F c o s \ ' s t e i n s( 1 9 9 8 ) l : . 1 , 1 . 1 - 4 . 1 8
l99t S|rjfgcr Vrrlng
ComplexAdaptive Systems
and the Evolution of Reciprocation
Karl Sigmund
Institute .for Matltenatics,
Universitv of )'ienna, Strttdlhofgasse 4, A-1090 \lienna, Austria
and IIASA. Laxenburg, Austria
AssrRecr
Complex adaptive systemsplay a major role in
the theory clf reciprocal altruism. Starting wirh
Axelrod's celebrated c(rlnpllter t()Llmarrents,
a wide variety of computer simulaticlns sholv
that cooperaticlncan evolve in populaticns ol
s e l f i s ha g e n t s ,b o t h w i t h d i r e c t a n d i n d i r e c t r e c i p r o cation.
I(ey
words:
reciprocal altruisrn; evolutionary
l l a m e s ;P r i s o n er ' s D i l e m m a g a m e ;e v o l u t i o n o f c o o p eration.
IurRonucrroN
Most of thc major transitions in evoh,rtionconsistin
assembling r-rnits of some type to fornt a new,
h i e r a r c l - r i c a l l yh i g h c r e n t i t y ( M a y n a r d S m i t h a n d
S z a t h r n a r y1 9 9 5 ) .A s a r u l e , t h e r e p r o d u c t i v ep o t e n t i a l i s m o n c l p o l i z e db y a m i n o r i t y o f t h e p r c v i o r - r s
u n i t s . T h i s i s m o s t n o t a b l y t h e c a s ew i t h m u l t i c e l l u l a r o r g a n i s m s ,w h e r e o n l y g e r m l i n e c e l l s t r a n s m i t
t h e g e n o m e , o r w i t h c o l o n i e s o f e u s o c i a la n i m a l s
having large proporticlnsof sterilewclrkers.This channeling ol reproductive opportunities entails a close
relatednessbetlveen elements of the higher-order
u n i t a n d e n a b l e sc o o p e r a t i o rbr a s c do n k i n s e l e c t i o n .
I n h u m a n s o c i e t i e sh, o w e v e r , r v h e r e c o o P e 1 . 1 1 i 1 1 n
i s u b i q u i t o u s , t h e r e a r e r e i n a r k a b l yf e w i n d i v i d u a l
i n e q u a l i t i e si n r e p r o d u c t i v ep o t e n t i a l .A f e w p o t e n t a t e s h a v e m a n a g e dt o o b t a i n a n a l m o s t u n l i m i t e d
control of their cclmmunity and to sire scveral
l-rur-rdredoffspring, br-rt these are exceptions that
occurreclat a late, and probably transient, stage of
cultural cvolution. Both in modern mass socicties
and in bar-rdsof hunter-gatherers. social rules tend
t o l e v e l r e p r o d u c t i v eo p p o r t u n i t i e sa n c l t o p r e v c n t
t h e e s t a b l i s h r n e not f a g l o b a l c o n t r o l i e r .A l t h o u g h a
large part of the servicesand tasksperformed within
households can be explained by kin selection, rhe
bulk of human cooperation is based,not on related-
R c c c i v c i l l : 1 A p r i l i 9 9 8 ; a c c c p t e dI 6 J u n e I 9 9 8
E-nrail: l<[email protected]
444
n e s s , b u t o n r e c i p r o c a t i o n .N o t g e n e t i c t i e s , b u t
e c o n o m i c e x c h a n g e s ,e x p l a i n t h e c o h e s i o n o f h u man communities.
T h i s i m p l i e s t h a t i n d i v i d u a l sc o l l a b o r a t eo n l y i f i t
is to their own good. The rnathernaticalframework
for str-rdyingthe economics of interacting egoistsis
ganrc theor)/: nlore precisell',that branch of game
theorl' lvhich is, somcwhat misleadingll', called
l-roncooperativegame theory [cf. Binn'rore (19941].
The ternr noncooperative
means in this context that
players cannot negotiate binding and enforceaLrle
a g r e e m e n t s .S u c h p l a y e r sc a n n e v e r t h e l e s sa c h i e v e
c o o p e r a t i o n ,n o t b y t h e c o m m a n d o f a c o n t r o l l e ro r
by cleferenceto the Lrenefitof the group, but by folIorving a myopic set clf mles evolved to optimize their
selfish interests. Thc challenge lies in showing how
they do it in spite of the ever-presentlllre of defection.
