ELECTRONIC SUPPLEMENTARY MATERIAL
1. ADDITIONAL METHODS FOR MODEL 1
Recursions for the frequencies of phenogenotypes in the next generation (assuming
oblique/horizontal cultural transmission of paternity beliefs) were calculated by summing
the probabilities that each phenogenotype will result from each of the 11 possible
genotype matings (columns c1-c14 in Supplementary Table S3), weighted by the
probability that each mating occurs (column c0 in Supplementary Table S3), the fitness
associated with that mating (from Table 1) and the sex ratio (SR, the proportion of the
population that is male, or 1-SR for females), divided by the total fitness of males (Wm) or
females (Wf), which represent normalising constants. For example, the frequency of MBp
males in the next generation (x1’) is given by:
Wm x1’ = [(1+s) Σc0c1 + 2(1+s/2) Σc0c3 + (1+2s) Σc0c7 + 2(1+s) Σc0c11] SR.
The full system of recursions was subject to extensive numerical analysis. The parameter
values were systematically varied across a range of biologically plausible values of each
(for s and r: 0, 0.001, 0.01, 0.05, 0.1, 0.2, 0.25; for b1 and b2: 0, 0.05, 0.1, 0.15, 0.2, 0.25).
Each parameter was varied both in isolation, and at each value of every other parameter.
For mathematical convenience, the fitness terms are associated with offspring rather than
parental clusters. This is based on the assumption that offspring beliefs are similar to the
beliefs of their parental cluster. Ideally the parameter r would apply directly to parental
clusters, particularly given that Bp offspring with one Bs parent currently (and
unrealistically) bear no cost of infanticide/male aggression. Assigning fitness to clusters
rather than offspring would, however, have rendered the model mathematically
intractable. The validity of this assumption was supported by the results of the agentbased Model 2, in which fitness is directly assigned to reproducing females, and in which
r had an identical effect as in Model 1.
Because the frequencies x1-x4 and y1-y4 were used to track changes in the
phenogenotype characteristics of the populations, the model dynamics indicated in the
two-dimensional Fig. 1 are more accurately represented in three dimensions (i.e. different
combinations of x1-x4 and y1-y4 can produce identical frequencies of M/m and Bp/Bs).
This third dimension, however, did not greatly change the results described here.
Furthermore, because populations rapidly converge on Bs or Bp, gene-culture ‘linkage
disequilibrium’ (Feldman & Zhivotovsky 1992) rapidly broke down.
Alternative versions of Model 1 were formulated and analysed, first with diploid genetics
and second with alternative cultural transmission assumptions. In the diploid model, MM
individuals behave monogamously, mm individuals behave polygamously and Mm
individuals behave monogamously half of the time and polygamously the other half. The
diploid model exhibited identical dynamics to the haploid model, so while humans are
not haploid, the simpler haploid model is presented here.
In the light of recent debate over the prevalence of vertical and oblique/horizontal
cultural transmission (R. Boyd, personal communication), a variant of Model 1 was
analysed that featured vertical cultural transmission of paternity beliefs, where offspring
beliefs are given by the frequencies of beliefs in the offspring’s parents. Here, offspring
of two Bp parents are Bp, offspring of two Bs parents are Bs, and offspring of one Bp and
one Bs parent inherit beliefs according to the biases b1 (favouring Bs in MxM matings)
and b2 (favouring Bp in mxm matings). Offspring of MBpxMBs parents are Bs with
probability (0.5+b1), and offspring of mBpxmBs parents are Bp with probability (0.5+b2).
Offspring of genetically different and culturally different parents have an equal
probability of being Bs or Bp. Each parameter individually and combined has the same
effect under the assumption of vertical cultural transmission as for oblique/horizontal
cultural transmission.
2. ADDITIONAL METHODS FOR MODEL 2
Each of 150 male or female agents possess one of the four previously described
phenogenotypes (MBs, MBp, mBs or mBp). A population size of 150 is representative of
traditional societies (Beckerman & Valentine 2002; Dunbar 1993). Non-overlapping
generations of agents undergo cycles of mating and then reproduction. Reproduction
continues until all 150 slots of the next generation are filled. Each generation undergoes
the same mating/reproduction cycle for 5000 generations (fewer than Model 1 due to
greater computational demands. Analyses proceeded as for Model 1.
Conformist cultural transmission (Boyd & Richerson 1985; Henrich & Boyd 1998) was
also simulated in Model 2 using the parameter k (0<k<1), which specified the probability
that an offspring adopts the majority belief in the entire parental generation, rather than
inheriting beliefs from parents only (vertical cultural transmission) or proportionally to
the frequency of beliefs in the parental generation (oblique/horizontal transmission).
Hence where k=1, all cultural transmission is conformist. Assortative cultural mating
(Cavalli-Sforza & Feldman 1981) was simulated using the parameter d (0<d<1), which
specified the probability that an agent will refuse to mate with an agent with a different
belief to themselves. Hence where d=1, only BsxBs and BpxBp matings are permitted. Both
k and d increased the likelihood of populations becoming fixated at Bs from Bs-dominated
regions, and at Bp from Bp-dominated regions, although this effect did not qualitatively
change the results.
SUPPLEMENTARY FIGURE CAPTIONS
Supplementary Fig S3. - Schematic illustrations of all allowed mating clusters in Model
1, and their associated mating system. The left hand circles of each cluster denote male
individuals, the right hand circles of each cluster denote female individuals.
SUPPLEMENTARY REFERENCES
Beckerman, S. & Valentine, P. (Eds.). 2002 Cultures of multiple fathers. Gainesville, FL:
University Press of Florida.
Boyd, R. & Richerson, P. J. 1985 Culture and the evolutionary process. Chicago:
University of Chicago Press.
Cavalli-Sforza, L. L. & Feldman, M. W. 1981 Cultural transmission and evolution.
Princeton: Princeton University Press.
Dunbar, R. I. M. 1993 Coevolution of neocortical size, group size and language. Behav.
Brain Sci. 16, 681-735.
Feldman, M. W. & Zhivotovsky, L. A. 1992 Gene-culture coevolution: Toward a general
theory of vertical transmission. Proc. Natl. Acad. Sci. USA 89, 11935-11938.
Henrich, J. & Boyd, R. 1998 The evolution of conformist transmission and the
emergence of between-group differences. Evol. Hum. Behav. 19, 215-241.