J. Social Biol. Struct. 1985 8,255 277
Towards a dynamicsof socialbehaviour:
Strategicand geneticmodels for the evolution
of animal conflicts
Peter Schuster* and Karl Sigmundl
rlnstitut
für Theoretische Chemie uncl Strahlenchemie undllnstitut für Mathematik
tler Universität Wien, A-1090 Wien and IIASA, Laxenburg' Austria
persistenceof behavioural traits in animal societiesrather than optimality of reproductive successis described in terms of the game-theoreticnotion of evolutionarily stable
strategy.Game dynamics provides a suitable framework to accomodate for the dynamic
urp..i, of permanenceand uninvadability, to model the evolution of phenotypes with
frequency äependent fitness and to relate the strategic models of sociobiology to the
mechanismsol inheritance in population genetics.
Introduction
investigations of animal populations exhibiting
provided
numerous
Classical ethology
'altruistic' behaviour. In this non-antropomorphic context,
on
based
social phenomena
an act performed by an amimal is called altruistic if it increasesthe fitnessof some other
animul at the cost of its own. Such behaviour was often explained by notions such as
'group selection' and 'benefit of the species'(for example, seeWynne-Edwards, 1962)'
But while there is no logical inconsistency in viewing groups or speciesas units of
selection.it seemswell establishedthat Darwinian evolution acts much more commonly
through selection of the level of individual organisms or eventually genes'The explanation of many altruistic traits in terms of individual selectionhas, therefore, to be viewed
as a major successof sociobiology.
Roughly speaking, there are three approacheswhich proved to be fruitful. They may
be designatedas genetic.economic and strategic,and attributed to Hamilton (1964)'
Trivers (1972) and Maynard Smith (1974), respectively,although the principles go back
to Haldane and Fisher, and arguably to Darwin himself.
(l) The geneticexplanation is basedon the notion of kin selection.A genecomplex
programming altruistic acts which benefit relativesmay well spread,sinceit occurs, with
a certain probability, within the relatives whose reproductive successis increased.
(2) The economic explanation relies on the notion of reciprocal altruism. If the
altruistic act is likely to be returned at some later occasion, then the behaviour may
become established,especiallyso if there is individual recognition between members of
the population preventing'cheaters'.
+23 $03.00/0
0l40 l750/85/030255
.O 1985 Academic Press Inc. (London) Limited
256
P. St'huster& K. Sismund
(3) The strategicexplanation,finally, usesthe panoply of game theory to show how
the fitnessof an individual may dependon what the other onesare doing. An act which
in certain situationswould be called altruistic neednot, in other situations,decreasethe
reproductivesuccessof the organism performing it.
In this paper.we shall deal exclusivelywith the third approachstressingits dynamical
aspects.It must, however, be stated at the outset that the game-dynamicapproach
described here constitutes only a fraction of the game-theoretical analysis of animal
behaviour. Our discussion is accordingly biased. The theory of games covers many
applicationsin biological evolution. Some of them do not easily fit into the mould of
ordinary differentialequations.For a balancedsurvey of thesetopics. we may refer to
a r e c e n tb o o k b y M a y n a r d S m i t h ( 1 9 8 2 ) .
within the static context of
Our paper consistsof two parts. The first one discusses,
game theory. the crucial notion of evolutionarilystablestrategy(ESS),which centreson
aspectsof permanencerather than optimality. The secondpart usesgame dynamicsto
obtain broader notions of permanence,to model the evolution of behaviouralstrategies
and to relate them to geneticmechanisms.
Beforewe start our discussionof strategicmodelsin sociobiologywe haveto comment
on the evolutionary aspectsof this approach.The actual mathematicalmodels used in
sociobiologyas well as those of population geneticsdiscussthe spreadingof mutant
genesin populations.Thecausesof mutations are consideredas externalfactorsand not
as part of the system under consideratiom.A proper description of the evolutionary
process,however.is completeonly if it includesthe mechanismof mutant formation. At
present. such complete theories of evolution exist for polynucleotidesin laboratory
systems(for example,seeEigen & Schuster,1979;Biebricheret al.l982; Küppers, 1983)
and eventually for simple viruses(Weissmann, 1914).The replication of bacteria follows
a much more complicatedmechanism.Already ten enzymesor more are required for
DNA copying. Although not all the mechanisticdetails of local mutations and DNA
rearrangementsin bacteriaare known yet, the basic featuresof bacterialevolution are
establishedin principle and we can expect a complete molecular theory to become
availablein the not too distant future.
In caseof higher. multicellularorganismswe encounternew sourcesof complications
which obscure the relation betweenmutations and their manifestationin phenotypic
properties.The major problem concernsthe unfolding of the genotype:higher organisms
obtain final form and shapethrough morphogenesis.This is a complicateddynamical
processwhich is not completelyunderstoodat presentand to which the genescontribute
markers and other regulatory signalsfor intercellularcommunication apart from the
ordinary duties in cell metabolismand cell division. Any completetheory of evolution
of higher organismshence has to deal explicitely with the influenceof mutations on
morphogenesis,which is far outside the bounds of present possibilities.Behavioural
traits are exceedinglycomplex phenotypic featureswhich apparently cannot be traced
'mutant genes'leadingto changesin behaviourhas
down to singlegenes.The notion of
to be understood as a metaphor. Mutations actually operate within complicatednetworks of gene actions which are convertedinto the properties of the phenotype by a
highly complex transformation during morphogenesis.
Evolutionarily stable strategies
'evolutionarily stable strategy'was introduced by Maynard Smith and Price
The term
(1913) to describe certain seneticallv determined traits of animal behaviour which are
Dynamics of sociul behaYiour
257
evolutionary stability in
uninvadableby mutants. The basic idea was to describesuch
was subsequentlyused
termsof the mathematicframework of gametheory. This method
betweenspecies'and can
to analysea wide varietyof biologicalconflicts.both within and
be righily consideredto be a cornerstoneof theoreticalbiology.
Game theorYand bblog.v
first publishedin
Ever sincevon Neumann and Morgenstern's(1953)classicaltreatise
of all kinds of
modelling
the
to
applied
been
have
1944.the methods of game theory
and plants'
animals
within
conflicts
to
applications
The
conflicts in human societies.
using game
forerunners
few
a
were
years.
There
ten
last
however. date back to the
one describedhere'
theoreticnotions in biology. but in a context quite differentfrom the
'games'were neglected
biological
that
astonishing
With the benefitof hindsilht it seems
there are at least
for so long in favour of political. military or economicones. Indeed.
'natural'
their
than
analyze
to
difficult
more
two reasonswhich make human conflicts
counterpartsin biologY:
money,
(l ) The payoff in human systemsis often doubtful. sinceit is hard to evaluate
evolutionarily
In
the
scale.
utility
a
single
on
social piestige,health and other factors
quantity. Although
weightedcontestsin biology, Darwinian fitnessis the ony relevant
and empirical
analysis
theoretical
for
this notron presents considerabledi{iculties
success'
reproductive
of
estimate
a
scalar
least,
measurementit providesin principle,at
at
'rationality
game
theorists
theoretical
and
empirical
axiom' frequently sets
(2) The
assumption'
an
such
by
is
unhindered
odds. The investigation of animal behaviour
the simplest
Biologicalconflictstend to be fairly straightforward.comparedwith all but
lunacy' etc'
fraud.
for
conspiracy,
possibilities
The
h,.,manconflictsof any real interest.
are greatly reduced
strategyIn
It is interestingto note. however,that the notion of evolutionarily stable
noncooperative
for
strategy
equilibrlum
Nash
biologicalcontestsis quite closeto that of
'Characterizationsof evolutionary
gamesbetweenrational players (cf. the section on
i s i n t e r p r e t e da s
! , u b l . r , . u , " g i e s( E S S ) ' b e l o w ) .I n P a r k e r & H a m m e r s t e i n( 1 9 8 4 ) ,t h i s
'quasi-ratioÄlity' of selection.In our view, the convergenceof the two conceptsis rather
'blind selection'and 'rational decision' lead to adaptative
due to the fäct that both
solutions.After all, rational behaviour itself is a product of selectron.
Strategies and Phenot vPes
conflictsentailsa shift
The shift in the applicationof gametheory from human to animal
'strategy'and 'payoff. 'Payoff, as we have
in the meaning of the two basic notions of
or-in subtler
said. now correspondsto Darwinian fitness,i.e. the number of offspring
generation'
next
the
to
[For an
genes
transmitted
situations-the number of copiesof
'Strategy'.in the
(
l9S2)']
Dawkins
to
refer
we
elaborateanalysisof the notion of fitness
of moves' This
context of chess.of war games,suggestsa nicely calculatedsequence
trait of fighting
innate
an
of
notion
the
by
replaced
aspectof plotting and schemingwas
viewpoint was
this
that
realized
was
soon
It
1973).
biauioui (Maynard Smith & Price.
of any clash
modelling
the
in
but
encounters.
useful.not only in the analysisof aggressive
of the
length
the
concerning
for
example,
ol interests:in the parent offspring conflict,
parental
in
share
respective
the
about
weaning period. or in the male female conflict
programmed way
investment ('I'flvers. 1g72).A strategy, thus, became a genetically
a
small fraction of
only
form
contests
But
such
of behaviour in parrwise contests.