Robert Trivers (l97 ll was the first to suggest
reciprocationas a basis lor mutr.ral assistance in
a n i m a l b e h a v i o r ( i n c l u d i n gh u m a n b e h a v i o r ; a n d t c l
discr-rss
it in terms of game theory. More precisely,
h e i n t r c l d u c e dt h e P r i s o n e r ' sD i l c m m a g a n r et o b r i n g
the problem into focus [seealso May (1987) and
B o y d ( 1 9 8 8 ) 1 .T l - r i sa p p r o a c hr v a sg r e a t l y e r p a n d c d
b y R o b e r tA x e l r o d a n d W i l l i a m D . H a m i l t o n ( 1 9 8 1 ) ,
who applied evolutionary game theory [seeMayn a r d S n r i t h ( 1 9 8 2 ) l t o t r t o p u l a t i o n so f p l a y e r s e n g a g e d i n m a n y r o l l n d s o f t h e r e p e a t e dP r i s o n e r ' s
D i l e m m a g a m e a g a i n s tr a n d o m l y c h o s c n c o p l a y e r s .
Already at this early stage, contpllter simulations
were esscntialto folkrw the comolex nonlinear
C t r r n p l e r A d a p t i v e S y s l e m sa n d t h e E v o l u t i o n o f R c c i p r o c a t i o n
clynamicsof the freqr..rencies
of strategiessubrnilled
t o n a t u r a l s e l e c t i o n( A x e l r o d 1 9 8 4 ) .
A ftrrther decisivc step was taken rvhen Axelrod
adapted the genetic algorithnrso[ John Holland
( 1 9 7 5 t t o s i m r - r l a t ct h c e f f e c t so f e v c l h , r t i o n a rtyr i a l
and error. This r'vasthe first application of gcnetic
algorithms to a genuine evolutionary problem
(rather than a tcchnical optimization problenr).It
p r o v e d p a r t i c u l a r l ] ' s r : c c e s s f r -brel c a u s et h e b i n a r l '
c o d i n g o f s t r a t e g i e s ,u s t r a l l y a m o o t p o i n t w i t h
S C n c t i ca l g o r i t h m s , w a s s t r a i g h t f o r w a r c l( A x e l r o d
1 9 8 7 ) [ s e e a l s o A x e l r o d ( 1 9 9 7 ) a n d E p s t e i na n d
Axtell (r99611.
T h i s s u c c c s sl e d t o m a n y f u r t h e r i n v c s t i g a t i o r . r s .
including the effectsof noise,extending the menory,
allowing for altcrnating moves, introducing rnore
c o m p l c x s t a t c - t r a s esdt r a t e g i e si ,n c r e a s i n gt h e n u m ber of players,changingthe payoff structure,restricti n g t h e i n t e r a c t i o n st o n e i g i r b o r so n l y , a d d r e s s i n g
c o n t i n u o u s v e r s i o r - r os f t h e g a n r e , a n d s ( ) o n . W e
present a short sllrvey of this field, r.vithpointers to
the biologicalmotivations behind these variants.
T h e b a s i c m e s s a g ei s t h a t d e t a i l s m a t t c r , b u t t h a t
cooperalion robustly emergeslronr a bottom-up
a p p r o a c h b a s e d o n r e c i p r o c a t i o n .W e f i n a l l y d c a l
witl-r indirect reciprocity, whicl-r is, accorcliuq tt'
R i c h a r d A l e x a n c l e r ( 1 9 8 7 ) , t h e b i o l o g i c a lb a s i s o f
h u m a n m o r a l s y s t e m s .I n t h i s s e t u p , a n a c t o f
assistancemay be returned, no1 to tl-rcdonor, but tcr
a third party. A model basedon thc statllsof thc
p l a y c r s s h o w s t h a t c o o p e r a t i o nc a n c m e r g e e v e n i f
a n y t w o i n d i v i c l u a l sn e v e r i n t e r a c tn t o r e t h a n O n c c .
T h i s b r i e f s u r v c y c o n c e n t r a t e so n p a p e r sc l e a l i n g
w i t h p a i r l v i s e i n t e r a c t i o n sa n d u s i n g e v o l u t i o n a r l
s i m u l a t i o n s .T h i s i s c e r t a i n l y n o t i n t e n d e d t o d e n l '
the importance clf other approachestctthe evolutior-r
o f c o c r p e r a t i o n[ s e eD u g a t k i n a n d o t h e r s 1 9 9 2 1 .
Drnncr RncrpRocATroN
T h c P r i s o n c r ' sD i l e m m a ( P D ) i s a t w o - p l a y e r g a r n e
where both playershavc the same two strarellies
a n d t h e s a m e p a y o f f s .T h e t w o s t r a t e g i e sa r e C ( t o
c o o p e r a t c )a n d D ( t o d e f e c t ) .l f b o t h p l a y e r su s e C ,
both gct tltc rewardR for mutual cooperation;if both
play D, both get rhe penal4tP for murr-raldefection; a
D player obtains the temptationpayoff f for unilaterally defecting against a coplayer who plays C,
whereas the coplayer receives Lhe sucker'spayoff S
l o r b e i n ge x p l o i t e dO
. n e a s s u m e sI > R > P > S a n d
>
*
2R
T
S. The first condition implies that D
d o n . r i n a t eC
s , i n t h e s e n s et h a t i t i s b e t t er n o n l a t t e r
what the other player chcloses,and the second
condition entails that joint cctclperationis better
than sharing the payoffs after a unilateral defection.
