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Game Theory and Animal Behavior

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G A M E T H E O R Y& A N I M A L B E H A V I O R

Edited byL E E A L A N D U G A T K I N andH U D SO N K E R N R E E V E

O X F O R D U N I V E R S I T Y P R E S SN ew Y o rk O x f o rd

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O x f o r d Uni v e r s i t y Pr e ssO x f o rd N e w Y o r kA t h e n s A u c k l a n d B a n g k o k B o g o ta B o m b a y B u e n o s A i r e sC a l c u t t a C a p e T o w n D a r e s Salaam D e l h i F l o r en c e H o n g K o n gI s t a n b u l K a r a c h i K u a l a L u m p u r M a d r as M a d r id M e l b o u r n eM exico Ci ty N airobi Paris Singap ore T aip ei T okyo T oronto Warsawand associated comp anies inB e r l i n I b a d a nC o p y r i g h t 1998 b y O x f o r d U n i v e r s i t y Pr e ss , Inc .First p ub l ish ed in 1998 by O xford Un iver si ty Press , I nc .19 8 M a d i s o n A v e n u e , N e w Y o r k , N e w Y o r k 1 00 16First issued as an O x f o r d U n i v e r s i t y Pr e s s p a pe r b a c k , 2000O x f o r d is a registered t r a d e m a r k o f O x f o r d U n i v e r s i t y P r e s sA l l r i g h t s reserved. N o p a r t o f t h i s pu b l i c a t i o n m a y b e r e p r o d u c e d ,stored in a re t r ieval system, or t ransmit ted, in any form or by any means,e l e c t r o n i c , m e c h a n i c a l , p h o t o c o p y i n g , r e c o r d i n g , o r o t h e r w i s e ,w i t h o u t t h e p r i o r p e r m i s s i o n o f O x f o r d U n i v e r s i t y Press.L i b r a r y o f Congress C a t a l o g i n g - i n - P u b l i c a t i o n DataG a m e t h e o r y a n d a n i m a l b e h a v i o r / e d it e d b y L e e A l a n D u g a t k i n a n d

H u d s o n K e r n R e e v e .p . c m .

I n c l u d e s b i b l i o g r a p h i c a l references a n d i n d e x .I S B N 0-19-509692-4I S B N 0-19-513790-6 ( P b k . )1. A n i m a l b e h a v i o r M a t h e m a t i c a l m o d e l s . 2 . G a m e t h e o r y .

I. D u g a t k i n , L ee A l a n , 1962- . I I . R e e v e , H u d s o n K e r n .Q L 7 5 1 .6 5 .M 3 G 2 5 19975 9 1 . 5 ' 0 1 5 ' 1 9 3 d c 2 1 96-29891

9 8 7 6 5 4 3 2 1Printed in the U n i t e d Slates o f A m e r i c aon ac id- free p ap er

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To Dana, (an, Alex, andAaron.They are al l too wonderful for words.

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Preface

I t is a lway s not or i ous ly diff icu l t to identify precisely what const i tutes ar evolut i onar ych ange wi t h i n a scientific d i sc i p l i ne (K uh n 1962) . In the f ie lds of ani mal b eh avi or( e t hol ogy) an d b eh avi or al eco logy , th e i nt r oduct i on o f i n c l u s i v e fitness m o d e l s ( H a m -ilton 1963, 1964) m ay q u a l i fy as s u c h a c h a n g e in that they reshaped th e m a n n e r inwh i ch b eh avi or al ecologists and ethologists think about near ly every quest ion theyaddress. The subject of this book, game theory, may rank second to inclusive f i tnessin t er ms of its effec t on the way ani mal b eh avi or i s t s cur r ent ly appr oach i ssues sur -r o u n d i n g th e evolut i on o f socia l b eh avi or . H owever , evolut i onar y game t h eor y ' s f u n-dament al pr i nc i p let h at ac t i ons t aken b y o n e i n d i v i d u a l h a v e effects on the f i tnesso f others and that all s u c h effects m u s t be a c c o u n t e d for w h e n e x a m i n i n g th e e v o l u -t ion of a t rai thas had a large enough impact that phrases l ike "payoff matr ix" and"evolut ionari ly s table strategies" are u s e d by v i r t u a l l y all pundi t s of behavioral ecol-o g y and anim al behavior and are amo ng the f i rs t conc epts ta u gh t in graduate c lasses.F ur t h er mor e , th e idea of an evolut ionari ly s table s trategy (ESS) h as surpassed th eb o u n d a r i e s o f ethology an d b eh avi or al eco logy an d c an often b e heard being ut teredby t he l ikes o f poli t ical scient is ts , mathematicians, an d psy ch ologi s t s . It is cer tainlyrare for mathematical terminology created by behavioral ecologists to be adoptedout s i de the field, and this s tands as a t es t ament to the i n f lue n c e o f e t h o logi cal gametheory.

H a r d l y a n i s s u e of s u c h j o u r n a l s as Animal Behaviour, Behavioral Ecology, Be-havioral Ecology and Sociobiology (to n a m e a few of the top j o u r n a l s in the field),or for t h at mat t er The Journal of Theoretical Biology is p u b l i s h e d in w h i c h at leas tone, and u su al ly mor e , ar ti c les cite some reference to evolut i onar y game t h eor y (mostoften M a y n a r d S m i th ' s 1 9 8 2 b o o k ,E volution and the Theory of Games). Y e t , to date,there is no p lace wh er e o n e c a n t u r n for a large-scale picture which n o t o n l y r evi ewsth e i m p a c t o f g a m e s o n behavioral ecology and ethology, b u t suggests direct ions fo rf u t u r e r esear ch . Th i s vo lume (and t h e 1995 National Animal Behavior Society s y m p o -s ium o n "Game T h e o r y an d A n i m a l B e h a v i o r " t h a t it is loosely based on) is anat t empt t o r emedy t h i s problem by gather ing some of the leading researchers in thefield and h avi ng t h em r evi ew (and somet i mes ext end) wor k i n t h e i r ar ea i n a manner

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viii Prefaceth a t i s a ccess ib le to th o se in ter es ted in ga mes a nd a nima l b eh a vio r , b u t no t necessa r -il y trained in the so met imes ted io u s ma th ema t ics of ga me th eo r y . T h is b o o k is in-tended for al l a d v a n c e d u n d e r g r a d u a t e s , g r a d u a t e s t u d e n t s ,an d pr o f ess io na l b io lo gis tsinter es ted in th e evo lu t io na r y a na ly s is o f a nima l b eh a vio r . A b a s ic ma th ema t ica l b a ck -gr o u nd in a lgeb r a ( a n d s om e e l e m e n t a ry c a l c u l u s ) w i l l b e suff icient to a l lo w a ccessto even th e mo st co mplex mo d els d iscu ssed in th is vo lu me. H o w ever , even in th ea b sence o f su ch t r a ining, th e va s t m a j o r i t y o f ma ter ia l w i l l b e a ccess ib le to th e in ter -ested reader.

A f t e r an opening chapter that provides a lu c id a t tempt to a nsw er th e q u e s t i o n"Wh a t is e v o l u t i o n a r y g a m e t h e o r y ? " ( H a m m e r s te i n ) , th e f o l l o w i n g to pics are re-view ed : th e impa ct o f ga me th eo r y a nd E S S th ink ing o n th e s tu d y o f so cia l f o r a ging(Giraldeau an d L ivo r e i l ) , co o per a t io n ( Du ga tk in) , a nima l co ntes ts ( R iech er t ) , co mmu -nica t io n ( Jo h nsto ne) , r epr o d u ct ive sk ew a nd nepo t ism w ith in gr o u ps ( R eeve) , s ib l ingr i v a l ry and parent-offspring c o n f l i c t ( M o c k e t a l . ) , a l ter na t ive l ife histories (Grossa nd R epk a ) , h a b i ta t se lec t io n ( B r o w n) , t r o ph ic- l eve l in ter a c t io n s (S i h) , l e a r n i n g (Ste-ph ens an d Clements) , an d h u ma n b eh a vio r ( Wilso n) . G o m u l k i e w i c z then reviews th er e la t i o n s a m o ng ga m e th eo r y , o pt im a l i ty , a nd q u a nt i ta t ive genet ics . I n a final chapter,w e br ief ly assess th e u t i l i t y o f g a m e - t h e o r e t i c r e a s o n i n g in the s t u d y o f socia l behav-io r ( R e e v e and D u g a t k in ) .

W e b e l i eve th a t ev o lu t io na r y ga m e th eo r y i s o ne o f th e m o st po w er f u l a na ly t ica lto o ls a va i la b le to behavioral ecologists an d ethologists today an d sincerely hope thatt h i s v o l u m e o p e n s th e d o o r to those readers w h o were tanta lized b y game theory, b u tu n t i l r ea d ing th is b o o k b e l ieved i t to be the d o m a i n s o l e l y o f th eo r e t ic ia ns .

ReferencesH a m i l t o n,W. D. 1963.The evolution of a l t r u i s t i c behavior.Am. Nat., 97, 354-356.H a m i l t o n , W. D. 1964.The genetical e v o l u t i o n of s o c i a l behaviour. I and II. J. Theor. Biol, 7,

1-52.K u h n , T. 1962.The Structure of Scientific Revolutions. Chicago: U n i v e r s i t y of Chicago Press.Maynard Smith, J. 1982.Evolution and the Theory of Games. Cambridge: Cambridge U n i v e r -

sity Press.

Spring 1997 L . A . D u g a t k i nU n i v e r s i t y o f L o u i s v i l l e

H . K . R e e v eCo r nel l Univer s i ty

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A ck n o wle d gm e n t s

F i r s t and f or emost , we wouldlike t o t h ank t h e cont r i b ut or s t o t h i s vo lume f or pr ovi d-in g out s t andi ng ch apt er s in a t i mely f ash i on. W e h a d very high expectat ions fo r thisb o o k ; an d t h a n k s to their talents an d effort , even o u r expectat ions were surpassed.

I n addition t o o u r o w n c o m m e n t s an d su gg es t i ons , each c h apt er in t h i s v o l u m ewas review ed by at least two e xternal referees. R eview ing for an edited boo k ist h ankless wor k , an d we are indeed indebted to the f o l lowi ng i ndi vi duals fo r theirefforts on t h i s f r ont : Rob er t Boy d,Phil Crowley, Larry Dil l , J imGilliam, Pet er Ham-mer ste in , D a v i d H a s ke ll , G e o f f H i l l , D o n H u g i e , L a u r e n t K el le r, S te v e L i m a , JeffL uc as , Peter N onac s , D av i d Pf enni g, D avi d Q uel ler, Peter R i ch er son, Jan S h el lm an-R e e v e , D a v i d S t e p h e n s , an d Geor ge U et z .

W e t h a n k D a n a D u g a t k i n for pr oof r eadi ng an d indexing this ent ire book. Danano w knows mor e ab out b eh avi or al eco logy an d game t h eor y t h an an y sane personou tside the f ield should . Final ly , w e are indebted to Kirk Jensen at O xford U nive rsi tyPress for al l his t ime, effort , an d e n c o u r a g e m e n t .W e a r e honored to h a v e t h i s v o l u m epub l i sh ed by t he o ldes t an d most r espect ed academi c pr ess in the w o r l d .

