Behavioral evolution and biocultural games: Oblique and horizontal cultural transmission

Z theor BioL (1989) 137, 245-269

Behavioral Evolution and Bioculturai Games: Oblique and Horizontal Cultural Transmission C. SCOTT FINDLAYt~:§ ROGER I. C. HANSELLt AND CHARLES J. LUMSDEN•

Department of Zoology, t University of Toronto, Toronto, Ontario, Canada M5S 1A1 and Department of Medicine~-, University of Toronto, Toronto, Ontario, Canada M5S 1A8 (Received and accepted 16 May 1988) We consider an evolutionary game model in which strategies are culturally transmitted among individuals rather than inherited biologically. In addition to vertical transmission (from parents to offspring), we investigate the effects of oblique (from members of the parental generation to offspring) and horizontal (among age peers) transmission on game dynamics. Our study yields two important results. Firstly, biocultural games show a greater diversity of dynamical behaviors than their purely biological counterparts, including multiple fully polymorphic equilibria. Secondly, biocultural games on average show greater equilibrium strategy diversity. These results suggest that cultural transmission in the presence of natural selection may be an important mechanism maintaining behavioral diversity in natural populations.

I. Introduction

The concept of an evolutionary game has proven successful in explaining many important aspects of behavioral evolution (reviewed in Parker, 1978; Maynard Smith, 1982). The approach is based on two key insights: (i) that the success of an individual using a particular strategy is, at least in part, contingent upon the strategies adopted by conspecifics; and (ii) that the invasability of a strategy is a good measure of its evolutionary success. This notion of invasability forms the basis for Maynard Smith & Price's (1973) concept of an evolutionarily stable strategy (ESS): a strategy which, if adopted by a l m o s t all members, renders a poulation uninvadable by any other strategy. Persistence and invasability imply an underlying dynamic, traditionally one based on natural selection. Under this assumption, one can derive the conditions a strategy must fulfil if it is to resist invasion (Maynard Smith & Price, 1973). While dynamics were not explicitly incorporated into the original ESS formulation, subsequent investigations (Bishop & Cannings, 1978; Taylor & Jonker, 1978; Zeeman, 1979) have shown that the standard ESS conditions in many cases do guarantee dynamic stability (but see Rowe et al., 1985). Moreover, although originally formulated strictly at the phenotypic level for infinite asexual populations, the ESS formulation can § Current address: Department of Biology, University of Ottawa, 30 Sommerset Street East, Ottawa, Ontario. Canada KIN 6N5 245

0022-5193/89/070245 + 25 $03.00/0

© 1989 Academic Press Limited

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still provide a useful characterization of behavioral evolution when such complicating factors as finite population size, sexual reproduction, mutational and environmental perturbations, and assortment of encounters are incorporated (reviewed by Hines, 1987). The problem with the ESS approach, at least as applied to behavioral evolution, is not with the concept of evolutionary stability per se, but rather with the assumption that the underlying dynamic is driven solely by natural selection. Cultural transmission of behavior, including imprinting, conditioning, observation, imitation and direct teaching, is prevalent in many species (Bonner, 1980). Furthermore, some of the best examples of cultural transmission in animals involve behavioral strategies that influence biological fitness (Goodall, 1986; Sutherland, 1987). These cases require a theory that incorporates the effects of both cultural transmission and differential reproductive success. Since previous studies indicate that the dynamics of behavior under cultural transmission is not simply biological evolution writ large (Cavalli-Sforza & Feldman, 1981; Lumsden & Wilson, 1981; Boyd & Richerson, 1985), classical evolutionary game theory is unlikely to yield accurate predictions about the fate of individual strategies. We suggest that for many (if not most) behaviors of interest, a natural paradigm for the study of evolution is the biocultural game, that is, a game in which both cultural and biological processes (and their interaction) figure prominently in the underlying dynamic. To this end, in a previous paper (Findlay et al., 1989), we introduced the concept of a bioeulturally stable strategy (BCSS): a strategy which, if adopted by almost all members, renders a population uninvadable by any other strategy under the combined influences o f both cultural and biological evolutionary processes. If/~ is a BCSS, then for all u ~ ~, the generalized BCSS conditions are: either F~,~,> F,~ or F~, = F,~, and F~,u > F,,, where F~,, is the bioeulturalfitness of an individual playing strategy p. against an opponent playing u. Strictly speaking, biocultural fitness is not a property of individuals but of strategies: in essence, it measures a strategies" ability to replicate itself in subsequent generations either through standard biological processes (e.g. mendelian inheritance) or cultural transmission. As formulated above, the BCSS conditions have the same form as the classical ESS conditions. The difference lies in what makes up a strategy's biocultural fitness F. In standard biological games, the fitness of a strategy is obtained directly from the biological payoff matrix or as a (usually linear) function thereof. By contrast, in biocultural games F depends not only on the biological payoff matrix but also on the particulars of cultural transmission, including the ease with which a strategy is learned or transmitted, its attractiveness and the type of cultural transmission involved (e.g. intrafamilial or extrafamilial). As a result, the biocultural fitness F of a strategy is often a highly non-linear function of its frequency in the population. The effect of these non-linearities on behavioral evolution can be considerable: even the simplest biocultural games exhibit dynamical behaviors not found in standard biological games (Findlay et al., 1989). In a previous paper (Findlay et al., 1989), we investigated biocuitural games involving vertical cultural transmission, that is, transmission from parents to offspring

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(Cavalli-Sforza & Feldman, 1981). While this is a common form of cultural transmission in both human and non-humans (Cavalli-Sforza et al., 1982; Hewlett & Cavalli-Sforza, 1986; Goodall, 1986), oblique transmission [from non-parental individuals of the parental (or older) generation to offspring] and horizontal transmission (among age peers) are also common (Hinde et al., 1985; Cheney et al., 1986; Smuts et al., 1987). Our objective in this paper is to explore the dynamics of biocultural games subject to these modes of cultural transmission and to derive the conditions for evolutionary stability. Throughout, we shall be concerned only with systems in which genetic differences among individuals do not influence the probability of particular strategies being adopted, a scheme referred to as pure cultural transmission (Lumsden & Wilson, 1981). The case of gene-culture transmission (Lumsden & Wilson, 1981) will be treated elsewhere.

2. Biological Game Dynamics Central to the concept of an evolutionary game is the evolutionarily stable strategy, or ESS. An ESS is a point in the strategy space of a species such that a population at the ESS can resist invasion by any mutant strategy and hence, is "evolutionary stable" (Maynard Smith & Price, 1973). Elements u = ( u ~ , . . . , uN) of the strategy space correspond either to the frequency of individuals using strategy i or, alternatively, define a (possibly mixed) strategy in which an individual adopting u plays pure strategy i with probability u~. Under the latter interpretation, Iz is an ESS if for all u ~/z, B . . > B,,.

(2.1a)

and

(2.1b)

or

Bu. = B..

