Grade Dynamics, Mixed Strategies and Gradient Systems

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

GAME DYNAMICS, MIXED STRATEGIES AND GRADIENT SYSTEMS

K. Sigmund February 1987 WP-87-21

Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS

A-2361 Laxenburg, Austria

PREFACE

Game dynamics, as a branch of frequency dependent population genetics, leads to replicator equations. If phenotypes corre- spond to mixed strategies, evolution will affect the frequencies of the phenotypes and of the strategies and thus lead to two dynamical models. Some examples of this, including the sex

ratio, will be discussed with the help of a non-Euclidean metric leading to a gradient system. Some other examples from popula- tion genetics and chemical kinetics confirm the usefulness of such gradients in describing evolutionary optimization.

Alexander B. Kurzhanski Chairman,

System and Decision Sciences Program

GAME DYNAMICS, MMED SJRATEGIES AND GRADIENT S E T E M S

Karl Sigmund

*

1. Introduction

The theory of frequency dependent selection received a strong boost, and in f a c t a new meaning, from game theory. The introduction of t h e notion of evolution- a r y stability by Maynard-Smith and P r i c e (1973) and t h e subsequent flourishing of evolutionary game theory must be viewed a s a major advance in theoretical biol- ogy. But like every new field, i t has met with its s h a r e of misunderstandings.

In particular, i t has probably been unavoidable t h a t the use of t h e term "stra- tegy" evoked hostile reactions. "Strategy" is closely associated with plotting and scheming: we don't expect much forethought from nonhuman brains. "Mixed s t r a - tegies", in particular, seem totally misplaced in t h e animal kingdom. Konrad Lorenz claims not to have met with a single one in all his life. Animals, a s o t h e r s have pointed out, do not have roulette wheels in t h e i r heads: s o how can they obtain probability distributions f o r different types of behavior?

But this i s a superficial view based on a semantic confusion. Indeed, in evolu- tionary games, a strategy i s a phenotype. The sex r a t i o is an example of a mixed strategy, and i t is common enough. Many species manage t o mix male and female offspring with nearly equal probability without playing roulette. Other c a s e s of mixed strategies a r e to be found in foraging, dispersal, parental c a r e e t c . We r e f e r t o Maynard-Smith (1982) f o r a thorough presentation of t h e biological aspects of this question. On t h e following pages, w e shall be more interested in t h e mathematical aspects, but stick to the sex r a t i o f o r illustration.

= ~ n t e r n a t i o n a l I n s t i t u t e f o r AppIied S y s t e m s Analysis, Laxenburg, A u s t r i a and t h e I n s t i t u t e f o r Mathematics, U n i v e r s i t y of Vienna, Austria.

2. The sex ratio

Already Darwin (1859) w a s puzzled by the prevalence of t h e s e x r a t i o

$.

I t is not, as we may f i r s t think, a n immediate r e s u l t of the s e x determination through

X-

and Y-chromosomes. In f a c t , at conception t h e r a t i o may be quite different, and subsequently shift t o yield the value

$

at b i r t h . What i s t h e evolutionary reason f o r this? After all, as animal b r e e d e r s know, a female biased sex r a t i o leads t o a higher o v e r a l l growth r a t e . Why a r e t h e r e s o many males around?

The ingenious explanation of Fisher (1930) anticipates, as Maynard-Smith notes, thinking in terms of game theory. We shall assume t h a t the number of chil- d r e n is not affected by t h e s e x r a t i o , and conclude t h a t t h e number of grandchil- d r e n is. Roughly speaking, if t h e r e w e r e more males than females, girls would have good prospects. Since t h e same holds vice v e r s a , this should lead t o a sex r a t i o of

i.

To check this, l e t us denote by p t h e sex r a t i o of a given individual, and by m t h e a v e r a g e sex r a t i o in t h e population. Let N1 be t h e population number in t h e daughter generation F1 (of which m All will be male and (1-)N1 female ) and N2 t h e number in the following generation F2. Each member of F2 h a s one mother and one f a t h e r : t h e probability t h a t a given male in t h e F1 generation i s i t s f a t h e r i s

- ^,

and t h e expected number of children produced by a male in t h e F1 genera- mN1

tion is t h e r e f o r e

-

N2 (assuming random mating). Similarly, a female in t h e F2- mN1

generation contributes a n a v e r a g e of N2

children. Since a p-phenotype ( 1

-

^IN1

produces male and female children in t h e r a t i o p t o 1 9 , i t s expected number of grandchildren will b e proportional t o

i.e., i t s fitness i s proportional t o

(we may clearly exclude the c a s e s m

=

⁰ and m

=

1 which lead t o immediate extinc- tion). For given m E (0,1), t h e function p w (p ,m ) i s affine l i n e a r , increasing f o r m

< k ,

decreasing f o r m

>

$ a n d constant f o r m

= i.

