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On evolutionarily stable strategies in populations with subpopulations having isolated strategy repertoires
263
BioSystems, 11 {1979) 263--268 © Elsevier/North-Hol:[and Scientific Publishers Ltd.
ON E V O L U T I O N A R I L Y STABLE STRATEGIES IN POPULATIONS WITH SUBPOPULATIONS HAVING I S O L N I E D S T R A T E G Y REPERTOIRES
HEINZ-JOACHIM POHLEY and B E R N H A R D THOMAS
lnstitut f~r Entwicklungsphysiologie der Uuiversitiit zu K6ln, D-5000 KSln, F.R.G. (Received November 14th, 1978) (Revision received February 20th, 1979)
The concept of a population with subpopulations that are isolated with respect to strategies is introduced and applied to a biologically important class of cases, showing that no proper mixed evolutionarily stable strategies can exist in such case.'~.
Introduction The concept of evolutionarily stable strategy (ESS) has been introduced by J. Maynard Smith and G.R. Price (1973) to analyse problems in evolutionary biology with game theoretical means. Animal conflicts are described in terms of such a game theoretical model, introducing a set H of behavioural strategies and a pay-off function E, which assigns the pay-off E(p,q) to any pair (p,q) of behavioural strategies. A strategy p is defined to be evolutionarily stable, if p is the best strategy to be adopted against itself and, in addition, if there is another strategy q, which does as well against p as p itself does, then p pays off better than q does against q. The biological interpretations of this game theoretical model essentially are statements concerning the stability of populations and hence are statements of theoretical population biology. Consider a model population A with a finite set I = {I1 . . . . , In} , the repertoire of elementary strategies (attributes). Let A be a primitive population in the sense that: (1) every individual a ~ A adopts just one elementary strategy I i ~ I; (2) the offsprings of a a d o p t the same strategy as their progenitor; (3) subsequent generations do n o t overlap; and (4) 1;he selective mechanism is
frequency depending selection (FDS-dynamics, see e.g., Lewontin (1958), Sperlich (1973)). Let x~ (i = 1 , . . . , n ) be the frequency of I i in the k-th generation, then the distribution of the frequencies of elementary strategies is x k = ( x ~ , . . . , X k n ) for the k-th generation. As x k has the properties of a probability vector over I, we can consider x k to be a mixed strategy adopted by the population A in the k-th generation (population strategy). Let the pay-off term E(ei,xk), e i being the i-th unit vector, determine the number of offsprings produced by a k-th generation individual adopting I i. Then the FDS-dynamics determine the frequency xk. ÷~ of individuals adopting I i m the subsequent generation. We have •
E(ei,x k) x k.~+ l = X ~ E ( x k , x k )
1
i= 1,...,n
where E(xk;x k) is the total pay-off for the k-th generation. If for some ~ the right hand side yields ~ again, ~ is called a fixed point. If the sequence of the x k within a certain neighb o u r h o o d of a fixed point ~, called the domain of attraction, converges towards as the limit distribution, ~ is called (asymptotically) stable. It should be noticed that in general stable population strategies do n o t coincide with optimal population strategies,
264 which are defined to maximize E ( x , x ) . Individuals of a non-primitive population A may adopt mixed strategies. Then it can be shown that A might be replaced by a primitive population A' for which the same results as for the original population can to some extent be obtained concerning the fixed points and stability. A population in a stable fixed point status cannot be invaded during subsequent generations by any deviating strategy within the domain of attraction of 37 and hence is evolutionarily stable in this sense. There may be more than one fixed point, however, each one with its own domain of attraction. Therefore, long term evolutionary changes can take place only if certain events are strong enough to actually drive the population out of one domain of attraction (catastrophe). In asymmetric conflicts the adopted strategy depends on the role an individual finds itself in and the individual roles in any such conflict are always different. In "The Selfish Gene", Dawkins (1976) constructed an instance of an asymmetric conflict, the "Battle of Sexes" and concluded that a certain combination of strategies in either sex is a mixed ESS. In this paper we are more generally concerned with populations that consist of subpopulations, which are isolated with respect to their strategy repertoires, and we derive conditions for the existence of a mixed ESS in such populations. Application to conflicts that are defined only between individuals of different subpopulations (e.g. in asymmetric conflicts) will show that no mixed ESS can exist in these cases. In addition, we state simple conditions for the existence of pure ESS's in such populations.
