Proc. NatL Acad. Sci. USA Vol. 78, No. 6, pp. 3976-3979, June 1981 Population Biology

Lumsden-Wilson theory of gene culture coevolution (human sociobiology/ethnography/epigenetic rules/social development) JOSEPH S. ALPER* AND ROBERT V. LANGEt *Department ofChemistry, University of Massachusetts, Boston, Massachusetts 02125; and tDepartment of Physics, Brandeis University, Waltham, Massachusetts 02154 Communicated by S. E. Luria, March 9, 1981

ABSTRACT A critique is presented of the Lumsden-Wilson ASSUMPTIONS OF THE THEORY theory [Lumsden, C. J. & Wilson, E. 0. (1980) Proc. Nati. Acad. Sci. USA 77, 4382-4386] of the transmission of cultural traits. An LW propose that we consider one cultural trait (called a "cul- analysis of the underlying assumptions and the mathematical na- turgen") at a time and study the mechanisms by which distri- ture of the theory clarifies its essentially reductionist and deter- butions of alternative forms of the trait in a society are estab- minist qualities. The mathematical functions governing the tran- lished. The central assumption of the LW theory is that there sition probability that an individual member of a group of a are genes that code for the rules that determine the probability specified size will switch from one trait to an alternative form of of changing from one alternative form of the trait to another. that trait is assumed to be genetically controlled although the sin- There is absolutely no evidence that any genes ofthis type exist gle independent variable of this function, the number of individ- and, as we shall argue more fully below, LW's claim that there uals characterized by each of the two forms of the trait, is envi- is evidence for the existence of such genes is invalid. The em- ronmentally determined. The model assumes that the cultural pirical evidence LW cite refers to the observed probabilities properties of a society are simply the sum of the properties of the that individuals prefer one alternative to another. Quite aside individuals; that each individual is equally influenced by every from whether these preferences are genetic, at least one can other member of the group; and that kinship structures, cultural show that there are differences among individuals with regard institutions, and historical factors can be neglected. to these preferences. No one has ever demonstrated that there Lumsden and Wilson (LW) have proposed a sociobiological the- are observable differences among people in the probabilities of ory that treats quantitatively the relationship between genetics their switching from one cultural trait to another, let alone that and the environment in shaping cultural behavior (1). Rather such differences might arise from differences in their genes. than assuming the existence of genes determining a particular Culturgens are assumed to be traits that are possessed and trait, as is usual in human sociobiology, LW hypothesize that transmitted by individuals, as opposed to behavior that makes the genes code for the epigenetic rules that determine the prob- sense only in a particular social context. LW have made the ability of changing from one form of a trait to another. Envi- reductionist assumption that the characteristics ofa society can ronmental influence is incorporated into the theory by assuming be understood as simply the sum of the characteristics of the that this probability depends not only on the genes but also on individuals of that society. There are, in this theory, no prop- the number of individuals already characterized by each of the erties that arise as a consequence of social interactions of the forms ofthe trait. LW then derive a formula for the probability group. Social institutions themselves, such as kinship structures distribution of individuals in the alternative cultural states (the and other groupings, as opposed to membership in these in- "ethnographic curve"). They find (ref. 1, p. 4382) that "the stitutions, cannot be included in the theory. It should be noted translation from individual epigenesis to social pattern is am- that all of the examples of what is called culture by LW (e.g., plified, to the extent that differences in [genetic] bias too faint preference for sugar, fear response ofstrangers, preferred level to be detected in ordinary developmental studies can generate of visual pattern complexity) are rather rudimentary. conspicuous variation in the ethnographic curves." LW then ssun.e that "enculturation is conducted not just We believe that a critical analysis of the LW paper is im- by the nuclear family, a common feature of some industrialized portant because sociobiologists claim far-reaching implications Western societies, but by a much broader array ofrelatives and of the theory as an explanation of human culture. In this cri- parent surrogates" (ref. 1, p. 4382). Wilson, in his textbook on tique, we focus our attention on three aspects ofthe work. First, sociobiology (2), presented a contrasting view: "The building the mathematical nature of the paper has forced the authors to block of nearly all human societies is the nuclear family. The state in an extremely clear fashion the assumptions underlying population of an American industrial city, no less