- Email: [email protected]
Evolution of Transmission Bias in Cultural Inheritance
J. theor. Biol. (1998) 190, 147–159
Evolution of Transmission Bias in Cultural Inheritance K T* Department of Biological Sciences, Graduate School of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan (Received on 28 February 1997, Accepted in revised form on 8 September 1997)
Evolution of transmission bias in cultural inheritance is investigated using simple models of cultural selection. Conventional models of cultural transmission describe cultural changes by incorporating transmission bias and non-vertical pathways into the ordinary population genetic framework. The methodology has been successful in understanding cultural changes in terms of natural selection, but it is difficult to see from the theoretical framework how biased transmission in favor of maladaptive traits might have evolved. To show that ordinary cultural processes lead at times to the evolution of a preference that favors a deleterious cultural variant, this study presents an alternative model of cultural transmission, where cultural elements are transmitted in a manner more like infections in epidemiological transmission. An ordinary equilibrium analysis indicates that, under certain conditions, runaway dynamics emerges and the coevolution of a maladaptive cultural variant and an associated preference in favor of the maladaptive variant is observed. If the preference of an individual does not change during its ontogeny (e.g., if it is transmitted genetically), however, then cultural selection alone does not produce such runaway dynamics, and only those preferences that favor adaptive variants should eventually evolve. Since cultural processes may at times result in a reduction in the fitness of individuals, simplistic adaptive interpretations of culture are unconvincing without detailed specification of the cultural processes involved. Moreover, cultural runaway of this kind may help to explain the existence of traits that are apparently maladaptive at the individual level but may be advantageous for the group. Inferences are also made regarding the observed differences between human and non-human social information transfer. 7 1998 Academic Press Limited
*E-mail: k32148.m-unix.cc.u-tokyo.ac.jp
1981; Durham, 1991), and has been regarded as a primary effect that may govern the dynamics of cultural changes (e.g. Durham, 1991; Dugatkin, 1992; Laland, 1994; Laland et al. 1995; Dawkins, 1976; Ormrod, 1992; Galef, 1995; Boehm, 1996). It has repeatedly been argued in behavioral evolutionary studies that cultures result from processes in the human brain that have been shaped through natural selection, and that resulting psychological mechanisms influence which variants observed in the surrounding cultural milieu are adopted (e.g. Lumsden & Wilson, 1981; Hinde, 1987; Richerson & Boyd, 1989; Alexander, 1990). These authors suggest that choices will be made according to the evolved algorithms of the relevant psychological mechanisms to enhance Darwinian (or inclusive) fitness of individuals (Alexander, 1979; Flinn, 1997). Their consequences on the phenotype are therefore likely to be adaptive, eventually producing transmission bias that usually
0022–5193/98/020147 + 13 $25.00/0/jt970541
7 1998 Academic Press Limited
Introduction Theorists have seen that when culture is modeled as a system of information inheritance, it is possible to maintain cultural traits that are maladaptive in terms of natural selection [see reviews in Cavalli-Sforza & Feldman, (1981); Boyd & Richerson (1985) and Durham (1991)]. This is true when biased cultural transmission favors maladaptive variants so as to overcome the countervailing effects of natural selection. Biased transmission is said to have occurred when individuals are predisposed to adopt some cultural variants over others in cultural inheritance. In the absence of strong natural selection against them, the favored maladaptive variants increase in frequency. This type of selective processes has been termed cultural selection (Cavalli-Sforza & Feldman,
148
.
favors adaptive cultural variants. In practice, however, this is not always the case (e.g. Hinde, 1987; Barkow, 1989; Richerson & Boyd, 1989, 1992; MacDonald, 1989; Blurton Jones, 1990; Logan & Qirko, 1996). This notion of maladaptive cultural practices goes back to Darwin (1871, chapters 19–21). Many contemporary researchers also recognize that maladaptive cultural practices do arise in human societies, as well as the transmission bias that underlies the adoption of such practices (e.g. Hinde, 1987; Barkow, 1989; Richerson & Boyd, 1989, 1992; Tooby & Cosmides, 1990; Logan & Qirko, 1996). Existence of predispositions that promote the acquisition of maladaptive variants has, therefore, been a crucial issue in the evolutionary studies of human behavior (e.g. Richerson & Boyd, 1989, 1992; Tooby & Cosmides, 1989; Logan & Qirko, 1996; Flinn, 1997). In this paper I provide simple mathematical models of cultural selection, and show that ordinary cultural processes do indeed lead at times to the evolution of a preference that favors a maladaptive cultural variant if that variant has sufficient attractiveness. Conventional models of cultural transmission describe cultural changes by incorporating transmission bias and non-vertical (e.g. oblique and horizontal) pathways into the ordinary population genetic framework (Cavalli-Sforza & Feldman, 1973; see also CavalliSforza & Feldman, 1981; Boyd & Richerson, 1985; Durham, 1991; Guglielmino et al., 1995). In this framework mathematical models are formulated to describe cultural dynamics by deriving frequency changes of cultural elements (analogous to genes or alleles in genetic transmission) in a generation where genetic reproduction should take place. While the methodology has been criticized for a number of reasons (see Tooby & Cosmides, 1989; Heyes, 1993; Sperber, 1994; Cronk, 1995; Galef, 1995), it has been successful in understanding cultural changes from the viewpoint of genetic adaptation and evolution. Indeed, although Cavalli-Sforza & Feldman (1981) deal entirely with purely cultural processes, they had earlier provided a theoretical framework for studies of gene-culture coevolution (Feldman & Cavalli-Sforza, 1976) on a range of problems as diverse as the origin and the initial evolution of cultural capacity (Cavalli-Sforza & Feldman, 1983a, b; Aoki & Feldman, 1987; Aoki, 1990, 1991; Takahasi & Aoki, 1995; Feldman et al., 1996), persistence of a sign language (Aoki & Feldman, 1991, 1994; Feldman & Aoki, 1992; see also Aoki, 1989; Aoki & Feldman, 1989), and the evolution of sex ratio (Kumm et al., 1994; Laland et al., 1995), just to name a few [see Feldman & Laland (1996) for a review]. It is difficult to see, however, from this theoretical framework why
transmission bias might evolve to prefer maladaptive states despite the counteracting effects of natural selection (Hinde, 1982, chapter 15). Contrary to the prevailing views, Dawkins (1976, chapter 11; see also Dawkins, 1982, chapter 6) argues that cultural changes should be understood in terms of the psychological appeal of each cultural element (or meme) irrespective of its effects on genetic adaptation or evolution (see also Sober, 1991). Indeed, recent memetic studies emphasize the formal analogies of cultureal transmission with epidemiological transmission rather than with genetic inheritance (Moritz, 1990; Lynch, 1991; Dawkins, 1993; see also Cavalli-Sforza & Feldman, 1981, chapter 1; Sperber, 1985, 1990, 1994; Dennett, 1995, chapter 12; Rogers, 1995). Unlike the conventional models of cultural transmission, memetic models do not incorporate genetic processes such as mating and reproduction (of genetic offspring) because they have nothing to do with purely cultural (or memetic) processes, which do not involve genetic changes. Following these memetic approaches in part, I develop in this paper simple mathematical models for the joint evolution of a cultural trait (i.e. a trait that may be modified through cultural transmission) that determines the attractiveness and the viability of an individual (trait B) and an associated trait that determines the preference of an individual for trait B (trait A). Evolutionary dynamics of the model system is investigated by following frequency changes of traits during an arbitrary short time period [rather than changes in a generation as in the conventional models; see Cavalli-Sforza & Feldman (1981)] where meeting and transmission of cultural elements among individuals take place. Similar lines of arguments are also found in the quantitative trait model of indirect bias discussed by Boyd & Richerson (1985, chapter 8). The Model Throughout I assume for simplicity that the traits are dichotomous and character states are described by subscripts 0 and 1 for both traits. Hence there are four culturally distinct phenotypes, A0B0, A0B1, A1B0, and A1B1, with their respective frequencies in the population denoted P00, P01, P10, and P11 (SijPij = 1). We investigate the evolutionary dynamics of the system by following frequency changes of the phenotypes during a short time period. Within a unit time, natural selection first acts at the level of differential viabilities of B0 and B1 individuals; B1 individuals have a per unit time viability of 1 − s relative to B0 individuals. On the other hand, the preference do not directly induce natural selection so
that A0 and A1 individuals are equally viable if they maintain the same state for trait B. Hence natural selection only acts indirectly on the preference trait, through non-random association between character states for the traits A and B. This non-random association is measured by the cultural analogue of genetic linkage disequilibrium, defined by D = P00P11 − P01P10 (see also Feldman & CavalliSforza, 1984; Feldman & Zhivotovsky, 1992). After natural selection, the phenotypic frequencies become P'00 = P00/(1 − sq1), P'01 = (1 − s)P01/(1 − sq1), P'10 = P10/(1 − sq1), and P'11 = (1 − s)P11/(1 − sq1), where a prime indicates a value after natural selection and qj = SiPij is the frequency of Bj individuals. Cultural transmission of the traits then follows, after the operation of natural selection in each time period. Each individual meets with other individuals and, according to its preference, selects a potential transmitter, an individual with whom the receiver interacts to receive cultural elements. Individuals with state A1 have a preference and tend to interact with individuals with state B1 by a degree of G ( − 1 Q G Q 1); i.e. the proportion of B1 individuals among the potential transmitters for A1 individuals is given by (1 + G)q'1 /[(1 − G)q'0 + (1 + G)q'1 ],
(1)
where q'j = SiP'i j is the frequency of Bj individuals after natural selection, with primes again indicating the values after natural selection. On the other hand, A0 individuals do not show a preference and randomly choose potential transmitters. Hence biased transmission occurs when the receiver is A1, while transmission is unbiased with A0 receivers. We see from eqn (1) that this model of selective interaction is frequency dependent, and is mathematically
8 8
Dp1 = 2c ·
equivalent to the relative fixed preference model of selective mating discussed by Kirkpatrick (1982; see also Takahasi, 1997). Through each interaction, the receiver changes its state for the trait B by a rate b ( r 0) if the chosen potential transmitter has a different state for that trait. In other words, among individuals with their initial state Bk who interact with Bl transmitters (l $ k), a proportion b of the receivers acquires the state Bl through the interaction; otherwise the original state is maintained. In a similar way, an individual changes its state for the trait A at a rate c ( r 0) through the interaction process; i.e., among Ak individuals who interact with Al transmitters (l $ k), a proportion c of the receivers acquires the state Al through the interaction. If the states of a trait are not altered through the interaction process, we have b = 0 or c = 0 and cultural transmission of the trait does not occur. Later in this paper it is inferred that this should be the case when the traits are transmitted genetically. Because of the selective interaction induced by the preference of A1 individuals, the above interaction processes may alone result in changes in phenotypic frequencies (see recursions below). Hence this is a model of selective meeting or interaction (see Lewontin et al., 1968; Eshel & Cavalli-Sforza, 1982) that may further lead to biased cultural transmission (Boyd & Richerson, 1985) or cultural selection (Cavalli-Sforza & Feldman, 1981; Durham, 1991). A further assumption of independent transmission of the two traits yields a model of cultural transmission as summarized in Table 1. Combining the effects of natural and cultural selection, we obtain the unit time changes in the frequencies of A1 (p1 = SjP1j ) and B1 (q1), and changes in the cultural analogue of genetic linkage disequilibrium (D = P00P11 − P01P10), as
9 9
1−s G[p1(1−sq1) − sD] D · −s · , 1 − sq1 1 − sq1 − G[q0 − (1 − s)q1] 1−sq1
(2a)
1−s G[p1(1−sq1) − sD] qq · −s · 0 1 1 − sq1 1 − sq1 − G[q0 − (1 − s)q1] 1−sq1
(2b)
Dq1 = 2b · and
149
DD = −[b(1 − c) + (1 − b)c]·
2G 1 (1−s)D + · (1 − sq1)2 1 − sq1 − G[q0 − (1 − s)q1] (1−sq1)3
× 4b(1 − c)·[p0(1−sq1) + sD]·[p1(1−sq1) − sD]·(1−s)q0q1 + bc[p1(1 − sq1) − sD]·[q0−(1 − s)q1]·(1−s)D + (1 − b)c ·(1−s)2D 25 −
(2c)
s[q0 − (1 − sq1)q1 ]D − (Dp1)·(Dq1). (1 − sq1 )2
.
