variant. Three major bias types are commonly studied:
- Content bias: Ideas may be intrinsically more memorable, intuitive, or emotionally appealing. Regardless of who holds them, these ideas are preferentially copied.
- Prestige bias: Individuals copy the most successful person in their social neighborhood. If one cultural variant is carried by successful individuals, it spreads.
- Conformist bias: Individuals are biased toward copying the most common variant in their neighbourhood (frequency-dependent transmission). This creates interesting nonlinear dynamics.
Let us focus on conformist bias, which leads to particularly rich dynamics. Suppose an individual samples $n$ role models at random and adopts their (modal) trait. Under conformist bias, the probability of adopting the more common variant exceeds the frequency of that variant.
Under conformist bias, let $p$ be the frequency of trait A. The expected frequency in the next generation is approximately:
$$p' = p + D \cdot p(1-p)(2p - 1)$$
where $D$ is the conformity strength (increasing with sample size and bias intensity). The term $p(1-p)$ vanishes at the boundaries (fixation states), and $(2p - 1)$ changes sign at $p = 1/2$.
Equilibria: Setting $p' = p$ yields three fixed points: $p^* = 0$, $p^* = 1/2$, and $p^* = 1$. The interior equilibrium $p^* = 1/2$ is unstable (a saddle point), while $p = 0$ and $p = 1$ are stable nodes.
Consequence: Conformism amplifies majorities. Any slight deviation from the 50/50 point drives the population toward fixation of whichever variant is already more common. This creates path-dependent cultural evolution: history matters, and cultural lock-in is possible.
Gene–Culture Coevolution
Genes and culture do not evolve independently. Humans carry both biological and cultural inheritance, and they interact. Cultural practices shape the selective environment faced by genes, and genetic predispositions influence cultural learning. This feedback loop is gene–culture coevolution.
A paradigmatic example is lactase persistence: the ability to digest lactose (milk sugar) as an adult. In most human populations, lactase expression switches off after weaning. However, in populations with a long history of pastoralism and dairy farming, mutations that maintain lactase expression have spread to high frequency. The cultural practice (dairy farming) changed the selective landscape, favouring alleles for lactase production. Simultaneously, the genetic ability to digest milk made dairy farming even more advantageous. The genes and the culture co-evolved.
The mathematical framework for gene–culture coevolution couples two dynamical systems: one for gene frequency (say, $q$, the frequency of the lactase-persistence allele) and one for cultural trait frequency (the prevalence of dairy farming, $p$).
The fitness of individuals depends on both their genotype and the cultural environment they inhabit. Individuals with genotype $L$ (lactase-persistent) gain an extra reproductive advantage in populations where dairy farming is prevalent. As dairy farming becomes more common, selection for the $L$ allele strengthens. Conversely, as the $L$ allele becomes common, the nutritional benefits of dairy farming increase, making the cultural practice more attractive to both individuals and communities.
The dynamical equations are of the form:
$$\dot{q} = q(1-q)(s \cdot p \cdot f_L - f_l)$$
$$\dot{p} = p(1-p) \sigma(q)$$
where $s$ is the selective advantage of lactase persistence in dairy-farming societies, $\sigma(q)$ is the cultural transmission bias (increasing in $q$, since individuals with the capacity to digest milk promote dairy farming). This coupling can generate rapid coevolutionary trajectories and create evolutionary lock-in where both the gene and the practice reach high frequency.
Feldman and Laland (1996) pioneered the mathematical study of such systems, and the concept of niche construction describes the general principle: organisms modify their environment, which then feeds back to shape selection.
Collective Narratives and Cooperation
One of the most powerful findings in recent cultural evolution research concerns the role of collective narratives — shared belief systems, stories, and symbolic frameworks — in facilitating cooperation. A key insight is that these narratives need not be morally grounded or even truthful to be effective. Arbitrary, fictional, nonsensical narratives can catalyse cooperation by providing a common reference frame for coordination.
Consider a population playing a coordination game known as the Stag Hunt. Two hunters can either cooperate (hunt stag: high payoff if both cooperate, zero if the partner defects) or defect (hunt hare: a guaranteed low payoff). The dilemma is that cooperation requires mutual trust. Without some signal of shared intention, rational individuals default to the safe (hare) strategy.