'subgames'of the strugglefor life. Differencesin resourceallocation. size of the litter'
way. In such cases'
disoersalrate etc.. affecittre fitnessof an individual in quite another
258
P. Schuster& K. Sigmund.
the successis determined,not through playing againsta particular opponent,or a series
of opponents,but through 'playing the field' (seeMaynard Smith, 1982).
A strategy, then, is any behavioral phenotype, or may be any phenotype at all. Indeed,
game theory has been successfullyapplied to desertshrubs in order to discussthe relative
merits of catchingrain water by spreadinghoizontal roots near the surface,or of taping
ground water by sendinga vertical root deep below: the successobviously dependson
what the neighboursare doing (seeRiechert& Hammerstein,1983).From thereit is only
a small step to the protectivecolouring of moths or the familiar featuresof Mendelian
peas.Thus, any phenotypemay be viewed as a 'strategy',as long as we wish Io analyze
it by game theoreticalmethods or, to put it less subjectively,as long as its fitnessis
frequencydependent,(which meansthat it dependson the distribution of phenotypesin
the population considered).
On the other hand not every strategyis a phenotype.In particular, a 'mixed strategy'
consisting in playing different strategieswith preassignedprobabilities-need not be
realizedas phenotype.In order to avoid confusion,we shall thereforekeep the distinction betweenphenotype and strategy, even if this implies some modifications in the
standard terminology of ESS theory.
Optimum phenotl'pesand genetic constraints
It is tempting.and often useful.to view selectionas an optimization process.In elementary situations,as for examplewith asexuallyand independentlyreplicatingunits, this is
indeed the case (seethe section on 'Asexual inheritance:optimization and the gamedynamical equation' below).
'optimum',
The term
of course,has meaningonly with respectto a given environment.
There is no reason to expect that under continuously changing conditions, selection
would lead to an optimum responseto the instantaneoussituation. Furthermore, since
the environment includesall other members of the population, adaptation to an environment will change this environment. It is easy to exhibit corresponding population
geneticalmodelswherethe frequencydependentfitnesswill not, on the average!increase.
This is actually the main theme of this paper.
But evenifeach phenotypehad constantfitness,one could not expect,in general,that
the phenotype with highest fitness would prevail. In particular, if this optimum
phenotypecorrespondsto a heterozygousgenotype,as may easily be the case,then at
leastone half of its offspring will be suboptimum. It is true that according to Fisher's
fundamental theorem. the average fitness will never decrease.and even increase,
provided the composition of the gene pool changesat all. However. the maximum
averagefitnessof the population may be much lower than the fitnessof the optimum
phenotype.Furthermore,a population may be 'trapped' in a statewherea local. but not
global maximum is attained. Moreover. the fundamental theorem rests upon the
assumptionthat fitnesscorrespondsto the probability of survival from zygote to adult
stageand that this probability dependson a singlegeneticlocus. For many-loci models
involving recombination, and for fecundity models where reproductive successis a
function of the mating pair, the averagefitnesswill not increase,in general(for example,
seeEwens. 1979).
Sel.fi
n t t,ra.s
t ttnI opt inti:a t ion
Apart from geneticconstraints,there may also be strategicobstaclesto an optimization
of the'good of the species',i.e. of the mean fitnessof a population. Selectionactsthrough
differential reproductive successof individuals, which may be quite opposed to the
Dynamics of social behaviour
259
welfare of the group. Even in human gamesallowing forethought and deliberation, the
self-interestof two players may in some caseslead without failure to an outcome which
is disastrousfor both.
'prisoners' dilemma'' Two
This is best shown by the well known paradigm of the
prisonerskept in separatecellsare askedto confessa joint crime. If both confess,both
will remain in jail for sevenyears. If only one confesses,he will instantly be freed, while
both will be detained
the other one getssentencedto ten years.Ifnone ofthem confesses,
Each prisoner,
obtained.
not
be
it
will
but
is
optimum
latter
solution
year.
This
for one
indeed. will confess: it is the better option, no matter what the other one is going to
decide.As a result, both will have to spend sevenyears in jail. [For a recent application
of the prisoners' dilemma to the problem of cooperative behaviour in biology see
Axelrod and Hamilton (1981).1
The prisoners' dilemma is a game between two opponents. Similar results hold for
gameswhere an individual is'playing the field'. The best known example,in this respect,
'tragedy of the commons', where the common meadow is ruined by
is Harding's
overexploitation, each villager trying to increasehis payoff. [We refer to Masters (1982)
'prisoners'
who bases his sociobiologicaldiscussionof political institutions on the
'tragedy of the commons'.]
dilemma' and the
Ev olut ionarily stable p henotypes
If adaption leads to an equilibrium at all, it will be characterizedby stability rather than
optimality. An evolutionarily stable phenotype, in Maynard Smith's definition, is a
phenotype with the property that if all members of the population share it, no mutant
phenotype could invade the population under the influence of natural selection.
One possibleway of formalizing this is the following one (Maynard Smith, 1982).Let
us denote by W(l,"r) the fitness of an individual /-phenotype in a population of
J-phenotypes and by pl + (1 - p)J the mixed population where p is the frequency of
.I-phenotypesand 1 - p that of ,I-phenotypes.A population of /-phenotypes will be
evolutionarily stable if, whenevera small amount of deviant "I-phenotypesis introduced,
the old phenotype 1 fares better than the newcomers ,I. This means that for all phenotypes J * I,
w(J,il
+ (l - {11 < w(1, e"I* (l - s)/),
(1)
for all e > 0 which are sufficientlysmall.
It must be mentioned that this definition has to be modified if the population is small.
In such a case,a deviant individual would change the composition of the population by
a notable amount, and not just by some small e. This situation has been analysedby
Riley (1979).In our context,we shallalwaysassumelargepopulation sizes,thus avoiding
this problem and, moreover, the effects of sampling errors.
By letting e - 0 we obtain from inequality (l) that
W(J,I) < W(1,D,
Q)
for all .I, which means that no individual fares better against a population of /-phenotypes, than the 7-phenotypeitself. The converseis not true in general:(2) does not imply
(l)
EvolutionarilY st able states
Now we restrict our attention to those biological gameswhich satisfy the following two
assumDtions:
260
P. Schuster& K. Sipmund
(l) They consistin one or repeatedpairwisecontests,rather than in'playing the field'.
This implies that the payoff lI/'is an affine function of its secondvariable. i.e. that for
any three phenotypesE, I and./ and any p e (0. l):
w(E,pJ + (l - p)1) :
pw(E, J) + (l - p)w(E. I)
(3)
Indeed the probability that an individual f-phenotype matches up against a Jphenotype,in the mixed populationpJ + (l * p)1. is just p; and its payoff. in this case.
is W(E. "/). Here. we have tacitly assumedrandom matching of opponents. In some
situations----e
.g. gamesagainstrelatives-this will be not valid. [For a discussionof this
casewe refer to Hines and Maynard Smith (1979).]
(2) There are only finitely many phenotypes E, , . . . , E, This assumption may look
innocuous. But there are situations where it is quite misleading,even as an approxi'war
mation. As an example,we mention the so called
of attrition', a contestwhere the
prize goesto the player who investsmore time-or energy-than his opponent.The set
of strategiesconsistsof a continuum of waiting times and its discretizationmay be as
ill-advisedas the replacement,say, of the bell shapedprobability density of body sizes
b y a f i n i t es u b d i v i s i o n .
The loss of generalityentailedby thesetwo assumptionsis compensatedin a way by
an increasein mathematicalconvenience.We may now describethe game in normal
form, with the help of a payoff matrix.
By ,t, we denote the frequency of the phenotype E, in the population. The point
( , t ,. . . . , n , ) d e s c r i b e s t h e s t a t e o f t h e p o p u l a t i o n . S i n c e t h e r , s a r e f r e q u e n c i e s , x l i e s
x:
1 l . B y o , ,w e d e n o t et h e
o n t h e u n i t s i m p l e xS , : { x : ( , t r , . . . , , r , ) : - r ): , 0 , L r , :
averagepayoff, i. e. the expectedchangein fitness,for an d-individual matched against
an,{-opponent. The n x n-maftix A : (a,,) is called the payoffmatrix. The average
payoff for the phenotypesd is
(Ax),
:
a,rt, -l ... I
t t i n X n:
I
(4)
u,*.t*.
k
If x and y describetwo populations,then the averagepayoff for a y-member opposed
to an x-member is
.y.Ar:,r1(Ax),
+...+,)',(Ax),
ll.r.o*,.t,.