150
The state of the system is uniquely determined by these three variables. To avoid mathematical complexities and to get a clear view of the system, we posit that the intensity s of natural selection and the transmission rates b and c are small, so that all quadratic terms in these may be ignored. The system (2) is then simplified as
$
Dp1 = 2c ·
$
Dq1 = 2b ·
%
Gp1 − s ·D, 1−G(q0 − q1)
%
Gp1 − s ·q0q1, 1−G(q0 − q1)
Henceforth, we focus on the approximate system (3). From the above eqns (3), we see that even if natural selection is not operating so that s = 0, a deterministic change in the frequency of B1 is observed (Dq1 $ 0) as long as A1 individuals interact preferentially (G $ 0) and the trait is culturally transmitted at a certain positive rate (b q 0). We also see by comparing eqns (3a) and (3b) that frequency changes in the preference trait A are induced only in a hitchhiking manner, which obviously requires D $ 0. Such correlated evolution has been previously observed in the models of (inter-) sexual selection, where the frequency of a (mating) preference allele changes only as a correlated response to changes in the frequencies of viability alleles (Kirkpatrick, 1982). It is worth mentioning that when c = b, we have, from eqns (3a) and (3b), Dp1 = Dq1·D/(q0q1); then either Dp1 = Dq1 = 0 or Dp1, Dq1 $ 0 holds. This case will be treated independently in case 1 below.
(3a)
(3b)
and DD = −(b + c)D +
2G 1 − G(q0 − q1)
× (bp0p1q0q1 + cD 2) − s(q0 − q1)D.
(3c)
T 1 Meeting table for the model of cultural selection Phenotypic distribution after interaction Receiver
Transmitter
Frequency of transmitter
A0B0
A0B1
A1B0
A 0B 0
A0B0
P'00
1
0
0
0
A0B1
P'01
1−b
b
0
0
A0B1
A1B0
A1B1
A1B1
A1B0
P'10
1−c
0
c
0
A1B1
P'11
(1 − b)(1 − c)
b(1 − c)
(1 − b)c
bc
A0B0
P'00
b
1−b
0
0
A0B1
P'01
0
1
0
0
A1B0
P'10
b(1 − c)
(1 − b)(1 − c)
bc
(1 − b)c
A1B1
P'11
0
1−c
0
c
A0B0
(1 − G)P'00 (1 − G)q'0 + (1 + G)q'1
c
0
1−c
0
A0B1
(1 + G)P'01 (1 − G)q'0 + (1 + G)q'1
(1 − b)c
bc
(1 − b)(1 − c)
b(1 − c)
A1B0
(1 − G)P'10 (1 − G)q'0 + (1 + G)q'1
0
0
1
0
A1B1
(1 + G)P'11 (1 − G)q'0 + (1 + G)q'1
0
0
1−b
b
A0B0
(1 − G)P'00 (1 − G)q'0 + (1 + G)q'1
bc
(1 − b)c
b(1 − c)
(1 − b)(1 − c)
A0B1
(1 + G)P'01 (1 − G)q'0 + (1 + G)q'1
0
c
0
1−c
A1B0
(1 − G)P'10 (1 − G)q'0 + (1 + G)q'1
0
0
b
1−b
A1B1
(1 + G)P'11 (1 − G)q'0 + (1 + G)q'1
0
0
0
1
1
q1
We now proceed to a more detailed analysis of the system (3). Since we are interested in finding the condition for the evolution of a preference that promotes the acquisition of maladaptive cultural variants, we focus on the cases in which the less viable state has the greater attractiveness and is hence more favored in biased transmission; otherwise the dynamic is uninteresting, which should be clear from eqns (3). Hence we may assume that G q 0 and s q 0 without loss of generality.
151
1 G q 0, s q 0, c = b. If the two traits are equally likely to be transmitted via the interaction so that c = b, we have, from eqns (3),
Dq1 = Dp1 ·
$
%
0
q0q1 Gp1 = 2b · − s ·q0q1. D 1−G(q0 − q1)
Then at equilibrium, we obtain 1−G b + ·px 1, 2G s
qx 1 = −
(4)
where a caret indivates the equilibrium value. Equation (4) indicates that a line of internal equilibria emerges and that Fisherian type of runaway dynamics for both the preference and the viability trait is observed as in the conventional models of sexual selection (Lande, 1981; Kirkpatrick, 1982; Laland, 1994; see also Boyd & Richerson, 1985, chapter 8). While the line of equilibria is inferred to be always stable in Kirkpatrick’s (1982) model of sexual selection, here the line may be unstable (see also Lande, 1981; Laland, 1994). Usual local stability analysis of the equilibria yields three eigenvalues, l1 = 1, l2 = 1 + 2b ·
0
1
G s · D − ·qx 0qx 1 , 1−G(qx 0 − qx 1) b
and l3 = 1 − 2b + (2sD /px 1) − s(qx 0 − qx 1). The possibility of the neutral stability of the equilibria is indicated by l1 = 1. If the other two eigenvalues are less than unity, the movement along the line of equilibria cannot be predicted from an ordinary linear analysis, and the equilibria are locally stable with respect to perturbations which move the population
p1
1
F. 1. The two-dimensional projected space in case 1. The vertical axis, q1, is the frequency of B1 individuals and the horizontal axis, p1, is the frequency of A1 individuals. In the figure, bold lines indicate the stable equilibria. Parameters are b = c = 0.025, s = 0.015, and G = 0.5. The figure also shows some results of the numerical iteration of the approximated system (3), plotted at intervals of 50 time units. For above parameter values, the approximated system (3) exhibits evolutionary dynamics very similar to the system (2). This holds also for the parameter values for Figs 2–4 (see below).
away from the equilibrium line. Since sign (l2 − 1) = sign(b/s − qx 0qx 1/Dx ), the line is locally stable only if b/s Q qx 0qx 1/D ,
(5)
and unstable otherwise. Noting that q0q1/D = Dq1/Dp1 [see eqns (3)] and b/s is the slope of the equilibrium line (4), this stability criterion (5) is analogous to the one found in a quantitative genetic model of sexual selection [see eqns (10) and (12) and figure 1 in Lande, 1981, p. 3723]. Hence as revealed by Lande (1981), if the condition (5) holds so that the equilibrium line (4) is stable, the system eventually converges to a point on the equilibrium lines irrespective of the initial state (see Fig. 1); limit cycles were never produced in the numerical iterations of the system (3). Moreover, once the system attains this neutrally stable state, no evolutionary effects arising from the assumption of the model cause further deterministic changes. This leaves open the possibility that relatively weak effects, such as random drift or natural selection at the interdemic level, may ultimately govern the evolutionary fate of the system (see Lande, 1981; Kirkpatrick, 1982; Boyd & Richerson, 1990; Koeslag, 1997). On the other hand if the condition (5) is not satisfied and the line (4) is unstable, the system moves away from
.
152
the line of interior equilibrium, until the boundary equilibria, given by qx 1 = 0 or qx 1 = 1, are attained. Additionally, when the inequality b/s Q (1 + G)/(2G)
(6)
holds, substitution of px 1 = 1 into eqn (4) yields qx 1 = b/s − (1 − G)/(2G) ( Q 1). Hence we obtain an equilibrium curve as in Fig. 2 that does not occur in the two-locus model of sexual selection (Kirkpatrick, 1982). Note that the inequality (6) is more likely to hold for stronger natural selection (larger s), less efficient cultural transmission of the traits (smaller b), and weaker cultural selection (smaller G). If these evolutionary effects are set to satisfy the condition (6), the system will then never reach the fixation of the maladaptive variant, while fixation may occur for the preference favoring that variant. 2 G q 0, s q 0, c $ b. The assumption c = b in the previous case is based solely on mathematical considerations. To posit generality, it seems reasonable to assume c $ b instead. This may represent the case when the preference of an individual is less prone to transmission compared with the viability trait so
that c Q b, or vice versa. Setting c $ b in eqns (3), we find that interior equilibria as observed in case 1 disappear and that only boundary equilibria are observed (see also Aoki, 1989; Laland, 1994). There are three distinct sets of equilibria and each can be locally stable under certain conditions. Two of the three comprise lines of equilibria, each corresponding to the fixation of one of the two states for the trait B (qx 1 = 0 or qx 1 = 1 with D = 0). These boundary equilibria always exist. The remaining equilibrium is located on a point given by px 1 = 1, qx 1 = b/ s − (1 − G)/(2G), and D = 0, and exists if (1 − G)/(2G) Q b/s Q (1 + G)/(2G).
(7)
Although other sets of parameter values, such as p1 = (1 + G)s/(2Gc), q1 = 1, and D = (1 + G) (b + c − s)/(2Gc), may seem to satisfy the equilibrium condition (Dp1, Dq1, DD) = (0, 0, 0) [see eqns (3)], such equilibria are not conceivable because they necessarily violate the definition of the linkage disequilibrium D = P00P11 − P01P10. For the first set of equilibria where the population is fixed on the state B0 (qx 1 = 0), the three eigenvalues that determine its local stability are l1 = 1, l2 = 1 + [2bGpx 1/(1 − G)] − s,
1
and l3 = 1 − (b + c + s) ( Q 1).
q1
Again l1 = 1 indicates the possibility of neutral stability. Since sign(l2 − 1) = sign(px 1 − (s/b)·[(1−G)/ (2G)]), it is locally stable with respect to perturbations which move the population away from the equilibrium line if px 1 Q (s/b)·[(1−G)/(2G)].
(8)
The corresponding condition for the fixation of B1 (qx 1 = 1) becomes px 1 q (s/b)·[(1+G)/(2G)] and b + c q s, since three eigenvalues are 0
p1
1
F. 2. The two-dimensional projected space in case 1 with the condition (6). As in the Fig. 1, the vertical axis is the frequency of B1 individuals and the horizontal axis is the frequency of A1 individuals. Bold lines indicate the stable equilibria. Parameters are b = c = 0.02, s = 0.015, and G = 0.5. Hence the inequality (6) is satisfied and the equilibrium line does not reach the fixation of the state B1 (q1 = 1). Some trajectories for the system (3) are plotted at intervals of 50 time units.
l1 = 1, l2 = 1 − [2bGpx 1/(1 + G)] + s, and l3 = 1 − (b + c) + s.
(9)
q1
1
0
p1
1
F. 3. The two-dimensional projected space in case 2 with b q c. In the figure, bold lines indicate the stable equilibria, and the two broken lines indicate the isoclines Dp1 = 0 and Dq1 = 0, respectively. The figure is drawn with parameters b = 0.025, c = 0.02, s = 0.015, and G = 0.5. Some trajectories are plotted at intervals of 50 time units.
The local stability analysis for the third equilibrium yields an eigenvalue 1 − bqx 0qx 142G/[1 − G(qx 0 − qx 1)]52 ( R 1), whereas the other two are the roots of a quadratic
153
the present situation. Then the correlated evolution of the preference trait as depicted above does not occur. I suspect that this should be the case if the preference trait A is transmitted genetically. Then it may be concluded that the genetic determination of the preference prevents the emergence of the runaway process, eventually producing only those preferences that favor a variant that improves the individual fitness. Since this conclusion is, strictly speaking, based on a model of within-generation transmission, and may only be extended to transgenerational processes in limited situations (see below), analyses of cultural selection models that explicitly incorporate (transgenerational) genetic inheritance of the preferences are required. This is done in the Appendix. As described in the Appendix, the genetic determination of the preference indeed precludes the emergence of the runaway dynamics, irrespective of which of the two modes of cultural transmission, oblique or vertical, of the viability trait are considered (see the Appendix). In both cases, we have sign(Dp1) = sign( − G), implying that only the transmission bias that favors the adaptive state (in this case, B0) may increase its frequency [see eqns (A.1a) and (A.3)]. The cultural determination of the preference, on the other hand, seems to allow runaway dynamics as discussed above.
1
f(l) 0 C0 + lC1 + l 2 = 0, where C0 = 1 + (b − c)[1 − (s 2qx 0qx 1 /b)] − (s/G)
Noting that both roots are less than unity if f(1) q 0 and df(1)/dl q 0, the stability condition for the equilibrium is obtained as c q b.
q1
and C1 = − 2 − b + c + (s/G).
(10)
The stability properties of the equilibria in case 2 are illustrated in Figs 3 and 4. We can see from the above conditions that the evolution of A1 as well as B1 becomes less likely for less efficient cultural transmission (smaller b and/or c), weaker cultural selection (smaller G), and stronger natural selection (larger s). In particular, when the trait A is not culturally transmitted at all so that individuals never change their preferences during the interaction processes (c = 0), Dp1 = −sD should always be negative unless D Q 0, which is unlikely in
0
p1
1
F. 4. The two-dimensional projected space in case 2 with b Q c. In the figure, bold lines and a large dot (W) indicate the stable equilibria, and the two broken lines indicate the isoclines Dp1 = 0 and Dq1 = 0, respectively. The figure is drawn with parameters b = 0.02, c = 0.025, s = 0.015, and G = 0.5. Some trajectories are plotted at intervals of 50 time units.
.