Evolutionarily stable strategies As was pointed out above, a mathematical description of ESS has to start from a (finite) repertoire I of elementary strategies in a population A. Individuals might adopt only
one of the I i (pure strategy) or use all of these according to a fixed distribution p. Hence we define mixed strategies by distributions p = (p~,..., p n ) m of probabilities over H
I(0~
p i = 1). 1
Pure strategies are included as unit vectors e i•
The set H of strategies and relations involving strategies can be treated completely in mathematical terms of an n-dimensional, real, normed linear space. The pay-off for a first contestant adopting I i against a second one applying /i may be denoted by aij = E(ei,e.i ) (i,j = 1 . . . . . n). thus giving rise to a pay-off matrix A = (aij). From this the pay-off for a mixed strategy p applied against q can be calculated as E ( p , q ) = p m A q . Maynard Smith (1974) showed how the existence of a pure ESS can easily be verified w i t h o u t taking into account ~he dynamics and just considering A: I k is a pure ESS, if and only if for all i ¢ /e we have alele >1 aik and, in addition, if = holds, we have aki > aii in turn As a consequence there might be more than just one pure ESS. Concerning a mixed ESS, we resume the conditions presented by Haigh (1975) in a slightly different formulation. Let us call t3 to be an equilibrium point (EP), if the payoff E ( e i ~ ) for all pure strategies, and hence for any strategy, applied against /5 gives the same value. Dynamically, this means that/5 is a fixed point and a population in equilibrium state will preserve this distribution. A slight deviation from/5, however, might destroy the equilibrium situation, unless/3 is stable. The condition given by Maynard Smith for/5 to be an ESS is that, for all alternative q, we have E(fa,q) > E ( q , q ) , in matrix notation: /ST A q > qTAq.
Define a*j = aij -- anj -- ain + ann c i = ain -- ann
i,j = 1 , . . . ,
n -- 1
Then the EP/5 is defined by the solution (if
265 unique) of A* 2 = - c *
(1) n
as[oi=Yci(i=
1,...,n--1),JOn=
1--
I
E
xi"
i=l
We call /5 an admissible EP, if/3 is i n d e e d a d i s t r i b u t i o n , and a p r o p e r EP, if n o n e o f the /3 i = O, i.e. all p u r e strategies are r e p r e s e n t e d in a p r o p e r EP. It can be s h o w n t h a t a p r o p e r EP/3 is an ES8, if and o n l y if A is negative d e f i n i t e or the m a t r i x C = !I (A* + A *'r) has o n l y negative eigenvalues, and t h a t this is the o n l y ESS in this case. This result cart be derived b y means o f a r e d u c t i o n o f t h e bilinear f o r m E ( p , q ) for distributions p,q, w h i c h can be generalized in a way to b e c o m e applicable to the case o f isolated s u b p o p u l a t i o n s .
S u b p o p u l a t i o n s with isolated strategy reper-
toires Our main i n t e n t i o n is t o discuss the exist e n c e o f ESS in p o p u l a t i o n s t h a t consist o f s u b p o p u l a t i o n s , w h e r e each m e m b e r has access t o a certain set o f strategies t h a t is assigned t o this s u b p o p u l a t i o n , b u t n o t t o a n y o f the others. Conflicts, w h e r e these strategies are involved, m a y o c c u r b e t w e e n m e m b e r s o f the same s u b p o p u l a t i o n and o f different subpopulations. As a case w h e r e " c o n f l i c t s " are d e f i n e d o n l y b e t w e e n m e m b e r s o f d i f f e r e n t subp o p u l a t i o n s , Dawkins c o n s i d e r e d the mating b e h a v i o u r o f males and females o f some species. Males c o u l d a d o p t t w o male strategies females had access t o the same n u m b e r o f f e m a l e strategies and, o f course, n o individual c o u l d switch t o the o t h e r sex t o m a k e use o f the o t h e r r e p e r t o i r e as well. T h e c o n f l i c t was designed as the m a t i n g o f a male and a female and p a y - o f f s for e i t h e r strategy were calculated f r o m a c o s t - b e n e f i t analysis. It was p o i n t e d o u t t h a t a certain c o m b i n a t i o n o f strategies in e i t h e r sex is e v o l u t i o n a r i l y stable.
