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International Journal of Game Theory, VoL 13, Issue 1, page 1-13.

A R e l a t i o n b e t w e e n P e r f e c t E q u i l i b r i a i n E x t e n s i v e F o r m G a m e s a n dP r o p e r E q u i l i b r i a i n N o r m a l F o r m G a m e s 1 )

B y E. van Damme, D e l f t 2 )

Abstract: The con cep t of quas i-perfec t equi libr ia for games in extensive form is in t roduc ed. I t i sshown th a t a prop er equil ibr ium of a norm al form gam e induces a quas i -per fec t equi libr ium ine ve ry e x te nsive f o r m g a m e ha v ing th i s no r m a l f o r m .

1 . I n t r o du c t i o nI n Selten [ 1 9 7 5 ] t h e c o n c e p t o f p e r f e c t e q u i l ib r i a w a s i n t r o d u c e d f o r g a m e s in e x -

t e n s iv e f o r m w i t h p e r f e c t r e c a ll in o r d e r t o e x c l u d e t h e p o s s i b i l it y t h a t d i s e q u i l i b r i u mb e h a v i o r is p r e s c r i b e d a t u n r e a c h e d p a r t s o f t h e g a m e t re e . S e l t e n a c h i e v e s e q u i l i b r i u mb e h a v i o r e v e r y w h e r e b y a s s u m i n g t h a t e v e r y p l a y e r w i ll m a k e m i s t a k e s w i t h a s m a l lp r o b a b i l i t y a n d , c o n s e q u e n t l y , w i l l t a k e e v e r y c h o i c e w i t h a p o s i t i v e p r o b a b i l i t y .H e n c e , in s u c h a " p e r t u r b e d g a m e " t h e r e a r e n o u n r e a c h e d p a r t s an d , t h e r e f o r e , e v e r yp l a y e r w i ll m a k e a n e q u i l i b r iu m c h o i c e e v e r y w h e r e . A p e r f e c t e q u i l i b r i u m i s t h e nd e f i n e d a s a n e q u i l i b r i u m w h i c h c a n b e o b t a i n e d a s t h e l i m i t o f a s e q u e n c e o f e q u i l ib r i a ,a s s o c ia t e d w i th a s e q u e n c e o f p e r t u r b e d g a m e s f o r w h i c h t h e m i s t a k e p a r a m e t e r g o e st o z e r o . S e l t en s h o w e d t h a t e v e r y f in i t e g a m e i n e x t e n s i v e f o r m w i t h p e r f e c t r e c a l l h a sa t le a s t o n e p e r f e c t e q u i l i b r i u m , s o i n d e e d t h e p e r f e c t n e s s c o n c e p t i s a u s e f u l r e f in e -m e n t o f th e N a s h e q u i li b r iu m c o n c e p t .

Selten [ 1 9 7 5 ] a ls o i n t r o d u c e d t h e p e r f e c tn e s s c o n c e p t f o r f i n it e g a m e s in n o r m a lf o r m . H o w e v e r , th e r e l a t i o n b e t w e e n p e r f e c t e q u i l ib r i a i n e x te n s i v e f o r m g a m e s a n dp e r f e c t e q u i l ib r i a i n n o r m a l f o r m g a m e s is n o t a s n i c e a s o n e ( p e r h a p s ) w o u l d l ik e i t t ob e . N a m e l y , a p e r fe c t e q u i l ib r i u m o f th e e x t e n si v e f o r m n e e d n o t b e p e r f e c t in t h e n o r -m a l f o r m ( e x a m p l e 4 ) a n d a p e r f e c t e q u i li b r iu m o f th e n o r m a l f o r m n e e d n o t b e p e r -f e c t i n th e e x t e n s i v e f o r m ( c f . t h e e x a m p l e s 1 - 3 ) .

1 The r e se a rc h f o r th i s pa pe r wa s done w he n the a u tho r wa s a t t he Un iver s ity o f Te c hno logyof Eindhoven in the Nether lands . Th e author w ould l ike to tha nk R einhard Se l ten , S te f Ti js , Janvan der Wal and Jaap W esse ls for m an y he lpful and s t imula t ing conversa t ions .2) Dr. E.E.C. van Damme, Dep t . of Mathemat ics and C om put ing Sc ience , Univers ity o f Tech no-logy, Delf t , The Nether lands .0020-7276/84/010001-1352.50 9 1984 Physica-Verlag, Vienna.

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2 E. van DammeIn Myerson [1978] ano th er undes i rab le p roper ty o f perfec t equ il ib r ia o f normal

fo rm g ame s is n o te d , n a m e ly , th a t a d d in g s t r i c tly d o m in a te d s tr a te g ie s ma y t r a n s fo rmimp erfec t equ i lib r ia in to p erfec t ones . There fore , Myerson in t ro duc ed a re f inem ent o fthe perfec tness concept : the p roperness concep t . Fur ther mo re , he showed tha t everynorm al fo rm game has a t leas t one proper equ i l ib r ium.