154 Discussion
The present study of the evolution of preferences in cultural selection suggests that there are cases where ordinary cultural processes lead to the evolution of both the deleterious cultural variant and the associated preference that favors it. The runaway process is observed only when the preference of an individual, together with its viability and attractiveness, can be repeatedly changed during its ontogenetic process, or in other words, only if it is culturally acquired (c q 0 in the above model). Genetic determination of the preference (c = 0) seems to preclude the emergence of the cultural runaway, and only the preferences that favor adaptive variants may eventually evolve. This is further supported by mathematical analyses of the models that explicitly incorporate genetic determination of the preference trait (see Appendix). This observation, that genetic preferences preclude the emergence of runaway dynamics (while culturally transmitted preferences do not), may give some insight into the nature of social learning behavior in (non-human) animals. In contrast to cultural processes in human socieities, which may often result in maladaptive cultural practices, most reported instances of social learning in non-human animals seem to promote the acquisition of adaptive behaviors (Galef, 1995, 1996; Laland, 1996). This interspecific difference in social information transfer may be understood if we consider how transmission bias is determined and transmitted between individuals in these species. Social learning in non-human animals is usually characterized by its transient property and learning mechanisms that are highly dependent on individual experiences (trial and error learning); on the other hand, human social learning enables stable transmission of information that may further lead to cumulative accumulation of cultural traditions (Galef, 1988; Whiten & Ham, 1992; Laland et al., 1993, 1996; Tomasello et al., 1993; see also Heyes, 1993). Then in non-human animals, with relatively simple forms of social learning mechanisms, transmission bias in social information transfer may be well modified and adjusted through trial and error learning to promote the acquisition of adaptive behaviors (Galef, 1995), and social interaction with other individuals should have only a small effect on the preference. If the preferences are indeed acquired through individual (rather than social) learning, then genetics should play a decisive role in the evolution of the preference (see Cavalli-Sforza & Feldman, 1983b), and hence in these species only those predispositions in favor of behaviors that improve Darwinian (or
inclusive) fitness might have evolved, as suggested from the above analyses. In contrast, human behaviors are so highly dependent on cultural processes that transmission bias may itself be transmitted culturally. In these situations, runaway cultural selection may give rise to the maladaptive cultural variants, as indicated in the present analyses. Such views on social information transfer in human and non-human animals are still questioned by many researchers (e.g. Tooby & Cosmides, 1989; Alexander, 1990; Ormrod, 1992; Sperber, 1994; Flinn, 1997), and the above consideration obviously deserves further investigations. As we have seen above, mathematical properties of the present model of cultural selection very much resemble to those of the conventional major gene models of sexual selection; for example, they both exhibit Fisherian type of runaway dynamics, if certain conditions are met (see Figs 1 and 2; see also Figs 3 and 4). This goes along with our intuition since biased transmission in cultural inheritance is formally analogous to selective mating in genetic transmission (see Richerson & Boyd, 1989), and the system (3) [or (2)] is mathematically very similar to the sexual selection models with two-locus haploid genetics (Kirkpatrick, 1982; see also Laland, 1994). Therefore, inferences from preceding studies of sexual selection may be fruitful in investigating cultural selection (Richerson & Boyd, 1989). The present study assumes that the preference in cultural selection, whether it is transmitted genetically or culturally, does not induce an additional viability cost. As can be inferred from the studies on costly mate preferences (Pomiankowski, 1987; Bulmer, 1989; Pomiankowski et al., 1991), an assumption of costly preferences should greatly alter the outcome of the present analyses. The resemblance is, however, merely of mathematical importance, and does not have any biological implications. For example, the evolution by cultural selection as revealed by the present study differs from the evolution by sexual selection, in that the runaway dynamics may be observed only when the preference is also culturally transmitted. This is not so in the models of sexual selection, where the system may exhibit runaway dynamics whether mating preferences are determined genetically or culturally (Kirkpatrick, 1982; Laland, 1994). It should be kept in mind that cultural selection is utterly a distinctive process that should clearly be discriminated from sexual selection. As clearly indicated in the present investigation, culture may at times reduce fitness of individuals. Therefore, simplistic adaptive interpretations of cultural traits are unconvincing without detailed
specificiation of the cultural processes involved. Possible examples of cultural runaway as presented in this study may include human body mutilations or deformations, such as piercing, tattooing, tooth removing, female circumcision, etc., that have been reported in many human populations (see Darwin, 1871, chapters 19–21; Boyd & Richerson, 1985, pp. 268–271; Logan & Qirko, 1996). Although appropriate data are lacking at present, I expect that sociological (or other) surveys on cultural changes in modern human populations should provide examples suitable for discussing the effects of cultural selection on sociocultural dynamics (see Durham, 1991; Laland et al., 1995; Boehm, 1996). Cultural runaway of this kind may further help to explain the existence of traits and preferences that are apparently maladaptive at the individual level but may be adaptive for the group, such as (non-kin, non-reciprocal) altruism (see Richerson & Boyd, 1992; Laland, 1994). As we have seen above, cultural selection may exhibit lines of equilibria that are themselves locally (or globally, as inferred from numerical iterations; see above) stable. The emergence of such neutral stability leaves open the possibility that relatively minor evolutionary effects, such as random drift or natural selection at the interdemic level, may play a crucial role in determining the course of evolutionary changes (see Lande, 1981; Kirkpatrick, 1982; Koeslag, 1997; see also Boyd & Richerson, 1990). Some authors have suggested that group selection among cultural variants, usually termed cultural group selection, may have had profound effects on sociocultural dynamics among human socieites (e.g. Campbell, 1975; Boyd & Richerson, 1982, 1985, 1990; Richerson & Boyd, 1989; Dugatkin, 1992; Findlay, 1992; Wilson & Sober, 1994; Soltis et al., 1995; Boehm, 1996; Wilson & Dugatkin, 1997). If this is indeed so, then certain types of social dilemma may be overcome by the joint effects of cultural runaway and natural selection at the interdemic level, since biased transmission in favor of altruists may dominate the effects of individual selection against them, further rendering intergroup processes sufficiently effective in promoting the evolution of cooperation (Richerson & Boyd, 1992). To show that simple cultural processes may at times lead to the evolution of a preference that favors a maladaptive variant, this study presents an alternative way to model cultural changes, where cultural elements are transmitted in a manner more like infections in epidemioloical (or parasitic) transmission (Moritz, 1990; Lynch, 1991; Dawkins, 1993; see also Cavalli-Sforza & Feldman, 1981,
155
chapter 1; Sperber, 1985, 1990, 1994; Dennett, 1995, chapter 12). A similar method has been employed in investigating the geographic spread of certain cultural practices such as farming (Ammerman & Cavalli-Sforza, 1987; Aoki et al., 1996; see also Aoki 1987; Rogers, 1995), where culturally different populations meet and exchange new cultural practices. Scholars of cultural changes seem to be more concerned with changes that occur within a relatively short time period compared with evolutionary time-scales. The present approach should have some benefits over conventional models in describing such rapid changes, because the conventional formulation is based on changes between generations and is difficult to follow the changes within a generation (see also Carotenuto et al., 1989). Studies of social learning in non-human animals, which appears to be characterized by highly horizontal social transmission (Laland et al., 1993, 1996), may also benefit by utilizing the method. Furthermore, although the present model neglects birth-death processes of individuals and hence may appear to fail in describing sociocultural changes that proceed over generations, above arguments also apply without modifications to such situations if individuals are replaced by newly born individuals of same phenotypes; e.g. if the mating is random and vertical cultural transmission is unbiased with complete linkage of traits. I expect that the approach presented here should prove useful when applying the theory of cultural transmission to other fields in social sciences. However, although the models of cultural change may describe the consequence of, say, a preference and an associated attractive character, they do not explain the causes of the preference nor the character being attractive; and usually, it is this very point that many social scientists regard as issues of interest (Sober, 1991). Obviously, there is more to culture than a system of inheritance. The author thanks K. Aoki, Carl Bergstrom, Marc Feldman, and Kevin Laland. Part of this research was performed during his stay at Stanford University, supported by NIH Grant GM28016 to M. W. Feldman. The research is also supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists, and Grant-in-Aid for Encouragement of Young Scientists from The Ministry of Education, Science, Sports and Culture, Japan.
REFERENCES A, R. D. (1979). Darwinism and Human Affairs. Seattle: University of Washington Press. A, R. D. (1990). Epigenetic rules and Darwinian algorithms. Ethol. Sociobiol. 11, 241–303.
156
.
A, A. J. & C-S, L. L. (1984). The Neolithic Transition and the Genetics of Populations in Europe. Princeton: Princeton University Press. A, K. (1987). Gene-culture waves of advance. J. Math. Biol. 25, 453–464. A, K. (1989). A sexual selection model for the evolution of imitative learning of song in polygynous birds. Am. Nat. 134, 599–612. A, K. (1990). A shifting balance type model for the origin of cultural transmission. In: Proceedings of the Fourth International Symposium in Conjunction with the Awarding of the International Prize for Biology: Population Biology of Genes and Molecules (Takahata, N. & Crow, J. F., eds) pp. 123–137. Tokyo: Baifukan. A, K. (1991). Some theoretical aspects of the origin of cultural transmission. In: Evolution of Life (Osawa, S. & Honjo, T., eds) pp. 439–449. Tokyo: Springer. A, K. & F, M. W. (1987). Toward a theory for the evolution of cultural communication: coevolution of signal transmission and reception. Proc. Natl. Acad. Sci. U.S.A. 84, 7164–7168. A, K. & F, M. W. (1989). Pleiotropy and preadaptation in the evolution of human language capacity. Theor. Popul. Biol. 35, 181–194. A, K. & F, M. W. (1991). Recessive hereditary deafness, assortative mating, and persistence of a sign language. Theor. Popul. Biol. 39, 358–372. A, K. & F, M. W. (1994). Cultural transmission of a sign language when deafness is caused by recessive alleles at two independent loci. Theor. Popul. Biol. 45, 101–120. A, K., S, M. & S, N. (1996). Traveling wave solutions for the spread of farmers into a region occupied by hunter-gatherers. Theor. Popul. Biol. 50, 1–17. B, J. H. (1989). The elastic between genes and culture. Ethol. Sociobiol. 10, 111–129. B J, N. G. (1990). Three sensible paradigms for research on evolution and human behavior? Ethol. Sociobiol. 11, 353–359. B, C. (1996). Emergency decisions, cultural-selection mechanics, and group selection. Curr. Anthropol. 37, 763–793. B, R. & R, P. J. (1982). Cultural transmission and the evolution of cooperative behavior. Hum. Ecol. 10, 325–351. B, R. & R, P. J. (1985). Culture and the Evolutionary Process. Chicago: University of Chicago Press. B, R. & R, P. J. (1990). Group selection among alternative evolutionarily stable strategies. J. theor. Biol. 145, 331–342. B, M. (1989). Structural instability of models of sexual selection. Theor. Popul. Biol. 35, 195–206. C, D. T. (1975). On the conflicts between biological and social evolution and between psychology and moral tradition. Am. Psychol. 30, 1103–1126. C, L., F, M. W. & C-S, L. L. (1989). Age structure in models of cultural transmission. The Morrison Institute for Population and Resource Studies Working Paper Series: Paper number 0016. C-S, L. L. & F, M. W. (1973). Cultural versus biological inheritance: phenotypic transmission from parents to children (a theory of the effect of parental phenotypes on children’s phenotype). Am. J. Hum. Genet. 25, 618–637. C-S, L. L. & F, M. W. (1981). Cultural Transmission and Evolution: A Quantitative Approach. Princeton: Princeton University Press. C-S, L. L. & F, M. W. (1983a). Paradox of the evolution of communication and of social interactivity. Proc. Natl. Acad. Sci. U.S.A. 80, 2017–2021. C-S, L. L. & F, M. W. (1983b). Cultural versus genetic adaptation. Proc. Natl. Acad. Sci. U.S.A. 80, 4993–4996. C, L. (1995). Is there a role for culture in human behavioral ecology? Ethol. Sociobiol. 16, 181–205. D, C. (1871). The Descent of Man, and Selection in Relation to Sex. London: John Murray.