than a band their theory of the genetic-environmental interaction. An ex- of hunter-gatherers in the Australian desert, is organized amination of these assumptions shows that they are not as self- around this unit." The assumption chosen by LW is advanta- evident as LW imply. Second, the mathematical model illus- geous for their theory because (i) the mathematical model is easy trates the extreme sensitivity of the predictions of the theory to realize and solve if every member of the population (15-75, to its assumptions. This sensitivity, rather than signifying the the cultural power ofthe theory to reveal the extreme sensitivity ofculture according to LW) exerts equal influence in affecting to genetic factors as LW argue, actually undermines it. Third, state of a given individual and (ii) as we shall see below, the the LW presentation suggests that, for sociobiology, genetic sensitivity of their model is guaranteed by this assumption. determinism and environmental determinism are compatible. There is no hard evidence supporting either the assumption of LW or the one held previously by Wilson. LW assume that cultur- The publication costs ofthis article were defrayed in part by page charge To make the mathematics tractable, payment. This article must therefore be hereby marked "advertise- ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. Abbreviation: LW, Lumsden and Wilson. 3976 Downloaded by guest on September 30, 2021 Population Biology: Alper and Lange Proc. Natl. Acad. Sci. USA 78 (1981) 3977 gens exist in one ofonly two possible discrete forms, e.g., pref- active decay, is characteristic ofrandom processes. LW give no erence or lack of preference for sugar. At various decision argumentjustifying this assumption forthe frequency ofcultural points, individuals choose between these two alternatives; they decision points. In fact they do not even mention that their either stick or switch. There are no gradual shifts from one al- choice of a first-order rate law constitutes an assumption and ternative to the other. It is assumed that these alternatives are that other rate laws are conceivable. always present in an unchanging form; development ofthe traits As a result ofall these assumptions, LW are then able to write themselves is specifically excluded. a differential equation relating the time rate ofchange ofP, the We now come to the assumptions governing cultural decision probability that in a population ofN individuals, nj and n2 will making. The model adopted by LW has its origins in the models possess culturgens 1 and 2, respectively, to the transition prob- used in the statistical mechanical description of irreversible ability functions and rate constants: processes in dilute gases and uses much the same formalism. aP(nl,n2,t) = (n1 + 1).v12(n1 + 1,n2 + 1, n2 - Lt) LW state that, in many cases, decision making is adequately -1)-P(nI described by a Markov process-i.e., the decision made at any + (n2 + 1)OV21(n1 - l,n2 + I).P(n1 - l,n2 + l,t) [1] given time either to retain or to switch cultural states depends - [ngv12(n1,n2) + n2v2j(nj,n2)]P(nj,n2,t) only on the state of that individual and on the state of all the surrounding individuals at that time. The previous history of In this equation, v. = riuij, where r, are the first-order rate the population can be ignored completely. For example, the constants. possibility that the environment was different the previous time This equation is the master equation ofirreversible statistical a behavioral transition occurred and that this could affect the mechanics (5). It can be solved numerically on a computer and current transition probability is not considered. In the language can even be solved exactly in the steady state for certain choices ofstatistical physics, the system has no memory. The social sci- of u... However, LW make the further assumption that (n2 - ences, dedicated to understanding the role ofthe past in shaping nl)/N can be treated as a continuous parameter, an assumption the present, become superfluous. true only for large values of N. Recall that N is the number of The Markov approximation isjustified only ifwe possess com- individuals in the population, all ofwhom have equal influence plete relevant knowledge ofthe state ofthe system. Only under in determining the probability that a given person will switch this condition does the present state determine all future states. from one culturgen to another. It has long been recognized that, even for physical systems, this condition is often not satisfied. In systems far from equilibrium CONCLUSIONS OF THE THEORY characterized by what is called the "mixing" property, slight changes in the initial state result in drastically different final In their discussion, LW present two conclusions. First, "even states. As the initial state for these systems cannot be known small differences in the epigenetic rules, reflected in the assim- with absolute accuracy, the final state cannot be predicted by ilation functions (the u. s) are magnified during social interac- using the methods of Markov processes (3). tions into the depenaent ethnographic patterns." Second, LW circumvent this difficulty by assuming that the state of "[e]ven when the underlying epigenetic