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C o n t e n t s

O N E

T W O

T H R E E

F O U R

F I V E

S I X

S E V E N

E I G H T

N I N E

Contributors xiiiW h a t I sEvolutionary Game Theory? 3Peter HammersteinGame Theory and Social Foraging 16Luc-Alain Gimldeau andBarbara LivoreilGame Theory and Cooperation 38LeeAlan DugatkinGame Theory andAnimal Contests 64SusanE. RiechertGame Theory andCommunication 94Rufus A. JohnstoneGame Theory, Reproductive Skew,and Nepotism 118Hudson Kern ReeveGame Theory, Sibling Rivalry, an d Parent-OffspringC o n f l i c t 146Douglas W .Mock, Geoffrey A. Parker, and P. L. SchwagmeyerGame Theory andInheritance in the Conditional Strategy 168Mart R . Gross and Joe RepkaGame Theory and Habitat Selection 188Joel S.Brown

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x ii ContentsT E N

E L E V E N

T W E L V E

T H I R T E E N

F O U R T E E N

G a m e T h e o r y a n dPredator-Prey R e s p o n s e R a c e s 2 2 1Andrew Sih

G a m e T h e o r y a n d L e a r n i n g 2 3 9David W .Stephens and Kevin C . ClementsG a m e T h e o r y an d H u m a n B e ha vio r 2 6 1David Sloan WilsonG a m e T h e o ry , O p t im i z a ti o n ,an d Q u a n t i t a ti v e G e n e ti c s 2 8 3Richard GomulkiewiczW h y W e N e ed E v o l u t io n a r y G a m e T h e o ry 3 04Hudson Kern Reeve and LeeAlan DugatkinI n d e x 31 2

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C o n t r i b u t o r s

Joel S . B ro w nD epar t ment of B i o l o g i c a l S c i e n c e sUniver s i ty of I l l inois at Chicago84 5 W. T a y lo r St.Ch i cago, IL60607United States

K evi n C. Clement sSchool of Bi o logi cal SciencesUn i v e rs i t y of N e b r a s k aL i n c o l n , N E 68588United States

L ee A l a n D u g a tk inD epar t ment of Bi o logyL i f e S ciences B ui ld i ngUniver s i ty of Loui svi l leL o u i s v i l l e , K Y40292United States

L u c - A l a i n G i r a l d e a uD epar t ment of B i o l o g yConcor di a U ni ver s i t y1455 Q u e s t B l v d . M a is o n n eu v eM ont real , Queb ec H 3G 1 M BCanada

R i c h a r d G o m u l k ie w ic zD e p a r t m e n t of G e n e t i c s and

C e l l B i o l o g yP.O. B ox644234Wash i ngt on S t at e U n i v e r s i t yP u l l m a n , W A 99164United StatesM a r t R . G r o s sD e p a r t m e n t of ZoologyUn i ve rs i t y of Tor ont o2 5 H ar b or d S tr ee tToron t o, O n ta rio M 5 S 3G 5C a n a d aPeter H am m er s te i nM a x - P la n c k - I n s t it u t fu r

V e rhal t e n s phys i ol ogi eA b t e i l u n g W i c k l e r82319 SeewiesenG e r m a n yR u f u s A . J o h n s t o n eDe p art me n t of ZoologyCamb r i dge U ni ver s i t yD owni ng S t r ee tCamb r i dge C B 2 3E JU n i te d K i n g d o m

xiii

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xiv Contr ibutorsB a r b a r a L i v o r e i lDe p art me n t o f B i ol o gyConcordia Univers i ty1455 Q u e s t B l v d . M a is o n ne u v eM on t re al , Que be c H 3G 1 M BC a n a d a

S u s a n E . R i e c h e r tDe p ar t me n t of Ec ol ogy &E v o l u t i o n a r y B i o l o g yUniver s i ty o f Te n n e s s e eK n o x v i l l e , T N37996U n i t e d States

D o u g l a s W . M o c kDe p ar t me n t o f Z o o l o g yUniver s i ty o f O k l a h o m aN o r m a n , O K ,73019U n i t e d StatesG e o f f r e y A. ParkerP o p u l a t i o n B i o l o g y G r o u pDe p ar t me n t o f E n v i r o n m e n t a l and

E v o l u t i o n a r y B i o l o g yUnivers i ty of LiverpoolLi v e rp ool , L 6 9 3 B XUn i t e d Ki n gdomJo e R e p k aDe p ar tm e n t of M at he m at ic sUniver s i ty o f Toron t oT oron to, O n t ar io M 5 S 3 G 3C a n a d a

P. L . S c h w a g m e y e rD e p a r t m e n t of ZoologyUniver s i ty o f O k l a h o m aN o r m a n , O K 73019U n i t e d StatesA n d r e w S ihDe p ar t me n t of Bi ol ogyUniver s i ty o f K e n t u c k yL e x in g t o n, K Y 40506U n i t e d StatesDavi d W. St e p he n sD e p a r t m e n t o f E c o l o g y , E v o l u t i o n

a n d B e h a v i o rU n i v e r s i t y of M i n n es o t a1 0 0 E c o l o gy B u i l d i n g1987 U p p e r B u f o r d Circ leSaint Paul , M N55108U n i t e d States

H u d s o n K e r n R e e v eN e u r o b i o l o g y a n d B e h a v i o rSeeley G . M u d d H a l lC orn e l l Un i v e rs i t yIthaca, N Y 14853U n i t e d States

Davi d S l oan Wi l s onD e p a r t m e n t o f B i o l o g yB i n g h a m t o n U n i v e r s i t yP.O. B ox 6000B i n g h a m t o n , N Y13902U n i t e d States

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P E T E R H A M M E R S T E I N

W h a t I s Evolut ionary G a m e T h e o r y ?

1 . 1 W h y E v o l u t i o n a r y G a m e T h e ory ExistsWhe n C har l e s Darwi n de ve l op e d hi s t he ory of n a t ura l s e l e c t i on , he c re a t e d a p i c t ureo f t he e v ol u t i on ary p roc e s s i n whi c h organ is mi c adap ta ti on was u l t i m at e l y c aus e d b yc om p e ti ti on for s u rv i va l an d re p rodu c t i on . T hi s biologica l "struggle for exis tence"bears considerable resemblance to the h u m a n s t r u g g l e b e t w e e n b u s i n e s s m e n wh o ar estr iving fo r e c o n o m i c s u c c e s s in c om p e t it ive m arke ts . L on g be fore D a r w i np u b l is h e dhi s work, s oc i a l s c i e n t i s t A dam Smi t h had a l re ady c on s i de re d t ha t i n bus i n e s s life,competi t ion is the dr i v i n g forc e be hi n d e c on omi c effic iency and adaptat ion. It isi n de e d v e ry s t r i k i n g how s i mi l ar t he i de as are on whi c h t he foun de rs of mode rnt he ory i n e vol u t i on ary b i o l ogy an d e c on omi c s have bas e d t he i r mai n t hought s .

Ideal ly, this s i m i lar i t y of i de as c ou l d have l e d t o a p e rman e n t i n t e rc han ge be -tween discipl ines after Darwi n ( 185 9) wrot e h is The Origin o f Species. I t s e e ms ,h o w e v e r , as if the theories o f e v o l u t i o n and of e c o n o m i c s first needed to m a t u r ei n d e p e n d e n t l y be fore s uc h an i n t e rdi s c i p l i n ary di a l ogue c ou l d be c ome ve ry f r u i t f u l .B i ol ogi s t s , on t he on e han d , had t o e xp l ore t he m e c han i s m s o f i n he r i t an c e an d toac hi e ve t he i r own s yn t he s i s of t he or i e s about p he n ot yp i c an d ge n e t i c e vol u t i on .E c o n -omi s t s , on t heother han d, had t o de ve l op t he mat he mat i c a l bac kbon e of t he i r c l as s i -c al t he ory of c omp e t i t i on . Thi s bac kbon e s ure l y i s t he t he ory of game s whi c h c ameinto e xi s t e n c e wi t h a f a m o u s book wri t t e n b y v o n N e u m a n n a n d M o r g e n st er n ( 1944) .The i n t e n s e di a l ogue be t we e n bi ol ogy an d e c on omi c s s t ar t e d a fe w de c ade s l a t e r .Bot h di s c i p l i n e s have s i n c e t r i e d t o s ys t e mat i c a l l y e xp l ore what t he i r c on c e p t s havein c o m m o n a nd h o w bi ol ogi s t s and economists c an share th e effort o f f ur t h e r theoryde v e l op me n t . The f i e l d of e v o l u t i o n a r y g a m e t h eo r y e m e rg e d as the m a j o r r e s u l t o fthis explorat ion.

W hat ini tiated the biologica l intere st in gam e theo ry? T he im portant event was ac han ge of p aradi gm re gardi n g t he l e ve l of aggre gat i on ( i .e . , species, p op ul a t i on ,g r o u p , o r i n d i v i d u a l ) at whi c h n at ura l s e l e c t i on s hows its s t ron ge s t effects. Un t i l th eearly 19 60 s, m an y bi ol ogi s ts had he l d t he v ie w t hat t he e vo l u t i on of an organ i s mi ctrai t can be explained by iden t i fying the t rai t 's benefi t to the species , or to other uni ts

3

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4 Peter Hammerste ina b o v e th e level of the i n d i v i d u a l . T h is v i e w w as then deeply shaken (Will iams 1966,M a y na rd Smith 1976). I t nei th er r epr esented Da r w in ' s o r igina l th o u gh ts pr o per ly , no rd id i t s ta nd u p to scr u t iny in th e u pd a ted th eo r y o f evo lu t io n ( b u t see Wilso n a ndD u g a t k i n , b o th in th is vo lu me, f o r a l ter na t ive view s o f th is su b j ec t ) .

We a r e no w u sed to th e id ea th a t na tu r a l se lec t io n tend s to a c t mo r e ef f ec t ive lyat th e level of ind ivid u a ls th a n at h igh er leve ls of aggregated enti t ies. Therefore, w eh a v e a strong inclination to look at na tu r a l se lec t io n " th r o u gh th e e y e s " of the ind i -v i d u a l s th a t ca r r y o u t th e Da r w inia n s t r u ggle f o r ex is tence . T h is h e lps u s to u nd er -stand t h e c o n c e p t u a l l i n k b etw een evo lu t io n a nd th e th eo r y o f ga mes . S imi la r to th eth eo r y o f ev o l u t io n a r y a d a ptat io n, th e latter theory is a lso deeply rooted in method-o l o g i c a l i n d i v i d u a l i s m . A f te r a l l , w e ex pect b u s inessmen to s t r ive f o r th e i r o w n su c-cess . Depend ing o n th e c i r cu msta nces , th is ma y o r ma y no t incr ea se th e w el l -b eingof so cie ty , ver y mu ch l ik e na tu r a l se lec t io n ma y h a ve a po s i t ive o r nega t ive effect o nth e o ver a l l per f o r ma nce o f a n i m a l g r o u p s o r p o p u l a t i o n s ( R i e c h e r t & H a m m e r s t e i n1983).