(5)
and in particular, the averagepayoff for contestswithin the x-population is
II
\pes1X1.
(6)
There is an alternativeinterpretationof distributionsof strategies.In game theory, E,
'pure
to A, are viewedas
strategies'.They correspondto the cornerse,of the unit simplex
we
By
e,
denote
the
lth
unit vector: all componentsare 0 exceptthe ith one which is
3,.
L We may interpret now a point x e S, as a 'mixed strategy'which consistsin playing
A , w i t h p r o b a b i l i t y . r ,( i : 1 , . . . , n ) . T h e e x p r e s s i o yn' A x i s t h e p a y o f f o f a y - s t r a t e g i s t
playing againstan x-strategist.In the sequel,we shall make use of both interpretations.
Points of ,S,will correspond,accordingto the context,either to strategiesor to statesof
the population.
Under our assumptions,the phenotyped is evolutionarily stableilT
e , ' A , ( t e ,+ ( l - 6 ) e i )< e , ' A ( t : e , * ( l for all .7 * i and all e > 0 which are sufficiently small.
e)e,),
(7)
261
Dvnamicsof social behaviour
One may extendthis definition,just as one extendsthe notion of a pure strategyto that
all
of a mixed one. A strategy (or state)p € S, is evolutionarily stable (or an ESS) iff for
p
.
J€S,withy *
(8)
- e)p) < p' A(ty + (l - e)p),
.
J A(sJ + (l
for all e > 0 which are sufficientlysmall. i.e smaller than some appropriate ä(y)'
Characterizationsof evolutionarill'-stable strategies ( ESS)
E q u a t i o n( 8 ) m a y b e w r i t t e n a s
( l - e ) ( p' A . p - y ' ^ p ) + E ( p ' L v - v ' A v ) > O '
(9)
This implies that p is an ESS iff the following two conditions are satisfied:
(l) equilibrium condition
(l 0 )
f o r a l l . 1€' S , ;
y' ^p < p'Ap
(2) stability condition
if
- y€ S , , y + p
y' ^p
and
:
p' Ap,
then
!' A! < p' Ly'
(l l)
This is the original definition from Maynard Smith (1914).The equilibrium condition
statesthat the strategypis a bestreply againstitself.It correspondsto the notion of Nash
e q u i l i b r i u mf a m i l i a r t o g a m e t h e o r i s t s .I n D a w k i n ' s ( 1 9 8 2 )t e r m s ' a n E S S " ' c a n b e
crudely encapsulatedas a strategythat is successfulwhen competing with copiesof itsell'
This property alone. however,does not guaranteeuninvadability,sinceit permits that
another itrategyy is an alternativebest reply top. In facts,if p is'properly mixed', i'e'
if p, > 0 for all i, then any J € S, is a best reply. The stability condition statesthat in
such a case,p fares better, against y, than y against itself. The same argument leads to
the following characterization: a phenotype d is evolutionarily stable iff
( I ) equilibrium condition
(r2)
ai12 a1,i for all A;
(2) stability condition
if
eii
ari
for
k + i,
then
a1, ) app'
(13)
A third equivalent definition of ESS,probably the most useful for computations' is the
following one (Hofbauer et al.,1979): p e S, is an ESS iff
(14)
p' Ax > x'Ax
for all x + p in some neighbourhood U of p in S,.
There are examplesof gamesadmitting no ESS, or severalones.If there is an ESS in
(14)
the interior of S,, however,then it is the only ESS.Indeed,in such a caseexpression
must be valid for all x # P.
The
'hawk-dove' game and mixed strategies
'hawk dove' game, a model devisedby Maynard Smith
Let us now consider the famous
price
(1973)to explain the prevalenceof ritual fighting in innerspecificconflicts'We
and
do not attempt to repeat here the considerable amount of work done on this model
(Maynard Smith, 1912; 1914; 1982;Zeeman, 1979:Schusteret al., l98l), but describeit
in its most rudimentary form in order to elaborate the distinction between mixed
strategiesand mixed populations.
262
P. Schuster& K. Sismund
until flight of the
We shall assumethat thereare two phenotypes.E, ('hawk') escalates
opponent or injury decidesthe outcome. E ('dove') is only preparedto fight in a ritual
way, without risking or inflicting injury. The prize of the contestcorrespondsto a gain
G in fitness.while an injury reducesfitnessby I C I*. The effort expandedin a ritual contest
costs I,El.A natural and self-explantoryorder relation among theseparametersis
G>O>E>C,
'hawk'has the samechanceto win or to get injured; a, is
Then a,, is (G + C)12----each
'dove' retreats
(G + E)12 sinceonly one of the 'doves'can win, a,, is G and a^ is 0-the
'hawk'.
when it meets a
Furthermore we restrict our discussionto the caselCl > G. Thus
G1
le+
A:l|
L''g#)
'doves' can be invaded
Clearly. no phenotypeis evolutionarily stable.A population of
'hawks' and vice versa.
by
'hawks'
and
The unique ESS of the population p : (pt, pr) is a mixture of p'
'doves',with
Pz:1-P'
E-G
v1
(15)
E+C
This follows from the fact that
(l 6 )
is positive for all x :
average fitness
(xr, xr) with xr * pr. Note that this state is not optimal. The
x'Ax
!(C 1 E
2Ex,+ :E + C;x,)
is largestfor x, : El(C + -E).A numerical exampleis shown in Fig. l.
What happensif we introduce a third phenotypeär, which plays the mixed strategy
correspondingto the ESS: p, et + p2e21(For simplicity, we set -E : 0). The payoff
matrix now is
G+C
A
('
9
G(G+C)
2c
G(G+c)
22C
(1 7 )
G(G + C) G(C - G) G(G + C)
2C
2C
2C
Conditions (12) and (13) show that E, is evolutionarily stable and cannot be
replacedby neither E1 nor E, phenotype. The corresponding state e3,however, is not an
E S S ,a n d n e i t h e ri s p , e 1 * p r e . , i n t h i s e x t e n d e dg a m e :c o n d i t i o n ( 1 1 ) i s n o t s a t i s f i e d
* We measureall parameters on the same value scale.Hence, gains like G are positive and losseslike C or
Ä negative quantities.
Dynamics of social behavtour
263
I
I
I
\l
i\
t\
!
t.
t\
- - - r - - - - - )\
---t--r\
lö
\
I
ro
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
o
Ol
o
o
-l
ä-z
-3
-4
-5
ol
i
I
I
-mox'
\r 8I '
Zt
g'
o.s
ESS :
'8' 8',
xl
' C : 1 4 ' E : 1 ' ä r a n d A 2a r e t h el r e q u e n c ' v
o vgea' m eG : 4,doves,,
',zä,rrrlrnt
F
i g .t . T h e ' h a x ' k - d:nÄ.tr'
pa1,offfor the total
v.hite@ is the average
p;
and
pa1,ofls
'
'dotes'
'hawks'
:
x1
|
x'
and
oJ'11
p'ipilation consisting
'Someexamples').Indeed,e3is not protectedagainstthe mixture and
(cf. also the section
vice versa.
'mixed' phenotypes corresponding to mixed strategies
we may consider other
(i'e' "x, I p')' Such pheno-rrer + -)r"e,which are not the ESS in the hawk-dove game
be invaded,either by'hawks"
types are nät evolutionarily stable:the population could
.doves'.Furthermore, such'mixed'strategiescould not invade E| All this is easy
or by
to check.
with severalphenotypes
continuing along theselines, one could analyze populations
'hawk' and the 'dove' strategies'There is no
corresponding to different mixtures of the
reason,however,tostopatafinitenumberofsuchphenotypes.Inthemostnaturalset
phenotypes competing with each
up, all possible mixed sirategieswould correspond to
shall not pursue this
other, but we pref'erto stick to a finite set of phenotypes here and
(1983)'
question any further, refering instead,to Zeeman,(1981)and Akin'
Howplausible,anyway,arephenotypescorrespondingtomixedstrategies?Thisisa
severalchapters' His condelicateproblem to which Maynard Smith (1982) devotes
'playing the field" e'g'
when
clusion is roughly that such phenotypesare most important
.sex ratio' game, but probably of rather minor importance in the pairwise contests
in the
we are considering here.
which are evolutionOn the other hand, does one find mixed statesof the population
are of course extremely
arily stable?Again, this is not an easy matter. Mixed states
of severalphenotypesfor
common, but we are looking for strategicequilibria, consisting
is obviously difficult to
which the fiequency depen-dentfitness values are equal. This
in Hamilton's, (1979)
verify. The saf-estempirical evidence,at present,seemsto be found
investigation of figwasPs.