D, R. (1976). The Selfish Gene. Oxford: Oxford University Press. D, R. (1982). The Extended Phenotype: The Gene as the Unit of Selection. Oxford: W. H. Freeman. D, R. (1993). Viruses of the mind. In: Dennett and His Critics: Demystifying Mind (Dahlbom, B., ed.) pp. 13–27. Oxford: Blackwell. D, D. C. (1995). Darwin’s Dangerous Idea: Evolution and the Meaning of Life. New York: Simon and Schuster. D, L. A. (1992). The evolution of the ‘‘con artist’’. Ethol. Sociobiol. 13, 3–18. D, W. H. (1991). Coevolution: Genes, Culture, and Human Diversity. Stanford: Stanford University Press. E, I. & C-S, L. L. (1982). Assortment of encounters and evolution of cooperativeness. Proc. Natl. Acad. Sci. U.S.A. 79, 1331–1335. F, M. W. & A, K. (1992). Assortative mating and grandparental transmission facilitate the persistence of a sign language. Theor. Popul. Biol. 42, 107–116. F, M. W., A, K. & K, J. (1996). Individuals versus social learning: evolutionary analysis in a fluctuating environment. Anthropol. Sci. 104, 209–232. F, M. W. & C-S, L. L. (1976). Cultural and biological evolutionary processes, selection for a trait under complex transmission. Theor. Popul. Biol. 9, 238–259. F, M. W. & C-S, L. L. (1984). Cultural and biological evolutionary processes: gene-culture disequilibrium. Proc. Natl. Acad. Sci. U.S.A. 81, 1604–1607. F, M. W. & L, K. N. (1996). Gene-culture coevolutionary theory. Trends Ecol. Evol. 11, 453–457. F, M. W. & Z, L. A. (1992). Gene-culture coevolution: toward a general theory of vertical transmission. Proc. Natl. Acad. Sci. U.S.A. 89, 11935–11938. F, C. S. (1992). Phenotypic evolution under gene-culture transmission in structured populations. J. theor. Biol. 156, 387–400. F, M. V. (1997). Culture and the evolution of social learning. Evol. Hum. Behav. 18, 23–67. G, B. G., J. (1988). Imitation in animals: field observations and laboratory analyses. In: Social Learning: Psychological and Biological Perspectives (Zentall, T. R. & Galef, B. G., Jr., eds) pp. 3–28. Hillsdale: Lawrence Erlbaum Associates. G, B. G., J. (1995). Why behaviour patterns that animals learn socially are locally adaptive. Anim. Behav. 49, 1325–1334. G, B. G., J. (1996). The adaptive value of social learning: a reply to Laland. Anim. Behav. 52, 641–644. G, C. R., V, C., H, B. & C-S, L. L. (1995). Cultural variation in Africa: role of mechanisms of transmission and adaptation. Proc. Natl. Acad. Sci. U.S.A. 92, 7585–7589. H, C. M. (1993). Imitation, culture and cognition. Anim. Behav. 46, 999–1010. H, R. A. (1982). Ethology: Its Nature and Relations with Other Sciences. Oxford: Oxford University Press. H, R. A. (1987). Individuals, Relationships and Culture: Links between Ethology and the Social Sciences. Cambridge: Cambridge University Press. K, M. (1982). Sexual selection and the evolution of female choice. Evolution 36, 1–12. K, J. H. (1997). Sex, the Prisoner’s Dilemma Game, and the evolutionary inevitability of cooperation. J. theor. Biol. 189, 53–61. K, J., L, K. N. & F, M. W. (1994). Gene-culture coevolution and sex ratios: the effects of female infanticide, sex-selective abortion, sex selection and sex-biased parental investment on the evolution of sex-ratios. Theor. Popul. Biol. 46, 249–278. L, K. N. (1994). Sexual selection with a culturally transmitted mating preference. Theor. Popul. Biol. 45, 1–15. L, K. N. (1996). Is social learning always locally adaptive? Anim. Behav. 52, 637–640.
L, K. N., K, J. & F, M. W. (1995). Gene-culture coevolutionary theory: a test case. Curr. Anthropol. 36, 131–156. L, K. N., R, P. J. & B, R. (1993). Animal social learning: toward a new theoretical approach. In: Perspectives in Ethology. Vol. 10. Behavior and Evolution (Bateson, P. P. G., Klopfer, P. H. & Thompson, N. S., eds) pp. 249–277. New York: Plenum. L, K. N., R, P. J. & B, R. (1996). Developing a theory of animal social learning. In: Social Learning in Animals: The Roots of Culture (Heyes, C. M. & Galef, B. G., Jr., eds) pp. 129–154. New York: Academic Press. L, R. (1981). Models of speciation by sexual selection on polygenic traits. Proc. Natl. Acad. Sci. U.S.A. 78, 3721–3725. L, R., K, D. & C, J. (1968). Selective mating, assortative mating, and inbreeding: definitions and implications. Eugen. Quart. 15, 141–143. L, M. H. & Q, H. N. (1996). An evolutionary perspective on maladaptive traits and cultural conformity. Am. J. Hum. Biol. 8, 615–629. L, C. J. & W, E. O. (1981). Genes, Mind, and Culture: The Coevolutionary Process. Cambridge: Harvard University Press. L, A. (1991). Thought contagion as abstract evolution. J. Ideas 2, 3–10. MD, K. (1989). The plasticity of human social organization and behavior: contextual variables and proximate mechanisms. Ethol. Sociobiol. 10, 171–194. M, E. (1990). Memetic science: I—general introduction. J. Ideas 1, 3–23. O, R. K. (1992). Adaptation and cultural diffusion. J. Geograph. 91, 258–262. P, A. P. (1987). The costs of choice in sexual selection. J. theor. Biol. 128, 195–218. P, A. P., I, Y. & N, S. (1991). The evolution of costly mate preferences. I. Fisher and biased mutation. Evolution 45, 1422–1430. R, P. J. & B, R. (1989). The role of evolved predispositions in cultural evolution: or, human sociobiology meets Pascal’s Wagner. Ethol. Sociobiol. 10, 195–219. R, P. J. & B, R. (1992). Cultural inheritance and evolutionary ecology. In: Evolutionary Ecology and Human Behavior (Smith, E. A. & Winterhalder, B., eds) pp. 61–92. New York: Aldine de Gruyter. R, E. M. (1995). Diffusion of Innovations, 4th ed. New York: Free Press. S, E. (1991). Models of cultural evolution. In: Trees of Life: Essays in the Philosophy of Biology (Griffiths, P., ed.) pp. 17–38. Norwell: Kluwer Academic. S, J. S., B, R. & R, P. J. (1995). Can group-functional behaviors evolve by cultural group selection? An empirical test. Curr. Anthropol. 36, 473–494. S, D. (1985). Anthropology and psychology: towards an epidemiology of representations. Man 20, 73–89. S, D. (1990). The epidemiology of beliefs. In: The Social Psychological Study of Widespread Beliefs (Fraser, C. & Gaskell, G., eds) pp. 25–44. Oxford: Oxford University Press. S, D. (1994). The modularity of thought and the epidemiology of representations. In: Mapping the Mind: Domain Specificity in Cognition and Culture (Hirschfeld, L. A. & Gelman, S. A., eds) pp. 39–67. New York: Cambridge University Press. T, K. (1997). Models of selective mating and the initiation of the Fisherian process. Proc. Roy. Soc. Lond. B 264, 839–844. T, K. & A, K. (1995). Two-locus haploid and diploid models for the coevolution of cultural transmission and paternal care. Am. Nat. 146, 651–684. T, M., K, A. C. & R, H. H. (1993). Cultural learning. Behav. Brain Sci. 16, 495–552. T, J. & C, L. (1989). Evolutionary psychology and the generation of culture, part I: theoretical considerations. Ethol. Sociobiol. 10, 29–49.
157
T, J. & C, L. (1990). The past explains the present: emotional adaptations and the structure of ancestral environments. Ethol. Sociobiol. 11, 275–424. W, A. & H, R. (1992). On the nature and evolution of imitation in the animal kingdom: reappraisal of a century of research. Adv. Study Behav. 21, 239–283. W, D. S. & D, L. A. (1997). Group selection and assortative interactions. Am. Nat. 149, 336–351. W, D. S. & S, E. (1994). Reintroducing group selection to the human behavioral sciences. Behav. Brain Sci. 17, 585–654.
APPENDIX In this section I analyse simple genetic models of cultural selection to argue that runaway dynamics as described in the text does not occur if preferences are transmitted genetically. Models developed here differ from those in the text in that they explicitly assume genetic transmission of the preference trait A, which produces the transmission bias toward the viability trait; other assumptions are as in the text. For the sake of simplicity, haploid inheritance of the trait is postulated. Two models with different assumptions are discussed. The first model assumes nonvertical (or, more precisely, oblique) transmission of the cultural viability trait B, and the second model assumes that the trait is transmitted vertically. To see if cultural selection could alone lead to the evolution of a preference in favor of a maladaptive cultural variant, it is required to preclude the possibility of other evolutionary effects such as sexual selection. Hence the mating is assumed to be random.
(i) Oblique Transmission of the Viability Trait B The first model assumes that the viability trait B is culturally transmitted via oblique pathway, from adults of the previous generation to genetically unrelated individuals of the offspring generation (Cavalli-Sforza & Feldman, 1981). Cultural transmission of the trait is assumed to occur after the reproduction but before the operation of natural selection in each generation. Before the cultural transmission of the viability trait, receivers (offspring) are assumed to be uncultured with respect to the trait. Hence, there are only two types of receivers, A0 and A1 . In addition, since the states of the preference trait of transmitters (adult of the previous generation) do not affect the oblique transmission process, the trait A of the transmitters need not be considered. Before the oblique transmission of the trait B, the genotypic frequencies of the trait A among offspring are simply given by the corresponding frequencies in the adults of the previous generation. This is because
.
158
T A1 Meeting table for the oblique transmission model Receiver
Transmission
Frequency of interaction
Phenotype of the receiver after interaction
A0
B0
p0q0
A 0B 0
B1
p0q1
A 0B 1
B0
(1−G)q0 p1 · (1 − G)q0 + (1 + G)q1
A 1B 0
B1
p1 ·
(1+G)q1 (1 − G)q0 + (1 + G)q1
A 1B 1
A1
Since the receivers (naive offspring) are assumed to be uncultured with respect to the viability trait B before the oblique transmission, there are only two types of receivers, A0 and A1. In addition, since the preference of transmitters (adult of the previous generation) has nothing to do when oblique transmission occurs from adults to offspring, their states for the preference trait A is not indicated in the table.
T A2 Mating table for the vertical transmission model Phenotypic distribution Mating
Mating frequency
A0B0 × A0B0
2 P00
1
0
0
0
A0B1
2P00P01
1/2
1/2
0
0
A1B0
2P00P10
1/2
0
1/2
0
A1B1
2P00P11
1/4
1/4
(1 − G)/4
(1 + G)/4
2 P01
0
1
0
0
A1B0
2P01P10
1/4
1/4
(1 − G)/4
(1 + G)/4
A1B1
2P01P11
0
1/2
0
1/2
2 P10
0
0
1
0
2P10P11
0
0
(1 − G)/2
(1 + G)/2
2 P11
0
0
0
1
A0B1 × A0B1
A1B0 × A1B0 A1B1 A1B1 × A1B1
A0B0
natural selection is assumed to act via differential viability (and not the fertility) of individuals, and the mating is assumed to be random. Hence the genotypic frequencies are unchanged through reproduction process. Oblique transmission as described in Table A1 follows. Note that the phenotype of a naive offspring with state Ai is modified to AiBj through cultural transmission process, if the offspring interacts with an adult with state Bj (see Table A1). Then the frequency of interaction between, say, a naive offspring with state Ai and an adult with state Bj as given in the Table A1 is just the frequency of AiBj offspring before natural selection. Viability selection then completes the life cycle. Following the ordinary methodologies of gene-culture coevolutionary models (Feldman & CavalliSforza, 1976; Feldman & Laland, 1996), the frequency change in the preference trait A between
A0B1
A 1B 0
A1B1
generations is derived as w ·Dp1=−2G ·
sp0p1q0q1 , (A.1a) (1−G)q0 + (1 + G)q1
where
$
w = 1 − s p0 +
%
(1 + G)p1 q (1 − G)q0 + (1 + G)q1 1
(A.1b)
is the average viability of individuals. Recalling that s q 0 (see text), we have from eqn (A.1a), sign(Dp1) = sign( − G), implying that only the transmission bias that favor the adaptive state (in this case, B0) may evolve. (ii) Vertical Transmission of the Viability Trait B In this model, cultural transmission of the viability trait B is assumed to occur from parents to their
offspring soon after, or simultaneously with, reproduction. To apply the biased cultural transmission model (1) in the text, it is assumed that two parents (mother and father) constitute the transmitters, while their offspring are the receivers. It is further assumed that the genetically transmitted trait A and the cultural trait B are transmitted independently. This assumption is equivalent to complete recombination (recombination rate r = 1/2) in genetic inheritance. With these assumptions, the mating table for the present model is obtained as in Table A2. Viability selection then follows, and the recursions for the phenotypic frequencies are derived from the table as
w ·P'00=(P00 + p0q0)/2,
(A.2a)
w ·P'01/(1 − s) = (P01 + p0q1)/2,
(A.2b)
w ·P'10=(P10 + p1q0)/2 − G(P10q1 + P11q0)/2,
(A.2c)
159
and w ·P'11/(1−s) = (P11 + p1q1)/2 + G(P10q1 + P11q0)/2,
(A.2d)
where w = 1 − sq1 − sG(P10q1 + P11q0)/2
(A.2e)
is the average viability of individuals. The difference equation for the genetic preference A becomes w ·Dp1= −s(P11 − p1q1)/2 − sGp0(P10q1 + P11q0)/2.
(A.3)
The first term, − s(P11 − p1q1)/2 = −sD/2, corresponds to the changes due purely to natural selection effects, and becomes zero at gene-culture equilibrium D = 0 (Feldman & Cavalli-Sforza, 1984; Feldman & Zhivotovsky, 1992). Assuming that D q 0, which seems reasonable for the present argument (see text), Dp1 q 0 requires G Q 0, implying that only the transmission bias that favor the adaptive state (B0) may evolve, as in the oblique transmission model as described above.