B . . > B .....

where B~j is the expected payoff, in units of biological utility, to an individual playing strategy i against an opponent in playingj (Maynard Smith & Price, 1973). Consider a set S of pure strategies, and let p = (p~ . . . . . PN) denote the state of the population, thought of as a probability distribution on the simplex A = {p ~ II~N; p~ -->0 and ~i P~ = 1}. Denote by 2x and OA the interior and boundary of A, respectively. A game dynamic is an at least C 2 map (generally a vector f i e l d ) f ( p ) , f: A~ff~ N. In the case of continuous reproduction, dp/dt =-f(p) for a standard biological game is given by

f(p)=-p~=p,(W,-l~'),

i = 1 . . . . . N,

(2.2)

where W, = ~j pjB~j is the expected biological fitness of strategy i and ~V = ~o pipjBo is the average fitness of the population. The dynamical behavior of this model is well understood (Taylor & Jonker, 1978; Zeeman, 1979). In the following sections, we introduce an extension of eqn (2.2) for the biocultural case involving oblique and horizontal cultural transmission.

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3. Biocultural Game Dynamics with Oblique Transmission With oblique transmission, juveniles learn from older individuals other than the parents, including grandparents, aunts, uncles and unrelated members of the parental generation (Cavalli-Sforza & Feldman, 1981). We consider three different functional forms:

Z ~ , ( p ) = p,[1 + ai(p, - 1)],

(3.1a)

Z~b,(p) = p,[1 - c~,(p,- 1)],

(3.1b)

Zd/~ (p) = p,/ [ 1 + exp ( - a , p i ) ],

(3.1c)

with 0 < •i ~ 1 and Z a normalization constant given by the sum over i of the right sides of eqns (3.1). Here, ~ is the probability of a juvenile adopting strategy i through oblique transmission, given that it was enculturated via oblique transmission. The quantity ai is a rate parameter that might, for example, depend on the complexity of strategy i (e.g. simpler strategies are transmitted or learned more easily than complex strategies). The assumption underlying eqns (3.1) is that the more common a strategy in the population, the more likely individuals are exposed to it during socialization, and hence the more likely it is to be adopted. (For discussion on empirical data pertaining to the "trend-watching" form of oblique transmission, see Lumsden & Wilson, 1981.) In addition, if no older individuals use strategy i, we assume it cannot be adopted by juveniles, whereas if all use it, all juveniles adopt it [hence, ~bi(0) = 0 , ~b,(1)= 1]. In formulating eqns (3), we envision a population in which unenculturated juveniles make contact with a number of older individuals other than parents. These older individuals act as teachers. We assume that the effects of individual contacts are not independent, i.e. there is not a constant probability of a juvenile adopting strategy i in contact with a teacher of type i (cf. Cavalli-Sforza & Feldman, 1981, p. 131). For example, eqn [3.1(b)] incorporates a saturating effect such that the weight accorded each contact with an i teacher declines with the number of previous i contacts. We further assume that the number of teachers contacted is sufficiently large such that stochastic effects arising from the finite nature of the teacher sample are negligible. Let B = [}Bjkll be the biological payoff matrix, with Bjk the payoff, in units of biological fitness, accrued to an individual using strategy j playing against an opponent using k. We consider the cases in which the fitness of a mated pair is given by

Wkl = Bo+ ~ pj(Bkj+ BIj) = Wlk,

(3.2a)

J

Wkl = Bo+ ~ pjBkj" ~ pjBtj = Wtk, J

(3.2b)

J

that is, by additive and multiplicative contributions by each member of the pair, with B0 a positive constant (see Findlay et al., 1989). This corresponds to situations in which strategy adoption influences fecundity (fecundity selection) rather than survivorship (viability selection). Clearly, when transmission is entirely oblique,

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differences in fecundity among mated pairs do not effect strategy frequencies. Changes in strategy frequencies depend only on the probability of strategy adoption via oblique transmission. The resulting game dynamic is then written as f(P)

(3.3)

=- lJ, = ( 0~ - P,) ff¢,

with I~" =~k.t pkptWk~ the mean biological fitness of the population. When there is vertical transmission (from parents to offspring) followed by oblique transmission, eqn (3.3) becomes =

ilk1+

1--

fl{I O~

,

(3.4)

k,I

with 4~kk = P ~ , d P k ~ = P ~ = & k i . In eqn (3.4), fl~ is the probability of an offspring inheriting strategy i through vertical transmission given that its parents use strategies k and l, so that 1 -F.~ fl{i is the probability of an offspring remaining unenculturated after vertical transmission. Assuming that all juveniles are eventually enculturated, this is also the probability of enculturation via oblique transmission. In both eqns (3.3) and (3.4), we assume that once a strategy is adopted (either through vertical or oblique transmission), the individual retains it. While this assumption may be an oversimplification in some cases, there is evidence from human populations that cultural traits often show considerable longitudinal stability (Cavalli-Sforza et at., 1982). Equations (3.3) and (3.4) show that, in biocultural games, the evolutionary fate of a strategy in general depends both on the biological payoff matrix B and the nature of the cultural transmission process as determined by the matrix of vertical transmission coefficients Tv ={fl{~} and the oblique transmission functions {~b~}. Together, these two components determine the biocultural fitness F of a strategy. For eqn (3.3) we have F~ = ~ , f f ' / p , ,

(3.5a)

whereas for eqn (3.4),

Using these definitions we can rewrite eqns (3.3) and (3.4) as f~, = p,(F~ - F ) ,

(3.6)

where P = ~j piFi = if" is the mean biocultural (and biological) fitness of the population. Putting eqns (3.3) and (3.4) in the form o f e q n (3.6) has two advantages. First, it immediately shows that any fixed point of a bioculturai game dynamic must satisfy F~ = P for all i. Second, by substituting W for F in eqn (3.6) we recover the standard dynamic for purely biological games (Taylor & Jonker, 1978). Note, however, that since Wi # Fi in general, points satisfying W,. = if' will not generally constitute valid equilibria for biocultural games. We turn now to the conditions for evolutionary stability in biocultural games. To do so, we allow individuals in the population to play both pure and mixed strategies,

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ET AL.

since stability implies resistance to invasion by both types (Maynard Smith, 1982, p. 185). We then interpret A as a continuous strategy set, points of which represent individual strategies rather than abundances of strategy types. Let ~b(u) be the probability that an individual chosen at random from the population plays a strategy lying on [u, u + d u ] . Define

FU(u, v)=- W,v~,/~b(/z), bt

~

(3.7a)

v

F~'(u, v)=- W,,o[fl,,~+(1 -~,o-fl,~o)~b~]/qS(/z),

(3.7b)

as the contribution of mating type u x v to the biocultural fitness of strategy/z under oblique and combined vertical and oblique transmission respectively. In the general case where strategies can be pure or mixed, we rewrite eqns (3.5a) and (3.5b) as

F,~ = ~a du do ~b(u)qS(v)F"(u, v),

(3.8)

with F,, the biocultural fitness of a strategy lying on I/Z,/Z +d/z]. Similarly, we can rewrite eqn (3.6) as ~(/Z) = 6(/Z)(F~ - P),

(3.9)

f P = j,, ,~(u)F~ du.