445
P l a y e r sh a v e t o c h o o s eD i f t l - r e 1w, a n t t o m a x i m i z e
their payoff, and this yieldsthe payoff P, rvhich is
l e s st h a n t h e r e w a r d R f o r m u t u a l c o o p e r a t i o n .
I n t h e r e p e a t e dP D g a m e , p l a y e r sh a v e t o c h c l o s e
s i m u l t a n e o u s l y i,n e v e r y r o u n d , w h e t h e r t o p l a y C
clr D. There is a constant probability w for another
r o u n c l , s o t h a t t h c a v e r a g el e n g t h o l t h e g a n r e i s
random. The total payoff is given by I,4,,w", where
. 4 , ,i s t h c p a y o f f i n r o u n d r . T h e s t r a t e g i e sf o r t h e
repeated PD can be arbitrarily cornplex, bLrt in a
b i o l o g i c a lc o n t c x t , i t o n l y m a k e s s e n s et o c o n s i d e r
strategiesgiver-rby simple knee-jerk rules, like Tit
For Tat (TFT which plays C in the first round and
t h e n s i r n p l y r e p e a t st h e c o p l a y e r ' sp r e v i o u s m c l v e ) .
The first lessonfrom Axelrod's compllter tournaments was that sr-rchsimple rules, ancl in particular
T F T ,m o r c t h a n h o l d t h e i r o w n a g a i n s tr n o r c s o p h i s t i c a t c d s t r a t e g i e sl b a s e col n s t o c h a s t i co p t i m i z a t i ( ) n ,
f o r i n s t a n c e) .
C o m p l e x a d a p t i v e s y s t e m sa r e r - r s e dt O s i m u l a t e
a r t i f i c i a is o c i e t i e so f p l a y e r sc n g a g e di n r e p c a t e dP D
g a m e s .I n A r e l r o d ( 1 9 8 7 ) , s t r a t c g i c sw e r c c o n s i d e r e d t h a t p l a y ' e dC o r D d e p c n d i n go n t h e o l r t c o m e
ilf the prcvious three rounds. Binary slrings ol
length 70 coded them. For every new generati()n,
tl-rcscstrings were subrnitted to point nlutation and
recombination.The playersthen engagedin a roundpayoff points that
robin tclurnancnt, accllrrrr-rlating
translated inlcl number of offspring forming the
n e x t g e n e r a t i o n .A n i n i t i a l i n c r e a s ci n d e f c c t o r sw a s
o l t e n f o l l o w c d b 1 , ar e e m e r g e n c eo f c c l o p e r a t i o n .
B e n d o r a n d c o l l e a g r - r e( 1s 9 9 1 ) , N o w a k a n d S i g nrund (1992\, and I(ollock (1993) havc consiclerecl
t h c i n l l u c n c e o f n o i s e ,w h i c h i s p a r t i c u l a r l yd i s r u p tive in a societyclominatedby TFT.When onc allows
only for reactivestratcgicsgiven by differer-rtpropens i t i e st o p l a y C , d e p e n d i n go n t h e l a s t m o v e o f t h e
c o p l a y e r ,o n e f i n d s t h a t c c l o p e r a t i o ne r n e r g e sb a s e d
on a tolerant rule called GTFT (Gencrous TFT),
w h i c h r e t a l i a t e so n l y l v i t h a c e r t a i np r o b a b i l i t ya f t e r
a D , b u t a l w a y s c o o p c r a t c sa f t e r a C ( u p t o r n i s t a k e s
in implementation).This rule callnot spreadin a
society of defectors, however; it is nccessarl' tirst
t h a t T F T i n v a d e sa n d p a v e st h e w a y , l i k e a p i o n e e r ing speciesin a plar.rtcomnunity, for GTFT 1o take
o v e r ( N c l r v a ka n d S i g m u n d 1 9 9 2 ) .
I f o n e c o n s i d e r ss t r a t e g i e st l ' r a t d e p e n d o n t h e
moves of both players in tl-reprevious round, one
f i n d s a r i c h c o l l e c t i o no f f a r - f r o m - e q u i l i b r i u md y namics even if only three clr four strategiesinteract
( N o w a k a n d S i g m u n c i 1 9 9 ) a ) . S r - r f f i c i e n t l yl o n g
m u t a t i o n - s e l e c t i o nc h r o n i c l e sr . r s u a l l yl e a d t o m e t a stablc cooperative regimes, horvever, based (in case
P + T < 2 R ) o n P a v l o v ,a r u l e w h i c h c o o p e r a t e si f
both playersused the same move in the prcviot-rs
446
I(. Signrrrntl
r o u n d . P a v l o v i s a w i n - s t a 1 ' ,l o s e - s h i t tr u l e : p l a l ' e r s
repeat their previous nove it it ied to a high payolf
(R or T) and try the other option if the payoff was
low (S or P). In a population of Pavlov players'
unilateral defections due t() uristakes cause one
round of mutual defection, after which both players
resume cooperation (Milinski l99l; Nowal< and
S i g m u n d I 9 9 3 b ) . P a v l o v e x p l o i t st t n c o n d i t i o n a lc o operators,preventing them {rom ir-rvadingand thus
o f f e r i n g t a r g e t s f o r e x p l o i t e r s .A g a i n , c o o l e r a t i ( ) I 1
c a n e m e r g e t l n l y a f t e r t h e i n v a s i o n o f s t e r t rr e t a l i a tory strategieslike TFT, which then are sr-rpcrsedcd
by the error-proof Pavlov.TFT acts in this senseas a
cataiystfor crtoPeration.