We shall describe such cases m o r e precisely b y i n t r o d u c i n g the t e r m o f " i s o l a t i o n with r e s p e c t to strategies". T h e following considerations given for the case o f t w o s u b p o p u l a t i o n s are easily e x t e n d e d t o m o r e c o m p l e x populations. Let r,s > 0, where r + s = 1, be the fixed frequencies o f s u b p o p u l a t i o n s AI and A2 in a p o p u l a t i o n A. We call AI and A2 to be isolated with r e s p e c t t o strategies, if t h e i r m e m b e r s dispose o f strategy r e p e r t o i r e s I~ = {J~ . . . . , Jn } and I: = {J~l , . . . , Jm }, respectively, w h e r e Ii and I2 are disjoint sets. Mixed strategies with r e s p e c t t o I~ arep~ , w h e r e E"i: ~ P li = r, and those with r e s p e c t to I2 are p : , w h e r e E J:~ m P 2 j = s, if we consider P l i and P H to be the frequencies o f the corresp o n d i n g strategies relative t o the c o m p l e t e p o p u l a t i o n A. An i m b e d d i n g can be p e r f o r m e d as follows: L e t I = I 1 u I 2 , giving I = { I 1 , . . . , I n . m } ' w h e r e I i = Ji, for i = 1 . . . . . n, and I n . i = ~., f o r ] = 1, . . . , m. D e n o t i n g ai]= £(ei, ej) , with (n + m ) - d i m e n s i o n a l unit vectors, we o b t a i n the c o m p l e t e p a y - o f f m a t r i x with r e s p e c t t o I: rAil A121 A = [_A21 A2~J
w h e r e the d i m e n s i o n o f A1, is n × n, o f A22 is m X m, o f A~2 is n × m, and o f A21 is m X n. Pl and P2 can be s u p e r p o s e d t o give (P l) , a d i s t r i b u t i o n over I. T h e e x p e c t e d P= P2 gain for pl~o2 in A~ and A2, resp., applied against ql ,q~ is 2
E ( p , q ) = p T A q = i,jE1 P i A i j q j
whereq=
(q')
q2 "
T h e t e r m s o f the sum can be r e d u c e d , i n t r o d u c i n g t h e r e d u c e d m a t r i c e s A*/ o f A i j and c*1 a p p r o p r i a t e l y d e f i n e d as a b o v e f o r
266 i,j = 1,2. Let A* be the composition of $ matrices A ii, giving a reduction of A, and
means that pay-offs are defined only between members of different subpopulations. Then C
c*-- i rc:' + sc :
takes the form C =
~rc~l + sc~! Then for the EP/3 =
(2) the first n - 1
and m - 1 components of/51 and/52, respectively, are given by the solution of
A*(~1) x2
=-c*
(3)
which is the analogue of (1). The notion of admissible and proper EP is appropriately extended to this case. Now, with a proper EP /5, the condition E{fg, q) > E ( q , q ) for/5 to be an ESS can be reformulated and expressed by means of the reduced matrices to give, after some calculation, T h e o r e m 1: (a) An admissible EP/3 is an ESS, if the symmetric matrix
C=
T
B~
consisting of B = ~ (A*2 + A*~ ) and B i = I(A*. + A *T ii), i = 1,2, has eigenvalues all negative, i.e. is negative definite. (b) A necessary condition for a proper EP/3 to be an ESS is t h a t C has only negative eigenvalues. The frequencies of the strategies relative to the corresponding subpopulations are obtained by division of the frequencies relative to A by r or s, respectively. Two important cases are to be considered: (1) A12 and A:i in A are zero-matrices (or can be shifted to this form). Then pay-offs are defined only between members of the same population. In this case C takes the form B
and has all eigenvalues nega-
tive, if and only if B1 and B2 have. Hence the subpopulations can be treated strictly separated. (2) A ~ and A22 are zero-matrices. This
[00 1 B-r
and it can be
shown that matrices of this type c a n n o t be negative definite. Thus we have T h e o r e m 2: If the pay-offs in a population with two subpopulations being isolated with respect to strategies are defined only between members of different subpopulations (i.e. give rise to a pay-off matrix A with A11 and A22 zero-matrices) then no proper mixed ESS can exist. This theorem applies to the example given by Dawkins in the "Battle of Sexes", where the corresponding matrix becomes
A =
i o o1 0 5 15
5-5 0 0 0 0
consisting of the pay-offs for female strategies against male strategies and those for male strategies against female strategies. Unaffected by any fixed ratio r : s of females and males we have by theorem 2 that no proper mixed ESS exists. For illustration we may verify this by calculation. From A we have the reduced
A*=
[10
--081andc * = ( - 1 ~ : )
after (2). As the solution of ( 3 ) w e obtain } , , = ~r, x21 = As and hence
,, giving the frequencies (5/6,1/6) and (5/8,3/8) relative to the corresponding subpopulations. The EP is a proper one and thus not evolu-
267 tionarily stable, as C takes the form C -[~
foriv~ i0
aio, n ~jo > ai, n * Jo
{4)
~1, w h i c h i n d e e d h a s e i g e n v a l u e s 2 a n d - 2 .