(3.10)

where

Suppose the population comprises a proportion 1 - e /z-strategists and e vstrategists with e small. Then, since F = ( 1 - e)F,~ + e f t , q~(/z)> 0 if, and only if, F , > Fo. That is, /Z can resist invasion by another strategy v if and only if its biocuitural fitness exceeds that of the mutant strategy. Hence, f o r / z to be a BCSS, it must be that for all v # / z ~ A,

a d u dw ¢b(u)~b(w)[F ~'(u, w) - FV(u, w)] > 0.

(3.11)

For ~b(tz)= 1 - e , th(v)= e and assuming Wwo = Wo,,, this becomes

ff e(1 - e) (4'~ - 1 + e) > 0 when there is only oblique transmission. Since if' and ( 1 - e ) e the condition (3.12) reduces to

(3.12) are both positive,

(%, + c~o)e - a~ < 0,

(3.13a)

(a, +ao)e-ao>0,

(3.13b)

o~, - e(o~o + a . ) > 0,

(3.13c)

when oblique transmission is of the form of eqns (3.1a-c) respectively. In the limit as e ~ 0, inequalities (3.13a) and (3.13c) always hold, while inequality (3.13b) never does. In the former case, individuals are not sufficiently sensitive to allow a mutant strategy to increase when rare, while in the latter case they are.

BIOCULTURAL

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251

F r o m eqns (3.13) we see that with fecundity selection and oblique transmission o f the form, given by eqns (3.1a) or (3.1c), any strategy which attains a high frequency in a p o p u l a t i o n (by whatever means) is always bioculturally stable irrespective o f its associated biological payoff matrix. By contrast, when oblique transmission takes the form o f eqn (3.1b), no single strategy is ever stable, since a p o p u l a t i o n m o n o m o r p h i c for the strategy can always be invaded. W h e n both vertical and oblique transmission occur, the situation is more complex. In particular, the conditions for evolutionary stability involve B as well as the various cultural transmission parameters. In the case o f additive pair fitnesses W~,~, = 2 [ ( I -

W~,o = (1 -

e)B~.. + eBb,o] + Bo,

e)(B~,~, + Bo~,)+ e(B~,o + Boo) + Bo,

(3.14a)

WvL,= 2[(1 - e)Bv~, + eBoo]+ Bo, while for multiplicative fitness, W~,~, = (1 -

e) 2B-~. + 2e(1-e)B~,oBuu +e2B~o+ Bo,

W~o=(1-e)2Bo,~B~,,., +e(1-e)[Bo,~B,~o+ B,~,~Bov]+e2B~,~Boo+ Bo, W ~ o = ( 1 - e ) -~B 2o~,+ 2e( 1 -

} (3.14b)

~ 2 Bo. e)Bo~,Bov + e-Boo+

To order e 3, the BCSS condition is then

-I~,s,A+e[A(C-~'

a~~.),-2~vD]+e" "

~'

2{oL[A~,~+AC+2~,t,(F+A)_213,~vD ,~

- 2 C ( F + A)+ AE-~°m,(4F + A)+ 2 ~ v ( G + D)+ 2 D H - K~°ov}+ O(e3)>O, (3.15) where A = Bo+ 2B~,~,, o

C = (fl~. + fl.u)(1 - a~) + o~, O~ = Otta + Otv,

D = Bo + B~,~.+ Bo., v

O=B~+Bo.-B~o-Boo, n = (t3~o + 3 ~ ) ( 1

- ~ , ) + ~,,,

K = Bo + 2B~,,, when fitnesses are additive and oblique transmission is o f the form given by eqn (3.1a). Expressing eqn (3.15) as a+be +ce2+ O(e3), the conditions for/.t to be a BCSS are, to order e 2, a = 0 and (i) b > 0 or (ii) b = 0 and c > 0 . N o t e that a = 0 implies/3~,~, = 0, so that unless the probability o f an offspring a d o p t i n g any alternate

252

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ET AL.

FINDLAY

strategy v # / z through vertical transmission given that both parents use /z is zero, v no single strategy can be a BCSS. When ~/B~I• = 1 for all k, l,/3~,, = 1,/3~u = 0 and u __ v /3,,v-/3v~,= 1/2 for all v ~ / z , the condition (3.15) reduces to the standard ESS conditions. The BCSS conditions under other forms of oblique transmission (3.1b, c) with multiplicative or additive pair fitness can be derived in the same manner. The resulting expressions are lengthy and will not be given here. Three general conclusions apply in all cases. First, /3~,, = 0 for all v # / z is always a necessary, but not sufficient, condition for /z to be a BCSS. The reason is that if / 3 ~ # 0, any mutant strategy will always increase initially due to vertical transmission, at least in the strictly deterministic case. Second, the standard ESS conditions are neither necessary nor sufficient for a strategy/z to be bioculturally stable. Third, since the BCSS conditions (3.15) are based on linear expansions in e, there may exist population states that are dynamically stable but do not satisfy the BCSS conditions. This point is discussed further below. We turn now to the case of pure strategies. Consider a population comprising two groups, one in state p the other in state q # p. Let F(q, P) = Z q,F~(p)

(3.16)

i

be the average biocultural fitness of a group in state q playing another in state p, with qi the proportion of individuals in the former playing pure strategy i. The state of the global population is p' = (1 - e) p + eq, e > 0. Then p is a bioculturally stable state if for all p' ~ p, fi(p, p) > F ( p ' , p) or fi(p, p) = fi(p', p) and F ( p , p') > F ( p ' , p') (Findlay et al., 1989) for e sufficiently small. If we regard p =/z and p ' = v as alternate (possibly mixed) strategies, these relations are equivalent to the condition F~, > Fv, exactly that used to derive the BCSS conditions above. However, since dynamics were not explicitly incorporated in deriving the BCSS conditions, the question naturally arises as to whether a state p = p. satisfying ,6 = 0 is dynamically stable under the dynamic eqn (3.6). Let p~ = p~ + sc~,Z. ~ = 0. Then p is a stable state if Q = ~, sC~F~(p)<0or Q = 0 and ~, sC,F~(p+ so)< 0. Since Y., ~:~= 0 a n d F ~ ( p ) = f i ( p ) if p is fixed point [eqn (3.6)], Q = 0 and the latter condition can be rewritten H = ~ sc~[F~(p+~¢)- P ( p + so)] < 0.

(3.17)

i

Substituting in eqn (3.6) and expanding to first order in ~' yields ~, ~ (s¢, + p,)[ F~(p + sc ) - F ( p + ~:)].

(3.18)

Let

Then for both p ~/k and p ~ OA, V(O) = 0 and V(sc) > 0 for all s¢ = O. Furthermore,

OV.