I t s h o u l d b e s t r e s s e dt h a t t h e s e s i m u l a t i o n s c l t
a r t i f i c i a lp o p u l a t i o n sb a s e do n e x t e n d e d m u t a t i c l u selection chronicles show a high degree ol history
d e p e n d e n c ea n c l o f t e n d i s p l a y p u n c l u a t e d e q u i l i b r i u m . T l ' r i si s p a r t i c u l a r l ) 'c l e a r i n L i n d g r e n ( 1 9 9 1 ) '
where extensions of the nemory were possible
(causedby mutations remit-liscentof gene dr'rplications that introduce strategiesbased r-rotjust on thc
last round, br.rt tlt-t a larger window of the past)'
led to the enlergcnce ot
Usually such sin-rr-rlations
ttl Pavlov, or ol varisimilar
strategies
cooperative
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cstaband
good at invading ptlpr.rlatitlnstlt defectors
l i s h i n g a s t a b l ec o o p e r a t i v er e g i m e. T h e s es t r a t e g i c s '
h o w e v e r , a r e i m m u n e o n l y a g a i n s te r r t l r s i n i r n p l e mentatioll, whereas the Pavlov rltle also avoicls
e r r o r s i n p e r c e p t i o n . L e i m a r 1 1 9 9 7 \ s h c x v e c tl h a t
there are huge numbers of linite-state aLltomata
l e a c l i n gt o l i m i t - E S S ( a v e r s i o n o f e v o l u t i o n a r i l y
s t a b l e s t r a t e g i e s ) .T h i s i m p l i e s p r o n t l t t n c c d p a t h
dependenceof evclltttiot'tarychronicles.
In the usr-ralPD setup, both players are sttl)poscd
to nlove at once. This is the case, fclr instatrce, in
predator-inspection games by sticklcbacksclr gttppies, where the fish take their mirror image lor a
coplayer (Milinski I987). In many situations ol
m u t u a l a i d , h o w e v e r ,p l a y e r sm o v e a l t e r n a t el y ' T h i s
is the case, for instance, when a well-ied vampire
b a t f e e d sa h t r n g r y c o n s p e c i f i c( W i l k i n s o n 1 9 8 4 ) o r
when a yollng male baboon diverts the attenlion trf
the dclminant male so that his pal can nlollnt an
e s t r o u s l e r t t a l e ( P a c k e r 1 9 7 7 \ . S i t l . t l t l a t i t l n os l t l l t :
a l t e r n a r i n g P D ( N t l r v a k a n c l S i g m r - r r l c1l9 9 4 ; F r c a t l
r
1 9 9 4 ; H a u c r t a n d S c h l r s t e r19 9 8) l e a c l a g a i t l t '
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P a v l o v .h o w c v e r , b t r t
d e f e c t so n l y a f t e r a n t l n w a r r a n t e d d e f e c t i o r lb 1 t h c
c o p l a y e r ) .E x p e r i m e n t sb y W e d e k i n d a n c l M i l i n s k i
1 1 9 9 6 t\ t l P p ( ) r t l t i s d i s t i r r c t i o bn e t w e c ns l r a l e g i e s
f o r t h e s i m u l t a n eo l t s a n d t h e a l t e r n a t i n gP D '
If players clo nclt interact at randcln, blrt only
r n r i t h i na c e r t a i n n e i g l ' r b o r h o o ds t r l t c t l t r c ,c o o p e r a t i o n b e c o t n e sn l t l c i l r l o r c r e a d i h ' e s t a b l i s h e de, v e n
f o r t h e o n c - s h o t P D [ s e e N c l r v a ka n t l N l a t \ 1 9 9 2 '
1 9 9 3 ) , S i g m r . r n c (l 1 9 9 2 1, H u L r e r n a t l a t r c l G l a l r c e
( I 9 9 3 ) , N o w a k a n c lo 1 l - r e r1s1 9 9 4 a1, 9 9 4 b ) . l - i r r t l 5 , r ' c r t
a n c l N o r d a h l ( 1 9 9 4 1, a n c l l ( i l l i n g b a c k a n c l l ) o e b e l i
( 1 9 9 6 ) a u c i , f o r a g e n e r a l i n l r c l d t r c l i o nt < l c c l l t r l a r
a u t o n a t a i n t h i s c o n t e x t ,D u r r e t t a n d L e v i n ( 1 9 9 4 ) l '
Again, if plal'ers have the possibility of cl.ttriccatld
of partners,cooperation becomesmuch easier
retr-tsal
t o a c h i e v e( S t a n l e ya n d o t h e r s 1 9 9 4 \ .l n b o t h c a s c s '