In th is paper , th e concep t o f quas i -perfec t equ il ib r ia fo r games in ex tens ive fo rm isin t roduce d . The d i f fe rence be tw een th is concept a nd Se l ten ' s pe rfec tness concept i stha t the la t te r requ ires tha t each p layer a t every in form at ion se t takes a cho ice whichis op t imal aga inst mis takes o f a l l p layers ( inc lud ing the p layer h imse lf ) , whereas thequasi-perfectness con cep t requires tha t a t every inf orm atio n set a choice is tak en w hichis op t imal aga inst m is takes o f the o ther p layers . Bo th the perfec tness and the quas i-perfec tness concept a re re f inem ents o f the sequent ia l equ i lib r ium concep t [Kreps/Wilson], bu t a quas i -perfec t equ i l ib r ium need no t be perfec t (c f . example 3 ) and aperfec t equ i l ib r ium need no t be quas i -perfec t (c f . example 4 ) . In sec t ion 4 o f the paper ,w e re la te the p roperness concept o f the norm al fo rm to th e quas i -perfec tness concep tof the ex tens ive fo rm . Our ma in resu l t i s tha t a p roper equ i l ib r ium of a norm al fo rmgame induces a quas i -perfec t equ i l ib r ium in every ex tens ive fo rm game hav ing th isnorm al fo rm . Hence , p roper equ i l ib r ia o f the norm al fo rm induce sens ib le behavior inthe ex tens ive fo rm .

In sec t ion 3 , we rev iew the reason w hy a perfec t equ i l ib r ium of the norm al fo rmdoes no t induce sensible behavior in the ex tensive fo rm. The exam ples g iven in th issec t ion also i l lus t ra te tha t the resu l t o f sec t ion 4 is the bes t w e can ob ta in : we ca nno texpec t th a t a cho ice p resc r ibed by a p roper equ i l ib r ium wil l be a bes t rep ly aga instmis takes o f the p layer h imse lf .Thr ough out the paper , the n o ta t io n used is the same as in Selten [1975] . For thec o n v e nie n ce o f th e re a d er s o me n o ta t io n a n d s o me d e f in it io n s f ro m th a t p a p e r a rereprodu ced in sec t ion 2 .

2. PreliminariesIn th is paper we cons ider noncoo pera t ive f in i te n -person games in ex tens ive fo rm

and the norm al fo rm games assoc ia ted wi t h them . A f in i te game in ex tensive fo rm isd e te rmin e d b y :( i ) A game t ree K , a t ree wi th f in i te ly m an y ver tices and edges ,( i i) fo r each ver tex of K, a spec ifica t ion of which p layer has to m ove a t th is ver tex ,Off) a spec if ica t ion of the in for m at ion a p layer has , wh en he has to m ove; th is

spec if ica t ion is p rov ided by the in fo rm at ion se ts ,(iv) a description of the possible choices a t each mo ve,(v) a spec if ica t ion Of the p robabi l it ie s tha t a re invo lved w i th the chance moves , and(v i) a spec if ica t ion of the rewards fo r each p layer .

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P e r f e c t E q u i l i b ri a i n E x t e n s i v e F o r m G a m e s a n d P r o p e r E q u i l i b ri a i n N o r m a l F o r m G a m e s 3

W e w i ll r e s t ri c t o u r s e lv e s t o g a m e s w i t h p e r f e c t r e c a l l [ K u h n ] , w h i c h m e a n s t h a t a t a n yi n s t a n t o f th e g a m e e a c h p l a y e r k n o w s w h a t h e p r e v i o u s ly h a s k n o w n a n d w h a t h ep r e v i o u s l y h a s d o n e . B e l o w , s o m e d e f i n i t io n s w i t h r e s p e c t t o a f i n i te g a m e I~ i ne x t e n s i v e f o r m w i t h p e r f e c t r e c a l l a r e g iv e n .

L e t u b e a n i n f o r m a t i o n s e t o f p l a y e r i i n F . A l o c a l s tr a t e g y b iu a t u i s a p r o b a b i l -i t y d i s t r i b u t i o n o n C u , w h e r e C u d e n o t e s t h e s e t o f c h o i c e s a t u . T h e p r o b a b i l i t y t h a tb iu a s si g ns t o c is d e n o t e d b y b i u ( c ) a n d B i u d e n o t e s t h e s e t o f al l l o c a l s t r a t e g i e s a tu . W e v i e w C u a s a s u b s e t o f B i u .

A b e h a v i o r s t r a t e g y b i o f p l a y e r i i s a m a p p i n g t h a t a s s ig n s to e v e r y i n f o r m a t i o n s e to f p l a y e r i a l o c a l s t r a t e g y . U d e n o t e s t h e s e t o f a ll i n f o r m a t i o n s e ts o f p l a y e r i a n dB i i s t h e s e t o f a ll b e h a v i o r s t r a t e g ie s o f th i s p l a y e r . A b e h a v i o r s t r a t e g y b i i scompletely m i x e d i f b iu ( c ) > 0 f o r al l u E U / a n d c E C u . I f b i E B i a n d b ~ u E B i u ,

!t h e n b i / b i u i s u s e d t o d e n o t e t h e b e h a v i o r s t r a t e g y w h i c h r e s u l t s f r o m b i i f b iu i sc h a n g e d t o b ' w h e r e a s al l o t h e r l o ca l s t ra t e g ie s r e m a i n u n c h a n g e d .iU 'A p u r e s t r a te g y ~ i o f p l a y e r i i s a m a p p i n g w h i c h a s s ig n s an e l e m e n t o f C u t oe v e r y u E U / . W e w r i t e ~ u f o r t h e c h o i c e w h i c h ~r as s ig n s t o u a n d I I d e n o t e s t h es e t o f a l l p u r e s t r a t e g i e s o f p l a y e r i .