264
P. Schtt:;ter& K. Sigmund
A s t'tnnteI r t( con t e.\ts
which are symmetric. in the sense that both
contests
have
considered
far
we
So
are asymmetrichowever:within the same
conflicts
play
role.
Many
same
the
opponents
betweenyoung and old. between
females.
males
and
between
have
conflicts
we
species
The
methods apply equally well to
so
on.
habitat.
and
of
a
intruders
owners and
preys.
If we assumeagain that the
predators
and
between
e.g.
interspecificconflicts.
we are led to bimatrix games.
phenotypes
finite.
numbers
of
pairwise
the
and
contestsare
T h u s l e t Ä r . . . . . d , d e n o t et h e p h e n o t y p e so f p o p u l a t i o nX . a n d F , . . . F , , t h o s eo f
population )'. By .v,we denotethe frequencyolE-,.and by l', that of {. Hence.x is a point
i n S , ,a n c l J ' i n S , , . I f a n d - i n d i v i d u a l i s m a t c h c da g a i n s ta n { - i n d i v i d u a l . t h e p a y o f f w i l l
be a,, for the former and h,, for the latter. Thus the game is describedby two payoff
matricesA : (a,,)and B : (ä,i).If the population X is in the state.r and the population
I in the state -y. then x'Ay and -l'' Bx are the respectiveaverage payoffs. If the
phenotypeä, is stablein contestsagainstthe population J, then it must do as well as all
c o m p e t i n gp h e n o t y p e sT. h u s e , ' A y ) e r ' A l f o r k : 1 . . . - . n . B u t w h a t i f e q u a l t i t y
holds? There is nothing. then. to prevent Ä^ from invading. Thus, in contrast to the
'best replys'. This leads to the lollowing
symrnetric case. we cannot allow several
d e f i n i t i o n .A p a i r o f p h e n o t y p e s( 8 . { ) i s s a i d t o b e e v o l u t i o n a r i l ys t a b l ei f f
e k i< l i t
for
k+i
(18)
hr,<h,,
for
k+i
(le)
and
Similarly, a pair of states(or strategies)(p. g) with p e $, and q € S-, is said to be
e v o l u t i o n a r l i l ys t a b l ei f
-r€S,.
x*P
p'Aq>x'Aq
forall
q'Bp > y'Bp
f o r a r l l J ' € , S , , , .) ' + q
(20)
and
(2l)
It is easyto seethat such a pair (p. q) must consistof pure strategies.Thus. in contrast
to the symmetriccase.a mixed strategycan neverbe evolutionarilystable.This has been
shown for a considerablywider class of asymmetric games by Selten (1980). It also
reflectson symmetric contestssince it is often quite possiblethat a small, seemingly
irrelevantdifferencecan break the symmetry betweenthe opponentsand transform the
o n g i n a l l ys y m m e t r i cc o n t e s ti n t o a n a s y m m e t r i co n e ( M a y n a r d S m i t h , 1 9 7 6 ;H a m m e r stein, 1979).This could explainwhy mixed strategiesare rather rare in pairwisecontests.
On the other hand. if the population X interacts,not only with the other population X,
but also with itself, then mixed ESSs become possibleagain (Taylor. 1919: Schuster
e t u l . . l 9 8 l a ,h )
Ita.shequilibria untl the
'cot'nessphilanderittg' game
A pair of strategies(p, q) (with p e S,, { € .S-) is called a Nash equilibrium pair if
p. Aq > x' ^q.
Vx e S,,
q'Bp > J' Bp.
Vl e S,,,.
(22)
Nash equilibria play an import role in classicalgame theory, sincefor rational players
there is no reasonto depart from the strategiesp and q. as long as their opponent sticks
of social behaviour
Dy,'nctmics
265
games.Nash equilibria are not
to it. we shall presentlysee,however.that in biological
invasion proof.
'coyness-philandering'game by
Indeed,let us consideranother famous example.the
an offspring increasesthe
Dawkins (1976).Let us supposethat the successfulraising of
will be entirely borne by the
fitnessof both parentsUy C. fne parental investmentlCl
strategyto counter
femaleif the male deserts.otherwise,it is sharedequally.The female
which costs
period.
engagement
long
a
upon
insistence
the
male desertionls'coyness',i.e.
and have
scale
value
same
the
on
parameters
all
measure
we
]El to both partners.Again
population X' namely E'
G > 0: C. E < 0. There are two phenotypesin the male
population I''
(.philandering')and ä, ('faithful') and two phenotypesin the female
namely F, ('coy') and F ('fast'). The payoff matricesare
o
I
A
t-o c.i.4
GI
fo*f*u".;l'B:1".. o*t_l "o'
population is an ESS
No pair of phenotypesis evolutionarilystable.and no stateof the
pair
of mixed strategles
unique
a
(we have or.,lyto check the pure states).There exists
p and q in Nash equilibrium, given bY
Pt
EC
E+C+G.
4r
ztr+cl
( 2s)
<lcl <2(G+ E)'
(with p:(p,,Pr)€S:,
Q : t 4 r , q r ) e S : ) p r o v i d e dl E l < G
say. the amount of
itris equilitrrium is not stabte.however. If a fluctuation decreases'
phenotypestill
philanäeringmales,then the payoff for the maleswill not change:each
One cannot expect
iras the samepayoff, which dependsonly on the femalepopulation.
for the femalepopulation will
the frequencyof philanderersto return to p,. The payoff
'fast' femalesgain more than 'coy' ones,sincetheir risk of being
actually increase:but
'fast' femalesincreasesthat the male
lt is only when the amount of
deserteddecreases.
.philanderers'gain more than the .faithful.
payoffs change.Again, they increase:but
'philanderers':hencemore 'coy' females;hence,less'philanderers"
nlut.r; hence.more
of game theory is
and so on. This looks like an oscillatingsystem.The static approach
no longer sulicient to deal with this situation'
Game dYnamics
present dynamical
Evolution is dynamrc, and hence any evolutionary model must
aspectsmay well
such
however.
persistence,
and
aspects.For the study of equilibria
a straight appliIndeed,
approach.
game
theoretical
remain implicit, as they do in the
may be quite
genetics
popullation
in
used
systems
cation of the inventory of dynamical
phenotyptc
is
essentially
theory
ESS
particular.
In
off the point, in ...tuin situations.
part
of sociomajor
a
that
quite
interesting
rather than genotypic.It is, incidentally,
into the
humanity'
including
everything,
biology-a sciencewhich reputedly delivers
genetlc
than
rather
strategic
a
taking
from
cluchesof selfishgenes-{erives its interest
point of view.
about the
The presentstateof knowledgedoes not allow anything definiteto be said
to
arbitrarily
trait
a
such
tie
to
and
trait;
giv-n
behavioural
geneticmechanismbehind a
could
machinery
Mendelian
The
things.
confuse
well
iome hypothetical genotyp. n1uy
P. St'huster& K. Sismund
regulation at the strategiclevel which one wants to
override the frequency-dependent
investigate.To avoid such interference,it will thereforebe quite appropriate to assume
asexualreproductionat leastas a first approximation. Most game-theoreticalmodels in
biology concernsexualpopulations,of course;but this may eventuallylead to irrelevant
complications based not on a necessaryconstraint but on speculationsabout the
inheritanceof behaviour.
Aserual inheritance: optinti:ation and the game-dlnamical equation
Let us considera population with r phenotypes.The point x(t) e S, denotesthe stateof
'like
begetslike'. The better the
the population at time t. Asexual reproduction means
phenotype{ is adapted.the higher its rate of relative increase
I
I d't'
ri
-x, dl
Ifthe fitnessofthe phenotypeE, is given by a constant L,, then the averagefitnessofa
p o p u l a t i o n i n s t a t e x i s g i v e n b y < D ( x))t:x t + . . . + ) . , , x , , . I I i s n a t u rt a
o la s s u m e t h a t
the rate of relative increase of d is given by the difference between the fitness ), of E,
and the average fitness <D(x).Thus the equation
(26)
restricted to the invariant set S. will describe the evolution of the distribution of
phenotypesin the population.
This equation plays an important role in the theory of Eigen and Schuster(1979)on
prebiotic evolution. The population, in this case,consistsof n types of selfreplicating
macromolecules,
RNA or DNA, in a flow reactor.It can easilybe shown that the average
'fitness', i.e. the mean reproduction rate. increasesmonotonically. Indeed,
the time
'fitness'
in the molecular population. The
derivative of O is just the variance of the
molecular types with lessthan averagefitnesswill be eliminated, and the averagefitness
thereby increased.In the limit, only the moleculeswith the highestfitnesswill remain.
