Two lines of inheritance

What humans receive from the previous generation is not only genes. Language, skills, food habits, how far to trust strangers and how to react under pressure can all pass down the generations through observation, imitation and teaching. Cavalli-Sforza and Feldman used the mathematics of population genetics to study this cultural transmission, distinguishing three channels: vertical transmission (from parents to children), oblique transmission (from non-parental adults of the previous generation to the young, such as teachers, elders and mentors) and horizontal transmission (among members of the same generation, such as peers) (Cavalli-Sforza and Feldman 1981). Boyd and Richerson treated genes and culture as two interacting lines of inheritance, proposing “dual-inheritance theory” and analysing when learning rules such as conformist bias and indirect bias (copying successful or prestigious models, later often called prestige bias) are advantageous (Boyd and Richerson 1985).

Borrowing these concepts for the family: a child’s expectations and automatic reactions about “what to do when stress comes” can be seen as a behavioural prior learned through several channels. The questions are how faithful each channel is, and how many generations a family’s habit can survive the influence of the surrounding community.

A simple transmission equation

# Vertical, oblique and horizontal transmission of a binary practice, after
# Cavalli-Sforza and Feldman (1981) and Boyd and Richerson (1985).
# x = chance a child in one lineage acquires it; q = community frequency.
#   x_next = V [f x + (1 - f)(1 - x)] + O q + H C(q),  V + O + H = 1,
#   C(q) = q + D q (1 - q)(2q - 1)   (conformist copying of peers)
# Deviations decay by lam = V (2f - 1) per generation. ILLUSTRATIVE values.
import math

def step(x, q, V, O, H, f, D=0.3):
    peers = q + D * q * (1 - q) * (2 * q - 1)
    return V * (f * x + (1 - f) * (1 - x)) + O * q + H * peers

def path(x0, q, V, O, H, f, gens=8):
    xs = [x0]
    for _ in range(gens): xs.append(step(xs[-1], q, V, O, H, f))
    return xs

MIXES = [("high-fidelity vertical", 0.8, 0.1, 0.1, 0.97),
         ("mixed channels", 0.5, 0.25, 0.25, 0.95),
         ("peers and mentors", 0.2, 0.4, 0.4, 0.95)]

if __name__ == "__main__":
    print("a family practising it (x0 = 0.90) settles where q = 0.20")
    print(f" {'channel mix':22s}  V    O    H    f  | lam  | half-life")
    for name, V, O, H, f in MIXES:
        lam = V * (2 * f - 1)
        print(f" {name:22s} {V:.1f} {O:.2f} {H:.2f} {f:.2f} | {lam:.2f} |"
              f" {math.log(0.5) / math.log(lam):4.1f} generations")
    print(" x in generations 1, 2, 3 and 8:")
    for name, V, O, H, f in MIXES:
        xs = path(0.9, 0.2, V, O, H, f)
        print(f"   {name:22s}: {xs[1]:.2f}, {xs[2]:.2f}, {xs[3]:.2f}, {xs[8]:.2f}")
    print("reverse: a family without it (x0 = 0.10) joins a community, q = 0.70")
    for name, V, O, H, f in MIXES:
        xs = path(0.1, 0.7, V, O, H, f)
        print(f"   {name:22s}: generations 1, 2, 3 -> "
              f"{xs[1]:.2f}, {xs[2]:.2f}, {xs[3]:.2f}")
    print("one new mentor in generation 1 only (O 0.10 -> 0.40, V 0.8 -> 0.5):")
    xs = [0.9, step(0.9, 0.2, 0.5, 0.4, 0.1, 0.97)]
    for _ in range(3): xs.append(step(xs[-1], 0.2, 0.8, 0.1, 0.1, 0.97))
    print("   x = " + ", ".join(f"{v:.2f}" for v in xs) + "  (compare row 1)")
# a family practising it (x0 = 0.90) settles where q = 0.20
#  channel mix             V    O    H    f  | lam  | half-life
#  high-fidelity vertical 0.8 0.10 0.10 0.97 | 0.75 |  2.4 generations
#  mixed channels         0.5 0.25 0.25 0.95 | 0.45 |  0.9 generations
#  peers and mentors      0.2 0.40 0.40 0.95 | 0.18 |  0.4 generations
#  x in generations 1, 2, 3 and 8:
#    high-fidelity vertical: 0.74, 0.62, 0.52, 0.31
#    mixed channels        : 0.52, 0.35, 0.28, 0.22
#    peers and mentors     : 0.32, 0.22, 0.20, 0.19
# reverse: a family without it (x0 = 0.10) joins a community, q = 0.70
#    high-fidelity vertical: generations 1, 2, 3 -> 0.24, 0.35, 0.43
#    mixed channels        : generations 1, 2, 3 -> 0.43, 0.57, 0.64
#    peers and mentors     : generations 1, 2, 3 -> 0.60, 0.69, 0.70
# one new mentor in generation 1 only (O 0.10 -> 0.40, V 0.8 -> 0.5):
#    x = 0.90, 0.54, 0.46, 0.41, 0.37  (compare row 1)

A family that practises the habit ($x_0 = 0.9$) settles in a community where only 20 % do:

• High-fidelity vertical transmission ($V = 0.8$, $f = 0.97$): $\lambda = 0.75$ and a half-life of about 2.4 generations; the probability in generations 1, 2 and 3 is about 0.74, 0.62 and 0.52, and in generation 8 it is still 0.31, above the community’s 0.2.
• Mixed channels ($V = 0.5$): $\lambda = 0.45$ and a half-life under one generation; by generation 3 it has fallen to 0.28.
• Mostly peers and mentors ($V = 0.2$): $\lambda = 0.18$; by generation 2 it is close to the community level.

In reverse, when a family without the habit ($x_0 = 0.1$) joins a community where 70 % practise it, the same channel structure sets how fast the family is “rewritten”: with mostly peers and mentors it reaches 0.60 in the first generation, while under high-fidelity vertical transmission it reaches only 0.43 by the third. The last line shows that, even with the family’s channel structure unchanged, introducing a new mentor in a single generation (the oblique weight rising from 0.1 to 0.4 for that generation) lowers that generation’s probability from 0.74 to 0.54, and every later generation stays lower.

The model’s conclusion can be put in two sentences: the more a channel is concentrated in the parents and the more precise the copying, the longer a family’s behavioural prior persists; the more it is spread across mentors and peers, the faster the community rewrites it. The same structure explains both the persistence of traditions and their rapid disappearance.

Evidence: persistence and rewriting

These two kinds of evidence correspond to the model’s two cases: attitudes such as trust are transmitted mainly within families and ethnic communities and persist longer, while language is used heavily among peers and in school, where community influence is strong, and shifts quickly. But both studies are observational and involve selection (who migrates, where they settle), so neither proves the model’s mechanism on its own.

The “channel” metaphor and its limits

Chapter summary

Cultural transmission has vertical, oblique and horizontal channels. The transmission equation shows that a lineage’s deviation from the community equilibrium is multiplied by $\lambda = V(2f-1)$ each generation: high-fidelity vertical transmission keeps a behavioural prior for several generations, while peer- and mentor-dominated channels rewrite it within one or two, and a new mentor in one generation changes the lineage’s course for good. Trust and language among immigrants’ descendants give observational examples of persistence and of rapid rewriting. The next chapter asks whether the same environment affects every child equally.