Let us now consider a phenotype with sex r a t i o q , and ask whether i t i s evolu- tionarily s t a b l e in t h e sense t h a t no o t h e r phenotype with sex r a t i o p c a n invade.

If such a deviant phenotype i s introduced in a small proportion E , t h e a v e r a g e sex r a t i o of t h e population i s r

=

^E p

+

( 1 - ~ ) q . The q-phenotype f a r e s b e t t e r t h a n t h e p-phenotype if and only if

This i s obviously t h e c a s e f o r e v e r y p when q

= $

(it i s enough t o note t h a t p and r a r e e i t h e r both smaller, o r both l a r g e r than

i).

For q

<

*, a s e x r a t i o p

>

q will do b e t t e r , however, and consequently, s p r e a d ; similarly, any q

> &

c a n b e invaded by a smaller p

.

Thus i s t h e unique "uninvadable" s e x ratio.

A similar game t h e o r e t i c a l analysis makes sense whenever t h e "payoff" f o r a given "strategy" ( o r t r a i t ) c o r r e s p o n d s t o i t s r a t e of i n c r e a s e i n t h e population.

In many examples from biology, this will mean t h a t t h e t r a i t is inherited, and i t s payoff Darwinian fitness

-

a fitness which, in general, will depend on what t h e oth- e r s a r e doing, and hence will b e frequency dependent. But examples where t h e t r a i t s p r e a d s through learning, o r o t h e r means, have also been discussed. We refer in p a r t i c u l a r toAxelrod (1984) and his computer tournaments between pro- grams f o r t h e Repeated P r i s o n e r ' s Dilemma, a striking approach t o t h e evolution of cooperation.

3. Game dynamics

Following Taylor and Jonker (1978), t h e evolution of t h e frequencies of t h e different phenotypes in t h e population will b e modeled by a game dynamics whose Ansatz consists in setting t h e growth r a t e of t h e frequency of a phenotype equal t o t h e difference between i t s payoff and t h e a v e r a g e payoff in t h e population.

For t h e s a k e of simplicity, w e shall assume t h a t t h e r e a r e only finitely many phenotypes El,

. . .

^,En with frequencies zl,

. . .

^,zn. The s t a t e of t h e population i s given by t h e point _z in t h e unit simplex Sn. By f i (z), w e shall denote t h e payoff f o r t h e phenotype Ei in a population in s t a t e _z. The a v e r a g e payoff in t h e popula- tion, then, i s

The r a t e of i n c r e a s e of t h e frequency of Ei i s

II.

The dynamics, then, i s given by 4

t h e

equation

on t h e s t a t e s p a c e Sn

.

(It is e a s y t o see t h a t t h e simplex Sn as well as i t s boundary consisting of f a c e s zi

=

0 i s invariant).

This " r e p l i c a t o r equation" h a s been d e r i v e d a n d analyzed in many different fields of evolutionary biology, as e.g. in population genetics, chemical kinetics and mathematical ecology (see, e.g., S c h u s t e r and Sigmund (1983) and Sigmund (1985)).

Of p a r t i c u l a r i n t e r e s t i s t h e case where t h e f i Cz) are l i n e a r : as shown by Hofbauer (1981), t h e r e p l i c a t o r equation t h e n i s equivalent t o t h e Lotka-Volterra equation

yi =

yi ( r i

+ zai,

^y,).

In many situations, t h e r e s t r i c t i o n t o finitely many phenotypes i s unnatural.

F o r t h e sex r a t i o , in p a r t i c u l a r , a l l values between 0 and 1 should b e allowed. I t i s e a s y t o d e r i v e t h e corresponding differential equation modeling t h e evolution of t h e frequency distributions. However, f o r o u r purposes t h i s will b e only of secon- d a r y i n t e r e s t : we shall stick t o d i s c r e t e approximations and assume t h a t only fin- itely many s e x r a t i o s pi may o c c u r .