In populationswith asexualreplicationand frequencyindependentfitness,selectionis a
global optimization process.
'error-free'
The aboveresultsare valid for
replication.In more detailedstudies(Eigen,
l 9 7 l ; T h o m p s o n & M c B r i d e , 1 9 7 4 ;J o n e s e t a l . , 1 9 7 6 ;S
' w e t i n aa n d S c h u s t e r ,1 9 8 2 )
mutations were taken into account.In this case.the averagefitness<Dis not, in general,
optimized.However, the statereachesa unique stableequilibrium, and this implies that
there existssome (Ljapunov) function which increasesmonotonically. Actually, Jones
(1978)displayeda function of this type which is closely related to the averagefitness.
Let us now return to the 'error-free'case,but assumethat the fitnessof phenotyped
is frequencydependent,i.e. a function of x. More precisely,let us assumethat it is given
'Evolutionary
by the game theoretical payoff (Ar), as derived in the section on
stable
states'above. The averasefitnessthen is
O(x)
r.Ax
accordingto equation (6). To be correct,(Ax), and x . Ax are not to be viewedas fitness
but as increasein fitnessresulting from the conflict. The differential equation (27) given
below, however, is invariant to the addition of constants to the columns of A and hence
remains unchanged when we replace differential fitness by total fitness.
Dynamics of social behaviour
The equation of
267
'game dYnamics
i, :
x , { ( A x ) ,- o ( x ) } '
i -- 1,"',n,
(27)
restricted to the invariant set S, (the unit simplex) describes the evolution of the
distribution of phenotypes (Taylor & Jonker, 1978). In writing down a differential
of
equation for game dynamicswe made two implicit assumptions:( 1) completemixing
in
this
mean
we
large
infinitely
By
size.
population
large
infinitely
generationsu"a 121
context so large that fluctuations can be neglected.
Equation (27) plays a central role for many models of selectionin fields as diverse as
prebiotic
evolution, population geneticsand mathematical ecology (see Schuster &
-Sigmr,nd,
1983).Its usefulnessin sociobiologyis a new facet of its widespreadapplicability.
<Dneed
In general,the averagefitness<Dwill not increasemonotonically' Maxima of
(27)
of
equation
the
dynamics
(27).
Moreover,
points
equation
of
not coincidewith fixed
in
Hofbauer
discussed
(examples
are
persistent
oscillations
i.e.
may admit limit cycles,
et at.. 1980),and strangeattrators, i.e. seeminglychaotic, highly irregular oscillations
with extremesensitivityto the choiceof initial conditions (seeHofbauer, 1981,Arneodo
Every game theoreticalequilibrium, i.e. every pointp on S, which satisfies
et a:.,19801).
(10)
is a fixed point of equation (27), and every ESS, furthermore, is asympequation
totically stable in the sensethat every state which is sufficiently close by will converge
towards it (Taylor & Jonker, 1978;Hofbauer et al., 1979;Zeeman' 1980)'
Thus, small perturbations of evolutionarily stable stateswill be offset by the dynamics
of the evolution.The converse,however,is not true: thereare asymptoticallystablefixed
points of equation (27) which cannot be found as an ESS by the game theoretical
approach (Hofbauer et al., 1979 Zeeman' 1979)'
Some eramPles
In a special case, namely when rz',: a;rholds for all i and j' the average fitness o will
alwayi increase.Games whose payoff matrix satisfiesthis condition are called partnership games:both playersalwayssharethe outcomefairly.Indeed,it can easilybe checked
that the rate of increaseof O corresponds,again, to the variation of the fitnessin the
population. It is, of course,no surprise that in partnership games,in contrast to the
'prisonersdilemma' and the like, an optimization principle holds.
' h a w k d o v e ' g a m e ,p :
- p')e, with Pr: (E - G)l
The ESS of the
Pr€r + (l
(E + C) as describedby equation (15) is globally stable.In the extended'hawk dove'
game given by Expression(17) with the phenotypeE correspondingto this ESS, there
existsa line F of fixed points through er and p (seeFig' 2)'
All orbits converge to 4 remaining on the constant level curves of the function
QG)
x.,r, P'x,
o'.
(28)
There is an argument (Hofbauer, private communication) that random drift will cause
the state to approach e.,,leading to fixation of the phenotype E.Indeed the constant
level curves of-q ut" concave. If the state of the population has approached some fixed
point 4 on F, and if a random perturbation sendsit away, then the differential equation
will lead it back to some point q' on -F. The probability I"haI q' is between 4 and e, is
slightlylarger than I /2 (seeFig. 2). In this way, a sequenceof small fluctuationswill drive
the state towards ej. This, however,dependsupon the assumptionof fluctuationswith
radial symmetry.
268
P. Schuster& K. Sigmund
e. lo u phcruttrpe
e, uul e- torrespondto'luvk' urul dore' plrcnttlt'pt'.t,
Fig. 2. Strutegie.s
plu.tingu ni.rt'tl strute,s,.t,
u('tingtt.s'hunk'nith prohuhilil.t'1t,tutdus'dovc'tith prohubilit.r
p . ( r h a r cp l u n t lp . u r e g i t ' e n h ) ' t l tEeS S p : ( p t e t + p . e . ) o / t h c ' h u w k d o t e ' g u m e ) . T h e
linc lront el to p (onsistso/ li-redpoint.s.Rundontdrilnnight leud the .\tuteof thc population,
throu.qhflttctuutittnsulong tlte line. tlttser to e.
We recall from equation (17) that phenotype E. is not evolutionarily stable.A morc
strictins examole for such a situation is obtaincd with
r0
A
[n
I']' :
Lt
I
il
( 2e)
( s e eF i g . 3 ) Ä , i s s t a b l ea g a i n s ti n v a s i o nb y E , a l o n e .o r b y E , a l o n e .b u t n o t i f b o t h
p h e n o t y p e si n v a d es i m u l t a n e o u s l y .
€z
€t
Fig. -l . The gunte-d.tnunitul
equuliott
(.:7) \'ith nnlri.r
givan ht
(29)
Dvnamicsof socialhehuviour
269
€1
Fig.4. The game-th'namicalequation (27) tt'ith mdtri't giv('nbt G0)
Another interesting example due to Zeeman (1981) is given by the matrix
A
u -ol
Io 0 s
l-3 3 o l I
(30)
L-l
In this case, equation (27) admits two asymptotically stable equilibria, namely
e r : ( 1 , 0 , 0 ) a n d m : ( 1 i 3 ,l i 3 , 1 / 3 ) ( s e e F i g . 4 ) . T h e f o r m e r p o i n t i s e a s i l y c h e c k e d t o
be an ESS. Hence the latter one, which lies in the interior of S,, cannot be an ESS'
Permanenceand uninvadlhilit.v in terms of dynamical svstems
As mentionedbefore, the orbits of a game-dynamicalequation (27) need not converge
to an ESS.They may convergeto an equilibrium which is asymptoticallystablebut not
a n e v o l u t i o n a r i l ys t a b l es t r a r e g yi n t h e s e n s eo f g a m e t h e o r y . o r t h e y m a y s e t t l et o a
persistentoscillatoryregime,or exhibit chaotic behaviour.In the two latter casesit can
be shown that the time averagesof the phenotypefrequenciesconvergeto an equilibrium
r t a1.,1981a).
v a l u e ( s e eS c h u s t e e
Of course.one cannot hope to measuresuch time averagesdirectly, sincethey would
have to includea largenumber of generations.If the population is subdividedinto many
suchpopulations(or demes)which oscillateout of phasewith eachother. then the mean
value (at a given instant) of the phenotype frequenciesof the different demes gives a
plausibleestimateof their time-averagesand henceof the quilibrium.
The game dynamical approach suggeststwo eventualities:
(l) the existenceof equilibria which, although not'evolutionarily stable',are neverthelessrelevantand persistentfeaturesof the model. either becausethey are asymptotically
stableand henceperturbation-proof,or becausethey correspondto the time averagesof
regular or irregular oscillations,and
(Zi ttre existenie of asymptotic regimes which, although not static. are nevertheless
robust featuresof the sYstem.
P. Schuster& K. Sigmund
270
Whether such phenomenaoccur in situationsof biologicalinterestis still open. but the
possibility shold be kePt in mind.