1.2 Wha t Is an Evolut ionary Game?A c la ss ica l ga me is a mo d el in eco no mic d ecis io n th eo r y d escr ib ing th e po tent ia linteractions of t wo or m o r e ind ivid u a ls w h o se in ter es ts do n ot entirely coincide. T h eterm "game" is c h o s e n b e c a u s e w h e n e v e r w e speci f y su ch a m o d e l , th is r e s e m b l e sthe process o f cr ea t ing a new pa r lo r ga me. W e h a ve to m a k e precise (a) who isinvo lved , ( b ) w h a t a r e th e po ss ib le a c t io ns , a nd ( c) h o w ind ivid u a l su ccess d epend so n the behav ior of a l l part icipan ts. O bviou sly , even a biologist w ho is not dealingw i t h decision theory, b u t w i t h fu n c t i o n a l a na ly s is o f a nima l o r p la nt in ter a c t io ns ,needs exact ly these three ingredients in order to describe th e ph eno ty pic scena r ioof co m pet it io n. T h er efo r e , th e s t r u c tu r e o f a ga me a rises n a tu r a l ly in evo lu t io na r ys tu d ies .

As f a r a s th e ma th ema t ica l r epr esenta t io n o f a b io lo gica l ga me is co ncer ned , th em o d e l e r has a choice o f sever a l f o r ms. For ex a mple , a very explici t description ofth e ph eno ty pic scena r io w o u ld b e given by a g a m e in extensive form (Selten 1983,1988). R o u gh ly spea k ing, th is is a ma th ema t ica l d ecis io n t r ee , th e b r a nch ing po ints( no d es) o f w h ich co r r espo nd to th e p l a y e r s ' o b j ec t ive d ecis io n s i tu a t io ns , a ndb r a nch es s ta nd f o r th e a l ter na t ive a c t io ns th a t a r e po ss ib le in su ch a s i tu a t io n. Asu per impo sed s t r u c tu r e d escr ib es th e po ss ib le inf o r ma t io n s ta tes of the players, an d astrategy is a " l is t of behavioral instructions" for a l l the di fferent inf o r ma t io n s ta tes( su b j ec t ive s i tu a t io ns) w h ich m a y a rise d u r ing a ga me. B y "b eh avio ra l ins t ru c t io n" iti s not necessari ly meant that in a given s i tu a t io n a single a l ternative has to be u sedw i t h probab il i ty 1 . T herefore, an instru c tion can be to use sev eral a l ternatives, eachwith pos it ive proba bil i ty . A strategy is ca l led "pure" if none of i ts instructions are ofthe la t ter type. In other words, in a pure strategy no randomization of act ion takesplace.H o w c a n o n e s implify a g a m e i n e x t e n s i v e f o r m ? A m o r e c o n d e n s e d w a y o fd escr ib ing su ch a g a m e is the so-cal led normal form (a lso referred to as the strategicform). I n th is d escr ip t io n, pu r e s t r a tegies a re n a m e d b y n u m b e r s an d th ei r ins t r u c t io nsa r e no t m a d e ex pl ic i t in th e mo d el . T h e n o r m a l f o r m o n l y c o n t a i n s i n f o r m a t i o n a b o u t

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What I s E v o l u t i o n a r y Game Theory ? 5h ow strategies and payoffs relate to each other. Suppose that there is a finite set ofal ternat ive pure s t rategies and that organisms interact pairwise in a symmetric game.Symme t ry me an s he re t ha t bot h "players" h a v e th e s a m e set of strategies and thatp ayo f f depends only on s t rategies and not on the qu est ion of who is p layer 1 andplayer 2 . A p ayoff mat r i x a=(aij) t he n de s c r i be s what a p l aye r woul d re c e i ve if heplays his r th pure s t rategy against another player who plays his y'th p ure s t ra t e gy. Themat r i x , a, is a l l one needs in order to specify a s y m m e t r i c g a m e in n o r m a l f o r m .Implic i t ly, however, there are more s t rategies than th e ones that define r o w s andc o l u m n s of this matrix. A general strategy .s is a probabi l i ty dis t r ibut ion over a p l ay-er's pure strategies. L et E (s, r) denote th e expected p ayo f f fo r p l a y i n g s u c h as t rategys against another player 's s t rategy r . B i ol ogi c a l l y s p e aki n g , t h i s f u n c t i o n describesh o w an i n di v i dua l ' s e xp e c t e d f i t n e s s i s c han ge d ac c ordi n g t o h i s p e rforman c e i n t heg a m e .W e are n ow e n t e r i n g t he di s c us s i on of t he dyn ami c c on t e xt i n whi c h an e vol u-t ionary game is imbedded. In order to analyze any kind of a game, one needs abackgroundtheory about theprocess thatgeneratesbehavior. This is the point wherebi ol og y an d c l as s i c a l e c on omi c s differ dramatical ly. Theoret ic ians in c lass ical eco-n o m i c s re l y on the p roc e s s of ra t ion al de c i s ion m aki n g. The y i de a li ze t he hu m anbrain as an apparatus with incredibly powerful cogni t ive ski l ls and with the dedica-t i on t o make t he be s t us e of t he m. In c on t ras t , e vol u t i on ary b i o l ogi s t s t e n d t o i n vokenatural se lect ion as the principal "decisionmaker." I n t he i r p i c t ure , in di v i du a l be h av-io r is governed byless p ot e n t me n t a l p roc e dure s whi c h are passed on from generat ionto generat ion via genet ic inheri tance. The biological theory of games is about thee v o l u t i o n of these procedures (s t rategies) . In this approach, sophis t icated behavioraladaptat ions of animals are t h o u g h tt o ref lec t th e c a l c u l a t i on p owe r of thee v o l u t io n a r yprocess , rather than cogni t ive ski l ls of the individual brain.T h e theory of the e v o l u t i o n a r y g a m e can b e based on fair ly di f fe re n t as s ump t i on sa b o u t th e m o d e of inheri tance, and its picture o f genetics c an be either more or lessexpl ic i t . In i t ial ly, evo lut ionary gam e theo ry w as considered to be a p h e n o t y p i c ap-p roac h t o f re que n c y-de p e n de n t s e l e c t i on i n whi c h ge n e t i c s had t o be ap p roxi mat e dvery c rude l y by t he as s ump t i on of e xac t as e xual i n he r i t an c e . Le t us have a br i e f l ookat such a se lect ion m o d e l for a p op ul a t i on wi t h di s c re t e n on ove r l ap p i n g ge n e ra t i on s .Suppose that n different strategies s^ , . .., sn are initial ly present in the p o p u l a t i o n .L et jc , denote the re lat ive frequency of s t rategy st in the populat ion, and le t x=( A : , , . . . ,xn) be the p op ul a t i on f re que n c y di s t r i bu t i on of s t rategies . Suppose that th eexpected fi tness wt of an individual "playing i" i s f re que n c y-de p e n de n t an d t ha t i tca n b e defined as a f u n c t io n w , - ( j c ) . L et w(x) d e n o t e th e p op ul a t i on me an fitness. T h e n ,after o n e generat ion t h e n e w populat ion s tate x ' =(x ,,..., x'n) is g i v e n b y t h e f o l -l owi n g di f fe re n c e e quat i on , kn own as t he discrete replicator equation with frequency-dependent fitness:

In orde r to l ink this repl icator equat ion with a p h e n o t y p i c g a m e , one has to bemore expl ic i t about th e n at ure of the fi tness f u n c t i o n w , - ( j c ) . Sup p os e t ha t in e ve ry

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6 Peter H a m m e r s t e i ngeneration, animals interact pairwise in a game-like si tuation, that pair formation isr a nd o m w ith r espect to strategies, and that a payoff matr ix describes how an individu-al ' s expected fitness is c h a n g e d by the co u r se o f actions in the g a m e . T h e f i tness o fstrategy st ca n th en b e defined as

w h e r e (s(,. j;) is the ga me pa y o f f fo r p la y ing s t r a tegyj; against s tra tegy .v;., and w ( ) isth e basic f i tness ex pecta tio n a n o r ga nism w o u ld h a ve i f it co u ld a vo id p la y ing th eg a m e at a l l .

T h e f i r s t m o d e l in e v o l u t i o n a r y g a m e t h e o r y ( M a y n a r d S m i t h & Price 1973) leftit to the reader ' s i n t u i t i o n t o i m a g i n e t h e d y n a m i c c o n t e x t o f t h e e v o l u t i o n a r y g a m e .H o w ever , i t is o b vio u s th a t e i th er th e d iscr e te r ep lica to r eq u a t io n ( 1) w a s w h a t M a y -nard Smith and Price (1973) had in m i n d , o r e lse a smo o th er ver s io n o f th is mo d el(Taylor & Jonker 1978) , in which the difference equation is replaced b y a c l o s e l yrelated di fferent ial equation (see a lso H o f b a u e r & S i g m u n d 1 9 8 8 ). M a n y b i o lo g i s tsfeel u nea sy w i th th ese eq u a t io ns , b e c a u s e th ey o nly d escr ib e ph eno ty pic ch a nge w i th -o u t k eeping t r a ck o f th e u nd er ly ing genet ics . I nd eed , o nce genet ics i s a d ded to th er epl ica to r eq u a t io n, th e evo lu t io na r y ga me ca n s t r o ngly ch a nge i t s d y na mic proper-t ies. Genet ics th en co nst r a ins th e co u r se o f ph eno ty pic evo lu t io n.U n d o u b t e d l y , th eo r e t ic ia ns h a ve to face a di lemma in this regard. Genetics isi m p o r t an t , but if one studies e v o l u t i o n a r y ga mes to geth er w i th th e u nd er ly ing genet -ics, this often b e c o m e s s u c h a ted io u s ta sk th a t th e th eo r y lo ses mo st of its h e u r i s t i cpo w er. T h e o nly t r a c tab le a ppr o ach es seem to b e o ne - lo c u s m o d els ( r eview ed inC r e s s m a n 1992 ) a nd mo d els o f quan t i t a t i v e genet ics , w h er e ma ny genes w i th ver ys m a l l ef f ec ts a r e co nsid er ed . B o th th ese a ppr o a ch es a r e b a sed o n s t ro ng a ssu m ptio nsa nd th er eb y c i r cu mvent a t lea s t pa r t o f th e d i lemma u nd er d iscu ss io n. Go mu lk iew icz( th is vo lu m e) gives a ver y nice r eview o f h o w f a r o ne gets w i th ev o lu t io na r y ga m eth eo r y in th e f r a mew o r k o f q u a nt i ta tive gene t ics . H o w ev er , a s w e sh a l l see la ter insect io n 1.4, th er e i s a lso a no th er ph i lo so ph y o f h o w to o v e r c o m e t h e m o d e l e r ' s d i -l e m m a . T h i s is th e ph i lo so ph y o f th e "streetcar ," w h ich w o r k s w el l even in th e di f f i -c u l t co ntex t o f general - l o c u s genetics, where genes are a l l o w e d to h a ve s t r o ngeffects o n ph eno ty pes .

1.3 N a s h E q u i l i b r i u m an d E v o l u t i o n a r i [ y S t a b l e S t r a t e g i e sCla ss ica l g a m e th eo r y ( L u ce & Rai f f a 1957, F u d enb er g & T irole 1991) has two dif-ferent b r a n c h e s . I n cooperative game theory, th e ph eno meno n o f cooperation is to acer ta in ex tent a ssu med a nd th u s no t su b j ec t to a complet e a na ly s is . Noncooperativegame theory, on the other hand, seeks to fu l ly explain cooperat ion as w e l l as n o n c o -operation. This branch is the one that matters in evolutionary biology. I t was estab-l i sh ed in c la ss ica l ga m e th eo r y b y N a sh ( 1951). H e su gges ted th a t o ne s h o u l d s t u d ya game by looking for a combination of s trategies (one for each player) with thef o l l o w i n g property. I f a l l players ac t ac co rding to thi s co m binatio n, then "everybody

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What Is E v o l u t i o n a r y G ame T heory? 7ac hi e v e s h is m a x i m u m p a y o f f a g a i n s t th e strategies of al l other players." T h is m e a n st h a t n o b o d y w o u l d h a v e an incentive to uni lateral ly deviate from s u c h a c ombi n at i on .T h e idea c a n b e re p hras e d in technical terms.