Evolution of Transmission Bias in Cultural Inheritance K T* Department of Biological Sciences, Graduate School of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan (Received on 28 February 1997, Accepted in revised form on 8 September 1997)
Evolution of transmission bias in cultural inheritance is investigated using simple models of cultural selection. Conventional models of cultural transmission describe cultural changes by incorporating transmission bias and non-vertical pathways into the ordinary population genetic framework. The methodology has been successful in understanding cultural changes in terms of natural selection, but it is difficult to see from the theoretical framework how biased transmission in favor of maladaptive traits might have evolved. To show that ordinary cultural processes lead at times to the evolution of a preference that favors a deleterious cultural variant, this study presents an alternative model of cultural transmission, where cultural elements are transmitted in a manner more like infections in epidemiological transmission. An ordinary equilibrium analysis indicates that, under certain conditions, runaway dynamics emerges and the coevolution of a maladaptive cultural variant and an associated preference in favor of the maladaptive variant is observed. If the preference of an individual does not change during its ontogeny (e.g., if it is transmitted genetically), however, then cultural selection alone does not produce such runaway dynamics, and only those preferences that favor adaptive variants should eventually evolve. Since cultural processes may at times result in a reduction in the fitness of individuals, simplistic adaptive interpretations of culture are unconvincing without detailed specification of the cultural processes involved. Moreover, cultural runaway of this kind may help to explain the existence of traits that are apparently maladaptive at the individual level but may be advantageous for the group. Inferences are also made regarding the observed differences between human and non-human social information transfer. 7 1998 Academic Press Limited
*E-mail: k32148.m-unix.cc.u-tokyo.ac.jp
1981; Durham, 1991), and has been regarded as a primary effect that may govern the dynamics of cultural changes (e.g. Durham, 1991; Dugatkin, 1992; Laland, 1994; Laland et al. 1995; Dawkins, 1976; Ormrod, 1992; Galef, 1995; Boehm, 1996). It has repeatedly been argued in behavioral evolutionary studies that cultures result from processes in the human brain that have been shaped through natural selection, and that resulting psychological mechanisms influence which variants observed in the surrounding cultural milieu are adopted (e.g. Lumsden & Wilson, 1981; Hinde, 1987; Richerson & Boyd, 1989; Alexander, 1990). These authors suggest that choices will be made according to the evolved algorithms of the relevant psychological mechanisms to enhance Darwinian (or inclusive) fitness of individuals (Alexander, 1979; Flinn, 1997). Their consequences on the phenotype are therefore likely to be adaptive, eventually producing transmission bias that usually
0022–5193/98/020147 + 13 $25.00/0/jt970541
7 1998 Academic Press Limited
Introduction Theorists have seen that when culture is modeled as a system of information inheritance, it is possible to maintain cultural traits that are maladaptive in terms of natural selection [see reviews in Cavalli-Sforza & Feldman, (1981); Boyd & Richerson (1985) and Durham (1991)]. This is true when biased cultural transmission favors maladaptive variants so as to overcome the countervailing effects of natural selection. Biased transmission is said to have occurred when individuals are predisposed to adopt some cultural variants over others in cultural inheritance. In the absence of strong natural selection against them, the favored maladaptive variants increase in frequency. This type of selective processes has been termed cultural selection (Cavalli-Sforza & Feldman,
148
.
favors adaptive cultural variants. In practice, however, this is not always the case (e.g. Hinde, 1987; Barkow, 1989; Richerson & Boyd, 1989, 1992; MacDonald, 1989; Blurton Jones, 1990; Logan & Qirko, 1996). This notion of maladaptive cultural practices goes back to Darwin (1871, chapters 19–21). Many contemporary researchers also recognize that maladaptive cultural practices do arise in human societies, as well as the transmission bias that underlies the adoption of such practices (e.g. Hinde, 1987; Barkow, 1989; Richerson & Boyd, 1989, 1992; Tooby & Cosmides, 1990; Logan & Qirko, 1996). Existence of predispositions that promote the acquisition of maladaptive variants has, therefore, been a crucial issue in the evolutionary studies of human behavior (e.g. Richerson & Boyd, 1989, 1992; Tooby & Cosmides, 1989; Logan & Qirko, 1996; Flinn, 1997). In this paper I provide simple mathematical models of cultural selection, and show that ordinary cultural processes do indeed lead at times to the evolution of a preference that favors a maladaptive cultural variant if that variant has sufficient attractiveness. Conventional models of cultural transmission describe cultural changes by incorporating transmission bias and non-vertical (e.g. oblique and horizontal) pathways into the ordinary population genetic framework (Cavalli-Sforza & Feldman, 1973; see also CavalliSforza & Feldman, 1981; Boyd & Richerson, 1985; Durham, 1991; Guglielmino et al., 1995). In this framework mathematical models are formulated to describe cultural dynamics by deriving frequency changes of cultural elements (analogous to genes or alleles in genetic transmission) in a generation where genetic reproduction should take place. While the methodology has been criticized for a number of reasons (see Tooby & Cosmides, 1989; Heyes, 1993; Sperber, 1994; Cronk, 1995; Galef, 1995), it has been successful in understanding cultural changes from the viewpoint of genetic adaptation and evolution. Indeed, although Cavalli-Sforza & Feldman (1981) deal entirely with purely cultural processes, they had earlier provided a theoretical framework for studies of gene-culture coevolution (Feldman & Cavalli-Sforza, 1976) on a range of problems as diverse as the origin and the initial evolution of cultural capacity (Cavalli-Sforza & Feldman, 1983a, b; Aoki & Feldman, 1987; Aoki, 1990, 1991; Takahasi & Aoki, 1995; Feldman et al., 1996), persistence of a sign language (Aoki & Feldman, 1991, 1994; Feldman & Aoki, 1992; see also Aoki, 1989; Aoki & Feldman, 1989), and the evolution of sex ratio (Kumm et al., 1994; Laland et al., 1995), just to name a few [see Feldman & Laland (1996) for a review]. It is difficult to see, however, from this theoretical framework why
transmission bias might evolve to prefer maladaptive states despite the counteracting effects of natural selection (Hinde, 1982, chapter 15). Contrary to the prevailing views, Dawkins (1976, chapter 11; see also Dawkins, 1982, chapter 6) argues that cultural changes should be understood in terms of the psychological appeal of each cultural element (or meme) irrespective of its effects on genetic adaptation or evolution (see also Sober, 1991). Indeed, recent memetic studies emphasize the formal analogies of cultureal transmission with epidemiological transmission rather than with genetic inheritance (Moritz, 1990; Lynch, 1991; Dawkins, 1993; see also Cavalli-Sforza & Feldman, 1981, chapter 1; Sperber, 1985, 1990, 1994; Dennett, 1995, chapter 12; Rogers, 1995). Unlike the conventional models of cultural transmission, memetic models do not incorporate genetic processes such as mating and reproduction (of genetic offspring) because they have nothing to do with purely cultural (or memetic) processes, which do not involve genetic changes. Following these memetic approaches in part, I develop in this paper simple mathematical models for the joint evolution of a cultural trait (i.e. a trait that may be modified through cultural transmission) that determines the attractiveness and the viability of an individual (trait B) and an associated trait that determines the preference of an individual for trait B (trait A). Evolutionary dynamics of the model system is investigated by following frequency changes of traits during an arbitrary short time period [rather than changes in a generation as in the conventional models; see Cavalli-Sforza & Feldman (1981)] where meeting and transmission of cultural elements among individuals take place. Similar lines of arguments are also found in the quantitative trait model of indirect bias discussed by Boyd & Richerson (1985, chapter 8). The Model Throughout I assume for simplicity that the traits are dichotomous and character states are described by subscripts 0 and 1 for both traits. Hence there are four culturally distinct phenotypes, A0B0, A0B1, A1B0, and A1B1, with their respective frequencies in the population denoted P00, P01, P10, and P11 (SijPij = 1). We investigate the evolutionary dynamics of the system by following frequency changes of the phenotypes during a short time period. Within a unit time, natural selection first acts at the level of differential viabilities of B0 and B1 individuals; B1 individuals have a per unit time viability of 1 − s relative to B0 individuals. On the other hand, the preference do not directly induce natural selection so
that A0 and A1 individuals are equally viable if they maintain the same state for trait B. Hence natural selection only acts indirectly on the preference trait, through non-random association between character states for the traits A and B. This non-random association is measured by the cultural analogue of genetic linkage disequilibrium, defined by D = P00P11 − P01P10 (see also Feldman & CavalliSforza, 1984; Feldman & Zhivotovsky, 1992). After natural selection, the phenotypic frequencies become P'00 = P00/(1 − sq1), P'01 = (1 − s)P01/(1 − sq1), P'10 = P10/(1 − sq1), and P'11 = (1 − s)P11/(1 − sq1), where a prime indicates a value after natural selection and qj = SiPij is the frequency of Bj individuals. Cultural transmission of the traits then follows, after the operation of natural selection in each time period. Each individual meets with other individuals and, according to its preference, selects a potential transmitter, an individual with whom the receiver interacts to receive cultural elements. Individuals with state A1 have a preference and tend to interact with individuals with state B1 by a degree of G ( − 1 Q G Q 1); i.e. the proportion of B1 individuals among the potential transmitters for A1 individuals is given by (1 + G)q'1 /[(1 − G)q'0 + (1 + G)q'1 ],
(1)
where q'j = SiP'i j is the frequency of Bj individuals after natural selection, with primes again indicating the values after natural selection. On the other hand, A0 individuals do not show a preference and randomly choose potential transmitters. Hence biased transmission occurs when the receiver is A1, while transmission is unbiased with A0 receivers. We see from eqn (1) that this model of selective interaction is frequency dependent, and is mathematically
8 8
Dp1 = 2c ·
equivalent to the relative fixed preference model of selective mating discussed by Kirkpatrick (1982; see also Takahasi, 1997). Through each interaction, the receiver changes its state for the trait B by a rate b ( r 0) if the chosen potential transmitter has a different state for that trait. In other words, among individuals with their initial state Bk who interact with Bl transmitters (l $ k), a proportion b of the receivers acquires the state Bl through the interaction; otherwise the original state is maintained. In a similar way, an individual changes its state for the trait A at a rate c ( r 0) through the interaction process; i.e., among Ak individuals who interact with Al transmitters (l $ k), a proportion c of the receivers acquires the state Al through the interaction. If the states of a trait are not altered through the interaction process, we have b = 0 or c = 0 and cultural transmission of the trait does not occur. Later in this paper it is inferred that this should be the case when the traits are transmitted genetically. Because of the selective interaction induced by the preference of A1 individuals, the above interaction processes may alone result in changes in phenotypic frequencies (see recursions below). Hence this is a model of selective meeting or interaction (see Lewontin et al., 1968; Eshel & Cavalli-Sforza, 1982) that may further lead to biased cultural transmission (Boyd & Richerson, 1985) or cultural selection (Cavalli-Sforza & Feldman, 1981; Durham, 1991). A further assumption of independent transmission of the two traits yields a model of cultural transmission as summarized in Table 1. Combining the effects of natural and cultural selection, we obtain the unit time changes in the frequencies of A1 (p1 = SjP1j ) and B1 (q1), and changes in the cultural analogue of genetic linkage disequilibrium (D = P00P11 − P01P10), as
9 9
1−s G[p1(1−sq1) − sD] D · −s · , 1 − sq1 1 − sq1 − G[q0 − (1 − s)q1] 1−sq1
(2a)
1−s G[p1(1−sq1) − sD] qq · −s · 0 1 1 − sq1 1 − sq1 − G[q0 − (1 − s)q1] 1−sq1
(2b)
Dq1 = 2b · and
149
DD = −[b(1 − c) + (1 − b)c]·
2G 1 (1−s)D + · (1 − sq1)2 1 − sq1 − G[q0 − (1 − s)q1] (1−sq1)3
× 4b(1 − c)·[p0(1−sq1) + sD]·[p1(1−sq1) − sD]·(1−s)q0q1 + bc[p1(1 − sq1) − sD]·[q0−(1 − s)q1]·(1−s)D + (1 − b)c ·(1−s)2D 25 −
(2c)
s[q0 − (1 − sq1)q1 ]D − (Dp1)·(Dq1). (1 − sq1 )2
.