a g g r e g a t i o no f c o o p e r a l o r s o c c l l r s t h r o u g h l o c a l
n r l e s . I n c o n t r a s t ,c o o p e r a t i o l lb e c t l m e sm u c h t l l t t r e
r . r n l i k e l yi f n r o r e t h a t l t w o p l a l ' e r s c l l l l a g , ci t l t l t c
i n t e r a c t i t t t tF
. o r e x t e n s i v es i m l t l a t i o n so l t h i s s e t l t p '
r , v er e f er t o H a L l e r ta n d S c h u s t e r( 1 9 9 7 )'
be stressedtl'rat tl"tcPD is certairlll' not
It si-ror-rld
the only game rtloclelingaspectsof cooperatiol't'lt't
p a r t i c l l l a r , S u g d en ( 1 9 8 6 ) h a s s t r e s s e dt h a t t h e
p a y o f f r a n k o r c l e r i n gf > R > S > P a l s t lm a k e ss c n s e
i n t h i s c o n t e x t .l n s t t c ha l l i l l t e r a c t i ( l n ,a p l a y c r f a c e c l
i,vith a defector Ivould neverthelc'ssprefer to plal' C'
This yields what garne theoristscall the Chickerr
s a l v k - D o v e ' R e p e a t e dC h i c k e r l
g a m c a n d b i c l l o g i s tH
n e e d n o t l e a d t o n l u t u a l c o o p e r a t i o n [ r ' t s e dt t t . t
rcciprocity; it is nrore likely 1[31 an asy]nmetr]'
b e c o m e se f l e c t i v ew i t h o n e p l a y e r a l w a y s p l a y i t l g C
a n c lt l - r eo t h e r D . T h i s c t x r l d s h e d s o m e l i g h t o n t h c
i n t r u c l c ' ri n s p e c l i o n sf y f c m a l e l i o t l : , r v l l e r e o n c c a n
o l t c n c l i s t i n g r - r i sl cha d c r sa n d l a g g a r c l [ss e eH c i t r : o h r r
a n d P a c l < e( r1 9 9 6 ) 1 .
IxnrRncr RncPRocATIoN
hr adclition to reciprtlcatitln based on rcl)eatcLl
intcractions lvithin a pair, tilcre cxists;tttothcr'
inclirect reciprclcity,wherc the donor dclesttot olrtain a retllrn frorn the recipienl, bttt frtlnl a third
recipicrlt ha\
llarty. Donors provide help if the
h e l p e d o t h e r si n t h e p a s t .T h i s w o r k s i l t l - t cc o s to f a n
a l t r u i s t i ca c t i s o f f s e t b y a r a i s e d" s c o r e , "o r s t a t u s '
w h i c h i n c r c a s es t h e c h a n c et o s t t b s e q n e l l t l yb e c o m e
t h c r e c i p i e n t o f a n a l t r n i s t i c a c 1 . C o o p e r a t i ( ) ni s
char-urelcdtoward the "valuable" members clf thc
comr-nunity.For Richarcl Alexander, "indirect reci-
C u m p l e x A d a p t i v e S 1 ' s t e r nasn d t h e E v o l r . r t i o no f R e c i p r o c a t i o n
p r o c i f i n v o l v c sr e p u t a l i o na n d s t a l u s ,a n c lr e s r . r l tisr r
everyone in tfre group continnally being assessecl
and re-assesscd."
N o w a k a n d S i g n - i u n d( 1 9 9 8 a , b )m o c ' l e l e di n d i r e c t
r e c i p r o c i t yi n a p o p u l a l i o n o l i n d i v i c l u a l sl v h o h a d
the options of helping or not helping another
i n d i v i d u a l . I n e a c h g e n e r a t i o n ,a n u m b e r o f P o l g n l i a l d o n o r - r c c i p i e r - rpt a i r s a r c c h o s c r . r a n d o n r l y : i f
help is providcd, there is a cosl c to the donor, a
I r c r r c l i1l ' l ( r l l t ( ' r t ' c i P i t ' r tat r. r dt l r c d o n o r ' ss c o r t ' i s
i r - r c r c a s cbdy l . T h e s c o r eo l a p l a y e r r e f u s i n gt o I r e l l r
is decreased by l. lnitially all scores arc' 0. Wc
c o n s i d e rs t r a t c g i c sg i v c n b y i n l e g c r sk : a p l a y e r w i t h
s u c h a s t r a t c g ) ,h e l p s i f a r . r coi n l y i f t h c s c o r c o f t h e
p o t c n t i a l r e c i p i e n t i s a t l c a s t / < .W e ' c a n f o l k r w t h c
f r e q u e n c i e s o I t h e s l r a t e g i e sl r o m g e n e r a t i ( ) n l o
g e n e r a t i o n ,a l l o w i n g f o r o c c a s i o n arl n u t a t i o n s .