1.3.1 N a sh E q u i l ib r iu mA Nash equilibrium is a combination of strategies for the players of a game, suchthat each player 's s t rategy is a best response to the other players ' strategies. A bestresponse i s a s t ra t e gy whi c h maxi mi ze s a p l aye r ' s e xp e c t e d p ayo f f against a f ixedcombinat ion of s t rategies played by the others .

I n order to i l lu s t rate this , le t us co nsider a gam e taking place in real life. In ap s y c h o l o g i c a l e xp e r i me n t , e n ve l op e s are dis t r ibuted to t hre e s ubj e c t s . The y are eachas ke d t o p ut an y amoun t of mon e y be t we e n 0 an d 100 un i t s i n t he i r envelope. N o-b o d y is given a c h a n c e to obs e rve what th e others c on t r i but e . T h e experimenter thenc ol l e c t s th e envelopes and p roc e e ds ac c ordi n g to the fol lowing r u l e w h i c h is k n o w nto everybody. A ll contribut ions are t hrown i n t o th e same box. I fthis bo xcontains 30mon e y un i t s or more , t he e xp e r i me n t e r h i ms e l f w i l l throw 15 addi t ional uni ts intot he box an d a l l t he ac c umul a t e d mon e y wi l l be s p l i t e qua l l y amon g t he t hre e s ubj e c t s .H o w e v e r , if the box contains less than 30 un i t s , all m o n e y g o e s to the experimenter.L e t u s treat this as a three-person game played b y t h e s u b j e c t s . W h a t w o u l d b ea N a s h e q u i l ib r iu m f o r t h em ? W e o n l y wan t t o as k he re for s ymme t r i c e qui l i br i a ,w h e r e e v e r y b o d y p l a y s th esame strategy and thus gives th e s am e a m o u n t . O b v i o u s l y ,(10, 10, 10) is a sy m m e t ri c N a s h e q u i l ib r iu m . T h e reason is t h a t no p l aye r has ani n c e n t i v e t o un i l a t e ra l l y de vi a t e from this solut ion: If the other two players each play10, the third player achieves h is m a x i m u m by a l s o p l a y i n g 10. I n t h i s N a s h e q u i li b -r i u m , the players cooperat ively exploi t their r e s o u r c e t h e e x p e r i m e n t e r a n dac hi e v e a net e q u i l i b r i u m p a y o f f of 5 u n i t s . W e n o w h a v e to ask whe t he r t h i s is theo nly s ol u t i on t o t he game . There i s in d e ed a n o th e r s y m m e t r ic N a s h e q u i l i b r i u m , (0 ,0 , 0 ) , i n whi c h a l l t hre e s ubj e c t s han d ove r e mp t y e n ve l op e s t o t he e xp e r i me n t e r .O b v i o u s l y , n obody has an i n c e n t i ve t o de vi a t e from t h i s s o l u t i on . Fur t he rmore , n ocooperat ion takesplace, t he re s ourc e re mai n s un e xp l oi t e d, an d t he e qui l i br i um payoffis zero.

This t yp e of game i s we l l kn own an d has be e n p l aye d i n n ume rous e xp e r i me n t s .I t i s presented here to give the reader a feel for the N as h e qu i l ibr i um . W e obs e rve ap h e n o m e n o n w h i c h is typical fo r game s , n ame l y t ha t t he re is more t han one s u c he q u i l i b r i u m . O n e w o u l d t h i nk t h a t th e p l a y e r s sh o u l d pl a y t h e coopera t ives o l u t io n .H owe v e r , i m ag i n e a pl aye r who worr ie s a bo ut t he ri s k in vol ve d i f o the r p laye rs c om eto a different c on c l us i on abo u t t he c hoi c e o f e q u i l i br i um . I f t h is p l aye r pl ays ze ro, heis re lat ively safe an d c an o n l y m i s s th e smal l cooperat ive payoff c o n s i s t i n go f 5 u n i ts .O t he rwi s e , he m i ght los e t he l a rge r am ou n t of 10 u n i t s i f he c oop e rat es an d an ot he rp l aye r fa i l s t o do s o . Thi s woul d be an argume n t i n favor of t he n on c oop e rat i veso lu t io n. O b v i o u s l y , i t c a n b e di f f i cu l t todecide be twe e n a l t e rn at ive N as h e qui l ibr ia .T u r n i n g b a c k n ow t o b i o l o g y , th e i m p ort an t s o l u t i on c on c e p t de ve l op e d b y M a y-n ard Sm i th an d P r ic e ( 197 3 ) , an d fore s hadow e d by H am i l ton ( 1967 ) , i s tha t of anev o l u t i o narily s t ab l e s t ra te gy ( E SS) . In t u i t i ve l y sp e akin g , E SS t he ory draw s t he m od-eler's at tent ion to populat ion s tates which are res is tant against the f o r c e s of se lect ionand m ut a t i on . A n ESS s e n s u M ayn ard Sm i th ( 19 82) is a s t ra t e gy wi t h the fo l l owi n g

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8 Peter H a m m e r s t e i np r o p e r t y : If a l l m e m b e r s o f a p o p u l a t i o n are g e n e t i c a l l y coded to play this strategy,any i n i t i a l l y r ar e m u t an t s t ra t e g y w o u ld re c e i v e n e g at i v e s e le c t io n pr essur e in thisp o p u l a t i o n .

L e t u s s e e w h a t t h i s m e a n s i n t h e f o r m a l c o n t e x t o f e q u a t i o n ( 1 ) . S uppose t h a tthe populat ion plays s trategy s{ an d t hat a m u t a n t s2 ar ises . Let the strategy f r e q u e n -c i e s be jt , = 1e and x2= e, so t h a t e is the m u t a n t f re q u e n c y . T h e e v o l u t i o n a r ystabi l i ty of 5] m e a n s t h a t for s u f f i c i e n t l y s m a l l e th e d i f f e r e n c e w](x) w2(x) m u s t bep o s i t i v e . If we are d e a l i n g w i t h r a n d o m p a i r w i s e i n t e r a c t i o n sand if fi tness is d e f i n e das in ( 2 ) , t h i s d i f f e r e n c e c a n b e w r i t t e n as f o l l o w s :

W e a r e n o w a b l e t o s e e h o w t h e N a s h e q u i l i b r i u m e me rge s i n e v o lu t i o n ar y b i o l -o g y . I f fo r s o m e m u t a n t s t ra t e g y the first s q u a r e b r a c k e t in (3) is n e g at i v e , t he nw t( x) w2(x) becomes negative for s u f f i c i e n t ly s m al l v a lue s of e. I norder to e x c l u d et h i s p o s s i b i l i t y , one has to r e q u i r e fo r a n E S S S L t hat it s h o u l d b e a best re s p on s e toi t se l f in the p h e n o t y p i c g a m e . T h e l a t t e r r e q u i r e m e n t c an be rephrased b y s a y i n g t h a tth e s y m m e t r i c p a i r o f s trategies ( S | , S ] ) s h o u l d b e a N a sh e q u i l i b ri u m .T hi s i s a n e c e s s ar y , bu t n o t s u f f i c i e n t , c o n d i t i o n f o r (3 ) t o be p o s i t i v e . S up p o s ethere is another s trategy s2 w h i c h is dif ferent from s,, b u t also a best re s p on s e to s^ .T h e f i r s t s quar e br ac k e t in (3) t h e n is zero and the second square bracket needs to bep o s i t i v e in orde r fo r select ion t o ac t ag ai n s t th e m u t a n t u n d e r c o n si d e ra t io n . C l e ar ly ,t h i s i s o n l y t h e c a s e if E(sl,s2)>E(s2,s2). I n o r d e r to e n s u r e e v o lu t i o n ar y s t abi l it y ,t h i s s e c o n d c o n d i t i o n n e e d s to ho ld for all s trategies s2 t h a t are a l t e r n at i v e be s t re-sponses to 5 , .

W e h a v e n o w r e c a p i t u l a t e d M a y n a r d S m i t h ' s o r ig i n a l t h o u g h t s u s i n g t h e l a n -g uag e o f g am e t he o r y . U s i n g t h i s lan g uag e ag ai n , h i s t e c hn i c a l def in i t ion o f an ES Sc an be d e s c r i be d as f o l lo w s . I t r e la t e s t o a s ym m e t r i c g am e i n s t r a t e g i c f o r m an d t ot h e d y n a m i c c o n t e x t o f ( 1 ) w i t h fi tn e s s f u n c t i o n (2 ) . A s trategy st is cal led evolution-arily stable if it sat isf ies th e f o l l o w i n gt w o c o n d i t i o n s :

1.3.2 Properties of an ESS1 . Property of a s y m m e t r i c N a s h e q u i l i b r i u m : . v , is a best response to s} . I n o t he rw o r d s , if an o p p o n e n t p lays t h i s s trategy, o n e r e c e i v e s th e hi g he s t p o s s i b le p ayo f f b yals o p layi n g t h i s s t r a t e g y.2. Stab i l i ty against a l tern at iv e best responses: I f a s t rategy s2 is d if ferent f rom s,

an d E( s2,s l)=E ( sl , s ] ) , t he n t he i n e q u a l i t y (s,,.s2) > E(s2,s2) ho ld s . I n o t he r w o r d s , i fanother s trategy also a c h i e v e s th e hi g he s t p ayo f f ag ai n s ts t h e n it is better to playagainst this o ther s trategy than to p l a y th e lat ter s t rategy s2 against i tse lf .

From an e c o n o m i s t 's p o i n t o f v i e w , it i s i n t e r e s t i n g t o n o t e t ha t w e o n l y hav e t os t ud y th e p h e n o t y p i c g a m e m o d e l in order to see w he t he r t he s e t w o E S S c o n d i t i o n sh o l d . T h e s e le c t i o n e quat i o n ( 1 ) is c o m p l e t e l y h i d d e n in the b a c k g r o u n d t h e o r y t h a tgave r ise to t h e se c o n d i t i o n s . N a s h h i m s e l f f o r e s a w th e p o s s i b i l i t y o f d y n a m i c i n t e r -

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W h a t I s E v o l u t i o n a r y Game Theory ? 9pretations of th e N ash eq u i l ib r i um w h en h e ment i oned i n h i s wor k th e "mass actioninterpretat ion" of his solut ion concept as an al ternat ive to decision-theoret ic interpre-t a ti ons . H owev er , it surely w as JVIaynard S m i t h ( 1 9 8 2 ) w h o first created s u c h a the-ory. Furthermore, the now f a m o u s s t a t e m e n t o f N a s h w as b u r i e d in his Ph.D. thesisan d c a n n o t b e f o u n d in his p u b l i c a t io n s . I t was only r edi scover ed w h en N ash r ece ivedthe N obel Prize in 1994.