A m i x e d s t r a t e g y q i o f p l a y e r i i s a p r o b a b i l i t y d i s t r i b u t i o n o n H . T h e p r o b a b i l i t yt h a t q i a s s i g n s t o 7 r i s d e n o t e d b y q i (Tr i ) a n d q i i s c o m p l e t e l y m i x e d i f q i ( T r i) > 0 f o ra ll 7r E H . T h e s e t o f a ll m i x e d s t r a t e g i e s o f p l a y e r i i s d e n o t e d b y Q i"

nL e t S / = B i U Q i a n d S = I I S / . E l e m e n t s o f S a r e c a l le d s t ra t e g y c o m b i n a t i o n s . Ai= 1s t ra t e g y c o m b i n a t i o n i s c o m p l e t e l y m i x e d i f al l o f i ts c o m p o n e n t s a re c o m p l e t e l y

! tm i x e d . I f s E S a n d s C S i , t h e n s i s d e n o t e s t h e s t r a t e g y c o m b i n a t i o n w h i c h r e s u lt sf r o m s i f s i s r e p l a c e d b y s~ a n d a ll o t h e r c o m p o n e n t s r e m a i n u n c h a n g e d . I f b i s ab e h a v i o r s t r at e g y c o m b i n a t i o n a n d b ' iu E Bz .u , t h e n b / b ~ u i s u s e d t o d e n o t eb / ( b i / b ~ u ) . F o r s E S a n d e v e r y v e r te x x o f t h e t r e e w e c a n c o m p u t e t h e p r o b a b i l i t yp ( x ; s ) t h a t x w i ll b e r e a c h e d w h e n s is p l a y e d . S i m i l a rl y , w e w r i t e p ( u ; s ) f o r t h ep r o b a b i l i t y t h a t t h e i n f o r m a t i o n s e t u w i ll b e r e a c h e d w h e n s i s p l a y e d . F o l l o w i n gK u h n [ 1 9 5 3 ] , w e c a ll a n i n f o r m a t i o n s e t u E U / r e l e v a n t w h e n p l a y i n g s i , i f u c a n b er e a c h e d w h e n s i s p l a y e d ( i .e . i f p ( u ; s / s i ) > 0 f o r s o m e s E S ) . T h e s e t o f a llu E U / w h i c h a r e r e le v a n t f o r s i s d e n o t e d b y R e l ( s i ) . F u r t h e r m o r e , w e d e f in e f o ru E U i a n d c E C u :

R e l ( u ) := ( ~ E I l i ; u E R e l ( ~ ) ) ,R e l ( c ) : = ( lr E R e l ( u ) ; Z r iu = c ) .

F r o m t h e a s s u m p t i o n o f p e r f e c t re c a ll i t f o l l o w s t h a t t h e p l a y e r s c a n re s t r ic t t h e m s e l v e st o b e h a v i o r s t ra t e g i e s [ K u h n ] . T o b e m o r e p r e c i s e : f o r e v e r y p l a y e r i a n d f o r e v e r y

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4 E. van Damm em i x e d s t r a te g y q i the re ex i s t s a behav io r s t r a tegy b i wh i c h is r e a l i z a t i o n e q u i v a l e n tto q i ' i .e. which satisfies

p ( x ; s / q i ) = p ( x ; s / b i ) f o r e v e r y s a n d x .I f a ll in fo rm at ion se t s a re r e levan t fo r q i (wh ich i s the case i f q i is c o mp l e t e l y mi x e d )the re i s a un ique behav io r s t ra tegy w hich i s r ea l i za t ion equ iva len t to q i" T h i s b e h a v i o rs t r a tegy i s g iven by

Z q i ( r r i)~ri~Rel(c)b i u ( c ) = Z q . ( r ri ) fo r a ll u e U/, c e C , (2 .1 )7ri~Rel(u) z

and i s ca l led the behav io r s t r a tegy i n d u c e d b y q i"As s o c i a te d wi t h e v e r y e n d p o i n t z o f K , t h e r e i s a p a y o f f f o r e a c h p l a y e r ; h i ( z )

d e n o t e s t h e p a y o f f t o p l a y e r i wh e n z i s r e a c h e d . I f t h e b e h a v i o r st r a t e g y c o mb i n a t i o nb i s p layed , the e x p e c t e d p a y o f f t o p l a y e r i i s H i ( b ) = Z p ( z ; b ) h i ( z ) . T h e n o r m a l

zf o r m assoc ia ted wi th F is the norm al fo rm G = 071 . . . . , l'In , H 1 . . . . . H n ) .I f p ( u ; b ) > 0 , t h e n t h e c o n d i t i o n a l p r o b a b i l i t y p ( x ; u , b ) tha t x wi ll be r eached ,

g iven tha t u has been rea ched an d b i s p layed i s we l l -de f ined fo r ev e ry ve r te x x o f K.We d e n o t e p l a y e r i ' s e x p e c t e d p a y o f f g i ve n t h a t u i s r e a c h e d a n d b is p l a y e d b yH i u ( b ) , h e n c e , H i u ( b ) = ~ p ( z ; u , b ) h i ( z ) .