The a v e r a g e s e x r a t i o i s

The fitness f i & ) of s e x r a t i o pi i s given by w (pi , m ) (see (1)). I t will b e useful t o write this i n a slightly d i f f e r e n t way. T h e r e are two "pure s t r a t e g i e s " in t h e s e x r a t i o model, namely "produce only sons" o r "produce only daughters", i.e. p

=

1 and p

=

0. The corresponding payoffs are w (1,m )

= -

1 ^and w (0,m )

= -

m

.

With

l-m F ( m )

=

w (1,m)

-

w ( 0 , m ) one obtains

?&I =zzifi ⁼

n r ~ ( m )

+ -

1-m 1 and

Similar equations o c c u r v e r y frequently in game dynamics. The c e l e b r a t e d

"Hawk-Dove" game from Maynard-Smith (1974) i s a c a s e in point. W e may assume t h a t an animal, when f a c e d with a fight, h a s two basic options: t o retreat o r t o escalate. The s u c c e s s of e a c h move depends on what t h e opponent i s likely to do.

To e s c a l a t e i s a good idea if t h e opponent will retreat: if not, i t might b e a f a t a l s t e p . I t i s conceivable t h a t individuals will display mixed s t r a t e g i e s and e s c a l a t e with a c e r t a i n probability.

Thus w e shall c o n s i d e r games satisfying two assumptions:

(a) T h e r e are two s t r a t e g i e s R 1 and R 2 . Each phenotype Ei will b e c h a r a c - t e r i z e d by i t s probability pi of using R 1 . Then m as given in (5) will b e t h e f r e - quency of R 1 i n t h e whole population.

(b) The payoffs A1 and Az f o r R 1 and R z depend only on m

.

With F ( m )

=

A l ( m ) - A 2 ( m ) , equations ( 6 ) and ( 7 ) will become f r @ )

=

p i F ( m )

+

A z ( m ) and

j @ ) =

m F ( m )

+

A z ( m ). Thus (8) will hold again.

In t h e s e x r a t i o game,

In t h e Hawk-Dove game, and more generally whenever t h e payoff depends on (one o r r e p e a t e d ) pairwise encounters, A l ( m ) and A 2 ( m ) are l i n e a r in m . Indeed, if a i j is t h e payoff f o r a n individual using Ri against a n individual using R j ( 2 , j

=

1 , 2 ) t h e n A l ( m )

=

a l l m

+

a 1 2 ( 1 - m ) , A z ( m )

=

a z l m

+

a z z ( l m ) and hence

Returning t o t h e g e n e r a l c a s e , we see t h a t f o r any t h r e e phenotypes Ei , Ej and E k , equation (8) admits a constant of motion, namely

This induces a foliation of t h e p h a s e s p a c e S, into one-dimensional invariant mani- folds.

The evolution of t h e a v e r a g e frequency m of R 1 i s given by

i

= F ( m ) Var

P

(12)

where P is the random variable taking the value pi with probability

xi.

If we neglect t h e degenerate situation Var P

=

0 (only one phenotype p r e s e n t in t h e population), w e obtain t h a t

i

has t h e sign of F ( m ) , i.e. t h a t the frequency of

R,

increases if and only if F ( m )

>

0. The set 12 E Sn :F(m )

=

0 j, which consists of linear manifolds, i s t h e set of rest points of (8).

1 3 4

W e sketch t h e situation f o r the sex r a t i o game and p

= -

₅ , p

=-

₅ a n d p a

= -

₅

in Figure 1. The invariants of motion corresponding t o (11) are t h e c u r v e s

12 =

const

z , ^zf

^{in Sn.} The o r b i t s converge along t h e s e c u r v e s t o t h e set m

= +

of equilibrium points. This s e t is evolutionarily stable, in t h e terminology of Tho- mas (1985).

In t h e sex r a t i o g a m e and many o t h e r situations, t h e r e holds a "law of dimin- ishing r e t u r n " in the sense t h a t the payoff f o r e a c h s t r a t e g y i s a decreasing func- tion of i t s frequency. In this c a s e (8) i s locally adaptive in t h e sense t h a t F ( m ) converges monotonically t o 0. W e shall presently see t h a t (8) i s a gradient and hence satisfies a global maximum principle.