rules and assimilation the system of N individuals is completely determined by the functions are rigidly constrained, they can generate wide cul- specification of the number ofindividuals, n1 and n2 = N - ni, tural diversity" (ref. 1, p. 4384). LW's major point is that im- in each ofthe alternative culturgens. In mathematical language, perceivable genetic differences can lead to widely varying social at each decision point, a probability transition function, which behavior. depends only on ni and n2, u12 (n1,n2), can be defined giving To support theircontention ofthe specificity ofthe epigenetic the likelihood that an individual will shift from culturgen 1 to rules, LW summarize in their table 1 approximate values ofthe culturgen 2. The Markov approximation dictates that this tran- relative assimilation probabilities for such traits as sugar pref- sition probability does not change with time-i.e., historical erence and color classification, which they estimate from ex- circumstances. The key assumption ofthe paper is that a func- perimental data. Their table 1, however, is labeled "Estimate tional form ofu and the parameters specifying it, the epigenetic of innate preferences" and the footnote to it states that "[t]he rules of individual behavior, exist and are genetically deter- preferred culturgen is arbitrarily designated as c2, and the es- mined or, in LW's words, are a product ofgenetic prescription. timated probability of choice of this culturgen as opposed to cl Two comments on this use of the Markov approximation are is denoted as u12." This definition ofu12 is inconsistent with the in order. First, in proposing a theory of behavior that depends definition of u., given in the body of the paper, where u12 is only on the present overt behavior of individuals in the popu- defined as the probability ofmaking a switch from cl to c2. For lation, LW adopt an extreme behaviorist view ofpsychology and example, although 80% ofthe individuals may prefer c2 and so, culture. Second, LW justify their assumption of the applica- according to table 1, u12 - U21 = 0.8 - 0.2 = 0.6, it may be bility of Markov processes by referring to a textbook in math- the case that each individual is satisfied with his or her choice ematical sociology by Coleman (4). Surprisingly enough, in view and so the probabilities of a switch from cl to c2 or from c2 to of their citation, Coleman sees little merit in the use of the cl are both zero. Markov approximation for modeling group behavior. For ex- This ambiguity in the meaning ofthe epigenetic rules appears ample, he writes (ref. 4, p. 38), "These models . have used throughout the paper. In table 1 and in the introduction, the the mathematics of the Markov process, and have taken this epigenetic rules are described as those that "affect the proba- mathematics about to the limit of its present development. bility of acquiring one culturgen as opposed to another (ref. 1, Unfortunately, these limits occur before much substantive com- p. 4382), while, in the methods section, they are described as plexity has been incorporated, and as a consequence, the "transition probabilities ui, from culturgen i to culturgenf' (ref. models are applicable only to extremely simple experiments." 1, p. 4383). To make contact with experiment, the definition LW go on to assume that the frequency ofthe decision points ofepigenetic rules as it appears in table 1 is necessary because, is described by a first-order rate law-i. e., that the probability as LW point out, cultural "responses have not been investigated that the next decision point will not occur in a time t after the with reference to their dependence on the behavior ofthe rest previous one is an exponentially decreasing function of t. This of society" (ref. 1, p. 4385), as would be required in evaluating frequency distribution, the same as the one describing radio- u12 (n1,n2) as defined in the body ofthe paper. The LW theory Downloaded by guest on September 30, 2021 3978 Population Biology: Alper and Lange Proc. Natl. Acad. Sci. USA 78 (1981)

cannot be related to experiment, as least at present, because that it represents the "simplest conceivable relationship" that the transition probability function, which is needed to quantify can be found to test their theory. This putative simplicity arises the master equation, is a totally unknown quantity and cannot, as a result of three assumptions: (i) incestuous behavior is gov- as LW imply, be estimated from the experimental data of table erned by rules influencing individual development; (ii) the tran- 1. It can only be concluded that the data of table 1 and the ex- sition probability function describing the propensity of an in- amples of biologically determined traits offered in the intro- dividual to change behavior with respect to incest is independent duction of the LW paper are irrelevant to the mathematical of the behavior of the other members of the society; and (iii) model for gene-culture coevolution. incestuous behavior can be subjected to a simple binary clas- Let us now examine LW's conclusion that small changes in sification, either its most extreme manifestation or its complete the assimilation function are magnified during social interac- absence. We will not discuss these assumptions here. tions. Their figure 3 displays the ethnographic curve P as a func- To implement their theory, LW need a means ofdetermining tion of( = (n2 - nl)/N, based on the transition probability func- uis (transition probabilities) from existing anthropological stud- tions u12 = aiexp(a2f) and u21 = a3exp(-a2f), where a,, a2, and ies. These studies, however, can provide only average fre- a3 are arbitrary parameters. For this particular choice of uj, quencies ofincest avoidance, in. By solving, in the steady state, there is no need to assume the continuity of f because the mas- the simple kinetic equation for the rate of change of n1, ter equation can be solved exactly in the steady state. We find hi =V21 (N - n1) - V12*1