g

We n o w t u r n t o t h e d e f i n i t i o n s o f p e r f e c t a n d p r o p e r e q u i li b ri a . L e t P b e a n e x -tens ive fo rm game and l e t e > 0 . A behav io r s t r a tegy com bina t ion b i s an e - p e r f e c te q u i l i b r i u m of P i f b is com ple te ly mixed and sat is fi es

i f H i u ( b / c ) < I I i u ( b / c ') , t h e n b i u ( c ) < ~ e for all i , u ; c , c ' E C u . ( 2 . 2 )A b e h a v i o r s t ra t e g y c o m b i n a t i o n i s a p e r f e c t e q u i l ib r i u m of P i f i t is a l imi t po in t ( ase t ends to 0 ) o f e -pe r fec t equ i l ib r i a .

L e t G b e a n o r ma l f o r m g a me a n d l e t e > 0 . A mi x e d s t r a te g y c o m b i n a t i o n q i s a ne -pe r fec t equ i l ib r ium of G i f i t is com ple te ly mixed and sat is fi es

i f H i ( q f i r i) < H i ( q f ir i ' ), t h e n q i 0 % ) < ~ e for all i, 7r , lr'i " ( 2 . 3 )A m i x e d s t r a t e g y c o mb i n a t i o n is a p e r f e c t e q u i l ib r i u m o f G i f i t i s a l i mi t p o i n t o fe -pe r fec t equ i l ib r i a o f G .

L e t G b e a n o r ma l f o r m g a me a n d l e t e > 0 . A mi x e d s t r a t e g y c o m b i n a t i o n q i s a ne - p r o p e r e q u i l i b r i u m of G i f i t is com ple te ly mixe d and sat is fi es

i f H i ( q / l r i ) < H i ( q /z r ~ ), t h e n q i Q r i )

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Perfect Equilibria in Extensive Form G ames and Proper Equilibria in Norm al Form Games 5R e m a r k s :1 . T h e d e f i n i t io n o f p e r f e c tn e s s g iv e n a b ov e is s o m e w h a t d i f f e r e n t f r o m S e l t en ' s

o r ig ina l de f in i t ion . We have g iven an equ iva len t de f in i t ion , d ue to M y e r s o n[ 1 9 7 8 ] . ( I t fo l l o w s f r o m L e m m a 5 a n d T h e o r e m 4 o f S e l t e n [ 1 9 7 5 ] t h a t i n d e e da lso fo r games in ex tens ive fo rm th i s def in i t ion i s equ iva len t . )

2 . S ince an e l -p rop er equ i l ib r ium i s a lso e2-proper i f e l < e2 , we have tha t , i f q i sa p r o p e r e q u i l i b r i u m o f G , t h e r e e x i st s f o r e v e r y e > 0 s o m e e - p r o p e r e q u i li b r iu mq e o f G su c h th a t q = l i m q e .e~ 0

3 . L e t F b e a n e x t e n s iv e f o r m g a m e w i t h n o r m a l f o r m G a n d l e t q b e a p r o p e r e q u il i-b r i u m o f G . F u r t h e r m o r e , l e t q e b e a n e - p r o p e r e q u i l ib r i u m o f G w i t hlim q e = q a n d l e t b e b e t h e b e h a v i o r s tr a t e g y c o m b i n a t i o n i n d u c e d b y q e . Sincee~0F i s f in i t e , the re ex i s t s a l imi t po in t (as e t ends to 0 ) o f {b e )e" Such a limi t po in tis called a l i m i t b e h a v i o r s t r a t e g y c o m b i n a t i o n i n d u c e d b y q . N o t e t h a t , i f b is al i m i t b e h a v i o r s t r a te g y c o m b i n a t i o n i n d u c e d b y q , t h e n b i i s r ea l iza t ion eq u iva len tt o q i ' f o r e v e r y i.

I n o r d e r t o f a c i li t a te c o m p a r i s o n b e t w e e n t h e p e r f e c t n e s s c o n c e p t a n d t h e q ua si -per fec tness con cep t , we g ive the fo l lowing charac te r iza t ion o f per fec t equ il ib r ia :L e m m a 1 : A b e h a v i o r s t r a te g y c o m b i n a t i o n b i s a p e r f e c t e q u i l i b r iu m o f a n e x t e n s iv ef o r m g a m e F i f a n d o n l y i f th e r e e x i s t s a se q u e n c e { b k } k ~ N o f c o m p l e t e ly m i x e dbehav ior s t ra tegy com bina t ions , converg ing to b , s u c h t h a t b iu is a l o c a l b e s t r e p l yagainst b k fo r ev ery L u and k , i .e .