4. Shahshahani gradients

A s shown by Shahshahani (1979) and Akin (1979), t h e metric most a p p r o p r i a t e f o r t h e r e p l i c a t o r equation (4) on Sn is not t h e Euclidean one. I t i s advantageous t o consider another Riemannian metric on t h e tangent s p a c e

For _z E int Sn and

l , ~

^{E T, Sn}

-

^, i t i s given by t h e i n n e r product

n

(while t h e Euclidean i n n e r product will be denoted by

- - t .

7

= ti

^qi).

i =1 In p a r t i c u l a r , t h e Fisher-Haldane-Wright selection equation

with symmetric matrix

M

i s a g r a d i e n t with r e s p e c t t o t h i s Shahshahani metric, with V k )

= 3s . ^M z

as potential. This implies Kimura's Maximum Principle: t h e a v e r a g e fitness

z . M z

i n c r e a s e s at maximal rate under t h e effect of selection.

In g e n e r a l , l e t V b e a real valued function defined on a n open set U in Rn con- taining t h e simplex Sn , a n d let

Dz

V a n ⁺ R b e i t s derivative at

s.

The v e c t o r field F:U ⁺ lRn i s t h e Euclidean gradient of.V, i.e. _F@)

=

g r a d V(z ) f o r a l l _z E U , if

-

holds f o r all 3 E T, Rn -Rn, while i t i s t h e Shahshahani gradient of V, i.e.

E k )

= G r a d V b ) f o r a l l y EintS,, if

holds f o r all

3

E T, Sn

.

If

-

j' = g r a d V i s a n Euclidean gradient, t h e n t h e v e c t o r field

- f

with com- ponents

i s t h e corresponding Shahshahani gradient, i.e.

- ^f

= G r a d V (cf. Sigmund, 1984).

For t h e c o n v e r s e direction, let u s define two v e c t o r fields

-

j' and g t o b e

-

equivalent,

- -

^j'

-

^{g , if}

^Pi

^{(g )}

^-

^{gi (g )} i s independent of i f o r all _z E Sn

.

I t i s easy t o see t h a t

-

^j'

- ^-

^{g implies}

- ⁼ - ⁶

on S n , and vice v e r s a . Now if

- f =

Grad V, then j'

-

^{g r a d V.} Thus t h e v e c t o r field

7

i s t h e Shahshahani gradient of V on int Sn if

- -

and only if t h e r e e x i s t s a r e a l valued function defined in a neighborhood of int Sn such t h a t

holds on int Sn

.

Equivalently, t h e r e p l i c a t o r equation (4) i s a Shahshahani gradient if and only

holds on i n t Sn f o r a l l pairwise different indices i

.

^{j ,k E t l ,}

^{. . .}

^{, n j. ( H e r e f f , j}

a f t

denotes t h e p a r t i a l derivative

-

^). This "triangular integrability condition",

a =,

t h e equivalent of t h e integrability condition f i l j

=

f j V i f o r Euclidean gradients, h a s been shown by Sigmund (1984) f o r l i n e a r v e c t o r fields f (z) = A

.

I t means t h a t t h e r e exist constants cj such t h a t t h e matrix with elements ail

-

^{c j i s sym-}

metric. The g e n e r a l c a s e w a s proved by Hofbauer (1985b), and w e shall presently s e e i t s usefulness. Let us note t h a t f o r n

=

2, condition (15) i s trivially satisfied and (4) t h e r e f o r e a gradient.

5. Mixed stategist games

Let us consider now a game with N p u r e s t r a t e g i e s R 1 t o R N , and n phenotypes E l t o En playing mixed s t r a t e g i e s : Ei plays s t r a t e g y R, with probability p:, and hence i s c h a r a c t e r i z e d by a v e c t o r z i E SN. If we denote t h e frequency of Ei in t h e population by x i , then

is t h e frequency of t h e s t r a t e g y Rk in t h e population. The s t a t e of t h e population i s given by

z

^ESN and t h e distribution of s t r a t e g i e s by

m

ESN. Let u s assume now t h a t t h e payoff depends on _z only through t h e frequency distribution _m E SN of t h e p u r e s t r a t e g i e s . (Thomas (1984) calls such games degenerating). If we denote t h e payoff f o r t h e p u r e s t r a t e g y R j by Aj (m), then t h e payoff f o r phenotype Ei i s given by

while t h e a v e r a g e payoff in t h e population i s

G a m e dynamics leads t o

which f o r N

=

2 yields ( 8 ) .