150
The state of the system is uniquely determined by these three variables. To avoid mathematical complexities and to get a clear view of the system, we posit that the intensity s of natural selection and the transmission rates b and c are small, so that all quadratic terms in these may be ignored. The system (2) is then simplified as
$
Dp1 = 2c ·
$
Dq1 = 2b ·
%
Gp1 − s ·D, 1−G(q0 − q1)
%
Gp1 − s ·q0q1, 1−G(q0 − q1)
Henceforth, we focus on the approximate system (3). From the above eqns (3), we see that even if natural selection is not operating so that s = 0, a deterministic change in the frequency of B1 is observed (Dq1 $ 0) as long as A1 individuals interact preferentially (G $ 0) and the trait is culturally transmitted at a certain positive rate (b q 0). We also see by comparing eqns (3a) and (3b) that frequency changes in the preference trait A are induced only in a hitchhiking manner, which obviously requires D $ 0. Such correlated evolution has been previously observed in the models of (inter-) sexual selection, where the frequency of a (mating) preference allele changes only as a correlated response to changes in the frequencies of viability alleles (Kirkpatrick, 1982). It is worth mentioning that when c = b, we have, from eqns (3a) and (3b), Dp1 = Dq1·D/(q0q1); then either Dp1 = Dq1 = 0 or Dp1, Dq1 $ 0 holds. This case will be treated independently in case 1 below.
(3a)
(3b)
and DD = −(b + c)D +
2G 1 − G(q0 − q1)
× (bp0p1q0q1 + cD 2) − s(q0 − q1)D.
(3c)
T 1 Meeting table for the model of cultural selection Phenotypic distribution after interaction Receiver
Transmitter
Frequency of transmitter
A0B0
A0B1
A1B0
A 0B 0
A0B0
P'00
1
0
0
0
A0B1
P'01
1−b
b
0
0
A0B1
A1B0
A1B1
A1B1
A1B0
P'10
1−c
0
c
0
A1B1
P'11
(1 − b)(1 − c)
b(1 − c)
(1 − b)c
bc
A0B0
P'00
b
1−b
0
0
A0B1
P'01
0
1
0
0
A1B0
P'10
b(1 − c)
(1 − b)(1 − c)
bc
(1 − b)c
A1B1
P'11
0
1−c
0
c
A0B0
(1 − G)P'00 (1 − G)q'0 + (1 + G)q'1
c
0
1−c
0
A0B1
(1 + G)P'01 (1 − G)q'0 + (1 + G)q'1
(1 − b)c
bc
(1 − b)(1 − c)
b(1 − c)
A1B0
(1 − G)P'10 (1 − G)q'0 + (1 + G)q'1
0
0
1
0
A1B1
(1 + G)P'11 (1 − G)q'0 + (1 + G)q'1
0
0
1−b
b
A0B0
(1 − G)P'00 (1 − G)q'0 + (1 + G)q'1
bc
(1 − b)c
b(1 − c)
(1 − b)(1 − c)
A0B1
(1 + G)P'01 (1 − G)q'0 + (1 + G)q'1
0
c
0
1−c
A1B0
(1 − G)P'10 (1 − G)q'0 + (1 + G)q'1
0
0
b
1−b
A1B1
(1 + G)P'11 (1 − G)q'0 + (1 + G)q'1
0
0
0
1
1
q1
We now proceed to a more detailed analysis of the system (3). Since we are interested in finding the condition for the evolution of a preference that promotes the acquisition of maladaptive cultural variants, we focus on the cases in which the less viable state has the greater attractiveness and is hence more favored in biased transmission; otherwise the dynamic is uninteresting, which should be clear from eqns (3). Hence we may assume that G q 0 and s q 0 without loss of generality.
151
1 G q 0, s q 0, c = b. If the two traits are equally likely to be transmitted via the interaction so that c = b, we have, from eqns (3),
Dq1 = Dp1 ·
$
%
0
q0q1 Gp1 = 2b · − s ·q0q1. D 1−G(q0 − q1)
Then at equilibrium, we obtain 1−G b + ·px 1, 2G s
qx 1 = −
(4)
where a caret indivates the equilibrium value. Equation (4) indicates that a line of internal equilibria emerges and that Fisherian type of runaway dynamics for both the preference and the viability trait is observed as in the conventional models of sexual selection (Lande, 1981; Kirkpatrick, 1982; Laland, 1994; see also Boyd & Richerson, 1985, chapter 8). While the line of equilibria is inferred to be always stable in Kirkpatrick’s (1982) model of sexual selection, here the line may be unstable (see also Lande, 1981; Laland, 1994). Usual local stability analysis of the equilibria yields three eigenvalues, l1 = 1, l2 = 1 + 2b ·
0
1
G s · D − ·qx 0qx 1 , 1−G(qx 0 − qx 1) b
and l3 = 1 − 2b + (2sD /px 1) − s(qx 0 − qx 1). The possibility of the neutral stability of the equilibria is indicated by l1 = 1. If the other two eigenvalues are less than unity, the movement along the line of equilibria cannot be predicted from an ordinary linear analysis, and the equilibria are locally stable with respect to perturbations which move the population
p1
1
F. 1. The two-dimensional projected space in case 1. The vertical axis, q1, is the frequency of B1 individuals and the horizontal axis, p1, is the frequency of A1 individuals. In the figure, bold lines indicate the stable equilibria. Parameters are b = c = 0.025, s = 0.015, and G = 0.5. The figure also shows some results of the numerical iteration of the approximated system (3), plotted at intervals of 50 time units. For above parameter values, the approximated system (3) exhibits evolutionary dynamics very similar to the system (2). This holds also for the parameter values for Figs 2–4 (see below).
away from the equilibrium line. Since sign (l2 − 1) = sign(b/s − qx 0qx 1/Dx ), the line is locally stable only if b/s Q qx 0qx 1/D ,
(5)
and unstable otherwise. Noting that q0q1/D = Dq1/Dp1 [see eqns (3)] and b/s is the slope of the equilibrium line (4), this stability criterion (5) is analogous to the one found in a quantitative genetic model of sexual selection [see eqns (10) and (12) and figure 1 in Lande, 1981, p. 3723]. Hence as revealed by Lande (1981), if the condition (5) holds so that the equilibrium line (4) is stable, the system eventually converges to a point on the equilibrium lines irrespective of the initial state (see Fig. 1); limit cycles were never produced in the numerical iterations of the system (3). Moreover, once the system attains this neutrally stable state, no evolutionary effects arising from the assumption of the model cause further deterministic changes. This leaves open the possibility that relatively weak effects, such as random drift or natural selection at the interdemic level, may ultimately govern the evolutionary fate of the system (see Lande, 1981; Kirkpatrick, 1982; Boyd & Richerson, 1990; Koeslag, 1997). On the other hand if the condition (5) is not satisfied and the line (4) is unstable, the system moves away from
.
152
the line of interior equilibrium, until the boundary equilibria, given by qx 1 = 0 or qx 1 = 1, are attained. Additionally, when the inequality b/s Q (1 + G)/(2G)
(6)
holds, substitution of px 1 = 1 into eqn (4) yields qx 1 = b/s − (1 − G)/(2G) ( Q 1). Hence we obtain an equilibrium curve as in Fig. 2 that does not occur in the two-locus model of sexual selection (Kirkpatrick, 1982). Note that the inequality (6) is more likely to hold for stronger natural selection (larger s), less efficient cultural transmission of the traits (smaller b), and weaker cultural selection (smaller G). If these evolutionary effects are set to satisfy the condition (6), the system will then never reach the fixation of the maladaptive variant, while fixation may occur for the preference favoring that variant. 2 G q 0, s q 0, c $ b. The assumption c = b in the previous case is based solely on mathematical considerations. To posit generality, it seems reasonable to assume c $ b instead. This may represent the case when the preference of an individual is less prone to transmission compared with the viability trait so
that c Q b, or vice versa. Setting c $ b in eqns (3), we find that interior equilibria as observed in case 1 disappear and that only boundary equilibria are observed (see also Aoki, 1989; Laland, 1994). There are three distinct sets of equilibria and each can be locally stable under certain conditions. Two of the three comprise lines of equilibria, each corresponding to the fixation of one of the two states for the trait B (qx 1 = 0 or qx 1 = 1 with D = 0). These boundary equilibria always exist. The remaining equilibrium is located on a point given by px 1 = 1, qx 1 = b/ s − (1 − G)/(2G), and D = 0, and exists if (1 − G)/(2G) Q b/s Q (1 + G)/(2G).
(7)
Although other sets of parameter values, such as p1 = (1 + G)s/(2Gc), q1 = 1, and D = (1 + G) (b + c − s)/(2Gc), may seem to satisfy the equilibrium condition (Dp1, Dq1, DD) = (0, 0, 0) [see eqns (3)], such equilibria are not conceivable because they necessarily violate the definition of the linkage disequilibrium D = P00P11 − P01P10. For the first set of equilibria where the population is fixed on the state B0 (qx 1 = 0), the three eigenvalues that determine its local stability are l1 = 1, l2 = 1 + [2bGpx 1/(1 − G)] − s,
1
and l3 = 1 − (b + c + s) ( Q 1).
q1
Again l1 = 1 indicates the possibility of neutral stability. Since sign(l2 − 1) = sign(px 1 − (s/b)·[(1−G)/ (2G)]), it is locally stable with respect to perturbations which move the population away from the equilibrium line if px 1 Q (s/b)·[(1−G)/(2G)].
(8)
The corresponding condition for the fixation of B1 (qx 1 = 1) becomes px 1 q (s/b)·[(1+G)/(2G)] and b + c q s, since three eigenvalues are 0
p1
1
F. 2. The two-dimensional projected space in case 1 with the condition (6). As in the Fig. 1, the vertical axis is the frequency of B1 individuals and the horizontal axis is the frequency of A1 individuals. Bold lines indicate the stable equilibria. Parameters are b = c = 0.02, s = 0.015, and G = 0.5. Hence the inequality (6) is satisfied and the equilibrium line does not reach the fixation of the state B1 (q1 = 1). Some trajectories for the system (3) are plotted at intervals of 50 time units.
l1 = 1, l2 = 1 − [2bGpx 1/(1 + G)] + s, and l3 = 1 − (b + c) + s.
(9)
q1
1
0
p1
1
F. 3. The two-dimensional projected space in case 2 with b q c. In the figure, bold lines indicate the stable equilibria, and the two broken lines indicate the isoclines Dp1 = 0 and Dq1 = 0, respectively. The figure is drawn with parameters b = 0.025, c = 0.02, s = 0.015, and G = 0.5. Some trajectories are plotted at intervals of 50 time units.
The local stability analysis for the third equilibrium yields an eigenvalue 1 − bqx 0qx 142G/[1 − G(qx 0 − qx 1)]52 ( R 1), whereas the other two are the roots of a quadratic
153
the present situation. Then the correlated evolution of the preference trait as depicted above does not occur. I suspect that this should be the case if the preference trait A is transmitted genetically. Then it may be concluded that the genetic determination of the preference prevents the emergence of the runaway process, eventually producing only those preferences that favor a variant that improves the individual fitness. Since this conclusion is, strictly speaking, based on a model of within-generation transmission, and may only be extended to transgenerational processes in limited situations (see below), analyses of cultural selection models that explicitly incorporate (transgenerational) genetic inheritance of the preferences are required. This is done in the Appendix. As described in the Appendix, the genetic determination of the preference indeed precludes the emergence of the runaway dynamics, irrespective of which of the two modes of cultural transmission, oblique or vertical, of the viability trait are considered (see the Appendix). In both cases, we have sign(Dp1) = sign( − G), implying that only the transmission bias that favors the adaptive state (in this case, B0) may increase its frequency [see eqns (A.1a) and (A.3)]. The cultural determination of the preference, on the other hand, seems to allow runaway dynamics as discussed above.
1
f(l) 0 C0 + lC1 + l 2 = 0, where C0 = 1 + (b − c)[1 − (s 2qx 0qx 1 /b)] − (s/G)
Noting that both roots are less than unity if f(1) q 0 and df(1)/dl q 0, the stability condition for the equilibrium is obtained as c q b.
q1
and C1 = − 2 − b + c + (s/G).
(10)
The stability properties of the equilibria in case 2 are illustrated in Figs 3 and 4. We can see from the above conditions that the evolution of A1 as well as B1 becomes less likely for less efficient cultural transmission (smaller b and/or c), weaker cultural selection (smaller G), and stronger natural selection (larger s). In particular, when the trait A is not culturally transmitted at all so that individuals never change their preferences during the interaction processes (c = 0), Dp1 = −sD should always be negative unless D Q 0, which is unlikely in
0
p1
1
F. 4. The two-dimensional projected space in case 2 with b Q c. In the figure, bold lines and a large dot (W) indicate the stable equilibria, and the two broken lines indicate the isoclines Dp1 = 0 and Dq1 = 0, respectively. The figure is drawn with parameters b = 0.02, c = 0.025, s = 0.015, and G = 0.5. Some trajectories are plotted at intervals of 50 time units.
.