1 .4 G e n e t i c s an d t he E v o l u t i o n a r y G a m eI n order to enr i ch evolut i onar y game t h eor y wi t h expl ic i t a s s u m p t i o n s a b o u t t he ge -net i cs of b eh avi or , one would h ave t o know wh at genes ar e i nvolved , h ow t h eyrecombine, and how phenotypes and genotypes relate to each other . I t often seemsl ike a hopeless task to t ry to accomplish this goal empir ical ly . Even if one sequencess t r a n d s o f D N A complet e ly for t h e i r molecular s t r uc t ur e , t h i s does not yet r evealh ow genot y pes ar e t r ans lat ed i nt o complex ph enot y pes b y development al pr ocesses .Therefore, it is t empt i ng to s t u d y th e evolut i on of p he n ot yp i c traits in p u r e l y p h e n o -t y pi c m odels . O r gani smi c b i o logi s t s ar e q u i te i mpr essed wi t h th e e xplana t or y powero f s u c h m o d e l s in their empir ical s tudies. This raises an i nt er es t i ng q ues t i on:W h y d ow e u n d e r s t a n d m a n y e x a m p l e s of evolut ionary adaptat ion so w e l l w i t h o u t k n o w i n gt h ei r genet i c b ackgr ound?

I n the remainder of this sect ion, I shal l review a theory which at tempts to answerthis quest ion. This is the theory of long-term evolut ion which had i ts or igin in publi-cat ions by E s h e l (1982) , Lessar d (1984), and most not ab ly b y E s h e l and Feldman(1984) . T h e development o f this theory and its applications is due to a n u m b e r offu r t h er pub l i cat i ons , i nc ludi ng th e w o r k o f L i b e r m a n an d F eldman (1986) , L i b e r m a n(1988), Lessard (1990), Eshel (1991, 1996), Eshel an d S a n s o n e ( 1 9 9 1 ) , M a t es s i an dEshel (1992) , M at ess i a n d D i Pasq uale (1996) , an d Weissing (1996). I n m y v i e w , I lanE s h e l h as p lay ed a par t i cu lar ly i mpor t ant role in the c o n c e p t u a l d e v e l o p m e n t of thet h eor y under d i scuss i on.

A streetcar metaphor ( H a m m e r s t e i n 1996) can be used to explain this theory(Fig. 1.1). T h e movi ng s t r ee t car is an evolvi ng populat i on. S uppose t h at in t h i s popu-lation an inherited trait is s u b j e c t to natural select ion and is coded for by m o r e t h a no n e gene. A s M or an (1964) an d K ar l i n (1975) h ave sh own, i t can easi ly happen thatphenotypic adaptat ion o f this trait is made i mposs i b le by the "r esh uf f l i ng" of g e n e sd u e t o r ecomb i nat i on. Th er ef or e , nat ur a l se lec t i on may dr i ve t h e populat i on t owar d ag e n o t y p e f r e q u e n c y e q u i l i b r i u m in w h i c h p h e n o t y p e s fail to m a x i m i z e fitness or toplay an E S S . T h i s is the first temporary stop of the streetcar an d we n ow dr aw ou rattention to new passengers that m ay ent er it .I t was the ingenious idea ofEshel and Feldm an (1984) to extend at this pointth e c lass i cal f r amewor k o f populat i on genet i cs and totake a par t i cu lar ly wi de r angeo f pot ent i a l mut at i ons (new passenger s) i n t o account . Th e gener al i dea wh i chemer ges from their approach is that if genetics is in the way of ph enot y pi c adapt a-tion, then it is possible to c o n c e i v e a t l e a s t m a t h e m a t i c a l l y a m u t a n tallele w h i c hw o u l d c a u s e fur ther p h e n o t y p i c e v o l u t i o n . I m a g i n e, fo r example , th e case o f sicklec e l l anemi a. I f a m u t a n tallele is able to p r o d u c e th e s a m e effect against malar i a asthe heterozygote, i t may spread and reduce the prevalence of s ickle cell anemi a i n

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1 0 Peter H a m m e r s t e i n

Figure I . I .Course of an evolvi ng populat ion in phe noty pe space. T he po pu la t i o ns to ps a t a gen o type f req u en c y eq u i l i br i u m . T em po ra ry s to ps a re t h en lef t after thep o p u l a t i o n is p e r t u r b e d by an appropriate n ew m u t a n t a l l e l e . A f i n a l stop is p h e n o -t y p i ca l l y s t a ble a g a i n s t gen et i c per t u rb a t i o n . A t s u c h a stop, ph en o t ypes a c t a c -c o rd i n g t o ga m e-t h eo ret i c pr i n c i ples . T h e s t reetc a r t h eo ry c o m pa res t h e c o u rs e o fan e v o l v i n g p o p u l a t i o n w i t h t h a t o f a s t reet c a r an d d ra ws th e m o d e l e r ' s attention toth e a n a l y s i s of f inal s t o ps (from H a m m e r s t e in 1 9 96 ).

the population. In other words, i f such a passenger enters the streetcar , the streetcarwill b e m o v i n g again. A t a sub sequ ent stop, anot h er n ew passenger m ay play th es a m e role. I n principle, this process may go on forever , or no stop may ever bereached.A streetcar has a final stop where it c o m e s to a permanent rest . At a f inal stop,new passengers may enter the streetcar , but i t wil l not move. Suppose that the popula-t ion has reached such a f inal stop,where no m utation would b e able toinitiate newph eno ty pic evo lu t io n. Wh a t ca n b e sa id a b o u t th e pr o per t ies o f su ch a f ina l s to p? T h e

a nsw er to t h i s q u e s t i o n w o u l d h a v e p le a se d C h a r l e s D a r w i n , b e c a u s e it r e v e a l s ano v e r w h e l m i n g e f f e c t of the ph eno ty pic f o r ces o f n a t u r a l s e l e c t i o n .1.4.1 S u r viva l of the Fittest

I f se lec tio n i s fr eq u enc y - ind e pend e nt , th en ph eno ty pic f itness m a x im iz a t io n i s to b eexpected at a final stop, an d ph eno ty pic o p t i m al i t y th eo r y a ppl ies (E sh el & F e l d m a n1984, L ib erma n 1988) u nd er a w id e ra nge o f co n d i t io ns . T h e tech nica l te r m u sed inth e o r igina l l i te r a tu r e ins tea d o f "final stop" is t h a t o f "external stability." A ny co n-

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W h a t I s E v o l u t i o n a r y Game Theory? I Ic e i v a b l e m u t a n t is here t aken i nt o account as a potential passenger , unless it c o u l dd is tu r b th ebasic mech ani sm of i nh er i t ance , such as a segr egat i on d i s t or t er would do .T h e reason not to consider th e lat ter type o f m u t a n t is t h a t an y s t o p b e it temporaryor f i n a l woul d be v u l n e r a b l e to invasion o f such genes . S uch gener al i nvas i b i l i t yi snot the cent r a l sub jec t of the t h e o r y u n d e r d i s c u s s i o n . E n v i r o n m e n t a l c h a n g e alsoe n s u r e s that no final stop is an eternal s top. E v e n tu a l l y , a streetcar wil l s tar t movingagain, and so d o e s an evolvi ng populat i on.

1.4.2 A S tre etc a r N a m e d N a s hW h e n selection is f r e q u e n c y - d e p e n d e n t an d t he phenotypic scenario is descr ibed b yr a n d o m pairwise interact ions in a s y m m e t r ic t w o - pe r so n g a m e in nor mal f or m, th ef o l lo w ing c an be s h o w n .T h e populat i on mean s t r at egy of any final stop is nec essar ilya b es t r esponse to itself in the p h e n o t y p i c g a m e . T h e r e f o r e , it cor r esponds to a s y m -m etr ic N ash eq ui l i b r ium ( th e f i rs t ESS condi t i on h o lds) . Th i s is an a l m o s t o b v i o u sresul t in light of the concept ual i nnovat i on byEshel and Feldman (1984).Togetherw ith a s i mple pr oof for the t wo- locus case o f viabi l i ty select ion, this resul t is f o r m u -lated i n sur vey art ic les b y H amm er st e in and S el ten (1994) and H am m er s t e in (1996) ,w h o s t udy mat h emat i cal pr ob lems with th e "interface" between th e theories o f l o n g -term evolut i on and of the e v o l u t i o n a r y g a m e . T h e y u s e , in par t i cu lar , th e f o l l o w i n gs u b t l e a d j u s t m e n t of the concept of external stability. I n evolut i onar y game t h eor y , itis inappropriate to general ly require that no mutantallele can invade at a f inal s top.T h e simple reason is that different genetic populat ion states m ay pr oduce th e s a m emean populat i on s t r at egy . Th i s means t h at a l le les may i nvade wi t h out leadi ng t o an ew phenotypic picture. Therefore, test ing for the "f inal s top property" means lookingat genetic per turbat ion an d aski ng wh et h er th e phenotypic t rajectory w i l l take th epo pu la t io n b a c k to its unper t ur b ed ph enot y pi c s t a t e . O n e c o u l d say t h at th e s t u d y o ffinal stops is that o f "phen otypic s tabi l ity against genetic per tu rbat ion" (H am m erstein1996).

1.4.3 A S t ree tcar N am ed E S SConsi der a f inal s t op wh i ch is ph enot y pi cally m o n o m o r p h i c in the sense that th esame strategy is played by al l the i n d i v i d u a l s o f a p o p u l a t i o n (at the genetic levelthere need not be f ixat ion of al l al le les) . Such a s top has the fol lowing strongerpropert ies . Under assumptionslike i n t h e pr evi ous par agr aph , a ph enot y pi cal ly mono-mor ph i c final stop necessar i ly satisfies t h e t w o E S S condit ions of M a y n a rd S m i th .Conver se ly , i f an ESS i s played b y a populat i on, no m u t a n tallele c an i nvade if it isu n a b l e t o gener at e t h e E S S wi t h t h e gi ven genet i c b ackgr ound (Hammer s t e i n & S el -te n 1994, H am m er s te i n 1996). Wei ss i ng (1996 ) s h o w s h o w t h e r e s u l t s a b o u t N a s he q u i l i b r i u m and t h e E S S can b e gener al i z ed t o n- locus models and t o nonl i nearg a m e s .