Whenever t h e r e are more phenotypes than s t r a t e g i e s ( N < n ) , t h e r e are non- trivial r e l a t i o n s

CcQi

^{= o .}

Since a l l

-

^pi belong t o S N , t h i s implies z c i

=

0 and hence

C ~ i ( Z ! ~ z ! k ) = o .

Thus one obtains from ( 2 0 ) t h e constant of motion x c i log xi

=

const

which c o r r e s p o n d s t o (11).

The mean _m s a t i s f i e s

where Cov P is t h e c o v a r i a n c e matrix of t h e random v a r i a b l e s

Pk

taking values p:

with probability zi

.

This c o r r e s p o n d s to (12).

Following Thomas (1985) w e c a n also consider t h e auxiliary game correspond- ing t o t h e p u r e s t r a t e g i e s . This "pure s t r a t e g i s t game" c o r r e s p o n d s t o ( 2 0 ) with n

=

N a n d p i t h e i -th unit v e c t o r of t h e s t a n d a r d basis in lRN. Denoting by yi t h e frequency of Ri , we g e t _m

= -

^{y and}

on SN.

Theorem: If t h e p u r e s t r a t e g i s t dynamics ( 2 4 ) is a Shahshahani gradient, then s o is t h e mixed s t r a t e g i s t dynamics ( 2 0 ) .

Proof. ( 2 0 ) i s a r e p l i c a t o r equation of t y p e (4) with

Hence

If ( 2 4 ) i s a gradient, t h e t r i a n g u l a r integrability condition ( 1 5 ) r e a d s

This implies t h a t t h e N X N-matrix D,,A can be written as S(7n )

+

C ( m ) , where S

b

⁾ i s symmetric and C

(z

⁾ h a s N equal rows. For some _C

=

^{_C (m ) E} R~ w e have

Clearly

-

^{p i}

.

S ( n )

-

^{p j}

⁼ -

^{p j}

.

S(7n ) p i

-

^and

-

^p

.

C& )

-

^{p f =_c}

-

p j . This t o g e t h e r with ( 2 6 ) implies t h a t ( 1 5 ) i s satisfied and hence t h a t ( 2 0 ) i s a Shahshahani gradient.

Equation ( 2 0 ) h a s actually t h e same potential as ( 2 4 ) . More precisely, if

- ^vt+

V ( y

-

⁾ is a Shashahani gradient f o r ( 2 4 ) , then _z1+ V(m ) i s a Shashahani gradient f o r ( 2 0 ) . Indeed, if

(cf. eq. ( 1 4 ) ) then

I r k )

^{= _ p i} . A & )

= C P

= C p f ( ~ ~ ( 7 n ) - + ( 7 n ) )

~ + *b)

j j

A s a corollary , we obtain t h a t (8) i s always a Shahshahani gradient: indeed, w e have only t o r e c a l l t h a t N

=

2 . Equation ( 1 4 ) , in t h i s c a s e , i s satisfied with

9 k ) =

A2(m ) and V a primitive function of F , since

av av

^am

( m )

=

( m )= p i F ( m )

-

a z i axi

For t h e Hawk-Dove game, F i s given by ( 1 0 ) and hence we may use

The expression in t h e s q u a r e b r a c k e t i s t h e difference between t h e actual f r e - quency of

R 1

in t h e population and i t s Nash equilibrium value in t h e Hawk-Dove game, provided i t l i e s in (0,l).

For t h e s e x r a t i o game, F i s given by (9) and hence w e may use

The product m (l-m ) of t h e f r e q u e n c i e s of males and females i n c r e a s e s t o i t s maxi- mal value, obtained f o r m

= &.

This principle, which w a s f i r s t formulated by Shaw and Mohler (1953). c a n now b e strengthened: t h e population evolves in such a way t h a t m (1 -m ) i n c r e a s e s a t a maximal rate.