A renarkalrli.' srnall number of lnteractions (for
l 0 a n d c - l , a n a v e r a g eo f t l v o i n t e r a c t i o r - rPse r
h
l i f e t i m e s u f f i c c s ,I o r i n s t a n c e) c a n l e a c lt o t h e c m e r g e n c eo l c o o p e r a t i v ep o p n l a t i o n sr v h e r em o s t r n e n r - 1 . I f t h e s i n u l a t i o ni s
berslrsck:0
or k:
c o n t i n u e d , s t r a t e g i e st h a t a r e l e s s c l i s c r i n r i n a t i n g
s p r c a d :p l a y e r sl v i t h k - - 3 , f o r i n s t a n c c ,w i l l r a r e l l '
e v e r r e f u s e t o h e l p , t h e i r s c o r e w i l l t h e r e l c l r ci n c r c a s el a s t e r t h a n a v e r a g c ,a n d h e n c e t h c t , l v i l l i n
t u r n b e h e l p e d n r o r e o l t c n . B L r ti f t h c f r c q r - r c n c yc l f
l e s sd i s c r i m i n a t i n gp l a y e r sr e a c h e sa c e r t a i n t h r e s h o l d , t h e r r d e f e c t o r s( p l a 1 ' e r rsv i t h k : 3 , f o r i n s l a n c e ,
ivho l'rardly er.,erprovicle I'rclp) can takc ove r, rvith
t h c r c s l r l tt h a t c o o p c r a t i o nd i s a p p e a r si n t h e p o p u l a t i o n . C ) n c et l ' r i sh a p p e n s ,t h e a v e r a g ek v a l u e s w i l l
a g a i n ,l e a d i n ge v e n t r - r a l lbya c kt o a c o o p c r a decrease
tive rcgirne of playcrs with r-naxin-ral
cliscrinrir-ration
( t h a t i s , k : 0 ) . T o s u r n n r a r i z e ,r a n d o m c l r i f t c a n
n fsd i s c r i m i n a 1 ael t r u i s l sb 1 ,i n d i s s u b v e r tp o p r - r l a t i c l o
c r i n r i n a t e a l t r u i s t s ; o n c e t h e i r l r e q r - r e n c yi s, l a r g e ,
d e f e c t o r sc a n i n v a d e ; b L l t a s s o o l r a s l h e d e l e c t o r s
h a v e r e c l u c c dt h e p r o p o r t i o n o l i n c l i s c r i m i n a t e
allrui s t s , t h e d i s c r i r . n i n a l ca l t n r i s t s c a n f i g h t b a c k a n c l
eliminate the clefectors.
Tl-risleadsagain to a cooperat i v c p o p u l a t i o n t h a t i s p r o o f a g a i n s td e f e c t o r s ,b u t
n o t a g a i n s ti n c l i s c r i m i n a t ca l t r u i s t s ,a n d s o o r r .
S r , r c hr r o d e l s s h o w t h a t i n d i r e c t r e c i p r o c i t yb a s c c l
on image scoringworks in principlc. It should be
s t r e s s e dt l ' r a t h e c h a n c eo f t w o p l a y e r se v e r r n c c t i n g
again is vanishingll' srnall. All that needs tcl bc
k n o w n i s t h e s c o r eo f t h c c o p l a y e r I s c c a l s o P o l k r c k
a n d D r . r g a t k i n\ 1 9 9 2 1 1E
. v e n t h i s i r n a g es c o r i n gn e e d
n o t b e p u b i i c k n o w l e d g e .O n e c a n m o d i f y t h e r r o d e l
t o i n c l u d e , f o r e v c r y p l a y e r ,a p r i v a t e a s s e s s m e not f
t h e o t h e r g r o u p m e m b e r s . h r p a r t i c u l a r ,o n c c a n
a s s u m et h a t a n i n t e r a c t i o r rb e t w e e n t l v o i n d i v i d r , r a l s
is only observedby a srr-rall
sr-rbset
of the popr-rlatiorr.
Onll' these onlookers will r-rpdatetheir score of the
447
c l o n o r .F o r l a r g e r g r o u p s ,i t l r e c o m e sm o r e d i f f i c t r l t
t o e s t a b l i s hc o o p e r a t i o n .
