Proc. Natl. Acad. Sci. USAVol. 86, pp. 568-572, January 1989Evolution

Behavioral evolution and biocultural games: Verticalcultural transmission

(evolutionary game theory/stable strategy/cultural transmission/biocultural game)

C. SCOTT FINDLAY*, CHARLES J. LUMSDENt, AND ROGER I. C. HANSELLt*Department of Biology, University of Ottawa, Ottawa, ON, Canada KiN 6N5; tDepartment of Zoology, University of Toronto, Toronto, ON,Canada M5S lA1; and tDepartment of Medicine, University of Toronto, Toronto, ON, Canada M5S 1A8

Communicated by Edward 0. Wilson, September 6, 1988

ABSTRACT We consider an evolutionary game model inwhich strategies are transmitted culturally from parents tooffspring rather than inherited biologically. Our analysis yieldstwo noteworthy results. First, biocultural games show a greaterdiversity of dynamical behaviors than their purely biologicalcounterparts, including multiple fully polymorphic equilibria.Second, biocultural games on average exhibit greater equilib-rium strategy diversity because of the countervailing influencesof cultural transmission and natural selection. Therefore,knowledge of a strategy's influence on Darwinian fitness is notsufficient to infer the evolutionary consequences of bioculturalgames. Further, our results suggest that cultural transmissionin the presence of natural selection may be an importantmechanism maintaining behavioral diversity in natural popu-lations.

1. Introduction

The conception of evolution as a formalized game hasilluminated many aspects of animal behavior (1-4). Yet littleattention has been focused on evolutionary games in systemsdriven by both biological and cultural forces ("bioculturalgames"). In most species behavior is not completely genet-ically determined. For humans and nonhumans alike, culturaltransmission during social development plays an importantrole in determining behavior (5-9). The dynamics of choosingbehaviors depends on various aspects of the cultural trans-mission process, including the ease with which differentstrategies are transmitted and learned (10). This is particu-larly true for psychologically advanced species, in which thecomplexity and attractiveness of individual strategies (andhence, their ease of transmission) is likely to vary across thestrategy set and for which there may be several complemen-tary modes of cultural inheritance (9, 10). Given the wide-spread occurrence of cultural transmission in animal popu-lations (6, 9, 10), biocultural games are likely to prove to beuseful extensions of the original game paradigm as a tool forunderstanding the evolution of social behavior (4, 11).

In this paper, we begin an investigation of evolutionarygames in systems subject to both cultural and biologicalevolutionary processes. We focus on the case of verticalcultural transmission-that is, transmission of behavior fromparents to offspring (10). Our rationale for doing so istwofold. First, studies ofboth human and nonhuman systemsindicate that many important behaviors [e.g., feeding strat-egies in birds (12) and monkeys (13), tool use by chimpanzees(14), and a host of human behaviors including choice ofvocation, religious affiliation, and hunting skills (15, 16)] aretransmitted vertically. Second, vertical transmission is theclosest cultural analog to standard biological inheritance, themode of transmission implicit to the original concept of

evolutionary stability (3) and to most applications of biolog-ical game theory to empirical data (4). Consequently, theanalysis provides useful information on the conditions underwhich the dynamics of biocultural games converge to thebiological case and facilitates comparisons between the two.