In any caseit is not dilhcult to describe,within the framework of population dynamics'
the notions of permanenceand uninvadabilityin a non-staticway which generalizesthe
concepr of evolutionary stability. Roughly speaking. the n phenotypesE' to E, are
pernlanentif there is some minimum level4 > 0 such that, if initially all phenotypesare
present.i.e. if .v,(0) > 0 for all i. then after sometime their frequenciesr'(t) will be larger
itrun q. Thesefrequenciescould oscillateor converge:the only relevantproperty, in this
context.is that random fluctuationswhich are small and occur rarely are not ableto wipe
out some of the PhenotYPes.
with respectto some lurther phenotypesd'*,
The societycan be termed unint,adahle,
to E,_r, if it is permanentand if all initial conditions with sullicientlylow frequencies
en d
' e r m a n e n ca
. r , , * , i 0 jt o . t , , , * ( 0 ) l e a d t o t h e u l t i m a t ev a n i s h i n go f t h e s ep h e n o t y p e sP
and
within
from
uninvadability, in this sense.mean protectednessagainstdisturbance
discussion
detailed
more
from without. We refer to Schusterand Sigmund (1984)for a
of thesenotions and their applicability to models of biological evolution.
rit' gamcs
D.rttuntit.sIor as.l'mntcl
Let us turn now to asymmetriccontests.i.e. to gamesdescribedby two payoffmatrices
'Asymmetriccontests'above).The sameargumentwhich led
A and B (seethe sectionon
to the game-dynamicalequation (27) now yields the differentialequation
-ii :
x , [ ( A Y )-, x ' A l ] .
i :
1""'n'
t, :
l,[(Bx); r'Br].
i :
1."'.m,
(31)
describingthe evolution of the population statesx(l) e S,' and y(t) e S,' (seeSchuster
et al.,1981b).
If one of the populations contains only one phenotype, equation (31) reducesto
equation (26) and henceto an optimization problem.
In the generalcase,selectionleads usually to the fixation of one genotype in each
population. This reflectsthe fact that there exist no mixed ESS for asymmetricgames.
In the casen : nlj however.it may also happen that the frequenciesoscillateperiodic'coynessphilandering' game of Dawkins
ally. This is the case, for example. in the
'Nash
e q u i l i b r i aa n d t h e c o y n e s s( S c h u s t e&r S i g m u n d ,l 9 8 l ; s e ea l s ot h e s e c t i o no n t h e
philandering game' above). Again, the time averagesconvergeto the game-theoretic
equilibrium. which is not evolutionarily stable.
D i:;cretegame dYnumics
tool to study infinite populations with distinct
appropriate
are
the
equations
Difference
generations.Thus. they apply to situations in which premise (l) for the validity of
equation (27) is not fulfilled. Often, blending of generationis preventedby the action of
someinternal or externalpacemakersfor the reproductivecycle.An obviousexamplefor
the latter caseis the periodicity of seasons.
The most straight-forward candidate for discrete-gamedynamics of symmetric contestsis the equation
(32)
Here r' : (ri. . . . ..rj) denotesthe state of the population in the next generation.It is
somewhatdisturbing that this classof differenceequations,in contrastto the differential
Dvnamics of .gocialhehaviour
271
equation
equationsdiscussedabove,does not seemto fit ESS theory very well. Indeed,
'hypercycle'
the
example
as
an
We
consider
cases.
(3)) behavesrather badly in several
gamewithn:3givenby
r, jl
A :
l0
0
(33)
Ll 0
Ir has rn : (l/3, l/3, l/3) as an ESS. The fixed point rr is not asymptoticallystable,
however.This is surprising,sincethe continuous analogueof expression(32), namely
r x,[(Ar), - x'Ax]
(34)
,ii : (x .Ar)
has the sametrajectoriesas the game-dynamicalequation (27). For asymmetricgames.
the difference equation
',t,(A.y)',
ri :
(r'Ay)
t; :
(Y'Bx) '1(Bx),
(35)
to the differentialequation
corresponds
ii :
(x'Ay)
' , t , [ ( A - Y-) ' r ' A Y ] .
j'i :
(-r 'Bx)
' - r 1 [ ( B x )-,
(36)
l''Bx],
which needno longer be equivalentto equation (3 1). In particular, the inner fixed point
(p, q) of the'coyness-philandering'gameis now asymptoticallystable (Hofbauer, see
Vuy"u.a Smith, 1982,appendixJ). This may be viewed either as a misleadingtrick of
the dynamics-after all, (p. 4) is not an ESS-or as a remarkable vindication of
Dawkins early claim (1916)that the strategiesdo convergeto (p, q)- Of course,thereare
no empirical dates to decidewhich of the two equations(31) or (36) is more correct.
An interestingapproach has been proposedby Eshel and Akin (1983).It consistsin
assumingthat the sign of ,t, (or. in the discretecase,of "ti x,) is that of the difference
.
like
equation(31) or (36).The
equation
explicit
an
down
writing
(Al ),
.r AJ, without
the basic traits of the
reflect
but
should
specified.
incompletely
dynamics,then, is only
and ESS has
equations
differential
equations,
difference
on
model; surely,the last word
yet.
not been said
'like
begetslike'. More sophisticated
So far, we have acceptedthe simplificationthat
of inheritance.
mechanism
Mendelian
the
discussionshave to take account of
Serual models
Although the strategicand the geneticpoint of view are to some degreedisjoint, it is
of interestto check their compatibility, even if this requestsassumptions
nevertheless
which are quite hypothetical.
It is obvious that a geneticconstraint can prevent the population from attaining an
ESS. In particular, an evolutionarily stable phenotype which is only realized by a
heterozygotegenotypecan neverbecomefixed. Such a caseof'overdominance' leadsto
different predictions of game theory and population genetics.So far only one locus
modelshavebeenstudied(Maynard Smith, l98l; Eshel,1982;Hines, 1980;Bomzeer a/..
1983).In this case it was always overdominancewhich led to divergencesfrom ESS
results.In many-locusmodels one may expectother geneticconstraintsas well.
272
P. Schuster& K. Sismund
Let us consider first the frequency independentcase.We assumethat there are k alleles
Ar to Ar in the genepool. Their frequenciesare denotedby p,to p*. The fitnessof the
genotypeA,A, is given by some constant w, which correspondsto the probability of
survival from the zygoteto the adult stage.Introducing the laws of Mendelian genetics
we obtain the classicalselectionequation of Fisher for the evolution of the gene
distribution
ii:
P , { t w P l ,- O ( P ) } .
(37)
l:1.....k.
The differential equation is invariant on S*, W : (w,i) is the 'viability matrix' and
O(p) : p 'Wp representsthe averagefitnessin a randomly mating population with the
g e n e d i s t r i b u t i o np : ( p , , . . . , p J . T h i s e q u a t i o n i s a s p e c i a lc a s eo f e q u a t i o n ( 2 7 ) .
Precisely,it correspondsformally to the (game)dynamicalversionof a partnershipgame
sincew, : nii. Accordingly,the averagefitnessO increases
monotonically.Selectioncan
be visualizedas optimization of the mean reproductivesuccess.
There is nevertheless
an
important differencewith respect to the nature of the optimization processin equation
(37) and in the asexualcasedescribedby equation (26). fn the latter caseoptimization
was global on S,.Here, selectiondoes not lead in generalto a global optimum of O.
Consider for example the particularly simple case r? : 2. The average fitness is of the
form
O
(h',, -
2 l r ' , 1* r r , , r , ) pf] 2 ( w r , -
t t , . r ) p ,I
w.2.
In case lr' > tl''r2and n,r, ) D'12,O has a minimum on the interval 0 < p, < 1 and
hence,thereare two optima coincidingwith the pure statespr : 0 and pr : 1. Depending on the initial conditions either allele A, or allele A, will be selected.
We turn now to the frequency dependentcaseand consider equations which combine
the game theoreticalmodel with popultion genetics.It will be appropriate to carry out
the analysison two levels,the phenotypicand the genotypicone. We shall assumethat
the population consistsof r phenotypesE, to En and that each genotypeA,A, corresponds to one of those phenotypes,or possibly to a probability distribution p(ry) :
whereprQ) is the frequencyof A,A,-individuals of phenotype E*.
Q,(D,...,p,(n),
The converse,however,need not be true: as it happensin the caseof dominance,two
or more genotypesmay give rise to the same phenotype.
The payoff is given as before by the n x n-matrix A. Now we have to specify how its
is relatedto reproductivesuccess.
This is a rather delicatepoint which cannot be decided
without a closerinspectionof the situation to be modelled.Two alternativesimplecases
are:
(l) The payoffis independentof sex, and the number of offspring of a given couple
is proportional to the product of parental payoffs.This situation correspondsto fights
which are not sex-specific,
like contestsfor food. Maynard Smith (1981)consideredsuch
a model.
(2) The expression of the genotype is sex-specificalthough the relevant genes are
carried by both sexes.The number of offspring of a given couple again is proportional
to contributions from both parents.The parent who carriesthe silentgenesnow contributes a constant factor. Such a model is appropriate for fights within the male population, e.g. contests for females, breeding grounds or other ressources.A detailed
discussionof this approach can be found in Hofbauer et al. (1982).