This result agrees with the result of the dynamical approach to ESS: If started with a distribution according to the calculated EP, dynamics preserve the relations. A small pertubation, however, will not be compensated but initiate an outward directed, spiral development which has no limit point. The statement of theorem 2 also holds for subproblems, i.e. finding an ESS in which only some of the strategies of either repertoire are involved. In general, there can be no mixed ESS in populations, where individuals have access to disjoint stral;egy repertoires, if conflicts always involve contestants which have access to different repertoires. The term "disjoint" has to be taken quite formally, since we find a typical example for this situation in asymmetric conflicts, i.e. the behaviour of contestants depends on the roles they find themselves in, and any two contestants always are in different role,', (Maynard Smith and Parker, 1976; Selten, 1!)78). The roles then define subpopulations like A j and A2, and adopting a strategy of the repertoire Ii as a member of AI means doing so in one role, whereas doing so in a second role means being a member of A::. Usually, in asymmetric contests the disjoint repertoires contain the same number of strategies and there are corresponding strategies in either repertoire that stana for the s a m e behaviour but displayed in different roles (see example below). Hence we have from theorem 2 that no mixed ESS can exist in asymmetric conflicts. It might be the case, however, that in populations as considered here a stable equilibrium hoMs, where the members of either subpopulation use just one strategy. Hence we might ask, whether a pair Ji , J[ of pure strategies can be evolutionari~y -tO . stable in case Alx and A2~ are zero-matrices. It can be shown that Ji and Jj form an evolutmnarlly stable pair o~ strategies, ff and only if •
.
.
0
.
O.
for j ¢ J0
an * .Jo, io ~ an ~J, io
As a consequence, we have the assertion that, if Ji together with some Jf forms an evolutlonarily stable pair of strategms, then no other pair containing Jio or JJ0 can be an ESS. Hence the number of different pairs that are ESS cannot exceed min { n , m ~for a given matrix• It can be easily checked that by this condition no such pair exists for the matrix given in the "Battle of Sexes", thus neither mixed nor pairs of pure strategies can be ESS's in this case. Finally, we shall apply our results to an asymmetric conflict described and analysed by Maynard Smith and Parker, from which their conclusions follow very easily• Contesting for a value V > 0, opponents may adopt strategy e ("escalate": fight until injured or opponent retreats) or d ("display": retreat, if attacked). Losing the contest when escalating will result in a negative value - D , e.g. from injury. Asymmetry, e.g. in fighting ability, has been introduced concerning the changes to win on escalation: a contestant might be "larger", i.e. wins with probability x > 0.5, or "smaller", i.e. wins with 1 - x. Concerning d, chances of winning V are 0.5, if both opponents display. In our terms, we have subpopulations A~ and A2 of "large" and "small" individuals, respectively. Strategies e and d for a member of AI define 1~ = (J~,J2 }, whereas for a member of A2 they define I2 = {all,d2 }, so I~ and I2 are formally disjoint repertoires. The pay-offs defined by Maynard Smith and Parker give rise to the following matrix in our notation: •
O
.
sO
Vx0
A =
V(1 - x ) - D x
.
D(10
V
.
_
o
v/2
0
x)
V" V/'
268 O f course, no m i x e d ESS can exist, as t h e o r e m 2 states. C o n c e r n i n g e v o l u t i o n a r i l y stable pairs of strategies, t h e c o n d i t i o n (4) shows t h a t ( J , , J ' , ) (escalate, if larger; escalate, if smaller) is the o n l y ESS pair, ifxD < (1 - x)V. If xD > (1 - x)V, t h e n (J,,J'2) (escalate, if larger; display, if smaller) is e v o l u t i o n a r i l y stable, a n d if, in a d d i t i o n , (1 - x)D > xV, t h e n even b o t h (J, J 2 ) and (J2,J~) (display, if larger; escalate, if smaller) are e v o l u t i o n a r i l y stable. This m e a n s t h a t if the e x p e c t e d loss, D(1 x), f o r the larger individuals on escalat i o n is larger t h a n their e x p e c t e d gain, x V, t h e r e are t w o stable situations in this p o p u l a t i o n : larger c o n t e s t a n t s escalate and smaller ones display, or larger individuals d o n o t escalate, while smaller o n e s do. F r o m t h e d y n a m i c a l p o i n t o f view, it d e p e n d s on t h e initial c o n d i t i o n s , w h i c h o f these situations will be realized.
References Dawkins, R., 1976, The Selfish Gene, Oxford. Haigh, J., 1975, Game theory and evolution, Adv. Appl. Probab. 7, 8--11. Lewontin, R.C., 1958, A general method for investigating the equilibrium of gene frequency in a population. Genetics 43,419--434. Maynard Smith, J., 1974, The theory of games and the evolution of animal conflicts. J. Theor. Biol. 47,209--221. Maynard Smith, J. and G.A. Parker, 1976, The logic of asymmetric contests, Anim. Behav. 24, 159-175. Maynard Smith, J. and G.R. Price, 1973, The logic of animal conflicts, Nature 246, 15--18. Selten, R., 1978, A Note on Evolutionarily Stable Strategies in Asymmetric Animal Conflicts, Working Papers Inst. Math. Econ. Univ. Bielefeld, Nr. 73. Sperlich, D., 1973, Populationsgenetik, Stuttgart.