In t he r e m a i n d e r o f this sect ion, tw o mat h emat i cal aspect s of the m o d e l u n d e rdiscussion are descr ibed, namely the select ion equation and the f i tness f un c t i o n . T h er e s u l t s about this model and i ts general izat ions are not presented here and can b ef o u n d in the l i terature ci ted above. L e t u s h a v e a look at the 2-locus case. W e c o n -sider th e s t andar d model descr i b i ng vi ab i l i t y se lec t i on in an infinite d i p l o i d p o p u l a -

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1 2 Peter H a m m e r s t e i nt ion wi t h nonoverlapping g e n e r a t i o n s , s e x u a l r e p r o d u c t i o n , a n d r a n d o m m a t i n g . C o n -s id er i n t h i s p o p u l a t i o n a n ev o l v i n g t ra it wh i c h i s c o d ed fo r b y t wo g en es , A a n d B .S u p p o s e t h a t t h e r e aren a l l e l e sA1;. . . ,An of g e n eA, and m allelesBh . . . ,Bmofg e n e B . A s s u m e t h a t r e c o m b i n a t i o n t a k e splace a t a f ixed rate r , w i t h 0

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What I s E v o l u t i o n a r y Game Theory ? 13

1 .5 Co n c l u d i n g R e m ar ksDoe s g ame t he ory have i t s p l ac e i n t he edifice of b i o l ogi c a l t hought , an d, i f s o , whe reis th i s p l ac e ? O bvi ou s l y , th e "me t hodol ogi c a l i n di v i dua l i s m" of game t he ory is thei mp ort an t "preadaptation" for i ts use in e vol u t i on ary b i o l ogy ( but see W i l s o n andD u g a t k i n , b o t h in t hi s vol ume , fo r al ternat ive views of t hi s s ubj e c t ) .A s pointed o u tin sect ion 1.1, the typical pic ture of phenotypic evolut ion is one in which nothingm att e rs m ore t han in di v i du a l p e rform an c e i n re p rodu c t i ve c o m p e ti ti on . T he re fore ,game models arise so natural ly in biology that i t even seems di ff icul t t o avoi d t he m.H owe v e r , wh at t he s e game s de s c r ibe is on l y th e re l a t i on s hi p be t we e n be havi or andf it n e s s i n org a n i s mi c i n t e rac t ion s . For an e vol u t ion ary a n al y s i s , m ore t han t h is n e e d sto be specif ied. The re fore , e vol u t i on ary game t he ory c on s i de rs th e p h e n o t y p i c g a m ein t he wi de r c on t e xt of a dyn ami c mode l of n a t ura l s e l e c t i on .

T h i s d y n a m i c m o d e l is bas e d o n as s ump t i on s about th e i n t e rac t i on s t ruc t ure of ap op ul a t i on , it s mat i n g s t ru c t u re , an d t he m e c h an i s m s of i n he r i tan c e . In s e c t i on s 1 .2and 1.3, w e di s c us s e d a part icular se lect ion model , in whi c h i n t e rac t i on p ar t n e rs areran dom l y c hos e n an d in he r i tan c e i s as sum e d t o be as e xu al an d e xac t . O bvi ou s l y , n om at in g s t ruc t ure n ee de d t o be s p e c i fi e d i n th i s c as e. We the n fo l l ow e d M ayn ard Sm i than d as ke d t he que s t i on of how t o c harac t e r i ze for t h i s mode l an ESS. I t t u rn e d outt hat s uc h a s t ra t e gy mus t b e a "best response to i tse lf" in the p he n ot yp i c game . I not he r words , e ve ry E S S c orre s p on ds to a p ar ti c u l ar N as h e qu i l ibr i um in this context .Thi s l as t s t a t e me n t may s oun d l i ke a ve ry t e c hn i c a l re mark, bu t i t has a ve ryinterest ing connotat ion: T h e mos t i mp ort an t c on c e p t of c l as s i c a l game t he ory de-s c r i be s th e mai n c harac t e r i s t i c of an E S S . O ne c ou l d hardl y i magi n e a s t ron ge r argu-m e n t in order to de fe n d th e i mp ort an t ro l e of game t he ory in bi ol ogy. H owe ve r , letu s n ot forg e t about a l l t he as s ump t i on s made . In re a l l ife, the interact ion s t ructurecan be far from ran dom, t he mode of re p roduc t i on i s u s u a l l y s e xual , an d mat i n g c anhave i ts assortat ive and disassortat ive aspects . Al l these aspects can, in principle ,c h a n g e th e characterizat ion of an E S S .

At the end of sect ion 1.4, we had a br i e f l ook a t an e vol u t i on ary game i n whi c hstrategies arecoded for by two re c ombi n i n g ge n e s in a dip loi d M e n d e l i an p op u l a t i on .Thi s mode l has a ran dom i n t e rac t i on s t ruc t ure an d mat i n g i s ran dom as we l l . A c r i t i cof e v ol u t i on ary game t he ory woul d c omp l ai n t ha t e ve n i n s uc h a ra t he r s i mp l e mode l ,genet ics c an impose severe constraints o n p he n ot yp i c e vol u t i on . From t h i s wel l -f o u n d e d point of view, i t seems naive to analyze an evolut ionary game merely at thep he n ot yp i c l e ve l . I f n ot hi n g c ou l d be he l d agai n s t t h i s v i e w, game t he ory w o u l dalready have lost i t s p lace in the edifice of biological thought . Fortunate ly, this i s notthe case. A s disc u ssed in sect ion 1.4, genet ic c on strain ts are u n l i k e l y to p lay a m a j o rrole in l o n g - t e r m e v o l u t i o n . T h e reason is t hat e vol u t i on no t on l y c han ge s p he n o-ty pes , but a l s o a l te rs the u n de r l y i n g ge n e t i c s . T hi s c aus e s a p roc e ss of s uc c e s s i veremoval of s uc h c on s t ra i n t s . I f this process ever comes to a p e r m a n e n trest in a g i ve ne n v i ron me n t , p he n ot yp e s wi l l be h i ghl y adap t e d t o t h i s e n v i r o n m e n t t h e s i tuat ion inwhi c h game t he ory mat t e rs .A C K N O W L E D G M E N T S Iwish to thank Jack Werren, an a nonymous referee, and the editors ofthis v o l u m e f o r a n u m b e r o f ver y he l p f u l c o m m e n t s on an earlier draft of this cha pter .

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159-170.L i b e r m a n , U . 1 9 8 8 . Exte rnal s tab i l i ty and ESS: Cri te r ia for ini t ia l incre ase of a ne w mutant

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34 ,613-653.M at ess i , C. & E she l , I . 1 9 92 . Se x ra t io in soc ia l hy m e nopte ra: a popula t ion g e ne t ic s tudy ofl o n g - t e r m e v o l u t i o n . Am. Nat., 139,276-312.M a y n a r d Smith , J . 1 9 76 . Group se le c t ion. Q . Rev. BioL, 51 ,277-283.

M a y n a r d S m i t h , J . 1 9 8 2 . E v o l u t i o n and the T h e o r y o f G a m e s . C a m b r i d g e : C a m b r i d g e U n i v e r -sity Press.M a y n a r d S m i t h , J. &Price, G . R . 1973. T h e logic of a n i m a l c o n f l ic t . Nature, 246, 15-18.

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N ash, J. F . 1951. N o n - c o o p e r a ti v e g a m e s . Ann. Math., 54, 286295.R i ec h er t , S . E . & H a m m e r s t e i n , P. 1983. G a m e t h e o r y in the e c o l o g i c a l c o n t e x t .Annu. Rev.Ecol. Syst., 14,377-409.S e l t e n , R . 1 9 8 3. E v o l u t i o n a r ys t a b i l i t y in e x t e n s i v e 2 - p e r s o n g a m e s . Math. Social Sciences, 5,269-363.

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L U C - A L A I N G I R A L D E A UB A R B A R A L I V O R E I L

GameTheory and Soc ia l Foraging

2 .1 IntroductionF o r a ging b eh a vio r i s centr a l to i ssu es su ch a s th e s t r u c tu r e a nd co mpo si t io n o f a nima lc o m m u n i t i e s ( S c h o e n e r 1 9 8 7 ) , th e a b u n d a n c e an d spat ia l d i s t r i b u t i o n o f o r g a n i s m s(Fretwell 1972) , th e ex tent to w h i c h s o c i a l i t y m ay evolve (Wilson 1975, pp.49-57),and intra- and in terspecif ic competi t ion. I t is not surprising that this realization in thel a te 1 9 60 s ( M a c A r t h u r & Pia nk a 1966) spa w ned an ex po nent ia l su r ge o f researchinto f o r a ging b eh a vio r ( S ch o ener 1987) . T h e a ppl ica t io n o f o pt ima l i ty mo d els to th es t u d y o f f o r a ging b eh a vio r led to a nu mb er o f s imple , ex pl ic i t , e c o n o m i c f o r a gingmo d els th a t ma d e q u a nt i ta t ive , tes ta b le pr ed ic t io ns . As a co nseq u ence , a f ie ld k no w ntoday as " F o r a g i n g T h e o r y " q u i c k l y e m e r g e d in the m i d - 1 9 7 0 s ( S t e p h e n s & K r e b s1 9 8 6 ) .

T h e u se o f s imple o pt ima l i ty mo d els h a s b een part icu lar ly s u c c e s s f u l in ad-d r ess ing tw o f o r a ging d ecis io ns: w h eth er to a t ta ck a n enco u nter ed pr ey ( pr ey mo d els)and w h eth er to per s is t ex plo i t ing a pa tch ( pa tch mo d els ; S teph ens & Krebs 1986).S imple o pt ima l i ty mo d els , h o w ever , are a ppl ica b le s t r ic t ly to si tuations where a be-havioral a l ternative can be assigned a payoff independently of the use of the same o rdifferent b eh a vio r a l a l ter na t ives b y o th er po pu la t io n memb er s . I n sh o r t , th ey d o no tpertain to s i tu a t io ns o f f r eq u ency -d epend ence th a t ch a r a cter iz e ma ny o f th ep o p u l a t i o n - l e v e l ph eno mena f o r a ging th eo r y pu r po r ted to a d d r ess . T o e m b r a c e thesep r o b l e m s , f o r ag i n g th eo r y mu st r e ly mo r e o n ga me th eo r y as an e c o n o m i c m o d e l in gtool. W e cal l this area of f o r a ging th eo r y "Social ForagingTheory."

2 .2 Soc ia l ForagingTheoryS o cia l f o r a ging d ecis io ns are ch a r a cter iz ed b y th e f r e q u e n c y - d e p e n d e n c e o f their pay-offs. A c o m m o n e x a m p l e o f a socia l f o r a ging d ecis io n is g r o u p m e m b e r s h i p . S o c i a l

1 6

2

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G a m e Theory an d Social Foraging 1 7car ni vor es an d some social spiders , fo r instance, at tack their prey in smal l gr oups( G i ra l de au 1988) . I n some cases , th e success of the at t ack depends on t he n u m b e r o fi ndi vi duals engaged in the at tack (Giraldeau 1988) . T he payoff that o n e ob t ai ns f r omj o i n i n g ah u n t i n g g r o u p d e p e n d s on i ts size an d h e n c e th e n u m b e ro f o t h er i ndi vi dualsthat h ave made t h e same deci s i on (S i b ly 1983; P u l l i a m & Car aco 1984; Clar k &M angel 1984; G i ra ldeau 1988; G i ra ldeau et al . 1994a) . Anot h er commonly s t udi edsoci a l f or agi ng deci s i on concer ns th e distr ibut ion of c o n s u m e r s o v e r a n u m b e r o fr esour ce pat ch es , a pr ob lem t h at h as b een appr oach ed us i ng Ideal Free D i s t r i b ut i ont he ory (Fretwell 1972, M i li n sk i & Parker 1991). Social f o r ag i n g decisions c an alsoi n v o l v e patch u se w h e n g r o u p s o f foragers exploi t a pat ch concur r ent ly (Par ker1978). T h e i n d i v i d u a lpayoffs for any patch residence t ime depend on the n u m b e r o fcompeti tors that have remained in or left th e patch (Parker 1978, Sjerps & H a c c o u1994) and s imple ver s i ons h av e b een tes t ed wi t h som e success (B eau ch am p and G i r -a ldeau 1997, L i vore i l and G i ra ldeau 1997) . T h e com peti ti ve pat ch explo i t a t i on gam eh as r ecent ly b een modeled as an -person continuous s trategy (i .e . , a war of attrition)between foragers (Sjerps & H a c c o u 1 9 9 4) . If the pat ch cont ai ns a n u m b e r o f differentprey types, then th e f or ager s m u s t also decide on t he b es t poss i b le comb i nat i on o ftypes, once agai n a pr ob lem t h at r eq ui r es a game t h eor e t i c analy s i s . F i nal ly , a com-m o n g r o u p f o r ag i n g deci s i on concer ns wh et h er tojo i n th e d i scover i es made b y fe l lowg r o u p memb er s . Th e dec i s i on h as b een associ at ed wi t h a game known as t h e pr o-ducer-scrounger (PS) gam e (B ar nar d & S i b ly 1981) , wh i ch h as b een modeled spe-cif ica l ly for a f or agi ng sy s tem only r ecent ly (V i cker y et al . 1991, Car aco &G i ra l d e au1 99 1) .In this chapter , we introduce readers to the general PS game, i ts applicat ion to asocial foraging decision, and some resul ts of experimental test ing of the game's pre-dict ions. We f irs t present two versions of the PS foraging game; onebased on f or ag-in g r ate m axi m i z ati on (V i ckery et al . 1991), th e other o n short fal l m i ni m i z at i on ( i.e. ,r i sk- sens i t i ve; Car aco & Gi r a ldeau 1991). For each, we review the resul ts of experi-ment al t es t s of t h e i r pr edi c t i ons . I n doi ng so, we p u r s u e tw o goals: (1) to demonst r at eth a t economi c model i ng wi t h i n f o r ag i n g t h eor y ext ends b ey ond pat ch an d prey deci-s i o n s and (2) to sh ow h o w game t h eor e t i c models c an m a k e n o v e l , q u a n t i t a t i v e ,an dtestable predict ions concerning social foraging behavior .