In any caseone is led to fecundity models.To set up the corresponding(discreteor
continuous) equationsfor the time evolution of genotypeor phenotype frequenciesis
immediate, but the analysis of the resulting systemsis in general difficult.
273
D I'namic.sof' soL'ialbehat'tour
lAol
?
AA. Aa and aa(corresponda typit:alevolutionof genott'pes
Fig.5 A tliagramdest,rihing
dependsondffirentiullbcunditv.The
irigtothecornerse,,e.,er)il'ttriirprortur'tilesr.r('('err
,t:it, u./th, pnpulatiotitont,ergn,,,;r.i.quitklt'to u'Hurdt' llefuberg'parahola,andevoltes
an equitibriumP [fbr
seletr'irriutn,igtheparaholatov'rtrd.s
thr1ughfriqiencv-ctependeni
rletuils.
seeHofbaueret al. (1982)l
In some cases.however. the investigationis simplified by the fact that a HardyW e i n b e r ge q u i l i b r i u mg e t se s t a b l i s h e di n: c a s e( l ) i f t h e m o d e l i s d i s c r e t e( n o n o v e r l a p ping generations)and in case(2) ifit is continuous(generationsblendinginto eachother.
,.. äg 5). Thus rhe frequencyof genotypeA,A, is given by the product of the frequencies oithe correspondinggenesA, and A,. (More generally.it often happensthat the
mating system-random or assortative-leai1sto relations between the genotype frequ..,.i..j. This allows to reducethe problem to the time evolution of genefrequencies'
The correspondingequationsare similar to equation (37). except that the l1'/rare now
in p, to p^. The averagefitnesswill not increasein general'
polynomials
An analysisof two-strategygames(seeMaynard Smith. l98l; Hofbauet et al., 1982
E s h e l .1 9 8 2 )s h o w s ,e s s e n t i a l l yt h. a t i f t h e g e n e t i cc o n s t r a i n t sa l l o w t h e E S St o b e r e a l i z e d
is
at all. then it will asymptoticallybe reachedindeed.If, for example.one of the alleles
of
the
considerations
game
theoretic
simple
by
the
predicted
dominant, then the outcome
'The'hawk dove' game and mixed strategies'above will also be obtained by
sectionon
t h e g e n e t i cm o d e l . ( I t i s w o r t h m e n t i o n i n gi n t h i s c o n t e x t t h a t B ü r g e r . 1 9 8 3 .u s i n g a
model of Sheppard.1965,has shown that dominancemay be established.if the selection
pressueis sufficientlyhigh, by the action of a secondarygene locus. Thus the genetic
mechanism itself may evolve towards a suppressionof the genotypic obstaclesto
phenotypic adaptation).
One may similarly investigategeneticmodelslor asymmetriccontests.In Bomzee'l a/.
(19S3);the'coyness philandering' model is discussedat some length.We briefly sketch
it. as an illustration. Let us assumethat two allelesA, and Ar determine the male
behaviour,and that P,,, P, and Pr are the probabilitesthat the male genotypesA'A'.
A,A, and A,A", respectively,are faithful. Then. if p denotesthe frequencyof A'.
6 :
p2{Po* P.-2Pt)+2p(P,-
P)+
P)
274
P. Schuster& K. Sismund
is the frequencyof faithful maleswithin the male population. Similarly,let the allelesB'
and B, determine female behaviour and let Qo, Q, and Q, be the probabilities that the
femalegenotypesBr Br, Br B, and BrB,, respectively.are fast. Then, if 4 is the frequency
of 8,,
q2(Qo
+ Q. - 2Q,)+ 2q(Q,- Q) + 2,
P :
'payoffs'
is the frequency of fast females, It is easy, now, to compute the
a) and a2 for
'payoffs'
faithful and philandering males, respectively,as function of B and the
a, and
a, for fast and coy females,respectively,as function of ä. If one assumesthat the payoff
correspondsto fertility, one can check that Hardy Weinberg relations hold. Simple
computations lead to the differential equations
- p)lp(Po * P, -
2P,) + P, -
Prl(a, -
ar)
i
Lp(l
s :
L q ( r - q ) l q ( Q*o Q , - 2 Q ) + Q , - Q . l ( a ,- a , ) .
The strategic component-i.e. the terms depending on the payoff matrices-reduce to
the factors at - az and a, - ar. This facilitates the analysisof the gene frequencies.We
shall only describethe interestingcaseof 'overdominance'.If P0 - P, and P. - P,have
(p, q)
the samesign,as well as Qo - Q, and Q, - Q,, then the orbit in the 'state-space'
will be periodic (see Fig. 6). There are four possible oscillatory regimes,depending on
initial conditons, but the time averagesof the strategieswill quickly convergeto the
equilibrium valuesobtained in Equation (25) by simple game theoreticconsiderations.
This may well be a typical situation: complicatedfeatureson the 'microscopic'level of
genefrequenciesyield a simple result on the 'macroscopic'level of phenotypes.(For a
detailed discussion see also Bomze et al., 1983). Asymmetric conflicts between two
different speciescan also be modelled in similar ways. The first paper in this direction,
which apparently found only little attention in the literature, seems to be due to
oo
O o
oo.sr
Fig. 6. A possibledynamicsfor the'coyns.s5*phfionderirg'game.
p and
Thegenefrequences
q governing male and.fbmale behaviour oscillate endlesslv, but the corresponding payofls
converge,in the time average, to the game theoretic equilibrium (25) (see Bomze et al.,
1983,t'or details)
Dynamics of .soe'ialbehaviour
275
Stewart(1971)and investigatesdimorphism in two coevolvingpopulationsof predators
'game' betweencats which can stalk or watch and mice which can run or
and prey. His
In its emphasison game
freeze.is similar in structureto the'coyness-philandering'game.
theoretic aspectsit is a remarkable forerunner of ESS theory: it also offers an analysis
of dynamical models basedon sexualand asexualgenetics,which fully agreeswith the
Nash equilibrium solution obtained from strategicconsiderationsalone.
Resultsof this type strongly suggestthat one may confidentlystick to the phenotypic
level. as long as there is no solid candidatefor the underlying genotypicmechanism.
Concluding remarks
The considerationsconcerning the role of intrinsic constraints on the evolutionary
optimization processcan be subsumedwith the help of Fig. 7. The mean reproductive
successin populations of independently and asexually reproducing individuals is optimized though evolution. This will be modified by two mechanisms:
(a) The rules of Mendelian geneticsrestrict the dynamicsof selectiononto the Hardy Weinberg surface or onto a'near Hardy Weinberg'submanifold (see Ewens, 1979).
Other distributions of genotypesare evolutionarily unstablefor geneticreasons.
(b) Direct interactionsof replicatingelementslead to constraintson the optimization
process.The result is a displacementof the evolutionarily stabledistribution from that
which is characterizedby the maximal reproductive success.
In populationsin which both constraintsare in operation,the geneticand the strategic
one, we observea superpositionof both effects.In the sexualmodels consideredso far,
many cases(e.g. dominance) lead to situations which are indistinguishablefrom the
correspondingasexual ones. This justifies a posteriori the various static or dynamic
asexualmodels.
Of course the situations analysed so far are characterizedby oversimplification. One
could introduce further complicationswithout end, but often without gaining lurther
insights.
Basically,the game dynamicalmodelsincorporatefeaturesof highly developedfields,
namely population geneticsand game theory. A further emphasison geneticaspects
G e n e t r cc o n s t r o i n l
C
;
c>
>o;
oo
I o!
Q
Asexuol
r e p lr c o ti o n
Globoo
l ptimizotion
of@
Sexuol
r e p l i c oi o
fn
R e s t r i c t i otno H - W s u r f o c e
O p t i m i z o t i oonf Ö
o
o
c
o
.c
o
>::
C
9öe
o
Y -! n
(-)
N o o p t i m i z o t i oonf @
R e s t r i c t i o tno H - W s u r f o c e
N o o p t i m i z o t i oonf 0
PöR3
[!
-
o
Fig. 7. Constraintson optimization. The geneticmechanismof se.rualreproductionleadsto
relationsbetweengeneand genotype.frequencies(e.g. the Hardy-Weinberg equation).This
reduces the state space of genot-Vpe
.frequenciesto a subset of lower dimension. FrequencT'
dependentJitnessparameterspreclude optimization through strategic constraints
276
P. Schuster& K. Sismund
leads to intricate frequency dependentpopulation genetics.and a stressingof game
theoreticaspectsto the subtleconstructionsof extensivegamesand subgames(seeSelten.