BIOCULTURAL

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253

which by eqn (3.17) is always negative. V is therefore a Liapunov function, so that if/z is a (possibly mixed) strategy satisfying the BCSS conditions, the corresponding population state p =/z is at least locally stable. Note, however, that since eqn (3.18) is only a first order expansion in s¢ while the Fi's are in general non-linear in p, there may well exist attractors which do not satisfy the BCSS conditions. Such points are stable, but not asymptotically stable, because the euclidian distance from the attractor to some point on a nearby orbit need not decrease monotonically with time. This point has also been made by Zeeman (1979) and Rowe et al. (1985) in the context of biological games. For games of low dimensionality, it is relatively easy to enumerate the stable classes (qualitatively different behaviors) for the associated dynamic. In the case of oblique transmission only, system equilibria are given by the admissable roots of the N (at most quadratic) equations ~ - p~ = 0. For the simplest case of N = 2, with Pl = P and P2 = 1 - p , the characteristic equations are

pZ(a~+ a2)-p(2a~+ a 2 ) + a~ = 0

(3.21a)

_p2(oq + %) + p ( 2 a 1+ t~2) - c~1 = 0,

(3.21b)

p ( a l + a2) - a2 = 0,

(3.21c)

all of which admit a single interior fixed point. In the case of eqns (3.21a) and (3.21c) this point is a repellor, in eqn (3.21b) an attractor. The situation is considerably more complex when both vertical and oblique transmission are involved. Here, p~ is of fifth order when pair fitnesses are additive (sixth order when fitnesses are multiplicative) and the number of stable classes increases dramatically. In Fig. 1, the stable classes for two-strategy games involving both vertical and oblique transmission with additive pair fitnesses are given for each representative biological pay-off matrix. This catalogue was produced by covering the parameter space {T v, To} = {/312 x/3 ~, ×/3 ~ ×/3 22 x a, × a2} with a 6-dimensional grid of mesh size 0.2 on each side, and tabulating the resulting flows. The search resolution is therefore r e a s o n a b l y - - b u t not inordinately--fine grained, so the resulting catalogue may not be complete. It does, however, suffice to illustrate two important points. First, for any B, the number of different outcomes (stable classes) in the biocultural case greatly exceeds that obtaining in the purely biological case. Second, even when /3~2 = / 3 ~ = 0 (the most restrictive condition) and pair fitnesses are additive, biocultural games can admit multiple interior fixed points--indeed, multiple interior attractors. When ill,_,_,/3~ > 0, we have observed up to five interior fixed points, of which three are stable. As the dimensionality of the game increases, the ratio of the number of stable classes in the biocultural case versus the biological case increases. Zeeman (1979) has identified 19 stable classes (up to flow reversal) for the three-strategy biological game. For biocultural games incorporating vertical and oblique transmission we have identified at least 50 more. As in the two-strategy case, a large number of these additional classes comprise games with multiple interior fixed points (Fig. 2), including multiple attractors. Cultural transmission serves not only to increase the number of stable classes relative to the purely biological case, but also to increase strategy equilibrium

254

C. S. F I N D L A Y

ET AL.

Biocultural

Biological

Case

:

a>c, d>b

C q

C

--

--

:

a>c~b>d

:

0

C

:

~

:

¢

:

:

0

4

C

0

-

-~

0 •

od

4

0

"~-

C

0

0 ~-

q

0

,-

~.

=

0

O 0

t t

Pu

FIG. 1. Stable classes for two-strategy biological and biocultural games involving vertical and oblique cultural transmission. The three cases correspond to possible arrangements of the biological pay-off matrix B = [ ~ dh]. In all cases, pair fitnesses are additive and /32: n -_/ 3 2it = 0. ( - - O - - O - - ) repellors, (----O---O--) attractors, ( ~ ) give the direction of flow. t Designates stable classes not found for games involving oblique transmission of the form given by eqn (3.1c).

o.~(~

.I \

\

/1~\

:~;~#~i~-':;~:~i~~;g#.s-~;:.:.'

~;~o--r5oqo o.~z:~;

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ii:::::::::::::::::::::::::::::::::::: ii

//@j

ii~o.sso.o o-o ij~ r t

f~o.3~o.o o.o ~!

o.eoo.35o.~5~;

~:~~: .:';:.~i~;~ ~:::-:.-~.-:' " :'##~!:

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._ \

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,,~

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FIG. 2. Biocultural games with vertical and oblique transmission. Vertices are P3, Pl, and P2 in clockwise order starting from the lower left. For each pair, the left triangle represents the biocultural game, the right triangle the corresponding biological game. The matrices are, in clockwise order around each pair starting from lower left: B, T~, = {/3[i}, T~,, T3v with T o in the center. Other symbols are the same as in Fig. I. In all cases, pair fitnesses are additive.

:::!i~:!~![!i~!ii?.ii:;~i~i~!:i:~:~::~ :::.:i

~1.9s-~.2

:,i:!o: OoOo:o

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t,J

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O"02,5

0"2

0"4

0.6

0-8

I-0

0.0

0-2

0-4.

0-225

. . . .

0-425

0"625'

.,3,, i .

0-025

.

0-425

Equil ibrium diversity

0-225

0-625

V0(2,3)

L~

0°025

..

0-225

0-425

0-625

ivoc3,_

l

FIG. 3. Distribution or equilibrium diversity scores D for two-strategy biocultural games with vertical and oblique transmission. Numbers in brackets refer to the vertical transmission class (I :/3~2=/3~ =0; 2:/3~2>0, /3~ =0; 3:,B~2, /3~ > 0 ) and oblique transmission function [1: eqn (3.1a); 2; eqn (3.1b); 3: eqn (3.1c)].

h

0.6

0.8

0-0

0-2

0-4

0.6

0-8

BIOCULTURAL

257

GAMES

diversity. The effect of cultural transmission was investigated by running a series of Monte-Carlo trials involving points selected at random from the space B (biological case) or {B x T v x To} (biocultural case). For each random sample we calculated the strategy diversity E

D=-

N

Y~ Y. cj(/~,ln/~,) i, j=l

i=1

a modification of the well-known Shannon information measure (Shannon & Weaver, 1949). In the above expression, (/~)j is the abundance of strategy i at the j t h stable equilibrium, E is the number of stable equilibria generated by the associated game dynamic, cj e [0, 1] is a weight proportional to the size of the basin of attraction of j, and N = 2 for two-strategy games. If, for example, N = 2 and the flow of the game associated with a random point ~ e {B X Tv X To} is (see Fig. 1 for symbols) C " ~, "- 0 " ~" "- 0 6

i

i

,

.,

.2

.~

i

then c, = x2 and c2 = 1 - x 2 . Thus, equilibria with larger basins of attraction are weighted more heavily than those with smaller basins, since a larger proportion of all possible trajectories will eventually end up at the former than at the latter. Repetition of this procedure with four different sets of random points indicates good convergence ( < 5 % discrepancy among runs) for the first four moments of the resulting distributions, so we have reasonable confidence that the sample statistics so obtained are representative of the system as a whole (Gillespie, 1975). The resulting distributions are shown in Fig. 3, with first and second moments summarized in Table 1. As shown in Table 1, cultural transmission increases strategy diversity irrespective of either the form of the oblique transmission function or the TABLE 1