2. Biocultural Game Dynamics with Vertical Transmission

Vertical cultural transmission involves the nongenetic trans-mission of behavior from parents to offspring (10). Forbiparental species, the process can be characterized by a setTv = {13} of transmission coefficients, Hi being the proba-bility that an offspring produced from a mating between amale using strategy k and a female using strategy I will adoptstrategy j, with j not necessarily equal to k or 1. In whatfollows we assume that selection takes the form of differentialfecundity rather than differential survival of strategy types(4), that cultural transmission is independent of both donorand recipient sex (i.e., PI, = Sk), and that vertical transmis-sion is complete (Xjg(3l = 1). Furthermore, although inbiological games it is usually assumed that offspring adopt(via standard Mendelian inheritance) the parental strategy, inthe case of pure cultural transmission, the probability of anindividual assimilating particular aspects of the socioculturalenvironment is independent of their genetic constitution (17).The extent to which socialization and cultural transmissionare mediated by genetic information (gene-culture transmis-sion) is a topic of considerable debate, both at the empiricallevel and at the theoretical level (9, 10, 17). The morecomplex case of biocultural games involving gene-culturetransmission will be treated in a subsequent report. For thepresent, we concentrate on pure cultural transmission. Wewill call this model the ''purely cultural case."Assume a large (effectively infinite) population and let WkI

be the biological fitness (number of offspring) of a k x Imating. If 4k is the frequency ofmating type k x 1, then underrandom mating, 'kl = 2pk.Pt if k + I and 40kk = Pk. with Pk thefrequency of k strategists in the population. Suppose that thefitness of a mated pair is given by

Wkl = Bo +Lpm(Bkm + Bim) = Wik.m

[2.1]

In Eq. 2.1 we assume that each parent engages in randominteractions with the population at large, with pair fitnessesgiven by the sum of the expected fitnesses of each member.We also assume that strategy adoption influences fecundityrather than survivorship (fecundity selection). Bo is a baselinefitness independent of participation in any games. [If inter-actions are not random, one can introduce an encounterfunction, Ek,(p), specifying the probability of an encounterbetween a k and I strategist-see ref. 18.] The average

Abbreviations: ESS, evolutionarily stable strategy; BCSS, biocul-turally stable strategy.

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biological fitness of the population is then W = Yk,l ckkWkI,and, in the case of continuous reproduction or weak selec-tion, the rate of change in the frequency of strategy i is

Pi = Z klWkI (Oil - Pi) =fi(P). [22.2kI

Equilibria J = (Pi) of Eq. 2.2 are obtained by solving thesystem ofN equations Pi = 0. Once derived, local stability ofp can be determined by first-order expansion, provided alleigenvalues of the Jacobian Df = [afil/ap] at J have nonzeroreal parts (19).From Eq. 2.2 we see that in the biocultural case, the

evolutionary success of a strategy is determined by itsbiological fitness and by its probability of adoption viacultural transmission. Together these two factors determinethe biocultural fitness R of a strategy. For vertical transmis-sion, we can write

Ri-Z4 klWkl Ski i = 1,..., N. [2.3]Pi kl

The definition 2.3 permits some useful insights into thesystem equilibria. In particular, when vertical transmission iscomplete (i.e., all offspring are enculturated), the meanbiocultural fitness of the population is just the mean biolog-ical fitness:

7-= >PiRi =W.and substitution in Eq. 2.2 yields

PiPi0i = p.Ri - W).

(2.4]

(2.5]

Hence, equilibria occur at points where the biocultural fitnessof all strategies is the same. The similarity between Eq. 2.5and the standard dynamic for continuous biological gameswith haploid inheritance Pij = pi(Wi - W) (20) suggests thatthe conditions for evolutionary stability can be cast in a formsimilar to the classical evolutionarily stable strategy (ESS)conditions, with the important difference that it is a strategy'sbiocultural rather than biological fitness that matters.THEOREM 1. Let A be a continuous strategy set, with qi (u)

the probability that a randomly chosen individual playsstrategy u lying on the interval [u, u, + du] and with Wu, thefitness ofpair type u x v. Let u, v, it E A and define

M(u, v) = WuvSyV/4(A) [2.6]

as the contribution of mating type u x v (= v x u) to thebioculturalfitness R of strategy A. Then ,u is a bioculturallystable strategy (BCSS) if / = 1 and, for all v 4 A butsufficiently close to ,/, either

MA'( A) > R(v, Wu [2.7a]

or

MA, A) = '(v, A) and ?(A, v) > I(v, v). [2.7b]

In terms of biological and cultural components, these con-

ditions are, in the case of vertical transmission,

2,Lv(B, + B + BO) > BVM + Bo [2.8a]

or

2P'-v(B, + BIAV + Bo) = Bv + Bo and

2plv(Bo + BMv + Bvv) > 2(Bvv - BvA)

-lv3(2Bv/A + BO) + Bo.