154 Discussion
The present study of the evolution of preferences in cultural selection suggests that there are cases where ordinary cultural processes lead to the evolution of both the deleterious cultural variant and the associated preference that favors it. The runaway process is observed only when the preference of an individual, together with its viability and attractiveness, can be repeatedly changed during its ontogenetic process, or in other words, only if it is culturally acquired (c q 0 in the above model). Genetic determination of the preference (c = 0) seems to preclude the emergence of the cultural runaway, and only the preferences that favor adaptive variants may eventually evolve. This is further supported by mathematical analyses of the models that explicitly incorporate genetic determination of the preference trait (see Appendix). This observation, that genetic preferences preclude the emergence of runaway dynamics (while culturally transmitted preferences do not), may give some insight into the nature of social learning behavior in (non-human) animals. In contrast to cultural processes in human socieities, which may often result in maladaptive cultural practices, most reported instances of social learning in non-human animals seem to promote the acquisition of adaptive behaviors (Galef, 1995, 1996; Laland, 1996). This interspecific difference in social information transfer may be understood if we consider how transmission bias is determined and transmitted between individuals in these species. Social learning in non-human animals is usually characterized by its transient property and learning mechanisms that are highly dependent on individual experiences (trial and error learning); on the other hand, human social learning enables stable transmission of information that may further lead to cumulative accumulation of cultural traditions (Galef, 1988; Whiten & Ham, 1992; Laland et al., 1993, 1996; Tomasello et al., 1993; see also Heyes, 1993). Then in non-human animals, with relatively simple forms of social learning mechanisms, transmission bias in social information transfer may be well modified and adjusted through trial and error learning to promote the acquisition of adaptive behaviors (Galef, 1995), and social interaction with other individuals should have only a small effect on the preference. If the preferences are indeed acquired through individual (rather than social) learning, then genetics should play a decisive role in the evolution of the preference (see Cavalli-Sforza & Feldman, 1983b), and hence in these species only those predispositions in favor of behaviors that improve Darwinian (or
inclusive) fitness might have evolved, as suggested from the above analyses. In contrast, human behaviors are so highly dependent on cultural processes that transmission bias may itself be transmitted culturally. In these situations, runaway cultural selection may give rise to the maladaptive cultural variants, as indicated in the present analyses. Such views on social information transfer in human and non-human animals are still questioned by many researchers (e.g. Tooby & Cosmides, 1989; Alexander, 1990; Ormrod, 1992; Sperber, 1994; Flinn, 1997), and the above consideration obviously deserves further investigations. As we have seen above, mathematical properties of the present model of cultural selection very much resemble to those of the conventional major gene models of sexual selection; for example, they both exhibit Fisherian type of runaway dynamics, if certain conditions are met (see Figs 1 and 2; see also Figs 3 and 4). This goes along with our intuition since biased transmission in cultural inheritance is formally analogous to selective mating in genetic transmission (see Richerson & Boyd, 1989), and the system (3) [or (2)] is mathematically very similar to the sexual selection models with two-locus haploid genetics (Kirkpatrick, 1982; see also Laland, 1994). Therefore, inferences from preceding studies of sexual selection may be fruitful in investigating cultural selection (Richerson & Boyd, 1989). The present study assumes that the preference in cultural selection, whether it is transmitted genetically or culturally, does not induce an additional viability cost. As can be inferred from the studies on costly mate preferences (Pomiankowski, 1987; Bulmer, 1989; Pomiankowski et al., 1991), an assumption of costly preferences should greatly alter the outcome of the present analyses. The resemblance is, however, merely of mathematical importance, and does not have any biological implications. For example, the evolution by cultural selection as revealed by the present study differs from the evolution by sexual selection, in that the runaway dynamics may be observed only when the preference is also culturally transmitted. This is not so in the models of sexual selection, where the system may exhibit runaway dynamics whether mating preferences are determined genetically or culturally (Kirkpatrick, 1982; Laland, 1994). It should be kept in mind that cultural selection is utterly a distinctive process that should clearly be discriminated from sexual selection. As clearly indicated in the present investigation, culture may at times reduce fitness of individuals. Therefore, simplistic adaptive interpretations of cultural traits are unconvincing without detailed
specificiation of the cultural processes involved. Possible examples of cultural runaway as presented in this study may include human body mutilations or deformations, such as piercing, tattooing, tooth removing, female circumcision, etc., that have been reported in many human populations (see Darwin, 1871, chapters 19–21; Boyd & Richerson, 1985, pp. 268–271; Logan & Qirko, 1996). Although appropriate data are lacking at present, I expect that sociological (or other) surveys on cultural changes in modern human populations should provide examples suitable for discussing the effects of cultural selection on sociocultural dynamics (see Durham, 1991; Laland et al., 1995; Boehm, 1996). Cultural runaway of this kind may further help to explain the existence of traits and preferences that are apparently maladaptive at the individual level but may be adaptive for the group, such as (non-kin, non-reciprocal) altruism (see Richerson & Boyd, 1992; Laland, 1994). As we have seen above, cultural selection may exhibit lines of equilibria that are themselves locally (or globally, as inferred from numerical iterations; see above) stable. The emergence of such neutral stability leaves open the possibility that relatively minor evolutionary effects, such as random drift or natural selection at the interdemic level, may play a crucial role in determining the course of evolutionary changes (see Lande, 1981; Kirkpatrick, 1982; Koeslag, 1997; see also Boyd & Richerson, 1990). Some authors have suggested that group selection among cultural variants, usually termed cultural group selection, may have had profound effects on sociocultural dynamics among human socieites (e.g. Campbell, 1975; Boyd & Richerson, 1982, 1985, 1990; Richerson & Boyd, 1989; Dugatkin, 1992; Findlay, 1992; Wilson & Sober, 1994; Soltis et al., 1995; Boehm, 1996; Wilson & Dugatkin, 1997). If this is indeed so, then certain types of social dilemma may be overcome by the joint effects of cultural runaway and natural selection at the interdemic level, since biased transmission in favor of altruists may dominate the effects of individual selection against them, further rendering intergroup processes sufficiently effective in promoting the evolution of cooperation (Richerson & Boyd, 1992). To show that simple cultural processes may at times lead to the evolution of a preference that favors a maladaptive variant, this study presents an alternative way to model cultural changes, where cultural elements are transmitted in a manner more like infections in epidemioloical (or parasitic) transmission (Moritz, 1990; Lynch, 1991; Dawkins, 1993; see also Cavalli-Sforza & Feldman, 1981,
155
chapter 1; Sperber, 1985, 1990, 1994; Dennett, 1995, chapter 12). A similar method has been employed in investigating the geographic spread of certain cultural practices such as farming (Ammerman & Cavalli-Sforza, 1987; Aoki et al., 1996; see also Aoki 1987; Rogers, 1995), where culturally different populations meet and exchange new cultural practices. Scholars of cultural changes seem to be more concerned with changes that occur within a relatively short time period compared with evolutionary time-scales. The present approach should have some benefits over conventional models in describing such rapid changes, because the conventional formulation is based on changes between generations and is difficult to follow the changes within a generation (see also Carotenuto et al., 1989). Studies of social learning in non-human animals, which appears to be characterized by highly horizontal social transmission (Laland et al., 1993, 1996), may also benefit by utilizing the method. Furthermore, although the present model neglects birth-death processes of individuals and hence may appear to fail in describing sociocultural changes that proceed over generations, above arguments also apply without modifications to such situations if individuals are replaced by newly born individuals of same phenotypes; e.g. if the mating is random and vertical cultural transmission is unbiased with complete linkage of traits. I expect that the approach presented here should prove useful when applying the theory of cultural transmission to other fields in social sciences. However, although the models of cultural change may describe the consequence of, say, a preference and an associated attractive character, they do not explain the causes of the preference nor the character being attractive; and usually, it is this very point that many social scientists regard as issues of interest (Sober, 1991). Obviously, there is more to culture than a system of inheritance. The author thanks K. Aoki, Carl Bergstrom, Marc Feldman, and Kevin Laland. Part of this research was performed during his stay at Stanford University, supported by NIH Grant GM28016 to M. W. Feldman. The research is also supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists, and Grant-in-Aid for Encouragement of Young Scientists from The Ministry of Education, Science, Sports and Culture, Japan.
REFERENCES A, R. D. (1979). Darwinism and Human Affairs. Seattle: University of Washington Press. A, R. D. (1990). Epigenetic rules and Darwinian algorithms. Ethol. Sociobiol. 11, 241–303.
156
.
A, A. J. & C-S, L. L. (1984). The Neolithic Transition and the Genetics of Populations in Europe. Princeton: Princeton University Press. A, K. (1987). Gene-culture waves of advance. J. Math. Biol. 25, 453–464. A, K. (1989). A sexual selection model for the evolution of imitative learning of song in polygynous birds. Am. Nat. 134, 599–612. A, K. (1990). A shifting balance type model for the origin of cultural transmission. In: Proceedings of the Fourth International Symposium in Conjunction with the Awarding of the International Prize for Biology: Population Biology of Genes and Molecules (Takahata, N. & Crow, J. F., eds) pp. 123–137. Tokyo: Baifukan. A, K. (1991). Some theoretical aspects of the origin of cultural transmission. In: Evolution of Life (Osawa, S. & Honjo, T., eds) pp. 439–449. Tokyo: Springer. A, K. & F, M. W. (1987). Toward a theory for the evolution of cultural communication: coevolution of signal transmission and reception. Proc. Natl. Acad. Sci. U.S.A. 84, 7164–7168. A, K. & F, M. W. (1989). Pleiotropy and preadaptation in the evolution of human language capacity. Theor. Popul. Biol. 35, 181–194. A, K. & F, M. W. (1991). Recessive hereditary deafness, assortative mating, and persistence of a sign language. Theor. Popul. Biol. 39, 358–372. A, K. & F, M. W. (1994). Cultural transmission of a sign language when deafness is caused by recessive alleles at two independent loci. Theor. Popul. Biol. 45, 101–120. A, K., S, M. & S, N. (1996). Traveling wave solutions for the spread of farmers into a region occupied by hunter-gatherers. Theor. Popul. Biol. 50, 1–17. B, J. H. (1989). The elastic between genes and culture. Ethol. Sociobiol. 10, 111–129. B J, N. G. (1990). Three sensible paradigms for research on evolution and human behavior? Ethol. Sociobiol. 11, 353–359. B, C. (1996). Emergency decisions, cultural-selection mechanics, and group selection. Curr. Anthropol. 37, 763–793. B, R. & R, P. J. (1982). Cultural transmission and the evolution of cooperative behavior. Hum. Ecol. 10, 325–351. B, R. & R, P. J. (1985). Culture and the Evolutionary Process. Chicago: University of Chicago Press. B, R. & R, P. J. (1990). Group selection among alternative evolutionarily stable strategies. J. theor. Biol. 145, 331–342. B, M. (1989). Structural instability of models of sexual selection. Theor. Popul. Biol. 35, 195–206. C, D. T. (1975). On the conflicts between biological and social evolution and between psychology and moral tradition. Am. Psychol. 30, 1103–1126. C, L., F, M. W. & C-S, L. L. (1989). Age structure in models of cultural transmission. The Morrison Institute for Population and Resource Studies Working Paper Series: Paper number 0016. C-S, L. L. & F, M. W. (1973). Cultural versus biological inheritance: phenotypic transmission from parents to children (a theory of the effect of parental phenotypes on children’s phenotype). Am. J. Hum. Genet. 25, 618–637. C-S, L. L. & F, M. W. (1981). Cultural Transmission and Evolution: A Quantitative Approach. Princeton: Princeton University Press. C-S, L. L. & F, M. W. (1983a). Paradox of the evolution of communication and of social interactivity. Proc. Natl. Acad. Sci. U.S.A. 80, 2017–2021. C-S, L. L. & F, M. W. (1983b). Cultural versus genetic adaptation. Proc. Natl. Acad. Sci. U.S.A. 80, 4993–4996. C, L. (1995). Is there a role for culture in human behavioral ecology? Ethol. Sociobiol. 16, 181–205. D, C. (1871). The Descent of Man, and Selection in Relation to Sex. London: John Murray.