BioSystems, 11 {1979) 263--268 © Elsevier/North-Hol:[and Scientific Publishers Ltd.
ON E V O L U T I O N A R I L Y STABLE STRATEGIES IN POPULATIONS WITH SUBPOPULATIONS HAVING I S O L N I E D S T R A T E G Y REPERTOIRES
HEINZ-JOACHIM POHLEY and B E R N H A R D THOMAS
lnstitut f~r Entwicklungsphysiologie der Uuiversitiit zu K6ln, D-5000 KSln, F.R.G. (Received November 14th, 1978) (Revision received February 20th, 1979)
The concept of a population with subpopulations that are isolated with respect to strategies is introduced and applied to a biologically important class of cases, showing that no proper mixed evolutionarily stable strategies can exist in such case.'~.
Introduction The concept of evolutionarily stable strategy (ESS) has been introduced by J. Maynard Smith and G.R. Price (1973) to analyse problems in evolutionary biology with game theoretical means. Animal conflicts are described in terms of such a game theoretical model, introducing a set H of behavioural strategies and a pay-off function E, which assigns the pay-off E(p,q) to any pair (p,q) of behavioural strategies. A strategy p is defined to be evolutionarily stable, if p is the best strategy to be adopted against itself and, in addition, if there is another strategy q, which does as well against p as p itself does, then p pays off better than q does against q. The biological interpretations of this game theoretical model essentially are statements concerning the stability of populations and hence are statements of theoretical population biology. Consider a model population A with a finite set I = {I1 . . . . , In} , the repertoire of elementary strategies (attributes). Let A be a primitive population in the sense that: (1) every individual a ~ A adopts just one elementary strategy I i ~ I; (2) the offsprings of a a d o p t the same strategy as their progenitor; (3) subsequent generations do n o t overlap; and (4) 1;he selective mechanism is
frequency depending selection (FDS-dynamics, see e.g., Lewontin (1958), Sperlich (1973)). Let x~ (i = 1 , . . . , n ) be the frequency of I i in the k-th generation, then the distribution of the frequencies of elementary strategies is x k = ( x ~ , . . . , X k n ) for the k-th generation. As x k has the properties of a probability vector over I, we can consider x k to be a mixed strategy adopted by the population A in the k-th generation (population strategy). Let the pay-off term E(ei,xk), e i being the i-th unit vector, determine the number of offsprings produced by a k-th generation individual adopting I i. Then the FDS-dynamics determine the frequency xk. ÷~ of individuals adopting I i m the subsequent generation. We have •
E(ei,x k) x k.~+ l = X ~ E ( x k , x k )
1
i= 1,...,n
where E(xk;x k) is the total pay-off for the k-th generation. If for some ~ the right hand side yields ~ again, ~ is called a fixed point. If the sequence of the x k within a certain neighb o u r h o o d of a fixed point ~, called the domain of attraction, converges towards as the limit distribution, ~ is called (asymptotically) stable. It should be noticed that in general stable population strategies do n o t coincide with optimal population strategies,
264 which are defined to maximize E ( x , x ) . Individuals of a non-primitive population A may adopt mixed strategies. Then it can be shown that A might be replaced by a primitive population A' for which the same results as for the original population can to some extent be obtained concerning the fixed points and stability. A population in a stable fixed point status cannot be invaded during subsequent generations by any deviating strategy within the domain of attraction of 37 and hence is evolutionarily stable in this sense. There may be more than one fixed point, however, each one with its own domain of attraction. Therefore, long term evolutionary changes can take place only if certain events are strong enough to actually drive the population out of one domain of attraction (catastrophe). In asymmetric conflicts the adopted strategy depends on the role an individual finds itself in and the individual roles in any such conflict are always different. In "The Selfish Gene", Dawkins (1976) constructed an instance of an asymmetric conflict, the "Battle of Sexes" and concluded that a certain combination of strategies in either sex is a mixed ESS. In this paper we are more generally concerned with populations that consist of subpopulations, which are isolated with respect to their strategy repertoires, and we derive conditions for the existence of a mixed ESS in such populations. Application to conflicts that are defined only between individuals of different subpopulations (e.g. in asymmetric conflicts) will show that no mixed ESS can exist in these cases. In addition, we state simple conditions for the existence of pure ESS's in such populations.