Now introduce a narrative — a shared belief system, story, or symbol. The narrative might be completely arbitrary: a flag, a saying, a mythological tale, a set of abstract rules. Individuals who share this narrative are more likely to coordinate with each other. The narrative acts as a correlation device, allowing agents with common beliefs to synchronize their actions.
Consider a two-stage process:
(1) Narrative spread: A narrative spreads through the population via cultural transmission. Narratives may spread via prestige bias (followers of successful individuals adopt their narrative), content bias (some narratives are more memorable), or conformist bias (people adopt the majority narrative).
(2) Coordination: In social interactions (specifically, coordination games like Stag Hunt), individuals sharing a narrative are more likely to cooperate. The narrative creates intersubjectivity — a shared mental model that enables trust and coordinated action.
Key mathematical finding: Let $x$ = frequency of cooperators, $y$ = frequency of narrative believers. Coupled replicator-like equations describe their evolution. The payoff matrix for the Stag Hunt is:
$$\begin{pmatrix} R & S \\ T & P \end{pmatrix} = \begin{pmatrix} b & 0 \\ b+\delta & c \end{pmatrix}$$
where $R = b$ (mutual stag hunting), $S = 0$ (cooperator exploited), $T = b + \delta$ (defector against cooperator), and $P = c$ (mutual hare hunting, $c < b$). The parameter $\delta > 0$ represents the bonus coordination payoff when both agents share the narrative.
The result: In both small and large populations, the emergence of any collective narrative — even arbitrary, fictional, amoral narratives — can shift the population from a defection equilibrium to a cooperation equilibrium. The narrative is not itself a moral norm; it is purely a coordination device. What matters is that it is shared.
This finding is reported in Gokhale, Bulbulia & Frean (2022), published in Humanities and Social Sciences Communications. The implications are profound: collective beliefs can facilitate cooperation not because they encode moral wisdom, but because they provide a focal point for coordination. In this sense, even myths, rituals, flags, and abstract symbols can have deep evolutionary significance.
Narratives in Structured Populations
The analysis above assumes well-mixed populations. Real human societies are structured: people live in villages, tribes, families, and networks. Group size varies. Individuals interact more frequently with in-group members than out-group members. Social and cultural identities shape who interacts with whom. How robust is the narrative-cooperation effect to realistic population structure?
Recent work extends the model to structured populations, showing that the catalytic effect of collective narratives persists across a wide range of population structures and group sizes. Importantly, narratives and actions can spread at different rates — beliefs may spread rapidly through cultural transmission, while cooperation spreads more slowly through the success of cooperators. This asymmetry creates rich dynamical behaviour.
The work of Fic & Gokhale (2024), published in npj Complexity, extends the narrative-cooperation model to structured populations with varying group sizes and heterogeneous interaction patterns.
Key findings:
- Collective beliefs catalyse cooperation even when beliefs lack moralising components. The narrative need not encode cooperative norms; it merely needs to provide coordination.
- The effect persists across different population structures: well-mixed, sparse networks, spatial grids, and scale-free networks all show similar patterns.
- Asymmetric spread rates: If narratives spread quickly (high cultural transmission fidelity) but cooperation spreads slowly (due to the Stag Hunt payoff structure), the population can transition from low-cooperation to high-cooperation regimes on a timescale set by the narrative spread, not the cooperation spread.
- Group size matters, but in a subtle way: larger groups face stronger problems of coordination, but narratives become more valuable as coordinators in larger groups.
These results suggest that the evolution of language, religion, ritual, and symbolic systems in human societies may be intimately linked to the problem of coordination in ever-larger groups.
Explore: Narratives & Cooperation
Social Learning and Innovation
Cultural evolution is the process by which populations explore and occupy a strategy space. Two complementary mechanisms drive this exploration: innovation (the generation of novel traits) and imitation (the copying of successful traits).