2 .3 J o i n i n g O t h e r I n d i v i d u a l s ' F o o d D i s c o v e r i e s an d t h e P S G a m eWh en i ndi vi duals sear ch fo r food in a gr oup, i nf or mat i on ab out th e locat ion o f a f oodpatch soon spreads to other grou p mem bers. T radit ional ly , g rou p forag ing behav iorhas beenmodeled as an i nf or mat i on- sh ar i ng process t h at assum es t h at a l l grou p m em-bers search fo r food independent ly , and, upon discovery b y o n e g r o u p m e m b e r , allcease searching and m o v e t o w a r d th e s u c c e s s f u l i n d i v i d u a l to gain a share of thefood (see, for instance, Clark & M angel 1984, M angel 1990, R ant a e t a l . 1993) . T h i si n format i on -s har i n g scenario h as b een developed most ly as an evolut i onar y explana-t ion for gr oup f or agi ng. Inf or mat i on- sh ar i ng models mer e ly assume t h at i ndi vi dualsj o i n all discoveries, b u t they d o n o t exami ne wh et h er it is pr of i t ab le t o d o s o . T h equestion remains whether selection w o u l d maintain this high frequency ofjoiningb e h a v i o r w i t h i n f or agi ng gr oups . W e propose to s tu d y j o i n i n gas a f o r ag i n g d e c i s i o n

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1 8 Luc-Alain Giraldeau Barbara Livorei lw h o s e ecologica l d e t e r m i n a n t s c a n b e u n c o v e r e d t h r o u g h m o d e l i n g o f i t s e c o n o m i cco nseq u ences . We a ssu me th a t w h en a gr o u p memb er d isco ver s f o o d , a l l o th er gr o u pm e m b e r s d e c i d e w h e t h e r t h e y w i l l pa r ta k e o f th e d isco ver y o r co nt inu e sea r ch ing f o rfo o d .

B a r n a r d a n d S i b ly ( 1981) w er e th e f i rs t to po int o u t th a t j o ining co nf o r m s to a na l ter na t ive-o pt io n scr a mb le th ey cal led the PS g a m e . T h e P S g a m e is an n-per so nga me ( a scr a mb le) b eca u se th e o u tco me o f a ny o ne p la y o f th e ga me d epend s o n th ecomposit ion of s trategies within the group, not the strategy used by a single oppo-n e n t . I t i s a n a l ter na t iv e o pt io n ga me, b e ca u se o n a ny o ne p la y o f th e ga m e th epla y er s h a ve o nly o ne a l ter na t ive , pr o d u cer (P) or scr o u nger ( S ) . I n t e r m s of g r o u pf o r a ging, p la y ing P mea ns sea r ch ing f o r f o o d w h i le never j o ining a nd S mea ns neversea r ch ing f o r f o o d w h i le a lw a y s j o ining , w h en a n o ppo r tu ni ty a r ises . T h e pr o po r tio nof th e po pu la t io n p la y ing P is q and the proportion playing S is (1q). T h e p a y o f f so f b o th P and S a l te r na t iv es a r e b o th a f un c t i o n of q: W(q\P) and W(q\S), respec-t ively.

Parker (1984) also identified th e ideal free distribution (1FD) as an n-persona l ter na t ive o pt io n ga me w h en th e d e c i s i o n i n v o l v e s th e d i s t r i b u t i o n o f n c o n s u m e r so ver tw o r eso u r ce pa tch es ( w h en j pa tch es are i n v o l v e d , it is an n-per so n y'-optiong a m e ) . H o w e v e r , t h e re a r e a n u m b e r o f f u n d a m e n t a l d i s t i n c t i o n s b e t w e e n IFD andPS g a m e s th a t a r e w o r th po int ing o u t here, i f o nly to ind ica te th e ex t r a co mplex i tyof a PS g a m e . In a PS g a m e , th e scr o u nger a l ter na t ive o nly ex is ts if there are ind ivid -u a l s p la y in g pr o d u cer . M o r eo ver , th e ini t ia l va lu e o f p la y ing S is not set, b u t dependson th e nu m b e r o f ind ivid u a ls p la y in g th e a l ter na t ive P. T h e m o r e th a t p la y P, th egreater the payoff for playing S, because of the greater ra te of appearance of exploita-t ion opportunit ies. In IFD, a l t e r n at i v e patches ( i .e . , s tra tegies) exist independently ofwh ether they are used by any player; and their va lu e is set by their basic suitabil i ty ,not the n u m b e r o f p l a y e r s u s i n g th e a l t e r n a t i v e .I n b o t h g a m e s th e p a y o f f s are nega-t i v e l y f r e q u e n c y - d e p e n d e n t ( d e c r e a s e w i t h i n c r e a s i n g f r e q u e n c y o f p la y er s u s ing th esa me a l ter na t ive) , b u t in th e PS ga me, tw o f a c to r s co ntr ib u te to th e f r eq u ency -d epend ence o f S p a y o f f s : (1 ) th e i n i ti a l n u m b e r o f i n d i v i d u a l s p l a y i n g th e a lternativeP a nd ( 2 ) th e nu mb er o f co mpet i to r s p la y ing S . T h i s " c o m p o u n d " f r e q u e n c y -d e p e n d e n c e character izes PS g a m e s an d n ot I F D, w h er e th e v a l u e of any a l ter na t ived epends o nly o n th e nu m b e r o f p lay er s u s ing th a t a l ter na t ive . I n a PS ga m e, th epayoff of p la y ing S is h i g h e s t w h e n p l a y i n g P is c o m m o n (q > 1) and l o w e s t w h e np l a y i n g S is c o m m o n (q- 0; Parker 1984; Fig. 2 .1 ) . A l t h o u g h the PS g a m e doesnot specify the detai ls of the producer f i tness f u n c t i o n , in a f o r a ging co ntex t th e mo stl i k e l y scenario is that the f u n c t i o n d ec l ines w i th incr ea s ing pr o po r t io n o f S w i t h i n th ep o p u l a t i o n ( Gir a ld ea u e l al. 1994b; Fig. 2.1).

In evo lu t io na r y ter ms, th e g a m e - t h e o r e t i c f o r a g i n g s c e n a r i o for the PS g a m e in -v o k e s s o m e a n c e s t ra l g r o u p w i t h i n d e p e n d e n t l y s e a r c h i n g i n d i v i d u a l s (P) . The f irstm u t a n t th a t p la y s S ca n ex plo i t th e f o o d d isco ver ed b y a l l o th er gr o u p memb er s .W h i l e g r o u p m e m b e r s p la y ing P are co nst r a ined b y their o w n i n t r i n s i c rate o f patche n c o u n t e r , th e m u t a n t S e n c o u n t e r s p a t c h e s m u c h m o r e q u i c k l y b y exploit ing everyfood pa tch th a t one of the P pla y er s f inds. B e c a u s e of the c o m p o u n d f r e q u e n c y -d e p e n d e n c e th a t c harac t e r i ze s th e pa y o f f o f p la y ing S , a s th e S alternative spreads inth e po pu la t io n, its p a y o f f is nega t ive ly a f f ec ted in two ways. First , as S g r a d u a l l y

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G a m e T h e o r y and S o c ia l F o r a g i n g 19

Fig. 2.1. T h e charac te r is t ic fitness f u n c t i o n s of a P r o d u c e r -s c r o u n g e r g a m e w i t h a s tab le so lu t ion i n v o l v i n g b oth a l te rna-tives(proportionofscroungersonxaxis=1 q).The f u n c -t ions me e t b oth condi t ions re quire d for a mixe d s tab le so lu-t ion. Conse que nt ly , w he n the proport ion of scroung e r w i th in aforaging g roup islow,the fi tness of scrou nger is high er thanthe f i tne ss of produce r . Conv e rse ly , w he n the proport ion ofscroung e r i s h ig h , the f i tne ss of scroung e r i s low e r than thef itness of produce r . Giv e n the se condi t ions , th e l ines wil lnec-e ssar i ly c ross , a n d h e n c e a n e q u i l i b r i u m point e xis ts . A d i s r u p -tion shifting th e proportion of s c r o u n g e r to the r i g h t of thee q u i l i b r i u m incre ase s the succe ss of the produce r a l te rnat iv e ,m o v i n g the proport ion of scroung e r b ack tow ard the e q u i l i b -r i u m . I f a d i s r u p t i o n mov e s the proport ion of scroung e r to theleft of the e qui l ib r ium, scroung e r i s fav ore d and the propor-t ion scrou ng e r incre ase s tow ard the point of e q u i l i b r i u m . T h ee qui l ib r ium, the re fore , i s s tab le . I f se le c t ion cause s thefre-q u e n c y of a l te rnat iv e s to c h a n g e , th e e q u i l i b r i u m is e v o l u t i o n -ari ly s tab le (E SS) . H ow e v e r , if i n d i v i d u a l s a d ju s t their u s e o fa l t ern a t i ves t h r o u g h l e a r n i n g , t h e n t h e e q u i l i b r i u m i s d e v el o p -me nta l ly stable ( D S S ) .

replaces P, there are fewer individuals to exploit and the "basic suitability" of the Salternative declines. Second, as the frequency of S increases, each food patch mustbe sharedwith a greater numberof Scompetitors. Of course, apopulation of pureSdoes very poorly, since no food is ever discovered. In a PS game, pure P can bestable under some conditions, but pure S is never stable. Assuming that all playershave equal competitive abilities (i.e., that the game is symmetric), a PS game has anequilibrium solution with proportion < 7A of Pwhenever

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2 0 Luc-Alain Giraldeau Barbara Livoreil

and

B o t h of the a b o v e c o n d i t i o n s m u s t h o l d if a PS g a m e is to h a v e a s ta b l e e q u i l i b r i u ms o l u t i o n (Parker 1984, Caraco & G i r a l d e a u 1 9 9 1 ; F ig . 2 . 1 ) . T h i s m i x e d s o l u t i o n c a nt ak e s e v e r a l f o r ms at the i n d i v i d u a l level (Fig. 2 .2) . T h e p o p u l a t i o n m a y b e ( 1 ) d i m o r -p h i c t h a t i s , c o m p o s e d o f special ized P and S i n d i v i d u a l s , (2 ) m o n o m o r p h i c ,w h e r e i n e a c h i n d i v i d u a l p l ays P an d S w i t h p r o b a b i l i t i e s qA and (1gA ) , respec-t i v e l y , o r ( 3 ) p o l y m o r p h i c , w h e r e i n e a c h i n d i v i d u a l d i s p l a y s i t s o w n f r e q u e n c y o f Pan d S, so l o n g a s w i t h i n th e p o p u l a t i o n th e av e r ag e p r o p o r t i o n o f P c o r r e s p o n d s to< 7 A (Parker 1984; but s e e O r z a c k & Sober 1994) . It is i m p o r t a n t to m a k e th ese a l t e r n a -t i v e m i x e d s o l u t i o n s t o t h e PS g a m e e x p l i c i t in order to avoid reject ion of the games i mp l y becausethe populat ion is not d i mo r p h i c w i t h respectto P and Sroles.