Evolutionarily stable strategies As was pointed out above, a mathematical description of ESS has to start from a (finite) repertoire I of elementary strategies in a population A. Individuals might adopt only
one of the I i (pure strategy) or use all of these according to a fixed distribution p. Hence we define mixed strategies by distributions p = (p~,..., p n ) m of probabilities over H
I(0~
p i = 1). 1
Pure strategies are included as unit vectors e i•
The set H of strategies and relations involving strategies can be treated completely in mathematical terms of an n-dimensional, real, normed linear space. The pay-off for a first contestant adopting I i against a second one applying /i may be denoted by aij = E(ei,e.i ) (i,j = 1 . . . . . n). thus giving rise to a pay-off matrix A = (aij). From this the pay-off for a mixed strategy p applied against q can be calculated as E ( p , q ) = p m A q . Maynard Smith (1974) showed how the existence of a pure ESS can easily be verified w i t h o u t taking into account ~he dynamics and just considering A: I k is a pure ESS, if and only if for all i ¢ /e we have alele >1 aik and, in addition, if = holds, we have aki > aii in turn As a consequence there might be more than just one pure ESS. Concerning a mixed ESS, we resume the conditions presented by Haigh (1975) in a slightly different formulation. Let us call t3 to be an equilibrium point (EP), if the payoff E ( e i ~ ) for all pure strategies, and hence for any strategy, applied against /5 gives the same value. Dynamically, this means that/5 is a fixed point and a population in equilibrium state will preserve this distribution. A slight deviation from/5, however, might destroy the equilibrium situation, unless/3 is stable. The condition given by Maynard Smith for/5 to be an ESS is that, for all alternative q, we have E(fa,q) > E ( q , q ) , in matrix notation: /ST A q > qTAq.
Define a*j = aij -- anj -- ain + ann c i = ain -- ann
i,j = 1 , . . . ,
n -- 1
Then the EP/5 is defined by the solution (if
265 unique) of A* 2 = - c *
(1) n
as[oi=Yci(i=
1,...,n--1),JOn=
1--
I
E
xi"
i=l
We call /5 an admissible EP, if/3 is i n d e e d a d i s t r i b u t i o n , and a p r o p e r EP, if n o n e o f the /3 i = O, i.e. all p u r e strategies are r e p r e s e n t e d in a p r o p e r EP. It can be s h o w n t h a t a p r o p e r EP/3 is an ES8, if and o n l y if A is negative d e f i n i t e or the m a t r i x C = !I (A* + A *'r) has o n l y negative eigenvalues, and t h a t this is the o n l y ESS in this case. This result cart be derived b y means o f a r e d u c t i o n o f t h e bilinear f o r m E ( p , q ) for distributions p,q, w h i c h can be generalized in a way to b e c o m e applicable to the case o f isolated s u b p o p u l a t i o n s .
S u b p o p u l a t i o n s with isolated strategy reper-
toires Our main i n t e n t i o n is t o discuss the exist e n c e o f ESS in p o p u l a t i o n s t h a t consist o f s u b p o p u l a t i o n s , w h e r e each m e m b e r has access t o a certain set o f strategies t h a t is assigned t o this s u b p o p u l a t i o n , b u t n o t t o a n y o f the others. Conflicts, w h e r e these strategies are involved, m a y o c c u r b e t w e e n m e m b e r s o f the same s u b p o p u l a t i o n and o f different subpopulations. As a case w h e r e " c o n f l i c t s " are d e f i n e d o n l y b e t w e e n m e m b e r s o f d i f f e r e n t subp o p u l a t i o n s , Dawkins c o n s i d e r e d the mating b e h a v i o u r o f males and females o f some species. Males c o u l d a d o p t t w o male strategies females had access t o the same n u m b e r o f f e m a l e strategies and, o f course, n o individual c o u l d switch t o the o t h e r sex t o m a k e use o f the o t h e r r e p e r t o i r e as well. T h e c o n f l i c t was designed as the m a t i n g o f a male and a female and p a y - o f f s for e i t h e r strategy were calculated f r o m a c o s t - b e n e f i t analysis. It was p o i n t e d o u t t h a t a certain c o m b i n a t i o n o f strategies in e i t h e r sex is e v o l u t i o n a r i l y stable.