P(6) = P(oN) (Nl)1 j( a3 ] they find, nl = N(1 + U12/U21) {exp - a2N(l- )1} [2] where they assume, without justification, that the average time The three curves ofLW's figure 3 correspond to three choices between decision points for outbreeders and inbreeders, T, and of a2. The uppermost curve, which arises on setting a2 = 0, is T2, are equal. not simply a slight adjustment ofa parameter. Instead it changes Because this equation provides ni as a function of the ratio the transition probability function uij, which varies exponen- of u21 and u12 (but not of u21 and u12 themselves), a further as- tially with f (the "trend watcher"), into one that is independent sumption is required. LW therefore make the additional as- of 4-i.e., independent of all influences from the other mem- sumption, again without justification, that u12 = u22 and u21 bers ofthe society (we might call it the loner). The term in braces = U11. The first equality states that the probability of making in Eq. 2 reduces to 1, and the expression becomes that of a ran- a transition into the inbreeding state from the outbreeding state dom walk (first bracket) that has a bias term (second bracket) is the same as the probability of staying in the inbreeding state (5). For a2 >0, the term in braces, which varies exponentially if the individual is already in that state. The second equality is with group size (25 in LW's example), is no longer equal to 1. the analogous statement for the opposite case. Thus, the prob- This exponential term totally dominates the random-walk term ability of making a transition to a particular state in the future as a2 increases. As a result, the lower two curves of LW's figure doesnoteven dependon the presentstate ofthe individual! By us- 3, characterized by a2 = 1.2 and 2.0, respectively, show qual- ingthis assumption, LWcan then determine thatu21 = 1N.- itatively different behavior from the uppermost one. LW next set up the master equation and solve it. Because It is evident, as even LW state, that the sensitivity of their the u..s are constants, the steady-state solution is given by Eq. results is due to this exponential variation with group size, 2 with a2 = 0, namely, a random walk that has a bias term. This which obtains for many choices of u... At least one of the sup- simple exact solution is true for all n1 and N. There is no need posed genetically determined parameters is multiplied by the for making, as LW do, the continuous f approximation, which large number N. Because this product appears as an exponent, results in a much more cumbersome and complicated formula. the sensitivity of the results to small changes in the parameter LW conclude that their theory "yields a correspondingly is guaranteed. LW maintain that this sensitivity is a strength clear and unsurprising relation between the degree of bias in ofthe theory because it shows how small changes in the genetic the epigenetic rules and the mode of the ethnographic curve" rules, the ui2s can result in large changes in the cultural patterns. and that "even fixed epigenetic rules, provided they do not favor This sensitivity can also be regarded as arising from the extreme one culturgen absolutely, can be expected to yield substantial sensitivity of the model to its fundamental assumptions. The cultural diversity" (ref. 6, p. 6250). These conclusions, however, magnification property arises because each individual is affected are vacuous. They arise from the fact that, as a consequence of to an equal extent by every other member ofthe population and LW's assumptions, h1 = u21N. Consequently, the average num- this population is fairly large (15-75). This model is similar to ber of outbreeders is directly proportional to the transition that for a dilute gas. A model based on, for example, that of a probability of switching from inbreeding to outbreeding be- liquid would assume that an individual is influenced to a great havior. Because the transition probability is assumed to be degree by its nearest neighbors (approximately six or eight for <1, h, is less than N-i.e., not everybody is an outbreeder. In a liquid), to a much lesser degree by its next-nearest neighbors, nonmathematical terms, this paper states that ifpeople are more and so on. Why should cultural influences be modeled better likely to switch from incestuous to nonincestuous behavior than by a gas, as LW propose, than, say, by a liquid? they are to make the reverse switch, then the society will