1983).In its presentstate,gamedynamicsis a compromisebetweenthesetwo directions,
trading elaboratesophisticationin one or the other direction for broadrangedcompatibility between both aspects.It aims at acquiring an intuitive insight into biological
conflicts.without too many technicaldetails.
It seemsvery difficult to derive from field data valid estimatesfor the parameters
involvedin the payoffmatrix. Someremarkableresultshave beenobtained,however[we
r e f e r t o M a y n a r d S m i t h ( 1 9 8 2 )a n d R i e c h e r ta n d H a m m e r s t e i n( 1 9 8 3 ) f o r s u r v e y s l .
Moreover. it has been shown in severalcasesthat the dynamics consistof a few types
only. which are valid for wide rangesof the parameters.Thus, diversesexualand asexual
'hawk-dove' conflict lead to a very small number of possible
modelsof variationsof the
outcomes(seeSchusteret at., 1981a.ä). It seemspossible,in suchcases,to relate(at least
qrnlitatively) empirical data with some of the few theoreticallypossiblecases.
Finally, we mention that the assumptionof geneticaldeterminationof behaviourcan
be relaxed. In particular. the effect of learning can be incorporated into the game
'developmentally
theoreticalmodels [for example.seeHarley (1981)and the notion of
stablestrategy'DSSby Dawkins (1980)1.It seemspossible.therefore,that the dynamical
approach sketchedin this paper can be adaptedto the modelling of complex behaviour
in highly developedsocial structures(for example.seeWilson. 1975.Masters. 1983).
Acknowledgements
'Fonds zur Förderung der
The work has been supported financially by the Austrian
Forschung'ProjectP 4506and P 5286.We would like to thank Prof.
Wissenschaftlichen
John Maynard Smith and Dr Peter Hammersteinfor fruitful discussions.
References
A k i n . E . ( 1 9 8 3 )C. a n .J . M a t h . 3 6 . 3 7 4 .
. h l ' sL. e t t . 7 9 . 2 5 9 .
, . & T r e s s c rC. . ( 1 9 8 0 ) P
A r n e o d oA
. . . C o u l l c tP
, . W . ( 1 9 8 1 )S. c l e n c e2,lrl . 1 3 9 0 .
A x e l r o dB
. . & H a m i l t o nW
C. K.. Diekmann.D. & Luce.R. (1982).J. math.Biol. 154'629.
Biebricher.
P. & Sigmund.K. (1983).J. theor.Biol. l0l. 19.
Bomze.I.. Schuster.
B ü r g e rR
. . ( 1 9 8 3 )J.. m a t h .B i o l . 1 6 . 2 6 9 .
Dasegoisrische
Deutsch'.
Dawkins.R. (1976).TheSelfishGere.Oxford:OxfordUniversityPress.
Gen.Berlin:Springcr-Verlag.
Oxford:Freeman.
Dawkins.R. (1982).TheErtendedPhenottpe.
58,465.
Eigen.M. (1971).Die Naturv,i.ssenscha,/'ten
Berlin:
of'NaturalSelfbrganizatiorr.
A Principle
The Hypercyclc'.
P. (1979).
Eigen.M. & Schuster.
Springer-Verlag.
Eshel.I. (1982).Theor.Pop.8io1.22.204.
Eshel.I. & Akin. E. (1983).J. math.Biol. 18. 123.
Berlin:Springer-Verlag.
Population
Genetics.
Ewens.W. J. (1979).Mathernatital
. iol.7.l.
H a m i l t o nW
, . D . ( 1 9 6 4 )J.. t h e o rB
New York:
in Insects.
Hamilton.W. D. (1979).In: (M. S. Blum& N. A. Blum.Eds)Competition
AcademicPress.
H a m m e r s t e iP
n ., ( 1 9 7 9 )A. n i m .B e h a v . 2 91, 9 3 .
. iol.89.6ll.
H a r l e yC
. . B . ( 1 9 8 1 )J.. t h e o rB
. iol.87,379.
H i n e sW
, . G . S . ( 1 9 8 0 )J.. t h e o r B
Hines.W. G. S. & MaynardSmith.J. (1979).J. theor.Biol.79. 10.
TMA 5, 1003.
Hofbauer,J. (1981).NonlinearAnalysis.
P. & Sigmund,K. (1979).J. tlrcor.Biol.8l,609.
Hofbauer,J., Schuster,
Dlnamic's oJ'soc'ialbehaviour
277
(1980).SIAM J' appl Math' 38' 382'
Holbauer. J.. Schuster.P.. Sigmund. K. & Wolff. R.
Biol' 11' 155'
rnath'
(1981)
J'
K'
Hofbauer. J.. Schuster.P. & Sigmund.
'
Hofbauer. J.. Schuster,P. & Sigmund. K (1982) Biol' Cvbern' 43' 51
1
6
9
'
6
.
J o n e s .B . L . ( 1 9 7 8 ) .J . m a t h . B i o l .
t o n . r . S . L . . E n n s . R . & R a g n c k a r .S ( 1 9 7 6 ) B u l l ' m a t h ' B i o l ' 3 8 ' 1 5
B. O. (19g3). Moleiular Theorl. of' Et,olution, Berlin: Springer-Verlag.
iäfp..r.
R.
D. (1982):J. sot'ialbktl. Struct' 5' 439
Maiie.s,
Press.
M;;;".ä Smith. J. (t9.72).On Evolution.Etinburgh: Edinburgh University
47.209'
Bbl'
(1914)J.
theor.
J.
Smith.
Maynard
M a y n a r d S m i t h , J . ( 1 9 7 6 ) .A m . S c i . 6 4 . 4 l '
M a y n a r d S m i t h , J . ( 1 9 8 1 ) .A m . N u t . 1 7 7 . 1 0 1 5 .
university Press.
vuynu.o smith. J. (lgszi. rrotutionart,Game lh.eor1"cambridge: Cambridge
l5
246'
(1973).
Nature
G.
Price.
&
J.
Smith,
Maynard
'
and Economic Behoviour
Neumann.. J. von & Morgensie.n,ö. ( t ssl) ' Theort' of Gumes
N.J.
Princeton
parker, G. A., Hammersrein.P., in Greenwood,P.. Slatkin, M. & Marvey. P. (1984).Evolution
University Press'
Essaysto the honour tt John Ma1'nurd snltft. Cambridge: cambridge
14,3'7'l
s)'stemarics.
Ecol.
Rev.
Riechert.S. & Hammerstein.P. (1983).Ann.
Riley, J. (1916).J. thettr.Biol.76' 109.
S c h u s t e rP. . & S i g m u n d .K . ( 1 9 8 1 ) .A n i m B e h a v '2 9 . 1 8 6 '
Schuster.P.,Sigmund.K..Hofbauer.J.&Wolff.R.(l98la):Biol.C).bern.40.1'
40' 9'
Schuster.P.. Silmund, K.. Hofbauer' J. & Wolff, R (1981Ö):Biol' Cyhern'
16.329
Chem'
(1982).
Biophl's
J.
Swetina.
P.
&
Schuster.
Schustcr,P. & Sigmund. K. (1983) J. theor' Biol 100' 533'
and Chaotic
P. & Sigmuno, i.'trssal. In: (P. Schuster.Ed.). stochastit'Phenomena
Schuster.
Springer-Verlag.
Berlin:
Vot.21.
in Cimpler S,rs/ens, Synergetics
Bi;ehayiour
Selten,R. (1980).-r. theor. Biol' 84' 93.
Selten.R. (1983).Math. soc. Scr. 5, 269'
i6.ppu.O. p. Nf . (f qSSt.Natural Selectionand Hereditt'.London: Hutchinson.
S t e w a r t ,F . M . ( 1 9 7 1 ) .T h e o r 'P o p B i o l - 2 , 4 9 3
Taylor, P. (1979).J. Appl Prttb. 16"76.
taytor. P. & Jonker. L' (1978).Math Biosci' 21, 121
Thtmpson, C. & McBridc, J. (1974).Math Biosci' 21' l2'7'
of man' London:
Trivers. R. L. (1972). tn: iCaäpUeil. Ed') Sexual selectionand the descent
Heinemann.
W e i s s m a n nC
, . ( 1 9 7 4 ) .F E B S L e t t . 4 0 ' 1 0
Cambridge' MA: Harvard University
Wilson, E. D. (1975). Socittbiologt'-The Nev' Svnthesis.
Press.
Behaviour. Edinburgh:
Wynne-Edwards. V. C. (1962). Animal Dispersion in Relation to Sot'ial
Oliver and Boyd.
Lecture Notesin Mathematics819'
of D.'namical s1'stems,
Zeeman.E. C. (1979).ln: Global Theor-v
Berlin: Springer-Verlag.
Z e e m a n .E . C . ( 1 9 8 1 ) .J . t h e o r 'B i o l . 8 9 ' 2 4 9 '