We shall describe such cases m o r e precisely b y i n t r o d u c i n g the t e r m o f " i s o l a t i o n with r e s p e c t to strategies". T h e following considerations given for the case o f t w o s u b p o p u l a t i o n s are easily e x t e n d e d t o m o r e c o m p l e x populations. Let r,s > 0, where r + s = 1, be the fixed frequencies o f s u b p o p u l a t i o n s AI and A2 in a p o p u l a t i o n A. We call AI and A2 to be isolated with r e s p e c t t o strategies, if t h e i r m e m b e r s dispose o f strategy r e p e r t o i r e s I~ = {J~ . . . . , Jn } and I: = {J~l , . . . , Jm }, respectively, w h e r e Ii and I2 are disjoint sets. Mixed strategies with r e s p e c t t o I~ arep~ , w h e r e E"i: ~ P li = r, and those with r e s p e c t to I2 are p : , w h e r e E J:~ m P 2 j = s, if we consider P l i and P H to be the frequencies o f the corresp o n d i n g strategies relative t o the c o m p l e t e p o p u l a t i o n A. An i m b e d d i n g can be p e r f o r m e d as follows: L e t I = I 1 u I 2 , giving I = { I 1 , . . . , I n . m } ' w h e r e I i = Ji, for i = 1 . . . . . n, and I n . i = ~., f o r ] = 1, . . . , m. D e n o t i n g ai]= £(ei, ej) , with (n + m ) - d i m e n s i o n a l unit vectors, we o b t a i n the c o m p l e t e p a y - o f f m a t r i x with r e s p e c t t o I: rAil A121 A = [_A21 A2~J
w h e r e the d i m e n s i o n o f A1, is n × n, o f A22 is m X m, o f A~2 is n × m, and o f A21 is m X n. Pl and P2 can be s u p e r p o s e d t o give (P l) , a d i s t r i b u t i o n over I. T h e e x p e c t e d P= P2 gain for pl~o2 in A~ and A2, resp., applied against ql ,q~ is 2
E ( p , q ) = p T A q = i,jE1 P i A i j q j
whereq=
(q')
q2 "
T h e t e r m s o f the sum can be r e d u c e d , i n t r o d u c i n g t h e r e d u c e d m a t r i c e s A*/ o f A i j and c*1 a p p r o p r i a t e l y d e f i n e d as a b o v e f o r
266 i,j = 1,2. Let A* be the composition of $ matrices A ii, giving a reduction of A, and
means that pay-offs are defined only between members of different subpopulations. Then C
c*-- i rc:' + sc :
takes the form C =
~rc~l + sc~! Then for the EP/3 =
(2) the first n - 1
and m - 1 components of/51 and/52, respectively, are given by the solution of
A*(~1) x2
=-c*
(3)
which is the analogue of (1). The notion of admissible and proper EP is appropriately extended to this case. Now, with a proper EP /5, the condition E{fg, q) > E ( q , q ) for/5 to be an ESS can be reformulated and expressed by means of the reduced matrices to give, after some calculation, T h e o r e m 1: (a) An admissible EP/3 is an ESS, if the symmetric matrix
C=
T
B~
consisting of B = ~ (A*2 + A*~ ) and B i = I(A*. + A *T ii), i = 1,2, has eigenvalues all negative, i.e. is negative definite. (b) A necessary condition for a proper EP/3 to be an ESS is t h a t C has only negative eigenvalues. The frequencies of the strategies relative to the corresponding subpopulations are obtained by division of the frequencies relative to A by r or s, respectively. Two important cases are to be considered: (1) A12 and A:i in A are zero-matrices (or can be shifted to this form). Then pay-offs are defined only between members of the same population. In this case C takes the form B
and has all eigenvalues nega-
tive, if and only if B1 and B2 have. Hence the subpopulations can be treated strictly separated. (2) A ~ and A22 are zero-matrices. This
[00 1 B-r
and it can be
shown that matrices of this type c a n n o t be negative definite. Thus we have T h e o r e m 2: If the pay-offs in a population with two subpopulations being isolated with respect to strategies are defined only between members of different subpopulations (i.e. give rise to a pay-off matrix A with A11 and A22 zero-matrices) then no proper mixed ESS can exist. This theorem applies to the example given by Dawkins in the "Battle of Sexes", where the corresponding matrix becomes
A =
i o o1 0 5 15
5-5 0 0 0 0
consisting of the pay-offs for female strategies against male strategies and those for male strategies against female strategies. Unaffected by any fixed ratio r : s of females and males we have by theorem 2 that no proper mixed ESS exists. For illustration we may verify this by calculation. From A we have the reduced
A*=
[10
--081andc * = ( - 1 ~ : )
after (2). As the solution of ( 3 ) w e obtain } , , = ~r, x21 = As and hence
,, giving the frequencies (5/6,1/6) and (5/8,3/8) relative to the corresponding subpopulations. The EP is a proper one and thus not evolu-
267 tionarily stable, as C takes the form C -[~
foriv~ i0
aio, n ~jo > ai, n * Jo
{4)
~1, w h i c h i n d e e d h a s e i g e n v a l u e s 2 a n d - 2 .