be characterized by a high proportion ofincest-avoiding individuals. APPLICATION OF THE THEORY Recently, Lumsden and Wilson (6) have applied their theory DISCUSSION to a discussion of"the relationship between the epigenetic rules Critiques of the type presented in this paper are often open to of brother-sister incest avoidance which operate during indi- the rejoinder that the assumptions made in the model are not vidual development, and the frequency of occurrence of this crucial to the success of the theory, that the assumptions are form of incest among cultures" (ref. 6, p. 6248). LW chose to made for ease in computation, and that relaxing them will not study this particular behavioral pattern because of their belief alter the results. For example, in their paper on incest, LW state Downloaded by guest on September 30, 2021 Population Biology: Alper and Lange Proc. Natl. Acad. Sci. USA 78 (1981) 3979 that the assumption that the uPs are constants can be relaxed struction would result in even greater changes in the ethno- to the assumption that the transition probabilities can be writ- graphic curves. These factors, such as the inclusion of memory ten in the form u..(f) = a, + bi,6, where a.. = u,. (0) and bij = effects, ofinternal states ofthe individuals, and ofthe influence duo,(O)/df. This, however, is a trivial modification. Because it of groups of individuals and cultural institutions, to name just is stated that Ib.I << la.I, the u., are essentially constants and so a few, could be incorporated as parameters to manifest them- it is not surprising that this relaxation does not affect the pre- selves as changes in what LW consider to be the genetic func- dictions of the theory. tional forms and parameters. Furthermore, as mentioned It is impossible to prove that a more realistic mathematical above, the increased complexity could lead to instabilities, so theory of culture cannot be constructed. Nevertheless, we re- that small changes in the relative importance of these factors gard such a possibility as highly unlikely. For example, for the could result in major changes in the curves. Such instabilities Markov approximation to be relaxed, it would be necessary to would result in a complete breakdown of the LW model; the incorporate memory effects into the theory. Since, as we have initial states cannot be determined with arbitrary precision, so seen, even within the Markov framework, the empirical data that the final curves cannot be predicted at all. do not provide enough information to calculate the uijs, it is LW constructed their theory in an attempt to overcome the difficult to imagine how the additional data needed for memory current limitations ofsociobiology. Instead ofthe biological de- terms could be generated. The basic difficulties lie in the un- terminism of postulating genes that code directly for various justified assumptions concerning the existence of genes gov- human behavioral traits, LW propose that genes code for tran- erning transitions from one trait to another and in the reduc- sition probability functions that depend on environmental pa- tionist belief that culture can be explained by a mathematical rameters. However, because the assumptions ofthe theory are theory incorporating a few behaviorist empirical parameters. so severe and because the environmental parameters are lim- It is our belief that the simplifications inherent in any such ited to a single one (the number of individuals in a particular mathematical treatment are irremediable. cultural state), the LW theory does not appear to offer any new We conclude with some remarks about the determinism of understanding of the coevolution of genes and culture. the theory. Aspects of both genetic and environmental deter- minism play an important role. The epigenetic rules-i.e., the We thank J. Beckwith, E. H. Egelman, R. C. Lewontin, and P. Tay- functional forms ofthe transition probabilities-are genetically lor for their criticisms and comments. determined subject to alteration only by processes of natural 1. Lumsden, C. J. & Wilson, E. 0. (1980) Proc. Natl. Acad. Sci. USA selection. These rules are functions of the numbers of people 77, 4382-4386. 2. Wilson, E. 0. (1975) Sociobiology: The New Synthesis (Harvard in the relevant cultural states, an environmentally determined Univ. Press, Cambridge, MA), p. 553. quantity. This description ofthe distribution ofvarious cultural 3. Balescu, R. (1975) Equilibrium and Nonequilibrium Statistical behaviors is thus reduced to a genetically determined function Mechanics (Wiley, New York), pp. 695-729. ofa single behaviorist parameter. We have seen that the model 4. Coleman, J. S. (1964) Introduction to Mathematical Sociology is extremely sensitive to assumptions concerning the precise (Free Press, New York), pp. 38; 460; 528. 5. Reif, F. (1965) Fundamentals of Statistical and Thermal Physics nature of the functional form of the uijs. It seems likely that (McGraw-Hill, New York), pp. 7-11; 548-550. corrections of the failings of the theory resulting from the ex- 6. Lumsden, C. J. & Wilson, E.G. (1980) Proc. Natl. Acad. Sci. USA treme behaviorist assumptions that are necessary for its con- 77, 6248-6250. Downloaded by guest on September 30, 2021