Modcls thal are even more simplified help to
c r p l a i n a n a l y t i c a l l l ,t h e c y c l i n g b e h a v i o r , w i t h i t s
krng bouts of cooperation intcrspersed by short
p e r i o c l so f d e f c c t i o n ,w h i c h i s r e m i n i s c e n to f t h e l a c k
of stabilitl, near a critical statc. Sclmewhat sr.rrprisingly, cooperation is more robust if the society is
challengedrnclrelrequently by invasion attempts ot
d e f e c t o r sO
. ne can computethe rninimal amount of
discrirninators.the minirnal number of ror-rndsper
g e n e r a t i o n ,a n d t h e m a x i m a l s i z eo f t h e s o c i e t y .f c l r
i n d i r e c t r e c i p r o c i t vt o w o r k . T h i s y i e l d sa s n e c e s s a r y
of nc,ludinconclitiorrftrr cooperation that the deylree
probability
player
knows thc
that a
tancesltip(the
score of thc coplal,er) is larger than thc cost-benefit
to Hamilton's rule
ratio c/b. This resr,rltis anakrgor.rs
(the prclbr,vlrichstatesthat the degreeof relatedness
p
l
a
y
e
r
'
s
g
e
n
o
me is also
ability that an allele in tl're
p r es e n t i n t h e c o p l a y e r )m u s t e x c ee d c / & .
DrscussroN
S u c c e s si n a n a l y z i n gt h e i t e r a t c dP D s h o u l d n o t h i d e
the fact that variants r'vitl-rrnorc than two players
lead onl1' rarcl)' to a cooperativeoutcome Isee
H a L r e ra
t n c lS c h u s t e r( 1 9 9 71 1 .S u c h N - p e r s o ng a m e s
h a v e a t t r a c t e dn r u c h a t t e n t i o n , f c l r i n s t a n c e ,a s t h e
F r e e R i d c r P r o b l e m o r a s t h e T r a g e d yo f t h e C o m n'rons (Harclin 1968). The lattcr name sLlggesls
a l r e a c l ya, s a m a j o r a p p l i c a t i o n ,t h e m a n a g c m c n l o l
ecosvstelTls.
M o r e g e n e r a l l y ,h o w c a n i n d i v i d u a l r e s t r . r i n ti n
t h e e x p l o i t a t i o no f a c o m m o n r c s o u r c ee m e r g ea n d
es t a t r l i s hi t s e l f i n t h e a b s e n c eo f a g l o b a l c o n t r o l l e r ?
T h i s p r o b l e m r - r n drel i e st h e e v o l u l i o n o f v i r u l e n c eo r
t h e c o n c e p to f t h c p m c l c n t p r e d a t o r .I n t h e h u m a n
c o l ) t e x l ,o r l n o r e g e n e r a l l ya m o n l l i n d i v i d u a l sr e c o g r.rizingeach other, the obvior,rssolr-rtion-the punishnrcnl of selfish exploiters-leads to anotirer dilelnma. To punish clefectors is a costly, possibly
dangeror-rsactivit),, and it is tempting to leave it to
o t h e r s . B u t t h i s c o n s t i t u t e sa s e c o n d - o r d e rc ' l e f e c 1 i o n ,r v h i c h o u g h t a l s o t o b e p n n i s h e d ,a n d s o o n .
S e ' n ' e r aalL r l h o r s( B o y d a n c l R i c h e r s o r r1 9 8 8 ; G l a n c e
a n d H t r b e r n r a n1 9 9 4 ) h a v e a c l d r e s sdet h i s p r c l b l e m .
I t s e e n r sp l a r . r s i b lteh a t t h e i d e a o f i n d i r e c t r e c i p r o c i t i t ; r t h t ' . r p p l i c ti rl rt h i s( ( ) n l c x l .
RE FERE NC E S
A l c r a n r l c r R D . 1 9 8 7 . T h c h i o l o g v o f r r t o r a l s ) ' s t c m s .N c l v Y o r k :
Altiine de Gr-ur,1er.
A r e l r o t l R . 1 9 8 4 .T h e e v o l u l i o n o l c o o p e r a t i o nB
. a s i cB o o k s ,N e r v
Y o r k I r c p r i n t c t l I 9 8 9 , H a r n r o r r c ] s r . v o r tP
h :c n g r i i nl .
448
I(. Signrrurcl
A r e l r O c i R . 1 9 E 7 . T h c e r o l r r r i L r no l s r r a r c g i e si r t h c i t c r a t c t i
P r i s o n e r ' sD i l e r n n r a .I n : L a r v r c n r e D , e d i t o r . G c n c t i c a l g o r i t h n r s
and simulatctlanncaling.London: Pitnrar. p l2-,f 1.
Axelrod R. 1997. Tl-rc complcritv of coopcration. Princcton:
P r i n c e t o nU n i v e r s i t ) 'P r e s s .
A r c l r o d R , H a r n i l t o nW D . l 9 E l . T h e e v o l i r t i o no f c o o p e r a t i o r r .
S c i c n c t '( W a s h i n g t o n ) 2 1 I : I 3 9 0 - 6 .