Moreover, ifPA. < 1, then no single strategy is bioculturallystable, and a population consisting ofa single strategy typecan always be invaded.

Proof: In the general case where i = v can be pure ormixed, Eq. 2.3 is rewritten as

R-,, = | du dv q(u)l(v)j'v(u, v)A

[2.9]

on substitution from Eq. 2.6. Here, 1,, is interpreted as theexpected biocultural fitness of a strategy lying on the interval[A, tL + dpi. Similarly, we can rewrite Eq. 2.5 as

[2.10]

Suppose that the proportion of i strategists in the popu-lation is 1 - E, with e small. Then since p = (1 - E)Ry + ER,,4(,u) > 0 if and only if Rg > al. Let Ak = qi(v)f" - O(u)f3UW.Then for A to be uninvadable, it must be that for all v + Iu,> Rv, or

1 du dw il(u)il(w)WAv > 0. [2.11]OAr)jOMiv)

By setting 4i(A) = 1 - e and +(v) = E and assuming W~,v = WVAand f38v = 8A , this becomes

(2B, + Bo)(p3. - 1)+ E{BgA[6(1 - alg) + 2Pyv] + 2[BiLV(P8. - 1)

+ (J3Bv - 1)] + Bo(l - 2I3,8 + 2Pf3v)}+ E2{Bs,,[6(1,3A - 1) - 4 l3yv] + Bvu[4(l - v) + 2PfL]

+ BIIV[4(1 - PEB) + 2 }3vj + 2 Bvv(P3/1 - 1)+ BO(PA -2P/,, + PA )} > 0 [2.12]

to order e2. Inequality 2.12 is particularly interesting, sincefor Pl, < 1 the limit as E -O 0 is

(2B,, + Bo)(p,8. - 1) < 0.

Hence, if P',L < 1, no single strategy-pure or mixed-isevolutionarily stable. If,3,. = 1 so that when both parents usethe same strategy, all offspring inherit it, the first term ofinequality 2.12 vanishes, and the stability conditions aregiven by Eqs. 2.8. By contrast, when tB = 1, l3 VA= =1/2, and f3P = 0 for all v # ,, these become either B > BvAor B,, = BvA and B,.V > B, the standard ESS conditions (4).This is not surprising because the model is now identical tothe case of haploid genetic inheritance (10), the transmissionscheme under which the standard ESS conditions wereoriginally derived (4).Under vertical cultural transmission, whether or not the

corners of A are valid equilibria depends on the matrix ofcoefficients Tv. If P3A. = 1 and f3A = 0 for all pu j u and v,then all points corresponding to fixation of a strategy areequilibria. But there is nothing in the nature of the culturaltransmission process to necessitate that these conditionshold. Indeed, it is likely that in many cases parents haveknowledge of a number of different strategies even thoughelecting to use only one. If the full array of alternatives istransmitted, offspring may well adopt a strategy other thanthat actually used by either parent (10). Moreover, unless PU3= 1 for all u E A, no single strategy is a BCSS. Note furtherthat evolutionary stability is in part determined by thebaseline biological fitness Bo, a result not included in theoriginal ESS formulation (4) but nonetheless equally appli-cable to purely biological games (21). The BCSS conditions(Eqs. 2.8) can also be derived for other forms of the pairfitness functions, including multiplicative fitness (22).

Evolution: Findlay et al.

OW = qj(tL)(R., .).

