Variable cultural acquisition costs constrain cumulative cultural evolution

Abstract

One of the hallmarks of the human species is our capacity for cumulative culture, in which beneficial knowledge and technology is accumulated over successive generations. Yet previous analyses of cumulative cultural change have failed to consider the possibility that as cultural complexity accumulates, it becomes increasingly costly for each new generation to acquire from the previous generation. In principle this may result in an upper limit on the cultural complexity that can be accumulated, at which point accumulated knowledge is so costly and time-consuming to acquire that further innovation is not possible. In this paper I first review existing empirical analyses of the history of science and technology that support the possibility that cultural acquisition costs may constrain cumulative cultural evolution. I then present macroscopic and individual-based models of cumulative cultural evolution that explore the consequences of this assumption of variable cultural acquisition costs, showing that making acquisition costs vary with cultural complexity causes the latter to reach an upper limit above which no further innovation can occur. These models further explore the consequences of different cultural transmission rules (directly biased, indirectly biased and unbiased transmission), population size, and cultural innovations that themselves reduce innovation or acquisition costs.

Figure 1. Exponential growth in scientific knowledge.

(A) Empirically-derived exponential growth in mathematical knowledge as measured by the number of published abstracts in mathematics from 1868–1965. The curve shown is a best-fit to data reported in May , regression equation n = 1400e^{0.025(t-1880)}. (B) Price argues that exponential increases in scientific output such as those documented by May (dashed line) are actually the initial part of a logistic growth rate (solid line), eventually reaching a saturation point due to constraints on cumulative cultural evolution.

Figure 2. Evidence for increasing cultural acquisition costs.

(A) Individual ontogeny recapitulates cultural history for mathematical knowledge: children learn mathematical concepts in the same order that they were first invented historically. The line is a best-fit logarithmic function with R² = 0.97. See Text S1 and Table S1 for sources. (B) Jones' maximum likelihood functions of the probability of a scientist or inventor producing a significant scientific or technological innovation (as measured by the awarding of a Nobel prize or entry in prominent technological almanacs) as a function of the innovator's age, separately for the years 1900 and 2000. Over this 100-year period the peak age of innovation has increased by approximately 6 years, and overall innovation rates have decreased. Functions are derived from equation (3) and Table 2 in ref. , recreating that paper's Figure 4.

Figure 3. Cultural accumulation over successive generations in Henrich's original unconstrained model (Eq. 1) and in the constrained Model 1 (Eq. 2).

Parameters: N = 100, α = 0.2, β = 0.05.

Figure 4. Time series of mean cultural complexity over time in Model 2 indicating that complexity reaches a maximum equilibrium _max.

Three alternative copying rules are shown: unbiased transmission (new individuals copy the cultural traits of a randomly selected member of the previous generation), indirectly biased transmission (new individuals copy the cultural traits of the member of the previous generation who had the highest individual fitness) and directly biased transmission (new individuals copy the cultural traits with the highest trait fitnesses across the entire previous generation). Parameters: N = 100, c_i = 10, c_s = 5, λ = 1000, µ_i = µ_s = 0; all results are the average of 100 independent runs.

Figure 5. Interaction between population size and cultural complexity in Model 2.

(A) Time series of mean cultural complexity over time in Model 2 at different population sizes, N, under the assumption of direct bias. (B) Relationship between maximum cultural complexity _max and N for direct bias, indirect bias and unbiased transmission, and with N plotted on a logarithmic scale. For direct bias, maximum cultural complexity _max increases logarithmically with N up to N = 100 (line is a logarithmic best-fit with R² = 0.991 for N≤100), after which it plateaus. For indirect bias, a similar logarithmic increase followed by a plateau occurs to that under direct bias, but the values of _max are lower and the plateau occurs at higher values of N (around N = 1000; line is a logarithmic best-fit with R² = 0.994 for N≤1000). For unbiased transmission, N has no effect on _max. Parameters: c_i = 10, c_s = 5, λ = 1000, µ_i = µ_s = 0, all results are the average of 100 independent runs.

Figure 6. The effect on maximum cultural complexity _max of varying (A) lifetime effort budget λ (with constant c_i = 10, c_s = 5), (B) innovation cost c_i (with constant c_s = 1, λ = 100), and (C) copying cost c_s (with constant c_i = 10, λ = 100).

Circles indicate direct bias, crosses indicate indirect bias and triangles indicate unbiased transmission. Lines show best-fit functions, in (A) showing a positive linear relationship between _max and λ for direct bias (R² = 0.999), indirect bias (R² = 0.991) and unbiased transmission (R² = 0.998); in (B) showing a negative linear relationship between _max and c_i for direct bias (R² = 0.995) and unbiased transmission (R² = 0.999), but for indirect bias only for larger c_i values (for c_i>30, R² = 0.990), with lower c_i values generating lower _max values than expected (no best-fit line drawn); and in (C) showing an inverse power law relationship between _max and c_s for direct bias (R² = 0.997), indirect bias (R² = 0.997) and unbiased transmission (R² = 0.992). Other parameters: N = 100, X = 100, µ_i = µ_s = 0, all results are the average of 100 independent runs.

Figure 7. Time series of mean cultural complexity when c_i and c_s decrease in proportion to according to proportionality constants µ_i and µ_s respectively.

In (A) transmission is directly biased, in (B) transmission is indirectly biased. Other parameters: N = 100, c_i = 200, c_s = 100, λ = 1000, all results are the average of 100 independent runs.