D, R. (1976). The Selfish Gene. Oxford: Oxford University Press. D, R. (1982). The Extended Phenotype: The Gene as the Unit of Selection. Oxford: W. H. Freeman. D, R. (1993). Viruses of the mind. In: Dennett and His Critics: Demystifying Mind (Dahlbom, B., ed.) pp. 13–27. Oxford: Blackwell. D, D. C. (1995). Darwin’s Dangerous Idea: Evolution and the Meaning of Life. New York: Simon and Schuster. D, L. A. (1992). The evolution of the ‘‘con artist’’. Ethol. Sociobiol. 13, 3–18. D, W. H. (1991). Coevolution: Genes, Culture, and Human Diversity. Stanford: Stanford University Press. E, I. & C-S, L. L. (1982). Assortment of encounters and evolution of cooperativeness. Proc. Natl. Acad. Sci. U.S.A. 79, 1331–1335. F, M. W. & A, K. (1992). Assortative mating and grandparental transmission facilitate the persistence of a sign language. Theor. Popul. Biol. 42, 107–116. F, M. W., A, K. & K, J. (1996). Individuals versus social learning: evolutionary analysis in a fluctuating environment. Anthropol. Sci. 104, 209–232. F, M. W. & C-S, L. L. (1976). Cultural and biological evolutionary processes, selection for a trait under complex transmission. Theor. Popul. Biol. 9, 238–259. F, M. W. & C-S, L. L. (1984). Cultural and biological evolutionary processes: gene-culture disequilibrium. Proc. Natl. Acad. Sci. U.S.A. 81, 1604–1607. F, M. W. & L, K. N. (1996). Gene-culture coevolutionary theory. Trends Ecol. Evol. 11, 453–457. F, M. W. & Z, L. A. (1992). Gene-culture coevolution: toward a general theory of vertical transmission. Proc. Natl. Acad. Sci. U.S.A. 89, 11935–11938. F, C. S. (1992). Phenotypic evolution under gene-culture transmission in structured populations. J. theor. Biol. 156, 387–400. F, M. V. (1997). Culture and the evolution of social learning. Evol. Hum. Behav. 18, 23–67. G, B. G., J. (1988). Imitation in animals: field observations and laboratory analyses. In: Social Learning: Psychological and Biological Perspectives (Zentall, T. R. & Galef, B. G., Jr., eds) pp. 3–28. Hillsdale: Lawrence Erlbaum Associates. G, B. G., J. (1995). Why behaviour patterns that animals learn socially are locally adaptive. Anim. Behav. 49, 1325–1334. G, B. G., J. (1996). The adaptive value of social learning: a reply to Laland. Anim. Behav. 52, 641–644. G, C. R., V, C., H, B. & C-S, L. L. (1995). Cultural variation in Africa: role of mechanisms of transmission and adaptation. Proc. Natl. Acad. Sci. U.S.A. 92, 7585–7589. H, C. M. (1993). Imitation, culture and cognition. Anim. Behav. 46, 999–1010. H, R. A. (1982). Ethology: Its Nature and Relations with Other Sciences. Oxford: Oxford University Press. H, R. A. (1987). Individuals, Relationships and Culture: Links between Ethology and the Social Sciences. Cambridge: Cambridge University Press. K, M. (1982). Sexual selection and the evolution of female choice. Evolution 36, 1–12. K, J. H. (1997). Sex, the Prisoner’s Dilemma Game, and the evolutionary inevitability of cooperation. J. theor. Biol. 189, 53–61. K, J., L, K. N. & F, M. W. (1994). Gene-culture coevolution and sex ratios: the effects of female infanticide, sex-selective abortion, sex selection and sex-biased parental investment on the evolution of sex-ratios. Theor. Popul. Biol. 46, 249–278. L, K. N. (1994). Sexual selection with a culturally transmitted mating preference. Theor. Popul. Biol. 45, 1–15. L, K. N. (1996). Is social learning always locally adaptive? Anim. Behav. 52, 637–640.
L, K. N., K, J. & F, M. W. (1995). Gene-culture coevolutionary theory: a test case. Curr. Anthropol. 36, 131–156. L, K. N., R, P. J. & B, R. (1993). Animal social learning: toward a new theoretical approach. In: Perspectives in Ethology. Vol. 10. Behavior and Evolution (Bateson, P. P. G., Klopfer, P. H. & Thompson, N. S., eds) pp. 249–277. New York: Plenum. L, K. N., R, P. J. & B, R. (1996). Developing a theory of animal social learning. In: Social Learning in Animals: The Roots of Culture (Heyes, C. M. & Galef, B. G., Jr., eds) pp. 129–154. New York: Academic Press. L, R. (1981). Models of speciation by sexual selection on polygenic traits. Proc. Natl. Acad. Sci. U.S.A. 78, 3721–3725. L, R., K, D. & C, J. (1968). Selective mating, assortative mating, and inbreeding: definitions and implications. Eugen. Quart. 15, 141–143. L, M. H. & Q, H. N. (1996). An evolutionary perspective on maladaptive traits and cultural conformity. Am. J. Hum. Biol. 8, 615–629. L, C. J. & W, E. O. (1981). Genes, Mind, and Culture: The Coevolutionary Process. Cambridge: Harvard University Press. L, A. (1991). Thought contagion as abstract evolution. J. Ideas 2, 3–10. MD, K. (1989). The plasticity of human social organization and behavior: contextual variables and proximate mechanisms. Ethol. Sociobiol. 10, 171–194. M, E. (1990). Memetic science: I—general introduction. J. Ideas 1, 3–23. O, R. K. (1992). Adaptation and cultural diffusion. J. Geograph. 91, 258–262. P, A. P. (1987). The costs of choice in sexual selection. J. theor. Biol. 128, 195–218. P, A. P., I, Y. & N, S. (1991). The evolution of costly mate preferences. I. Fisher and biased mutation. Evolution 45, 1422–1430. R, P. J. & B, R. (1989). The role of evolved predispositions in cultural evolution: or, human sociobiology meets Pascal’s Wagner. Ethol. Sociobiol. 10, 195–219. R, P. J. & B, R. (1992). Cultural inheritance and evolutionary ecology. In: Evolutionary Ecology and Human Behavior (Smith, E. A. & Winterhalder, B., eds) pp. 61–92. New York: Aldine de Gruyter. R, E. M. (1995). Diffusion of Innovations, 4th ed. New York: Free Press. S, E. (1991). Models of cultural evolution. In: Trees of Life: Essays in the Philosophy of Biology (Griffiths, P., ed.) pp. 17–38. Norwell: Kluwer Academic. S, J. S., B, R. & R, P. J. (1995). Can group-functional behaviors evolve by cultural group selection? An empirical test. Curr. Anthropol. 36, 473–494. S, D. (1985). Anthropology and psychology: towards an epidemiology of representations. Man 20, 73–89. S, D. (1990). The epidemiology of beliefs. In: The Social Psychological Study of Widespread Beliefs (Fraser, C. & Gaskell, G., eds) pp. 25–44. Oxford: Oxford University Press. S, D. (1994). The modularity of thought and the epidemiology of representations. In: Mapping the Mind: Domain Specificity in Cognition and Culture (Hirschfeld, L. A. & Gelman, S. A., eds) pp. 39–67. New York: Cambridge University Press. T, K. (1997). Models of selective mating and the initiation of the Fisherian process. Proc. Roy. Soc. Lond. B 264, 839–844. T, K. & A, K. (1995). Two-locus haploid and diploid models for the coevolution of cultural transmission and paternal care. Am. Nat. 146, 651–684. T, M., K, A. C. & R, H. H. (1993). Cultural learning. Behav. Brain Sci. 16, 495–552. T, J. & C, L. (1989). Evolutionary psychology and the generation of culture, part I: theoretical considerations. Ethol. Sociobiol. 10, 29–49.
157
T, J. & C, L. (1990). The past explains the present: emotional adaptations and the structure of ancestral environments. Ethol. Sociobiol. 11, 275–424. W, A. & H, R. (1992). On the nature and evolution of imitation in the animal kingdom: reappraisal of a century of research. Adv. Study Behav. 21, 239–283. W, D. S. & D, L. A. (1997). Group selection and assortative interactions. Am. Nat. 149, 336–351. W, D. S. & S, E. (1994). Reintroducing group selection to the human behavioral sciences. Behav. Brain Sci. 17, 585–654.
APPENDIX In this section I analyse simple genetic models of cultural selection to argue that runaway dynamics as described in the text does not occur if preferences are transmitted genetically. Models developed here differ from those in the text in that they explicitly assume genetic transmission of the preference trait A, which produces the transmission bias toward the viability trait; other assumptions are as in the text. For the sake of simplicity, haploid inheritance of the trait is postulated. Two models with different assumptions are discussed. The first model assumes nonvertical (or, more precisely, oblique) transmission of the cultural viability trait B, and the second model assumes that the trait is transmitted vertically. To see if cultural selection could alone lead to the evolution of a preference in favor of a maladaptive cultural variant, it is required to preclude the possibility of other evolutionary effects such as sexual selection. Hence the mating is assumed to be random.
(i) Oblique Transmission of the Viability Trait B The first model assumes that the viability trait B is culturally transmitted via oblique pathway, from adults of the previous generation to genetically unrelated individuals of the offspring generation (Cavalli-Sforza & Feldman, 1981). Cultural transmission of the trait is assumed to occur after the reproduction but before the operation of natural selection in each generation. Before the cultural transmission of the viability trait, receivers (offspring) are assumed to be uncultured with respect to the trait. Hence, there are only two types of receivers, A0 and A1 . In addition, since the states of the preference trait of transmitters (adult of the previous generation) do not affect the oblique transmission process, the trait A of the transmitters need not be considered. Before the oblique transmission of the trait B, the genotypic frequencies of the trait A among offspring are simply given by the corresponding frequencies in the adults of the previous generation. This is because
.
158
T A1 Meeting table for the oblique transmission model Receiver
Transmission
Frequency of interaction
Phenotype of the receiver after interaction
A0
B0
p0q0
A 0B 0
B1
p0q1
A 0B 1
B0
(1−G)q0 p1 · (1 − G)q0 + (1 + G)q1
A 1B 0
B1
p1 ·
(1+G)q1 (1 − G)q0 + (1 + G)q1
A 1B 1
A1
Since the receivers (naive offspring) are assumed to be uncultured with respect to the viability trait B before the oblique transmission, there are only two types of receivers, A0 and A1. In addition, since the preference of transmitters (adult of the previous generation) has nothing to do when oblique transmission occurs from adults to offspring, their states for the preference trait A is not indicated in the table.
T A2 Mating table for the vertical transmission model Phenotypic distribution Mating
Mating frequency
A0B0 × A0B0
2 P00
1
0
0
0
A0B1
2P00P01
1/2
1/2
0
0
A1B0
2P00P10
1/2
0
1/2
0
A1B1
2P00P11
1/4
1/4
(1 − G)/4
(1 + G)/4
2 P01
0
1
0
0
A1B0
2P01P10
1/4
1/4
(1 − G)/4
(1 + G)/4
A1B1
2P01P11
0
1/2
0
1/2
2 P10
0
0
1
0
2P10P11
0
0
(1 − G)/2
(1 + G)/2
2 P11
0
0
0
1
A0B1 × A0B1
A1B0 × A1B0 A1B1 A1B1 × A1B1
A0B0
natural selection is assumed to act via differential viability (and not the fertility) of individuals, and the mating is assumed to be random. Hence the genotypic frequencies are unchanged through reproduction process. Oblique transmission as described in Table A1 follows. Note that the phenotype of a naive offspring with state Ai is modified to AiBj through cultural transmission process, if the offspring interacts with an adult with state Bj (see Table A1). Then the frequency of interaction between, say, a naive offspring with state Ai and an adult with state Bj as given in the Table A1 is just the frequency of AiBj offspring before natural selection. Viability selection then completes the life cycle. Following the ordinary methodologies of gene-culture coevolutionary models (Feldman & CavalliSforza, 1976; Feldman & Laland, 1996), the frequency change in the preference trait A between
A0B1
A 1B 0
A1B1
generations is derived as w ·Dp1=−2G ·
sp0p1q0q1 , (A.1a) (1−G)q0 + (1 + G)q1
where
$
w = 1 − s p0 +
%
(1 + G)p1 q (1 − G)q0 + (1 + G)q1 1
(A.1b)
is the average viability of individuals. Recalling that s q 0 (see text), we have from eqn (A.1a), sign(Dp1) = sign( − G), implying that only the transmission bias that favor the adaptive state (in this case, B0) may evolve. (ii) Vertical Transmission of the Viability Trait B In this model, cultural transmission of the viability trait B is assumed to occur from parents to their
offspring soon after, or simultaneously with, reproduction. To apply the biased cultural transmission model (1) in the text, it is assumed that two parents (mother and father) constitute the transmitters, while their offspring are the receivers. It is further assumed that the genetically transmitted trait A and the cultural trait B are transmitted independently. This assumption is equivalent to complete recombination (recombination rate r = 1/2) in genetic inheritance. With these assumptions, the mating table for the present model is obtained as in Table A2. Viability selection then follows, and the recursions for the phenotypic frequencies are derived from the table as
w ·P'00=(P00 + p0q0)/2,
(A.2a)
w ·P'01/(1 − s) = (P01 + p0q1)/2,
(A.2b)
w ·P'10=(P10 + p1q0)/2 − G(P10q1 + P11q0)/2,
(A.2c)
159
and w ·P'11/(1−s) = (P11 + p1q1)/2 + G(P10q1 + P11q0)/2,
(A.2d)
where w = 1 − sq1 − sG(P10q1 + P11q0)/2
(A.2e)
is the average viability of individuals. The difference equation for the genetic preference A becomes w ·Dp1= −s(P11 − p1q1)/2 − sGp0(P10q1 + P11q0)/2.
(A.3)
The first term, − s(P11 − p1q1)/2 = −sD/2, corresponds to the changes due purely to natural selection effects, and becomes zero at gene-culture equilibrium D = 0 (Feldman & Cavalli-Sforza, 1984; Feldman & Zhivotovsky, 1992). Assuming that D q 0, which seems reasonable for the present argument (see text), Dp1 q 0 requires G Q 0, implying that only the transmission bias that favor the adaptive state (B0) may evolve, as in the oblique transmission model as described above.