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3. Bioculturally Stable States and Strategies

Consider two groups, one in state p and the other in state q.Let

a(q, p) E qiai(p) [3.1]

be the average biocultural fitness of a group in state q playinganother in state p, with qj the proportion of individuals in theformer playing pure strategy i. Consider now a globalpopulation p' = (1 - E)p + Eq, E > 0. Then p is a bioculturallystable state if for all p' E A, p' p,

90p p) > R(p, p) [3.2a]

or

R(p, p) = R(p', p) and (p, p') > R(p', p') [3.2b]

for E sufficiently small. Note that Eqs. 3.2 are not equivalentto R(p) > a(p'), since as in the purely biological case, themean biocultural fitness need not increase monotonically(23). If we regard p = ,u and p' = v as alternate (possiblymixed) strategies, fulfillment of conditions 3.2 is required for

-,, > jv, so 3.2 is exactly the condition used to derive theBCSS criteria 2.8. Hence any BCSS ,u corresponds to abioculturally stable state p = u satisfying conditions 3.2.THEOREM 2. Ifu is a BCSS, then a state p = ,u satisfying

p = 0 is at least locally stable.Proof: Let p! = pi + {i, with {i small. Substitution of Eq.

3.1 into Eqs. 3.2 yields Q = Xi fiRi(p) < 0 or Q = 0 and

H = i + ) - '(p + 0] < 0, [3.3]

since X (i = 0 and pi(p) = R(p) for all i ifp = 0. Substitutingpi + (i for pi in Eq. 2.5 yields

(i(6 + pi) [(p + P) - R(p + +)] [3.4]

to first order. Let

V() = + pi log( ). [3.5]

4. Multiple Interior Equilibria and Equilibrium Diversity ofBiocultural Games

As with the purely biological case, for games of low dimen-sionality, the fixed points of the dynamic 2.2 determine thenumber of qualitatively different outcomes (stable classes).Fig. 1 enumerates the nontrivial stable classes for two-strategy biocultural games and their corresponding biologicalanalogs. When 01h = 1822 = 1, P22 = Pu2 = 0, and pair fitnessesare additive, there are no differences between the two withrespect to their stable classes. If either ,11 + 1 or 1322 # 1 orif pair fitnesses are multiplicative, there may exist multipleinterior equilibria in the biocultural case but not in the purelybiological case. Hence, biocultural games may admit elemen-tary catastrophes in A (24) in addition to the classic Hopf andexchange of stability bifurcations characterizing parameter-ized biological games (23). As the dimensionality of the gameincreases, there is an increase in the ratio of the number ofstable classes in biocultural versus biological games. For N> 2, even if Pi = 1 and 1j3k = 0 for all i, j, k (i + j, k) andfitnesses are additive, there may exist multiple interiorequilibria. Fig. 2 gives examples of biocultural games withtwo interior repellors, one attractor and one saddle, and twoattractors, respectively. None of these flow structures arepossible in simple biological games (23) with no mutation andconstant payoff matrices.The general effect of vertical cultural transmission on

equilibrium strategy diversity in a two-strategy game isshown in Fig. 3. In each case, 50,000 points were chosen atrandom from the parameter space B, B x 8112, B x 1312 X P111,B x 13 X 1312 X 822 (Fig. 3 A-D, respectively), correspondingto biocultural games with decreasing restrictions on thecultural transmission coefficients. For each point, the equi-librium strategy diversity D = X jc1 ( Ai In i)j was calculatedat all stable equilibria for both the biological (Fig. 3A) andbiocultural (Fig. 3 B-D) cases, where (Pi) is the frequency ofstrategy i at the jth equilibrium, and Cj is a normalizedweighting coefficient proportional to the size of the basin ofattraction ofj. Repetition ofthis procedure with four differentsets of random points indicates good convergence (<5%discrepancy among runs) for the first four moments of theresulting distributions, so we have a reasonable confidence inthe validity ofthe results (25). Unless transmission is absolute(that is, when ,111 = 1832 = 1), two-strategy biocultural gamesin general show greater equilibrium diversity than theirbiological counterparts. In the case of absolute transmission

For any p E A, V(O) = 0, and V(f) > 0 for all 0.