This result agrees with the result of the dynamical approach to ESS: If started with a distribution according to the calculated EP, dynamics preserve the relations. A small pertubation, however, will not be compensated but initiate an outward directed, spiral development which has no limit point. The statement of theorem 2 also holds for subproblems, i.e. finding an ESS in which only some of the strategies of either repertoire are involved. In general, there can be no mixed ESS in populations, where individuals have access to disjoint stral;egy repertoires, if conflicts always involve contestants which have access to different repertoires. The term "disjoint" has to be taken quite formally, since we find a typical example for this situation in asymmetric conflicts, i.e. the behaviour of contestants depends on the roles they find themselves in, and any two contestants always are in different role,', (Maynard Smith and Parker, 1976; Selten, 1!)78). The roles then define subpopulations like A j and A2, and adopting a strategy of the repertoire Ii as a member of AI means doing so in one role, whereas doing so in a second role means being a member of A::. Usually, in asymmetric contests the disjoint repertoires contain the same number of strategies and there are corresponding strategies in either repertoire that stana for the s a m e behaviour but displayed in different roles (see example below). Hence we have from theorem 2 that no mixed ESS can exist in asymmetric conflicts. It might be the case, however, that in populations as considered here a stable equilibrium hoMs, where the members of either subpopulation use just one strategy. Hence we might ask, whether a pair Ji , J[ of pure strategies can be evolutionari~y -tO . stable in case Alx and A2~ are zero-matrices. It can be shown that Ji and Jj form an evolutmnarlly stable pair o~ strategies, ff and only if •
.
.
0
.
O.
for j ¢ J0
an * .Jo, io ~ an ~J, io
As a consequence, we have the assertion that, if Ji together with some Jf forms an evolutlonarily stable pair of strategms, then no other pair containing Jio or JJ0 can be an ESS. Hence the number of different pairs that are ESS cannot exceed min { n , m ~for a given matrix• It can be easily checked that by this condition no such pair exists for the matrix given in the "Battle of Sexes", thus neither mixed nor pairs of pure strategies can be ESS's in this case. Finally, we shall apply our results to an asymmetric conflict described and analysed by Maynard Smith and Parker, from which their conclusions follow very easily• Contesting for a value V > 0, opponents may adopt strategy e ("escalate": fight until injured or opponent retreats) or d ("display": retreat, if attacked). Losing the contest when escalating will result in a negative value - D , e.g. from injury. Asymmetry, e.g. in fighting ability, has been introduced concerning the changes to win on escalation: a contestant might be "larger", i.e. wins with probability x > 0.5, or "smaller", i.e. wins with 1 - x. Concerning d, chances of winning V are 0.5, if both opponents display. In our terms, we have subpopulations A~ and A2 of "large" and "small" individuals, respectively. Strategies e and d for a member of AI define 1~ = (J~,J2 }, whereas for a member of A2 they define I2 = {all,d2 }, so I~ and I2 are formally disjoint repertoires. The pay-offs defined by Maynard Smith and Parker give rise to the following matrix in our notation: •
O
.
sO
Vx0
A =
V(1 - x ) - D x
.
D(10
V
.
_
o
v/2
0
x)
V" V/'
268 O f course, no m i x e d ESS can exist, as t h e o r e m 2 states. C o n c e r n i n g e v o l u t i o n a r i l y stable pairs of strategies, t h e c o n d i t i o n (4) shows t h a t ( J , , J ' , ) (escalate, if larger; escalate, if smaller) is the o n l y ESS pair, ifxD < (1 - x)V. If xD > (1 - x)V, t h e n (J,,J'2) (escalate, if larger; display, if smaller) is e v o l u t i o n a r i l y stable, a n d if, in a d d i t i o n , (1 - x)D > xV, t h e n even b o t h (J, J 2 ) and (J2,J~) (display, if larger; escalate, if smaller) are e v o l u t i o n a r i l y stable. This m e a n s t h a t if the e x p e c t e d loss, D(1 x), f o r the larger individuals on escalat i o n is larger t h a n their e x p e c t e d gain, x V, t h e r e are t w o stable situations in this p o p u l a t i o n : larger c o n t e s t a n t s escalate and smaller ones display, or larger individuals d o n o t escalate, while smaller o n e s do. F r o m t h e d y n a m i c a l p o i n t o f view, it d e p e n d s on t h e initial c o n d i t i o n s , w h i c h o f these situations will be realized.
References Dawkins, R., 1976, The Selfish Gene, Oxford. Haigh, J., 1975, Game theory and evolution, Adv. Appl. Probab. 7, 8--11. Lewontin, R.C., 1958, A general method for investigating the equilibrium of gene frequency in a population. Genetics 43,419--434. Maynard Smith, J., 1974, The theory of games and the evolution of animal conflicts. J. Theor. Biol. 47,209--221. Maynard Smith, J. and G.A. Parker, 1976, The logic of asymmetric contests, Anim. Behav. 24, 159-175. Maynard Smith, J. and G.R. Price, 1973, The logic of animal conflicts, Nature 246, 15--18. Selten, R., 1978, A Note on Evolutionarily Stable Strategies in Asymmetric Animal Conflicts, Working Papers Inst. Math. Econ. Univ. Bielefeld, Nr. 73. Sperlich, D., 1973, Populationsgenetik, Stuttgart.