I t i s interesting t o consider a l s o t h e case of N

>

2 mating types. E v e r y "sex"

Ri can mate with any o t h e r

Rj,

j

#

^i. Since mi denotes t h e frequency of Ri and l - m i t h e frequency of i t s possible mates, t h e frequency of Ri-matings i s

and t h e payoff f o r a n

Ri

-individual, i.e. i t s s h a r e in matings, i s

I t i s e a s y t o check t h a t

satisfies t h e t r i a n g u l a r integrability condition (27). This implies t h a t t h e sex r a t i o game with N mating types i s a g r a d i e n t system. Kow

is of t h e form -(m

av

)

+

^{+(m ),} with

a"'i

i.e., since Emrc

=

^1. with

Hence (36) i s t h e potential f o r t h e N-type sex r a t i o game. The state

~ ( t )

evolves in s u c h a way t h a t c o n v e r g e s at a maximal r a t e towards 0. In equili-

brium, t h e N s e x e s will b e equally r e p r e s e n t e d in t h e population. (This i s somewhat disappointing. To quote Fisher (1930): "No p r a c t i c a l biologist i n t e r e s t e d in sexual reproduction would b e l e t t o work out t h e detailed consequences e x p e r i e n c e d by organisms having t h r e e o r more sexes: y e t what else should h e do if h e wishes t o understand why t h e s e x e s a r e , in f a c t , always two". I t would have been nice t o find out t h a t t h e model with N s e x e s leads t o t h e extinction of all but two of them.)

6. Discussion

(a) Gradient systems with r e s p e c t to a non-Euclidean metric o c c u r in s e v e r a l fields of t h e o r e t i c a l biology. In p a r t i c u l a r , Hofbauer (1985a) discussed t h e mutation-selection equation of Hadeler (1981) and r e l a t e d models. Such systems are Shahshahani g r a d i e n t s if and only if t h e mutation r a t e from allele A, t o allele Ac does not depend on j. In Hofbauer (1985b) one c a n find some more gradient systems: f o r example, f e r t i l i t y equations with two alleles, o r with additive f e r t i l i t y contributions of t h e p a r e n t s , o r with multiplicative but sex-independent contribu- tions.

In S c h u s t e r a n d Sigmund (1985) i t i s shown t h a t competition of autocatalytic r e a c t i o n s may lead in important c a s e s to gradients and hence t o maximum princi- ples.

(b) Our discussion of t h e s e x r a t i o neglected many bioIogical a s p e c t s . For example, w e assumed t h a t t h e "costs" f o r producing male and female offspring are t h e same: i t frequently happens, however, t h a t they differ. F i s h e r ' s argument i s still valid: i t s a y s now t h a t t h e total (life time) e f f o r t in producing sons and d a u g h t e r s must b e equal (see, e.g. Charnov (1982) and T r i v e r s (1985)). W e have f u r t h e r m o r e failed t o consider t h e c a s e t h a t competition f o r mates Is l o c a l r a t h e r t h a n global: this c a s e may lead t o e x t r a o r d i n a r y s e x r a t i o s (Hamilton (1968)).

Another a s p e c t which w e neglected i s t h e genetic basis of t h e s e x r a t i o (this i s a common t r a i t of "phenotypic" game dynamics f o r frequency dependent selection).

T h e r e i s a considerable amount of work on genetic models, which seem independent of F i s h e r ' s argument but lead again t o t h e s e x r a t i o (see, e.g., Eshel and Feldman (1982) and Karlin and Lessard (1986)). Sex linked meiotic d r i v e may again lead t o e x t r a o r d i n a r y s e x r a t i o s (Hamilton, (1968)).

The outcome of both genetic and game t h e o r e t i c models i s a prediction on t h e t o t a l s e x r a t i o in t h e population, and not on t h e s e x r a t i o of individuals. The popu- lation may just as w e l l consist of a unique phenotype with s e x r a t i o o r of two equally r e p r e s e n t e d phenotypes with s e x r a t i o s 0 and 1, say. Why i s t h e r e in a c t u a l populations a p r e v a l e n c e f o r individual s e x r a t i o s close ta *? Numerical simulations by Poethke (1986) suggest t h a t t h i s i s due t o t h e finite size of t h e popu- lation.

Finally, we mention t h a t evolutionary dynamics seems w e l l o n t h e way t o invade classical game t h e o r e t i c a l fields. F o r example, Samuelson (1985) d e a l s with a s e x r a t i o model as a link between modes of thought in economics and biology.

A c k n o w l e d g e m e n t s . I t i s a p l e a s u r e t o thank Academician Kurzhanski f o r suggesting t h e model with

N

mating types, and Dr. Hofbauer f o r discussions on g r a - dient systems. This work h a s been supported financially by t h e Austrian Forschungsf6rderungsf onds, P 5994P.