Equilibrium strategy diversity D in two-strategy biocultural games with vertical and oblique transmission versus purely biological games. Each entry is the mean (variance) obtained from a Monte Carlo sample of 50 000 points from the relevant parameter space. The three classes of biocultural games (quadratic 1, quadratic 2, exponential) correspond to the different forms of oblique transmission given in eqns (3.1a-c) respectively Vertical transmission class I

_

2

/3z2 - / 3 ~ = 0 /3~2 > 0,/3~, = 0 ill_,,_,/3~, > 0

Biological game

quadratic 1

Biocultural Game -" quadratic 2

O. 121 (0.060) 0"121 (0-060) 0"121 (0.060)

O. 163 (0.072) 0.443 (0-070) 0"551 (0'031)

0.312 (0.089) 0"530(0-056) 0.621 (0"012)

exponential 0.200 (0-081) 0'474(0"069) 0-588 (0"0"021)

258

C. S. F t N D L A Y

ET AL

constraints on the vertical transmission coefficients J~jk" Analysis of higherdimensional games (e.g. N = 3) suggest that as N increases, cultural transmission results in greater equilibrium strategy diversity relative to the purely biological case. 4. Biocultural Game Dynamics with Horizontal Transmission

Horizontal cultural transmission involves the communication of behavioral information among age peers (Cavalli-Sforza & Feldman, 1981). As with oblique transmission, we assume that horizontal transmission is similar to infection: the more contacts an individual has with peers using strategy i, the greater the probability qbi that they will adopt it. Under this assumption, we have qbi = gi(p') where p' is the state of the population after vertical and oblique transmission, and g is given by eqns (3.1a-c) substituting p' for p. Write pl=-hi(p). If, for example, horizontal transmission takes the form of eqn (3.1b) (see, for example, Cavalli-Sforza & Feldman, 1981, p. 151), the corresponding game dynamic is

t~i= hi(p){l + ~i[1-h(P)]}-pi{ l + ~ ~ihi(p)[1-h~(P)]}

(4.1)

with 0 <- ~i <- 1. Now hi(p) =

piFff I~,

(4.2)

with F~ as given in eqns (3.5a) or (3.5b). Letting F~-=-~

l+~'i 1

(4.3)

we obtain eqn (3.6) in the form

1~ = pi( F l -

F'),

(4.4)

with PI=~,p~FI and F~ interpreted as the biocultural fitness of strategy i under vertical, oblique and horizontal transmission. Equation (4.4) can be derived for any of the functions specified in eqns (3.1) by substituting suitable expressions for FI in eqn (4.3). The BCSS conditions for a game dynamic of the form of eqn (4.1) can be derived in the same manner as for vertical and oblique transmission. Let u, v, /.L~ A be (possibly mixed) strategies, and consider the case where vertical, oblique and horizontal transmission all occur. Then F~=

(

1+~'~, 1

~

/],

(4.5)

with

F~, = fA du dv ~b(u)dp( v)F~'( u, v),

(4.6)

"l~'= fa du dv d~(u )qb(v) Wuv,

(4.7)

BIOCULTURAL

GAMES

259

where ~b(u) is the probability of a randomly chosen individual having a strategy lying on [u, u + d u ] and F~'(u, v) is as given in eqn (3.7b). Rewriting eqn (4.4) as ~(p.) = q~(/~)(F~ - P ) ,

(4.8)

it is evident t h a t / z is a BCSS if and only if F~,> F'. If ~b(/z) = 1 - e and qb(v) = e, this requires F;, > F " for all v # tz given e sufficiently small, or 2 2 ff'[F~, (1 + ~r) - Fv(1 + ~ro)]- (1 - e)F~,~, + eFv~o > 0.

(4.9)

Even to order e 2 this inequality is lengthy when written explicitly in terms of B, Tv, T o and Tn, and so will not be given here. We note, however, that as in the case of oblique and vertical transmission, no single strategy/~ can be bioculturally stable unless for all v #/~, /3~,~,=0. It goes without saying that the standard ESS conditions are neither necessary nor sufficient for stability. Game dynamics under vertical, oblique and horizontal transmission may be very complex indeed. A quick glance at eqn (4.1) shows that since hi(p) may be of the fifth or sixth order, /Ji can be a polynomial of high degree. Even in the simplest case of two strategies, the number of stable classes is at least 50% greater than in the case of vertical and oblique transmission only. In the three-strategy case generating planar flows, one finds examples of all possible classes of non-wandering sets (see Guckenheimer & Holmes, 1983), including multiple interior fixed points, limit cycles, and homo- and heteroclinic orbits (Fig. 4). As with vertical and oblique transmission, horizontal transmission on average increases equilibrium strategy diversity relative to the purely biological case (Figs 5 and 6, Table 2). Behavioral diversity is a common characteristic o f many species (West-Eberhard, 1985; Clark & Ehlinger 1987). A large body of evolutionary theory is concerned with explaining how such variation arises and persists. A favorite explanation is that phenotypic variation is "adaptive" (in the Darwinian sense) insofar as it reduces intraspecific competition (Schoener, 1986). Frequency- or density-dependent effects may also give rise to adaptive polymorphisms (Wilson & Turelli, 1986; Hedrick, 1986), as is the case in standard biological games. Underlying all these explanations is the equivalence of biological fitness among behavioral variants: polymorphisms arise and are maintained because the spatial (e.g. across niches or habitats) or temporal (e.g. in the case of density-dependence) average biological fitness of all types are about equal. In effect, alternative strategies are simply different means of achieving the same fitness end. From an empirical perspective, the question is whether or not these mechanisms are sufficient to account for the behavioral variation observed in populations. As indicated in Tables 1 and 2, sampling of the space of all possible biological pay-off matrices yields (on average) low diversity measures. (Note that this does not imply that particular biological games cannot generate high D measures, only that before the payoff matrix B has been determined, the expectation is that any biological game will generate a relatively low D.) By contrast, sampling of the space of all possible biocultural games yields higher D estimates (Tables 1 and 2). A priori then, the expectation is that such systems will tend to show relatively higher diversity. Note, however, that this variation may not be "adaptive" in the biological sense. Rather, it is adaptive in the biocultura! sense: at stable polymorphisms, strategies

,,~:~.8o0.350-55.".-~

,L~ )c \

A

\\ ~.55

c

oi3o.7 ::~,/

/

A

I \

jl~ / \

\

i::~o.7~o.~oouo~

~i:::o.oa o.o

..........

~E~O,O 0-0 0,15-:.": ~:-2,8-0"8 2,9E~ ::::::::0-20 0-15 0-2~i:: ::i:: 3"7 -7-2 2.2:::::: : K~:::: ~:~i:::::i::i::i::ii ::ii i:::~::i:;::i ::::::::~:!: ~ ! ~ ! !: i E E ~~:E E i i::: :~:i:i::~:~::i:

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: .......................