H i u ( b k / b i u ) = m a x 1 -1iu ( b k / b ~ u ) fo r all i, u, k.b .u~ B iuP r o o f . T h i s i m m e d i a t e l y f o l lo w s f r o m t h e t h e o r e m s 4 a n d 7 o f S e l t e n [1975] . [ ]

3 . E x a m p l e sIn th i s sec t ion , we g ive some ex amp les to i l lus tra te the d i sc repan cy be twee n per fec t -

n e ss i n t h e n o r m a l f o r m a n d p e r f e c t n e s s i n t h e e x t e n s iv e f o r m [ a ls o se e S e l t e n , sec t ion13]. Th e d i scuss ion o f the exam ples wil l exh ib i t the re la t ion be tw een prope rness in thenorm al fo rm a nd quas i -per fec tness in the ex tens ive fo rm. In exam ple 3 we il lus t ra tew h a t k i n d o f i n f o r m a t i o n c a n n o t b e r e g a i n e d b y c o n s i d e ri n g p r o p e r e q u i li b ri a .

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6

E x a m p l e 1 .3 1 0 01 0 0 1

extensive form I"

E . van Damme

L 2 R 2

L l r

R l r

11 0

00 1

22 2

22 2

Corresponding norma l form G iT h e u n i q u e s u b g ame p e r f e c t ( h e n c e , u n i q u e p e r f e c t ) e q u i l i b r iu m o f I " i s

( L 1 , L 2 ) . This equ i l ib r ium i s al so a pe r fe c t equ i l ib r ium of G, bu t i t is no t the on lyone: a lso ( R x r , R 2 ) i s p e r f e c t i n G . N a me l y , f o r e v e r y e E ( 0 , 1 / 3 ) , we h a v e t h a tqe = (qe l, qe ) w ith q~ = (e , e , e , 1 - 3 e) and q~ = ( e , 1 - e) is a 3 e-p erfec t equi l i -b r i u m o f G wi t h l im q e = ( R l r , R 2 ) .e*0I n o r d e r t o i n v e s ti g a te w h y a p e r f e c t e q u i l i b ri u m o f G n e e d n o t b e p e r f e c t in P , l e tu s c o m p a r e e - p e r f e c t e q u i li b r ia o f P w i t h e - p e r f e c t e q u i li b r ia o f G . N o t e t h a t ( 2 . 2 ) e x -p resses th a t , in an e -pe r fec t equ i l ib r ium o f F , eve ry non-o p t im a l cho ice has to bechosen w i th a sma l l p robab i l i ty . How ever , i f q e i s an e -p e r fec t equ i l ib r ium o f G c loset o ( R l r , R 2 ) , t h e n

qe l (L l r )>~qe l (L l l ) , i f e i s s m all, ( 3 . 1 )s ince R2 m us t b e a bes t r ep ly aga ins t q~ i f e i s sma ll . T here f o re , we have fo r theb e h a v i o r s tr a t e g y b~ i n d u c e d b y q ~ :

be l v ( r )>~ b e ( l ) i f e i s sm a ll.lv ( 3 . 2 )He nce , p laye r 1 i s r equ i red to choo se r w i th a pos i t ive p roba b i l i ty a t h i s second in fo r -m a t ion se t , wh ich obv ious ly i s i r r a t iona l s ince l s t r ic t ly dom ina tes r . Th e reason tha ti r r a t iona l behav io r i s p resc r ibed a t p laye r l ' s s e cond in fo rm at io n se t i s tha t th e loca ls t r a te g y a t t h i s i n f o r m a t i o n s e t is c o m p l e t e l y d e t e r mi n e d b y t h e m i s ta k e p r o b a b i li t ie sq ~ ( L 1 / ) a n d q ~ ( L 1 ) , w h i c h m a y b e c h o s e n a r b i t r a r y i n a n e - p e r f e c t e q u i li b r iu m.

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Perfect Equilibria in Extensive Form Gam es and Proper Equilibria in Norm al Form G ames 7No t i c e t h a t , i f q e i s a n e - p r o p e r e q u i li b r iu m o f G , t h e n ( 3 . 1 ) c a n n o t b e s a t is f ie d

and , the re fo re , the o n ly p ro pe r equ i l ib r ium of G_ s (LI l , L2 ) . H ence , in th i s exam ple ,r a t io n a l b e h a v i o r i n t h e e x t e n si v e f o r m c a n b e d e t e c t e d a l r e a d y i n t h e n o r m a l f o r m .Ho wev er , in the exam ples 2 and 3 we wi ll show th a t th ings a re no t a lways a s n ice a sin th i s example .E xample 2

I o L ~

2 ~ i r

R s

R ri

221

extensive form F norm al form GL r i s a p r o p e r e q u i l ib r i u m o f G , wh i c h is n o t p e r f e c t i n P ( o n l y L1 is p e r f e c t i n P ) .

He n c e , a p r o p e r e q u i l i b ri u m o f t h e n o r m a l f o r m m a y p r e s c r ib e i r ra t i o n al ( n o n ma x i -miz ing) behav io r a t an in fo rm at ion se t wh ich i s no t r e levan t when th i s equ i l ib r iumi s p l a y e d . Ho w e v e r , r a t io n a l b e h a v i o r a t s u c h a n i n f o r m a t i o n s e t c a n b e o b t a i n e d b ycons ide r ing l imi t beh av io r s t ra teg ies ( c f . r emark 3 o f sec t ion 2 ) . N am ely , i f qe is ane -p roper equ i l ib r ium w i th l im qe = Lr , t h e ner

qe (R r)