T h e P S g a m e , a s m o s t g a m e m o d e l s in b e h a v i o r a l e c o l o g y , w a s o r i g i n a l l y f o r m u -l a t e d in t e r ms o f s e l e c t i o n ac t i n g o n a l t e r n at i v e g e n e t i c s t r a te g i e s w h o s e s o l u t i o n s ,t h e r e fo r e , w e r e c a ll e d e v o l u t i o n a r i l y s t a b le s tr a t e g ie s ( E S S ) ( M a y n a r d S m i th 1 9 82 ).H o w e v e r , f ixed use of an a l t e r n a t i v e is u n l i k e l y in f o r a g i n g g a m e s , w h e r e f r e q u e n t l yc h a n g i n g c o n d i t i o n s f a v o r f l e x i b i l i t y ( C a r ac o & G i r a l d e a u 1 9 9 1 ) . A d j u s t m e n t s o f gAw i t h i n a g r o u p l i k e l y i n v o l v e i n d i v i d u a l l e a r n i n g i n r e sp o n se t o th e a m o u n t o f re -w a r d s o b t a in e d t h r o u g h e a c h f o r a g i n g a l t e r n a t i v e . L e a r n e d s o l u t io n s to g a m e s h a v ebe e n te r me d d e v e l o p m e n t a l l y s t ab l e s t r a te g i e s ( D S S ; D a w k i n s 1 9 8 0 ) o r b e h a v i o r a la s s e s s m e n t ESSes ( D a v i e s 1 9 8 2 ; se e S t e p h e n s & C l e m e n t s , t h i s v o l u m e , fo r m o r e o nl e a r n i n g a n d game t h e o r y) . I w i l l u s e D S S w h e n s o l u t i o n s to g a m e s a re r e ac h e db y l e a r n i n g . D e s p i t e some i ni t i a l e n t h u s i a s m f o r t h e p r o b le m o f o pt im a l l e a r n i n g o fe q u i l i b r i u m a l l o c a t i o n to a l t e r n a t i v e s ( H a r l e y 1 9 8 1 , R e g e l m a n n 1984, M i l i n s k i 1 98 4,H o u s t o n & S u m i d a 1 9 8 7) , th e r e h a s be e n v e r y l i t t l e r e c e n t w o r k d e v o t e d to s t u d y i n gh o w i n d i v i d u a l s r ea c h a D S S .

2.4 A Rate M aximizing P S Foraging GameI n t h i s s e c t i o n , w e d e t e r m i n e th e c o n d i t i o n s u n d e r w h i c h p o p u l a t i o n s a re e x p e c t e d tob e p u r e P a n d t h e n a s k h o w c h a n g e s in f o r a g i n g c o n d i t i o n s a f f e c t th e expected e q u i -l i b r i u m proport ion o f p r o d u c e r ( < J A ) . T h e m o d e l is a t w o - s t r a t e g y v e r s i o n o f V i c k e r yetal.'s ( 1 9 9 1 ) t h r e e - s t r a t e g y PS f o r a g i n g g a m e ( G i r a l d e a u e t a l. 1994b) . I tapplies tos i t ua t i o n s w h e r e s e v e r a l f o r ag e r s c a n e x p l o i t th e s a m e f o o d c l u m p c o n c u r r e n t l y , a sopposed to w h e n i n d i v i d u a l s d i sp l ac e e a c h o t h e r from food discoveries .T h e h y p o t h e s i z e d c u r r e n c y of f i tness is the m e a n gross e n e r g y i n t ak e (/) m e a -s ur e d o v e r some t i me h o r i z o n T . T h e m o d e l ' s d e c i s i o n v ar i ab l e is a p o p u l a t i o n ' spr o po r t io n o f p r o d u c e r q(Q

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Fig. 2 . 2 . Three theoretical ly poss i b le f r equency dis t r ibut ions of p r o d u c e r ands c r o u n g e r a l ter na t ives within a g r o u p of twel ve f or a ger s when < ?A = 0.5. I n A ,th e population is d i m o r p h i c w i t h six i n d i v i d u a l s u s i n g o n l y p r o d u c e r and sixother s using onl y scr ounger . I n this ca se , th e f r e q u e n c y of i n d i v i d u a l s of onea nd the other type within th e popul a t ion must cor r espond t o t h e E S S o r D S S .I n B ,the population is monomorphic.Herealltwelveindividuals use bothp r o d u c e r and scrounger a l ternatives in the s a m e w a y . I n a m o n o m o r p h i c p o p -ul a t ion a l l individua l s must be using f r equencies of pr oducer a nd scr oungertha t cor r espond to the ES S or DS S . In C , the popul a t ion is pol ymor phic wi t heach i n d i v i d u a l e x h i b i t in g i ts own c o m b i n a t i o n of f r e q u e n c i e s of p r o d u c e rand scr ounger a l ter na t ives . A s i n A , t h e p o p u l a t i o n f r e q u e n c y of p r o d u c e r ands c r o u n g e r m u s t correspond t o t h e E S S o r D S S .

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2 2 L u c - A l a i n G i r a l d e a u & B a r b a r a L i v o r c i li n d i v i d u a l s w i l l u s e t h e D S S gA . T h e p r e d i c t i o n i s t h a t t h e p o p u l a t i o n w i l l e x h i b i t #A ,a n d a n y c o m b i n a t i o n o f i n d i v i d u a lus e o f P an d S t h a t l e a d s toqA at the p o p u l a t i o nl e v e l i s s u i t a b l e (see Fig. 2.2).H o w e v e r , n o t a l l s o l u t i o n s m a y b e e q u a l l y s t a b l e(C r a w f o r d 1989) n o r e q u a l l y a c c e p t a b l e as e v i d e n c e o f a d a p t a t i o n (O r za c k & Sober1 9 9 4 ) .

A g r o u p ofG i n d i v i d u a l s f o r ag e s s u c h t h a t theGs= (l~q)G i n d i v i d u a l s p l a y i n gS a l w a y s d e t e c t a n d e x p l o i t a l l o f t h e f o o d c l u m p s u n c o v e r e d b y t h eGv=q Gi n d i v i d -u a l s p l a y i n g P . ( N o t e t h a t G here r e p r e s e n t s th e n u m b e r o f i n d i v i d u a l s m a i n t a i n i n gs e n s o r y c o n t a c t ; in r e a l i t y t h i s m a y b e a s u b s e t of an a c t u a l f o r a g i n g g r o u p ) . T h i sa s s u m e s t h a t t h e qG f o r a g e r s s e a r c h f o r p a t c h e s i n d e p e n d e n t l y , t h a t t h e t i m e r e q u i r e dt o e n c o u n t e r p a tc h e s i s l o n g r e l a t i v e t o p a t c h e x p l o i t a t i o n t im e , a n d t h a t p a t c h e s a r en e v e r e x p l o i t e d c o n c u r r e n t l y ( v i o l a t i o n s to th e s e a s s u m p t i o n s h a v e l i t t l e q u a l i t a t i v eeffect . ) A l l f o o d p a t c h e s c o n t a i n F i t e m s , e a c h o f w h i c h i s e a t e n w h o l e a n d c a n n o tb e s h a r e d . U p o n d i s c o v e r i n g a p a t c h , a P p l a y e r a l w a y s o b t a i n s a ( w h e r e 0

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G a m e T h e o r y an d S o c i a l F o r a g i n g 2 3

F i g . 2 . 3 . T h e s u r fa c e pred i c t i n g t h e s ta ble pro po rt io n o f pro d u c er u s i n g a two-strategyvers i o n o f V i c k e r y e tal.'s ( 1991) d et erm i n i s t i c ra t e-m a x i m i z i n g PS g a m e . T h e proport ionof pro d u c er i n c rea s es wi t h t h e f i n d er ' s s h a re bu t decreases w i t h i n c r e a s i n g gro u p s i z e .T h e f l a t po r t i o n on the top r i gh t h a n d side is the p r o d u c e r - o n l y s u r f a c e w h e r e n os c r o u n g e r c o u l d e c o n o m i c a l l y exist . A t small group sizes, al l p r o d u c e r is possible for awi d e ra n ge of finder's s ha res . H o wev er , as gro u p s i z e i n c rea s es , th e r a n g e s h r i n k so ffinder 's shares for w h i c h pro d u c er o n ly i s s t a ble .

b e c o m e a v a i l a b l e . T h u s , th e pr o po r t io n u s ing P, and c o n s e q u e n t l y th e proportion u s-in g S, are predicted to h a ve o nly tw o determinants, group size ( G ) an d t he f i n d er ' sshare fa/F; eq u a t io n (3)] .Figure 2 .3 depicts how the equil ibr ium proportion of producer (c /A ) is affectedb y the f inder 's share and g r o u p size. T h e area of the graph fo r w h i c hqA= 1 i s t h e P-only su r f a ce . T h e greater th e P - o n l y su r f a ce , th e greater th e range o f co nd i t io ns u nd erw h i c h pur e P is a DS S . N o te th a t th e P-o nly su r f a ce is larger fo r s m a l l G a n d shrinkswith increasing G . T h i s is b eca u se as G increases, an i n d i v i d u a l p l ayi n g P m u s tmonopolize an increasingly large fraction of the pa tch (alF) in order to prevent Sfrom pr o vid ing an e c o n o m i c advantage. I t m e a n s , fo r instance, that in a si tuation

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2 4 L u c - A l a i n G i r a l d e a u & B a r b a r a L i v o r e i lw h e r e an i n d i v i d u a l f o r a g e s w i t h tw o partners ( i .e . , G = 3) , p u r e P wil l be a D S S i f i tc o u l d o b t a i n 2 / 3 o r m o re o f t h e i t em s i n t h e e l u m p s b efo re t h e a r r i v a l o f s c ro u n g ers .Fo r G = 10, the f r a c t i o n increases to9/10o r m o r e , m a k i n g i t i n c r e a s i n g l y u n l i k e l y t oo b s erv e p u re P a s G in c r e a s e s ( V i c k e r y e t a l . 1 9 9 1 ) .

W h e n f o r m a l i z e d as a f o r a g i n g m