\

~:~i~:i~:~~:~:i:i :~i~i'i i:7.!:':.:.~i~:~:~::,:';.:.~

A

A.

o.o ~::::: /

i::iiO-O,ouo o-p.~!~ ::~::g.d.~..~,. 0...0..::::::::

~:~~-~-~-:~ ~:-i~:,:~i~;~'~ 0'75 0"0 i~ ~0-~0 0.02 o. i ~

::~o.o

i~o.o o-o o-8~i~ :iiiii~ ~-T ~-o-2-~iii~

/

:~8:::;:!:7~:~:~::~:~ ~;!;k:;::::::;:.?.~:~:7 !:::~:~::::

~:~o-o o-6.~0.o ::~C ~E;o.ro o~o o • ic~:~ A !ilio-6s 0-95 o-2o~i i.'!.;~iO'60 0-0 0.0 i!~:. / \

\

d

I \ :~:~0.75 0.6 0-7 ~i~i~:: /

\

iiiii:O- 5 0.7

0.4i:~::

~!Fo:~ ' ~':"~~:6::~:

A /

\.

i

~i~6:bb:~50:0 ~:::::~::

:~:]!!0-0 0'0 0-25!P:i iiii0"50 0"25 0o60P:P:~ !:::.::!:.!~i:ili~:i:i:i:.!:~:i:i~i~!:i~i:M:i:i:i~::.i:.i:.::.P.ii i i

..............................,....................................

::::::::i::O.O0-10 0-0 i::~::::

~::~:?:o-6~o.ea o. o

::::::::::::::::::::::::::::::::::::::::::::::::::::

f:b::~:o.oo.o o.~5~C

\

::~:!:i:::~:i:i:~:i:i:::ii !:{;~8::::i:~:~:~:i:::g<.%:

/ V oooo:

A

/

FIG. 4. Biocultural games with vertical, oblique and horizontal transmission. Format is the same as in Fig. 2 except that the first row of the center matrix give T o , the second r o w T M. The oblique and horizontal transmission functions used are (a) (2, 2); (b) (2, 3); (c) (2, 2); (d) (1, 2), with 1 corresponding to eqn (3.1a), 2 to eqn (3.1b) and 3 to eqn (3.1c). Figure 4(a) shows a biocultural game with two interior attractors, one inside a homoclinic orbit. In Fig. 4(b), the interior line segment parallel to the (p~, p~) face is pointwise fixed. Figure 4(c) shows a periodic orbit, and Figure 4(d) a heteroclinic orbit.

i!:!~4 .+.~:::~:~:!:~;~::i ,3~: :-0"05 4.~~:~!:'..:!~,~:~:::~::;:~ -3.~::: 8 -I-7 -0"05 0-T:$: !::ii...~:.0 -2.9 ...... .Q 5::i::i:

//

i$~::::$:~-~::]::,:::-'I-',':~:~:K':':~:~:~:" ~.~.~:~:,,'-

~]i.O.-.tpR:p ~:0 !ii:l ~ \

iiiiio.loo.o o-o ~ii::: / \

~;.:~:: ~:~~:::~!::~..<~a: ~~::::~~:~ :i ~ ~i~O-eJ()( ~ - 1 0 0-I~:}:::i

i•B ::.:':'::

0.76 -I-11::::

1-15 0-0

::2.7

~o-:~ o.o o-o fi

bJ

0-225

0.425

VH(3, I)

0-625

.___..J

0-025

0-2

0.4

0.6

0.8

1.0

O.OL

0.2

0-4

o-6

VH(2,1)

0.025

0-425

Equilibrium diversity

0.225

VH(2,2)

0-625

0.025

0.225

0-425

0-625

Ftc;. 5. Distribution of equilibrium diversity scores D for two-strategy biocultural games with vertical and horizontal transmission. Numbers in brackets refer to the vertical transmission class and horizontal transmission function (see Fig. 3),

tl.

o>,

0-8

1.0

0-0

0.2

0.4

0.6

0-8

i-,.)

I11

0

> E-

c

c E-

6 c~

0 .0 2 5

0-2

0.4

0-6

0.8

0"0

0.2

0"4

0-225

0.425

0- 62 5 0"025

0"425

Equilibrium diversify

0- 225

VOH(5,2)

VOH(2,2)

0"625

0"025

0"225

0"4'25

0- 625

FIG. 6. Distribution of equilibrium diversity scores D for two-strategy biocultural games with vertical, oblique and horizontal transmission. N u m b e r s in brackets refer to the oblique and horizontal transmission functions: the three sets correspond to vertical transmission classes 1, 2, and 3 respectively.

U-

0.6

0.8

0-0

0.2

0.4

0.6

0.8

b.~ ~x I,J

OI

b_

0"025

0.2

0-4

0.6

0.8

1.0

0-0

0.2

O.4

~0.6

0.8

0 , 0

o.2

0 -4

0-6

0.8

0.425

VOH(3,1)

-

0-625

~ . ~ ~ j

0.225

, L~Zr'JF1

VOH(I,])

-

I

0.025

0.425

FIG. 6--continued

Equilibrium diversity

0.225

VOH(2,2)

0"625

-

0-025

0"225

0"425

0'625

t~a

0.6 o-q

I-0 0.8

0.0

0.625

_'~-~_;2~--'--~--

VOH(3,I)

__~

VOH(2,P)

0-225 0.425

I.. _~_

0-025

0.2

0.8 0.6 0.4

1.0

0.0

0.2

" k.

g

VOH(I,I)

°'2ttm.~i=i=,~al~ll

0"4

0-6

I'0 0.8

0-025

0.425

FIG.

6--continued

Equilibrium diversify

0.225

VOH(3,2)

0.625

i,_.jvoHc 2,

0.025

i

0.225

0.425

VOH(3,3)

0-625

L VOH(I,3) I

>.< -d

{--

7

bJ

265

B1OCULTURAL GAMES

TABLE 2

Equilibrium strategy diversity D for biocultural games involving vertical and horizontal (VII) and vertical, oblique and horizontal (VOH) transmission compared with purely biological games (BIO). Each entry is the mean (variance) obtained from a Monte Carlo sample of 75 000 points from the relevant parameter spaces. Numbers in brackets refer to the type of horizontal transmission (VII) or oblique and horizontal transmission ( VOH): 1 corresponds to eqn (3.1a), 2 toeqn (3.1b) and 3 to eqn (3.1c) Vertical Transmission Class /312>0 , /3~, = 0

Game type

f112=fl~z = 0

VH(I) VH(2) VM(3)

VOH(1, 1) VOH(I, 2) VOH(I, 3) VOH(2, 1) VOH(2, 2) VOH(2, 3) VOH(3, 1) VOH(3,2) VOH(3, 3)

0.013(0.006) 0-184 (0.076) 0'035 (0.017) 0"057 (0.031) 0"270 (0-094) 0.118 (0.057) 0'101 (0.050) 0"446 (0"078) 0.226 (0-083) 0'066 (0"036) 0"342(0'095) 0-141 (0"065)