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8 E. van Damme

T h e u n i q u e p e r f e c t e q u i l ib r i u m o f P i s L I (p laye r 1 has to choo se L s ince he hast o i n c o r p o r a t e t h e p o s s ib i l it y o f ma k i n g m i s ta k e s a t h i s s e c o n d i n f o r m a t i o n s e t ) . Ap r o p e r e q u i l ib r i u m o f G i s R I a n d t h e u n i q u e l i mi t b e h a v i o r s tr a t e g y i n d u c e d b y R l is R Ii t s e lf . Hen ce , a p rope r equ i l ib r ium may p resc r ibe a cho ice which is no t op t ima laga inst mis takes mad e by the p laye r h imse l f . How ever , i f p laye r 1 is su re tha t he wi llm a k e n o mi s t a ke s , h e c a n s a f e l y p l a y R a t h i s f ir s t i n f o r m a t i o n s e t . T h e r e f o r e , we c a ltR I a quas i -pe r fec t equ i l ib r ium o f F .

Fo r a general n -pe r son gam e in ex tens ive fo rm w i th pe r fe c t r eca l l , a quas i -pe r fec tequ i l ib r ium i s de f ine d as a behav io r s t r a tegy com bina t ion w hich p resc r ibes a t ev e ryi n f o r m a t i o n s e t a c h o i c e wh i c h i s o p t i ma l a g a i n st mi s t a k e s o f t h e o t h e r p l a y e r s . Be f o r eg iv in g t h e f o r m a l d e f i n i ti o n , we i n t r o d u c e s o me n o t a t i o n . F o r a n i n f o r m a t i o n s e t u ,we wr i t e Z ( u ) f o r t h e s e t o f a ll e n d p o i n t s o f t h e t r e e c o m i n g a f t e r u a n d i fu , v E U/ , we w ri te u ~< v i f Z (u) D Z (v) . As usual , u < v s tands fo r u ~< v and u r v( h e n c e , u < v me a n s v c o m e s a f t e r u ) . N o t e t h a t t h e r e l a t io n ~< p a r t i a l ly o r d e r s U ,s ince the game has p e r fec t r eca l l. I f h i , b~ E B i and u ~ U/ , the n we use b i / u b ~ o de -n o t e t h e b e h a v i o r s tr a t e g y b~ ' d e f i n e d b y

b';v =b ~. i fv > ~ u,IVb v o the rwise .

F u r t h e r mo r e , i f b E B a n d b~ E Bi , t h e n b / u b ~ d e n o t e s t h e b e h a v i o r s tr a t e g y c o mb i n a -t i o n b/ (b i /u b~) .De f in i t ion I . L e t P b e a n n - p e r s o n g a me i n e x t e n s i v e f o r m wi t h p e r f e c t r e c al l. Ab e h a v i o r s tr a t e g y c o mb i n a t i o n b is a quas i -per fec t e qu i l ibr ium o f P i f t h e r e e x i s ts a

{ b k } k ~ N o f c o mp l e t e l y m i x e d b e h a v i o r s t r a t e g y c o mb i n a t i o n s , c o n v e r g in ge q u e n c eto b , such tha t fo r a ll L u and k

H i u ( b k / u b i ) = m a x Hiu (bk /ub~) .b ~ . ~ B i

( 3 . 3 )

I n e x a m p l e 3 , we h a v e a l r e a d y s e en t h a t a q u a s i - p e rf e c t e q u i l ib r i u m n e e d n o t b ep e r f e c t . T h e n e x t e x a m p l e s h o ws t h a t a p e r f e c t e q u i li b r iu m n e e d n o t b e q ua si-p e r f e c t .

Th e un iq ue quas i -pe r fec t equ i l ib r ium of F i s (L 1 /, L : ) . Th i s equ i l ib r ium i s a lsop e r f e c t , b u t i t i s n o t t h e o n l y o n e : a n o t h e r p e r f e c t e q u i l i b r iu m i s ( R 1 /, L z ). No t e t h a tt h i s e x a mp l e a ls o sh o ws t h a t a p e r f e c t e q u i li b r iu m o f P n e e d n o t b e p e r f e c t in G ( t h eu n i q u e p e r f e c t e q u i l i b ri u m o f G i s ( L 1 /, L 2 ) ) .

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Perfect Equilibria in Extensive Form G ames and Proper Equilibria in Normal Form Games 9E x a m p l e 41 0 1 01 0 1 0

1extensive form I"

L 2

1L l l 1

0 'L l r L 0

1R I Z

1R l r

normal form G

R 2

, 1 ' l0 o i0 "

1 ! o !

I t c a n e a s i l y b e s h o w n t h a t e v e r y q u a s i -p e r f ec t e q u il i b ri u m o f t h e e x t e n si v e f o r mi n d u c e s a p e r f e c t e q u i l i b ri u m i n t h e n o r m a l f o r m ( t h e c o n v e r s e is n o t t r u e , c f . e x a m p l e1). The fo l lowing propo s i t ion i s easy to sho w e i ther [c f . Krep s / lC i l so n , Propos i t ion 5 ] ,

P r o p o s i t i o n 1 . Eve ry quas i -per fec t equ i l ib r ium i s a sequen t ia l equ i l ib r ium.T h e c o n v e r s e o f t h i s p r o p o s i t i o n is f a ls e , si nc e f o r n o r m a l f o r m g a m e s t h e s e t o f

per fe c t equ i l ib r ia co inc ides wi th the se t o f qu as i -per fec t equ i l ib r ia and s ince fo rsuch a game no t every sequen t ia l (= Nash) equ i l ib r ium i s per fe c t .4 . The M ain Resul t s

I n t h i s s e c t io n , w e w i ll p r o v e t h a t e v e r y p r o p e r e q u i l i b ri u m o f a n o r m a l f o r m g a m einduces a quas i -per fec t equ i l ib r ium in eve ry ex tens ive fo rm gam e hav ing th i s norma lform . Fu r the rm ore , severa l charac te r iza t ions o f quas i -per fec t equ i l ib r ia wil l be der ived .