Furthermore,

V=>jE i = H, [3.6]

which by Eq. 3.3 is always negative for f sufficiently small.Hence, V is a Liapunov function and p is at least locallystable, irrespective ofwhether it lies in the interior A or on theboundary aA of the game space A.As a corollary to Theorem 2, we note that point attractors

of the dynamical system 2.2 need not be BCSSs becauseconditions 3.2 specify only asymptotically stable equilibria.Hence, there may be points that attract all nearby orbits butare not BCSSs (see refs. 21 and 23 for similar resultspertaining to biological games).We come now to a discussion of the dynamical behavior of

Eq. 2.2. In summarizing our observations, we will focus onimportant properties of biocultural games not found in simplebiological games. Particular attention is given to the existenceof multiple interior equilibria and to the effects of culturaltransmission on behavioral diversity.

FIG. 1. The stable classes for two-strategy biocultural (Left) andbiological (Right) games. A lists the classes when pair fitnesses areadditive, and A and B list the classes when fitnesses are multiplica-tive. Line segments represent the interval [0, 1], with endpointscorresponding to fixation of one strategy and extinction of the other.Solid dots are attractors, open dots are repellors, and arrows give thedirection of flow.

A

p~~~~~~~

-

B

C4C_-p

A

0 0~~~0

40

B

, C* 0 C0* *

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Proc. Natl. Acad. Sci. USA 86 (1989) 571

FIG. 2. Biocultural games with multiple interior equilibria. Symbols are the same as in Fig. 1. The matrices are, in clockwise order (startingat the upper left-hand corner) around each triangle, B, Tt = {olf3, TI, and T,. Starting from the lower right, the vertices are, in clockwise order,P3, pl, and P2- In all cases, pair fitnesses are additive.

(Fig. 3B), the reduced diversity shown by biocultural gamesis due principally to the asymmetry introduced by theone-dimensional game dynamic: random choice of 012 maytransform a biological game with an interior ESS ofarbitrarilyhigh diversity into a game where one of the fixation equilibriais the only attractor (hence, D = 0), but in cases where choiceof /312 leads to a stable interior BCSS although the corre-

sponding biological game has no interior ESS, the BCSSgenerally has low diversity. Analysis of the three-strategycase suggests that for N > 2, even biocultural games with 13

= 1 for all i show greater average equilibrium diversity thantheir purely biological counterparts.

0.8 -

0.6

0.4 -

0.2

C.)

a)

a-a1)U-

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6

5. Discussion

The nongenetic transmission of behavioral information fromparents to offspring is perhaps the simplest form of culturaltransmission and the one most closely resembling Mendelianinheritance. However, there are important differences. Inparticular, it may be that information about the strategy setis transmitted rather than the individual strategies them-selves. Consequently, offspring may adopt a strategy that isnot used by either parent. Furthermore, it is not simply thepresence or absence of a strategy in the parental complementthat determines its probability of adoption. Strategies in

0.8 0.0 0.2 0.4 0.6 0.8

0.0 0.2 0.4 0.6 0.8 0.0 0.

Equilibrium diversity (D)|2 0.4 0.6 0.8

FIG. 3. The effect of cultural transmission on equilibrium behavioral diversity. Sample statistics are based on 50,000 Monte Carlo trials ofpoints drawn randomly from B (A), B X ,12 (B), B x 8h1 X ,12 (C), and B x 8h1 X 1832 X 22 (D). The first is representative of purely biologicalgames, and the remaining three, of biocultural games. For further details, see the text.