0.190(0.074) 0'379 (0'094) 0.250 (0"088) 0.314 (0'072) 0"505 (0-062) 0"404 (0'072) 0-348(0-080) 0-597 (0"032) 0"479 (0'068) 0"323 (0"075) 0.551 (0"050) 0"429 (0'073)

0.533(0.036) 0-623 (0"019) 0.582 (0-022) 0-423 (0-054) 0-597 (0-021) 0.513 (0"037) 0-476(0-051) 0.645 (0"006) 0"593 (0"019) 0.440 (0'054) 0-628(0"011) 0.548 (0.030)

BIO

0.121 (0.059)

0.121 (0.059)

0.121 (0.059)

/312, 131, > 0

need not have equal biological fitness, only equal biocultural fitness. Moreover, there is no guarantee that these are positively correlated (see, for example, CavalliSforza & Feldman, 1981; Lumsden & Wilson, 1981; Boyd & Richerson, 1985, for detailed discussion of this and related points). The concept of adaptive variation in biocultural versus purely biological or purely cultural systems is illustrated in Fig. 7. Let q~: A--, W x C :p,--~[qb~(p), ~Pc(P)] be the map from the strategy (phenotype) space A into W x C, the space of biological (W) x cultural (C) fitnesses, whose components qbs, qbc are biological and cultural selection operators. Furthermore, let p : W x C ~ F be a map from W x C into F, the space of biocultural fitnesses. Then f : A ~ F is such that A

~ F

WxC commutes. For purely biological games, ~ c = K (all strategies have equal (possibly zero) cultural fitness), c I ) : A ~ W x R and ~o= L(W), so that the biocultural fitness of a strategy is a strictly linear function L of its biological fitness. Here, "adaptive"

266

C . S . FINDLAY E T A L .

W

---e

A

%

L~

L~

cp W

A

L~

A

(a)

(b)

(c)

FIG. 7. Adaptive variation in biological (a), cultural (b), and biocultural (c) systems. In the top panel, both the biological (W) and cultural (C) adaptive landscapes are sharply peaked. Purely biological or purely cultural dynamics would then lead to an equilibrium population state characterized by low strategy diversity. Under a biocultural dynamic, however, the operator ~ generates a biocultural adaptive landscape that is relatively fiat, such that a number of different phenotypes in the phenotype space A have similar biocultural fitnesses. Hence, equilibrium strategy diversity increases. In the bottom panel the situation is reversed.

variation implies adaptive variation in biological fitness, perhaps due to one or more of the mechanisms described above. On the other hand, for purely cultural games qbB = K (all strategies have equal biological fitness), q b : h ~ C ×g~ and ~o= L(C), and "adaptive" variation implies adaptive variation in cultural fitness, as might, for example, result from frequency-dependent biases in cultural transmission (see Boyd & Richerson, 1985, pp. 213-226, and above). But it is entirely possible for variation to be "non-adaptive", both in terms of biological and cultural fitness, and still be adaptive in terms of biocultural fitness (Fig. 4, top panel). Conversely, variation may appear adaptive at either the biological or cultural level, yet be non-adaptive at the biocultural level (Fig. 4, bottom panel). Which of these results obtains depends explicitly on the operator ~p and the form of the biological qbB and cultural qbc adaptive landscapes. We have confined ourselves to the analysis of low-dimensional biocultural games (i.e N = 2 and N = 3). Since the corresponding flows are at most planar, the possible classes of dynamical behavior are limited. For N-> 4, the highly non-linear nature of biocultural fitnesses (and hence, the associated evolution equations) permits very complex dynamics, including chaotic motion. Investigation and characterization of these dynamics is an important task for future research. 5. Discussion

Our study is motivated by a body of data indicating that, in many species, behavioral transmission includes a cultural component. Our intent is twofold:

BIOCULTURAL

GAMES

267

(i) to extend standard evolutionary game theory in such a way that it can be usefully applied to biocultural systems, and (ii) to compare and contrast the results from biocultural games with their purely biological analogs. In constructing the former, we assume that natural selection takes the form of differential fecundity. While there are many examples of traits which do influence fecundity (reviewed by Lumsden & Wilson, 1981; see also Vining, 1986), differential survivorship is also likely to play an important role in strategy dynamics. The model considered here incorporates only the former: the consequences of the latter for biocultural games are currently unknown. The results described here and in an earlier study (Findlay et al., 1989) lead to three important conclusions: (i) the conditions for invasability (and hence, evolutionary stability) in biocultural games reduce to the standard ESS conditions only under exceptional circumstances and are otherwise considerably different; (ii) the former admit a larger number of qualitatively different dynamical behaviors than the latter; and (iii) in biocultural games, the effects of cultural evolutionary forces (e.g. the ease with which strategies are learned and transmitted) can counterbalance and even reshape the effects of natural selection, leading to the establishment of stable polymorphisms that would otherwise not obtain. These findings are all of considerable importance to evolutionary biologists, for reasons discussed below. It is reasonable that the conditions for stability differ between the two classes of games--they do, after all, induce different dynamics. However, the implication for empirical investigators is that predictions about the evolutionary fate of strategies based solely on the biological payoff matrix B may be inaccurate if cultural transmission influences behavioral dynamics. This is particularly true if cultural transmission is horizontal or oblique, because unlike vertical transmission, these modes have no biological analog. It is also worth noting that because in the pure cultural case with viability selection, evolution under either oblique or horizontal transmission is more or less independent of between-family variation in reproductive success (at least, is less dependent than in the case of biological inheritance or vertical transmission), there is the potential for very rapid changes in strategy frequencies. If, for example, one adult is responsible for the teaching of a large number of offspring, a single strategy can come to dominate the population in just a few generations (Cavalli-Sforza & Feldman, 1981). If we consider the number of stable classes in a game to be a crude measure of its complexity, then biocultural games are more complex than their purely biological counterparts. For empirical investigation, this increased complexity is both a boon and a problem. On the one hand, the differences may aid in the initial stages of hypothesis testing; if, for example, observations on a natural system are consistent with a stable class of behaviors not admitted by purely biological games, then one can be reasonably sure that it is not just differential biological fitness driving the system. On the other hand, the greater complexity of biocultural games means that accurate predictions will be considerably harder. In a purely biological game, predictions about the equilibrium composition of the population require only that the entries o f B be known (e.g. Gross & Charnov, 1980). By contrast, if strategies

268

c.s.