W e s ta r t w i t h a d e f i n it i o n . L e t b b e a c o m p l e t e l y m i x e d b e h a v i o r s t r a te g y c o m b i n a -b 'i o n a n d l e t iu b e a l o c a l s tr a t e g y a t t h e i n f o r m a t i o n s e t u o f p l a y e r i . W e s a y t h a t

b'iu is a q u a s i - b e s t r e p l y aga ins t b i f the re ex i s t s some b~ E B i which p resc r ibes b'iu a t us u c h t h a t

(b / b ""i u ( b / u b ~ ) = m a x H i u " , u i )"b "~B .l lW e h a v e t h e f o l l o w i n g l e m m a :L e m m a 2 . b i s a quas i -per fec t equ i l ib r ium o f F i f and o n ly i f the re ex i s t s a sequence( b k ) k ~ N o f c o m p l e t e l y m i x e d b e h a v i o r s tr a t e g y c o m b i n a t i o n s c o n v e r g i n g t o b s u c ht h a t biu is a quasi-best rep ly against b k fo r ev ery i, u an d k .P r o o f . I t i m m e d i a t e l y f o ll o w s f r o m t h e d e f i n t i o n s t h a t a q u a s i - p e r fe c t e q u i l ib r i u m h a st h e p r o p e r t y o f th e l e m m a .

Converse ly , assume tha t b a nd { b k ) k ~ N s a t is f y t h e c o n d i t i o n o f t h e l e m m a . F o r ap laye r i and u E Ui , def ine

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10 E. van Damm eS ( u ) : = ( v ~ U i ; v > u ) ,$1 (u ) := (v E S (u ) ; no t the re ex i s t s w E S (u ) wi th v E S (w)} ,Z1 (u ) := (z E Z ; no t the re ex i s t s w E S (u ) wi th z E Z (w ) ) .

S (u) ( resp. S 1 ( u ) ) i s the se t o f a ll in fo rm at ion se t s com ing a f te r u ( r esp . comingimm edia te ly a f te r u ) . F ix k ~ N. We wil l p rove (3 .3 ) by us ing backw ards induc t ion int h e g a m e t r e e .

I f S ( u ) = 0 , t h e n ( 3 . 3 ) f o l lo ws im m e d i a t e l y f r o m t h e d e f i n i t io n o f q u a si -b e s treplies.

N ex t , a s sume (3 .3 ) has a l ready been p ro ved fo r a ll in fo rm at ion se t s in S (u ) . Then ,f o r e v e r y b I E B i , w e h a v e * )

H i u ( b k /u b i / b i u ) == Z p ( v ; u , b k / u b ~ / b i u ) I -l iv ( b k / u b ~ / b i u ) +v ~ S l ( u )+ ~ p ( z ; u , b k / u b ~ / b i u ) h i ( z )z ~ Z 1 (u )

p ( v ; u , b k / u b i ) I t i v ( b k / vb ~ ) +v ~ S x ( u )+ Z p ( z ; u , b k / u b i ) h i ( z )z~ Z 1 (U)< . Z p ( v ; u , b k / u b i ) I t i v ( b k / v b i ) +v ~ S x ( u )+ ~ p ( z ; u , b k / u b i ) h i ( z )z ~ Z 1 (u )= H i u ( b k / u b i ) ,

wh e r e t h e i n e q u a l i ty f o l lo ws f r o m t h e i n d u c t i o n h y p o t h e s i s . C o m b i n i n g t h is w i t h t h ef a c t t h a t b iu i s quasi-best rep ly against b k , we see tha t fo r a ll b i ' E B i

~ u ( b k u b l ) < < ' ~ u (b X u b i ) 'wh i c h p r o v e s ( 3 . 3 ) a n d c o m p l e t e s t h e p r o o f o f t h e l e m m a . [ ]

We wil l need the fo l lowing chara c te r iza t ion o f quas i -bes t r ep lie s :4) bk /ub} /b iu = bk /u ( b~ /b iu ) = ( bk / ub} )/b iu so no confusion can result.