Ilenclor .1, Kramcr RM, Stout S. 1991. Whcn in iloubt...:
c o o p e r a t i o n i n a n o i s v P r i s o r t c r ' sD i l c n u r t a . J C o n f l i c t R c s o h . r r i o n l 5 : ( r 9 l - 71 9 .
. cxington(NlAl:Hca1ll.
1 3 i n n r o rK
c ( ; . 1 9 9 4 .F r r na n t l g a n r c s L
B o c r i i j s t, \ 1 , N o r v a l i N l A , S i e r r u n d K . 1 9 9 7 . T h e l o g i c o l c o n t r i t i o n . . JT l t t ' o rB i o l I E 5 : 2 E l 9 1 .
l ' ] o v d R . 1 9 E 8 .I s t h e r e p e a t r d P r i s o n r r ' s D i l e m n r a a q o o d n r o d t ' l
t r [ r e c i p r o c aal l t r r - r i s n ?
E t h o l S o c i o b i o 9i : 2 l I 2 2 .
t s o 1 d R . R i c h e r s o nP 1 9 8 E . T h e r e v o l u t i o n o f r e c i p r , r . i t ) i n
5 i 7 , 1 l r l 1 ' q ; r r 1I 1
TP
l t rtr. r r B i , r lI l 2 : I ] 7 - 5 r r .
D r r g a t k i nL A , M e s t e r t o n - G i t r t r o nMs, H o u s l o n A I . 1 9 9 2 . B e v o n t l
t h c P r i s o n e r ' sD i l e r t r n t a :t o n ' a r d n r o d el s t o r l i s c r i n r i n . r t ca u r o n g
n r e c h a n i s n rosl c o o p e r a t i o ni n n a l r r r e .T R E E 7 : 2 0 2 I
D u r r e t t R , L c v i n S A . 1 9 9 4 .T h c i m p o r t a n c eo l b e i n g c l i s c r e t lea n d
s p a t i a l) . T h e o r P o p t r l B i o l 4 6 : 3 6 l - 9 4 .
E p s t e i n J i \ ' t . A r t e l l R . 1 9 9 6 . G r o r v i n g a r t i l i c i a i s o c i e l i e s .C - . i n r b r i d g e ,N I A : B r o o k i n g sI n s t i l u t i o nP r e s s .
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H a u e r t C h . S c h L l s l cH
r C l . 1 9 9 7 .E f l e c t so l i n r r e a s i n ut h e n u r l b e r
o f p l a v c r s a n d l t t c r t t o r l ' s i z ci r t t i r e i t e r a t c d P r i s o n e r ' sD i l e m n r a :
a nunrericaa
l p p r o . r c hP. r o c R S o cL o n d I B ] 2 6 1 : 5 1 ) 9 .
L l a u c r tC h , S c l t r - r s 1Hc G
r . 1 9 9 8 .E r t c n d i n g 1 h t ' i t c r a l c t lI ' r i s { ) n e r ' s
D i l e n r n r a\ v i l I r o u t s y r t c l t r o r r r 'F. o r t h c o r n i n g .
H e i n s o h n R . P a c k e rC . 1 9 9 5 . C o r n p l e r c o o p e r a l i v es t r r t c g i t r i r r
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s .n n
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H r r b e m a n B A , G l a n c eN S . 1 9 9 3 . E v o l r r l i o n a r ls a n r es a n t l c o r n 1 r r - r t cs ri n r u l a l i o n sP. r o cN a t l A c a d S c iU S A 9 O : 7 7 l 2 5 .
s 1s p a t i a l
L i n d g r e nI ( , N o r r l a h lI l ( ; . I 9 9 + . l r o l u t i o n a r - tvl l , n a r n i c o
g a r n e sP
. h V s i c aI ) 7 5 : 2 9 2 l O 9 .
N l a t ' R N { . 1 9 8 7 . X f t r r c r v o l u t i o r t l l c o o p e r a l i o r tN
. alure {Lond)
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t ' v o l L r t i o n .O x f t r r d : W H F r e e n r a n .
N l i l i n s k i N l . 1 9 8 7 . T i t F o r T a t i n s t i c l < l e b a ca
knd tfre evolution o1
c o o l ) c r a t i o n .N a t u r e ( l - o t r d ) 3 2 5 : 4 3 4 - 5 .
N l i l i n s k i , \ 1 . 1 9 9 1 . C o o p e r a t i o r r r ' v i n s a n d s t a V s .N a t u r c ( L o n d )
1 6 4 :| 2 - 3 .
. p a t i a lg a n r e sa n d
N o n a k N I A , B o n h o c l l c r S , N l a v R M . 1 9 9 . 1 aS
t l l c n l a i n l c n a n c c o l c o o p c r a t i o n .P r o c N a t l A c a d S c i U S A
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l n t . l B i t u r c a t r o nC i r a o s1 : 3 1 - 5 6 .
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