FINDLAY

ET AL.

are transmitted culturally, accurate prediction requires not only estimates of B, but also of the relevant matrix of cultural transmission coefficients. These observations lead to two important questions: (i), what is it about culture in general, and cultural transmission in particular, that gives rise to the increased complexity of biocultural versus purely biological games; and (ii), are these properties unique to culture, or are they also present in biological systems? The answer to the first question appears to be that it is the frequency-dependence of cultural transmission that is responsible for the more complex dynamical behavior. In the games considered here, we assume that the more an individual is exposed to a particular strategy during socialization, the more likely they are to adopt it. In keeping with this assumption, adoption probabilities are expressed as non-linear functions of the strategy abundances. These non-linearities in turn give rise to evolution equations that are of a higher degree than the corresponding biological game dynamic. Ultimately it is this increased non-linearity that generates complex dynamics. For biocultural games involving only vertical cultural transmission, the adoption probabilities depend only on the parental phenotypes and are independent of strategy abundances in the general social environment. Frequency-dependent effects are therefore negligible, and the number of possible dynamical behaviors much smaller than for games involving oblique or horizontal transmission. For example, in the two strategy case there are only two stable classes (up to flow reversal) in the case of vertical transmission only (Findlay et aL, 1989), four in the case of vertical and oblique transmission (Fig. 1). Frequency-dependence is not, of course, unique to cultural systems. Indeed, evolutionary game theory is designed to deal with situations in which biological fitnesses are frequency-dependent (Maynard Smith, 1982). But as regards transmission of strategies, frequency-dependent effects in purely biological systems are negligible because transmission is solely from parents to offspring. This does not imply that the effects of frequency-dependent transmission as manifested in biocultural games cannot be simulated by other, purely biological, processes. The difference is that frequency-dependent transmission is a general property of biocultural systems, whereas the prevalence and importance of these hypothetical biological analogs is currently unknown. The analyses presented here and earlier is a small step towards understanding game dynamics in biocultural systems. Still outstanding is a general evolutionary theory of the relationship between biological fitness and cultural success (see, for example, Vining, 1986). While some excellent preliminary work has been done on the evolution of cultural transmission and social interactivity at the populational level (Cavalli-Sforza & Feldman, 1983; Aoki & Feldman, 1987), these studies have not addressed what is perhaps the most critical issue: the moulding of rational thought and decision-making--and in particular, the rationale for c h o i c e - by the evolutionary process. At least in Homo sapiens (and probably other species as well), strategy choice is often an act of conscious deliberation and assessment of the utility of each option (Lumsden & Wilson, 1981). Construction of powerful theories of human behavioral evolution requires that we identify these utilities, the mechanism for their assessment, and how they stand in relation to biological fitness.

BIOCULTURAL GAMES

269

We t h a n k Ken Aoki, Mart Gross, Anatol R a p o p o r t and David Sloan Wilson for helpful discussion, two a n o n y m o u s reviewers for constructive criticism, and A n n e H a n s e n for careful preparation o f the manuscript. S u p p o r t e d in part by Population Biology Grants to CJL and R I C H from the Natural Sciences & Engineering Research Council o f C a n a d a ( N S E R C ) , an N S E R C doctoral fellowship to CSF and by a grant from the Cray Research Fund o f the University o f Toronto. Simulations used the resources o f the C R A Y X / M P s u p e r c o m p u t e r at the Ontario Centre for Large Scale C o m p u t a t i o n . CJL is a Career Scientist o f the Medical Research Council o f C a n a d a ( M R C ) .

REFERENCES AOKI, K. & FELDMAN, M. W. (19873. Proc. natn. Acad. Sci. U.S.A. 84, 7164. BISHOP, D. T. & CANNINGS, C. (19783. J. theor. Biol. 70, 85. BONNER, J. T. (1980). The Evolution of Culture in Animals. Princeton: Princeton University Press. BOYD, R. & R1CHERSON, P. J. (19853. Culture and the Evolutionary Process. Chicago: University of Chicago Press. CAVALLI-SFORZA,L. L. ~¢.FELDMAN, M. W. ( 1981 ). Cultural Transmission and Evolution: A Quantitative Approach. Princeton: Princeton University Press. CAVALLI-SFORZA, L. L. & FELDMAN, M. W. (1983). Proc. natn. Acad. Sci. U.S.A. 80, 2017. CAVALLI-SFORZA, L. L., FELDMAN, M. W., CHEN, K. H. & DORNBUSCH, S. M. (1982). Science, N.Y. 218, 19. CHENEY, D., SEYFARTH, R. • SMUTS, B. (1986). Science, N.Y. 234, 1361. CLARK, A. B. & EHLINGER, T. J. (1987). In: Perspectives in Ethology (Pateson, P. P. G. & Klopfer, P. H., eds.) New York: Plenum Press. FINDLAY, C. S., LUMSDEN, C. J. & HANSELL, R. 1. C. (1989). Proc. natn. Acad. Sci. U.S.A. 86, 568. GILLESPIE, D. T. (19753. The Monte Carlo Method o.fEvaluating Integrals. Springfield: NWC Technical Publication 5714. GOODALL, J. (19863. The Chimpanzees of Gombe. Cambridge: Harvard University Press. GROSS, M. R. & CHARNOV, E. L. (19803. Proc. nam. Acad. Sci. U.S.A. 77, 6937. GUCKENHEIMER, J. & HOLMES, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag. HEDRICK, P. W. (1986). A. Rev. EcoL Syst. 17, 535. HEWLE'r'T, B. S. & CAVALLI-SFORZA, L. L. (1986). Am. Anthrop. 88, 922. HINDE, R. A., PERRET-CLERMONT, A. & STEVENSON-HINDE, J. (eds) (19853. Social Relationships and Cognitive Development. Oxford: Oxford University Press. HINES, W. G. S. (1987). Theor. Pop. Biol. 31, 195. LUMSDEN, C. J. & Wl LSON, E. O. ( 1981 ). Genes, Mind and Culture. Cambridge: Harvard University Press. MAYNARD SMITH, J. (19823. Evolution and the Theory o f Games. Cambridge: Cambridge University Press. MAYNARD SMITH, J. & PRICE, G. R. (19733. Nature, Lond. 246, 15. PARKER, G. A. (19783. Nature, Lond. 274, 849. ROWE, G., HARVEY, I. F. & HUBBARD, S. F. (1985). J. theor. BioL 125, 269. SCHOENER, T. W. (19863. In: Resource Partitioning (Kikkawa, J. & Anderson, D. J. eds.). Melbourne: Blackwell. SHANNON, C. E. & WEAVER, W. (19493. The Mathematical Theory of Cummunication. UrbanaChampagne: University of Illinois Press. SMUTS, B., CHENEY, D., SEYFARTH, R., WRANGHAM, R. t~ STRUHSAKER, T. (eds) (19873. Primate Societies. Chicago: University of Chicago Press. SUTHERLAND, W. J. (19873. Nature, Lond. 325, 483. TAYLOR, P. D. & JONKER, L. B. (1978). Math. Biosci. 40, 145. VINING, D. R. (1986). Behav. Brain Sci. 9, 167. WEST-EBERHARD, M. J. (1985). Proc. natn. Acad. Sci. U.S.A. 83, 1388. WILSON, D. S. & TURELLI, M. (19863. Am. Nat. 127, 835. ZEEMAN, E. C. (19793. In: Global Theory of Dynamical Systems (Nitecki, Z. & Robinson, C. eds). New York: Springer-Verlag.