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Perfect Equilibria in Extensive Form Gam es and Proper Equilibria in Norm al Form Gam es 11L e m m a 3 . L e t / ~ b e a c o mp l e t e l y m i x e d b e h a v i o r st r a te g y c o m b i n a t i o n a n d l e t b iube a loca l s t r a tegy a t u E U/ . Then b iu i s a quas i -bes t r ep l y aga inst b i f and on ly i ff o r a ll c E C :U

i f b i u ( c ) > O , the n c is a quasi-best reply agains t/~.tP r o o f . T h i s i mm e d i a t e l y fo l l o ws f ro m t h e o b s e r v a ti o n t h a t f o r e v e r y b i E B i

t t i u ( b / u b ~ / b i u ) = ~ t t i u ( b / u b ~ / c ) b i u ( c )c ~ C u

( b / u b i / c ) d e p e nd s o n l y o n w h a t b i p r e sc r ib e s a t t h e i n f o r m a t i o nn d t h e f a c t t h a t H / u " 's e t s wh ich com e a f te r c (he re again we use tha t the game has pe r fec t r eca ll ) . [ ]L e t u s c a ll a s t r a te g y c o m b i n a t i o n b a n e - q u a s i - p e r f e c t - e q u i l i b r iu m ( e > 0 ) i f i t

i s co m ple te ly mix ed an d sa tisf ies fo r a ll i, u and c , c ' E C u :i f m a x t t i u ( b / u b l / e ) < m a x t t iu ( b /u b ~ / c ') , t h e n b i u ( c ) < ~ e .b ' . E B . ' ~t l b i B i

Th en i t is an easy conseq uence o f Lem m a 2 and I_emma 3 tha t quas i -pe r fec t equ i l ib r iacan be ch a rac te r i zed as fo l lows :P r o p o s i t i o n 2 . A behav io r s t r a tegy com bina t ion i s a quas i -pe r fec t equ i l ib r ium i f andon ly i f i t i s a l imi t po in t ( a s e t en ds to ze ro ) o f e -quas i -pe r fec t equ i l ib r i a .

Th e m a in re su l t o f th i s sec t ion i s:T h e o r e m 1 . L e t G b e a n o r ma l f o r m g a me a n d l e t q b e a p r o p e r e q u i l i b r iu m o f G .I f F i s a n e x t e n s iv e f o r m g a me w i t h n o r ma l f o r m G a n d i f b i s a l i mi t b e h a v i o r s t r a te g yc o m b i n a t i o n i n d u c e d b y q i n P , t h e n b is a q u a s i- p e r fe c t e q u i l ib r i u m o f P .P r o o f . L e t G a n d F b e a s i n t h e f o r m u l a t i o n o f t h e t h e o r e m . I n v ie w o f p r o p o s i t i o n 2 ,i t s u f fi c e s t o s h o w t h a t a b e h a v i o r s tr a t e g y c o m b i n a t i o n b e , i n d u c e d b y a n e - p r o p e requ i l ib r ium q e o f G, i s a x /~-quas i -pe r fec t equ i l ib r ium o f F , i f e i s su f f i c ien t ly sma ll .He n c e , l e t q e b e a n e - p r o p e r e q u i l ib r i u m o f G a n d l e t b e b e i n d u c e d b y q e in F . F i r s to f a ll , no t i ce th a t b e i s com ple te ly mixed . Ne x t , a s sume i, u , c and c ' a re such th a t

m a x H i u ( b e / u b ~ /c ) < m a xb i E B i b i E B i H i u ( b e / u b i /c ' ) . ( 4 . 1 )S i n ce t h e m a x i m u m i n ( 4 . 1 ) i s al wa y s a t t a i n e d b y a p u r e s t r a te g y , ( 4 . 1 ) is e q u i v a l en t t o

m a x H . ( bC l rc i) < m a xz r i~Re l (c ) m ~r~ Re l (c ' ) H i u ( b e / T r i ) " ( 4 . 2 )

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12 E. van Damm eLe t ~ i E R e l ( c ') b e s u c h th a t t t i u ( b e / # i ) i s equa l to the r igh t han d s ide o f (4 .2 ) . Th en

( b e / e i ) f o r a l l s g e l ( c ) ,and , the re fore

H i ( b e / T r i ) < H i ( b e / z r i / u ~ ) for a ll w E R el (c). (4 .3 )Since b e i s rea l iza tion equ iva len t to q e , (4.3) is equiva lent to

H i (qe /T r i)

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Perfect Equilibria in Extensive Form G ames and Proper Equilibria in Normal Fo rm G ames 13

H o w e v e r , a ls o t h e c o n v e r s e o f t h is m u c h s t r o n g e r v e r s io n o f T h e o r e m 1 i s f a l se : t h e r ee x i st q u a s i - p r o p e r e qu i li b ri a o f e x te n s iv e f o r m g a m e s w h i c h c a n n o t b e i n d u c e d b yp r o p e r e q u i l i b r ia o f t h e a s s o c i a t e d n o r m a l f o r m [ se e e. g. t h e g a m e o f f i g u re 6 . 5 . 2 o fv an D a m m e ] .

References

Damme, E.E.C. van: Refinements o f the Nash equilibrium concept. P h.D . thesis EindhovenUniversity of Technology, Eindhoven 1983.Kreps, D.M., and R. Wilson: Sequential equil ibr ia . Econo metr ica 50, 1982, 86 3-8 94 .K u h n , H. W.: Extensive games and th e problem of information. Annals of M athematics Studies28 , 1953 , 193-216 .Myerson, R.B.: Refinements o f the Nash equilibrium concept. Int. J . Game Th eor y 7, 1978,7 3 - 8 0 .Sel ten, R .: Reexam ination of the perfectness co nce pt for equilibrium points in extensive games.In t . J. Gam e T heory 4 , 1975, 2 5 -5 5 .

Received April 1981(revised version April 1983)