7 . R e f e r e n c e s

Akin, E., 1979: The geometry of population genetics. L e c t u r e Notes in- Biomathematics

2.

Springer-Verlag, Berlin-Heidelberg-New York.

Axelrod, R., 1984: The evolution of cooperation. Basic Books.

Charnov, E.L., 1982: The t h e o r y of s e x allocation. Princeton University P r e s s , Princeton, N e w J e r s e y .

Darwin, Ch., 1859: The origin of s p e c i e s by means of n a t u r a l selection.

Cambridge-London (1964): Reprint H a r v a r d University P r e s s .

Eshel, I. and M. Feldman, 1982: On evolutionary genetic stability of t h e s e x r a t i o . Theor. Pop. Biol. 21,, 430-439.

Fisher, R.A., 1930: The genetical t h e o r y of n a t u r a l selection, Clarendon P r e s s , Oxford.

Hadeler, K.P., 1981: S t a b l e polymorphisms in a selection model with mutation.

SIAM J. Appl. Math.

41,

1-7.

Hamilton, W.D., 1967: E x t r a o r d i n a r y s e x r a t i o s . Science 1 5 6 , 477-488.

Hofbauer, J . , 1981: On t h e o c c u r r e n c e of limit cycles in t h e Volterra-Lotka equa- tion. Nonlinear Analysis TMA

3,

1003-1007.

Hofbauer, J., 1985a: Gradients v e r s u s cycling in genetic selection models, in 'Dynamics of Macrosystems", eds. J.P. Aubin, D. S a a r i and K. Sigmund. Lec- t u r e Notes in Economics a n d Mathematical Systems

m,

Springer-Verlag, Berlin-Heidelberg-New York.

Hofbauer, J., 1985b: The selection mutation equation. J. Math. Biol. 2 3 , 41-53.

Karlin, S. and S. Lessard, 1986: Theoretical Studies on Sex Ratio Evolution, Princeton University P r e s s , Monographs in Population Biology, Princeton, N e w J e r s e y .

Maynard-Smith, J. and G . P r i c e , 1973: The logic of animal conflicts. Nature 2 4 6 , 15-18.

Maynard-Smith, J., 1974: The t h e o r y of games and t h e evolution of animal con- flicts. J. Theor. Biol. 4 7 , 209-221.

Maynard-Smith, J., 1982: Evolutionary game theory. Cambridge University P r e s s , Cambridge, Massachusetts.

Poethke, H.J., 1986: Sex r a t i o polymorphism: t h e impact of mutation on evolution.

To a p p e a r in J. Theor. Biol.

Samuelson, P.A., 1985: Modes of thought in economics and biology. Amer. Econ.

Rev. 75, 166-172.

S c h u s t e r , P. and K. Sigmund, 1983: Replicator dynamics. J. Theor. Biol. J,QQ, 533- 538.

S c h u s t e r , P . and K. Sigmund: Evolutionary optimization. J.B

.

Bunsengesellsch

.

,

m,

^668-682.

Shahshahani. S.: A new mathematical framework f o r t h e study of linkage and selec- tion. Memoirs AMS =1(1979), Providence.

Shaw,

R.F.

and J.D. Mohler, 1953: The selective significance of t h e s e x r a t i o . Am.

N a t . 8 7 , 337-342.

Sigmund, K., 1984: The maximum principle f o r r e p l i c a t o r equations. In: Lotka- Volterra Approach t o Dynamic Systems. Ed. M. Peschel. Akademie-Verlag, Berlin.

Sigmund, K., 1985: A s u r v e y on r e p l i c a t o r equations. In: Complexity, Language and Life. Ed. J. Casti and A. Karlquist. L e c t u r e Notes in Biomathematics l6, Springer-Verlag, Berlin-Heidelberg-New York.

Taylor, P. and L. J o n k e r , 1978: Evolutionarily s t a b l e s t r a t e g i e s and game dynam- ics. Math. Bioscience

a,

^145-156.

Thomas, B., 1984: Evolutionary stability: s t a t e s and s t r a t e g i e s . Theor. Pop. Biol.

a,

^49-67.

Thomas, B., 1985: Evolutionarily s t a b l e s e t s in mixed s t r a t e g i s t models. Theor.

Pop. Biol. 28, 332-341.

Trivers, R., 1985: Social evolution. Benjamin Cummings, Menlo P a r k .