Innovation is risky: a novel idea might fail catastrophically. Imitation is safe: copy what is already known to work. This trade-off creates a paradox identified by Boyd and Richerson: in a well-mixed population at equilibrium, social learning (imitation) is only beneficial when it is rare. Why? Because the benefit of social learning is to avoid the cost of innovation; but if everyone social learns, no one innovates, and the population converges to a fixed set of strategies. Occasional innovation is essential to maintain cultural diversity and adaptability.
This is Rogers' paradox: Rogers showed that in environments that change over time, intermediate rates of innovation and social learning maximize population fitness. Too little innovation leaves the population maladapted to changing conditions; too much innovation wastes resources on failed experiments.
Consider a population in a fluctuating environment with strategies:
- Innovators: Each generation, attempt a novel strategy. With probability $\mu$, the innovation is advantageous (fitness $f_I^+$); with probability $(1-\mu)$, it fails (fitness $f_I^-$). Expected fitness: $\mu f_I^+ + (1-\mu) f_I^-$.
- Social learners: Copy the most successful individual in the previous generation. This guarantees fitness equal to the maximum in the previous generation, $f_{\max}^{(t-1)}$.
In a constant environment, social learners always do better, and innovation goes extinct. But in a fluctuating environment, the best strategy from yesterday may be worse today. Only the innovators discover new optima. The population benefits from a mixture: most individuals social learn (staying near current optima), but some innovate (searching for improvements). The equilibrium frequency of innovators depends on the rate of environmental change and the success probability of innovation.
Importantly, the connection to evolutionary game theory is direct: strategies in cultural evolution need not be genes; they are ideas, practices, technologies, and beliefs that spread via cultural transmission. The replicator equation applies just as well to memes as to alleles. The frequency of a cultural strategy increases if it has higher fitness (expected payoff) than the population average.
Exercises
Conceptual Questions
- Explain the concept of the meme and how it differs from a gene. What makes cultural transmission fundamentally different from genetic inheritance?
- Describe Rogers' Paradox: why does social learning (pure imitation) alone lead to a population that cannot adapt to environmental changes? What is the optimal balance?
- What is prestige bias and why might humans preferentially learn from high-status individuals? What cultural dynamics does this create compared to random or proportional imitation?
- In gene-culture coevolution, explain how a genetic predisposition (e.g., lactase persistence) and a cultural practice (dairy farming) can reinforce each other evolutionarily.
- What is an epidemiology of representations, and how does it explain why certain cultural traits (e.g., "earworms" or conspiracy theories) spread more readily than others?
Computer Problems
- Rogers' Paradox: Innovation vs Social Learning. Implement a population with $N$ individuals using either innovation (success probability $\mu = 0.1$) or social learning (copy best from previous generation). Use payoff $f_+ = 1$ (success) and $f_- = 0$ (failure). Simulate a constant environment and show that social learning goes to fixation. Repeat with a fluctuating environment (optimal strategy changes every $T$ generations) and show that intermediate frequencies maximize fitness.
- Prestige Bias and Cultural Dynamics. Model a population where individuals choose to learn from others based on success (prestige bias): if individual $i$ has payoff $w_i$, the probability of copying strategy $s_i$ is proportional to $w_i$. Compare this to random imitation and payoff-biased learning. Plot trait frequency over time for each mechanism.
- Gene-Culture Coevolution: Lactase Persistence. Implement a model with a genetic locus (lactase gene: $A$ allele confers persistence, $a$ does not) and a cultural trait (dairy farming yes/no). Give fitness bonus only if cultural+genetic match. Show how culture adoption drives allele frequency of $A$, which then favours further dairy practice.
- Conformist Bias and Runaway Dynamics. Implement a population where individuals adopt the majority strategy with increased probability (conformist bias). Use a continuous trait (e.g., clothing style on [0,1]). Show how conformism can create frequency-dependent selection that leads to rapid fixation or bistability depending on initial frequency.
- Meme Transmission Network Model. Implement an agent-based model on a social network where individuals hold beliefs/traits and share them with neighbours. Vary network structure (random, small-world, scale-free) and measure adoption time and final fixation probability. Show how network topology affects cultural evolutionary dynamics.