2.3 4.5 8.5 1.0 .25 .42 7.3 4.5 3.0 1.0 .20 .50 2.1 .55 -0.5 1.0 .35 .50-.5 2.8 9.0 .25 0.0 0.0 1.85 1.8 4.0 .20 0.0 0.0 1.1 -0.2 1.0 .35 0.0 0.0

2.0 -4.5 5.5 .42 0.0 0.0 7.0 2.5 -2.0 .50 0.0 0.0 1.9 -1.0 -0.4 .50 0.0 0.0

0.0 0.0 .58 0.0 .75 0.0 0.0 0.0 .50 0.0 .80 0.0 0.0 0.0 .50 0.0 .65 0.00.0 0.0 .80 .75 1.0 .20 0.0 0.0 .75 .80 1.0 .25 0.0 0.0 .80 .65 1.0 .20.58 .80 1.0 0.0 .20 0.0 .50 .75 1.0 0.0 .25 0.0 .50 .80 1.0 0.0 .20 0.0

AMean = 0.120

Variance = 0.059

- ~ ~~~ -f| X r

DMean = 0.568

Variance= 0.018

I

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general will differ in attractiveness or simply the ease withwhich they are learned. By contrast, under genetic transmis-sion the probability of an offspring inheriting a particularparental allele is the same for all alleles at a locus (barring, ofcourse, the complexities introduced by cytoplasmic inheri-tance, segregation disorders, and meiotic drive).Our comparison of a biocultural game model incorporating

vertical cultural transmission with the standard biologicalmodel (haploid inheritance and no mutation) yields threeimportant results: (i) the conditions for evolutionary stabilityin biocultural systems are different from those in the purelybiological game, so that points in the strategy space satisfyingthe standard ESS conditions need not be bioculturally stable,and BCSSs need not satisfy the standard ESS conditions; (ii)biocultural games can evolve to different equilibrium statesdepending on the initial conditions, with the number ofpossible states generally exceeding those found in simplebiological games; and (iii) equilibrium strategy diversity isgreater on average than that in the biological case. Thesedifferences arise because in biological games the strategydynamics depend only on biological fitness, whereas inbiocultural games both biological fitness and the probabilityof strategy adoption through cultural transmission determinea strategy's fate. Hence, the tendency of natural selection toincrease high-fitness strategies may be counteracted by theeffects of cultural selection influencing strategy adoption.Alternatively, strategies conferring low biological fitnessmay be culturally selected (9, 10, 17). Both scenarios can leadto the establishment of stable polymorphisms that wouldotherwise not obtain.The question arises as to whether the observed differences

between biological and biocultural games is a unique conse-quence of cultural transmission per se. In the case of verticaltransmission only, it may be possible to simulate the effectsof cultural transmission described above by introducingadditional complexities into purely biological games-e.g.,mutation, diploid inheritance, or several episodes of selectionover the course of an individual's lifetime. This is an impor-tant area for future research.

Transmission of behavior from parents to offspring is onlyone possible mode of cultural inheritance. Detailed investi-gations (10) suggest that the evolutionary consequences ofcultural transmissions are critically dependent on the natureof the transmission process. In the case of games involvingmore complex patterns of cultural transmission, some of theunique properties of the socialization process may becomemore evident. In particular, because in purely biologicalsystems the strategies are inherited by offspring from theirparents, strategy transmission is not frequency-dependent.However, in the case of oblique or horizontal culturaltransmission, the probability of an individual adopting astrategy depends on the individual's exposure to it duringsocialization (9, 10, 17). Moreover, while cultural transmis-sion is often Lamarckian, biological inheritance presumablyis not. These two features of culture learning are expected toplay an important role in the dynamics of biocultural games.

We thank Ken Aoki, Mart Gross, and David Sloan Wilson forhelpful discussion, two anonymous referees for useful criticism andsuggestions, and Anne Hansen for careful preparation of the manu-script. The work was supported in part by Population Biology Grantsto C.J.L. and R.I.C.H. from the Natural Sciences and EngineeringResearch Council of Canada (NSERC) and by a grant from the CrayResearch Fund of the University of Toronto. Simulations used theresources of the CRAY X/MP supercomputer at the Ontario Centrefor Large Scale Computation. C.S.F. is a Doctoral Fellow of theNatural Sciences and Engineering Research Council of Canada, andC.J.L. is a Career Scientist of the Medical Research Council ofCanada.

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