Revista de Ja Union Matematica Argentina Volumen 36, 1990.

CONSTRUCTING BIRATIONAL GAMES WITH. GIVEN EQUIUBRIUM POINTS

EZIO MARCHI and JORGE A. OVIEDO

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A B S T RA C T . Thi s work i s addr es s ed to the problem o f construct ing ) b irat ional games with predet ermined equil ibr ium po int s . We ) develop techn iques which g eneral i z e tho s e introduc ed for b ima -tr ix games .

A nece s sary and suff ic i ent cond i t ion for a pair of s trat eg i e s to be a un ique equ i l ibr ium po int of a b irat ional game is g iven .

K E Y W O R D S . Two -p er son game s . B irat ional games . Equ i l ibr iu� po int s . Construc t ing b irat ional games . Uniquene s s .

I . I NT RO D U C T I O N

This work i s concerned with pr e s ent ing t echnique s for cons tru£ t ing b irat ional games with pr edet ermined equ i l ibr ium po int s . The s e t echn ique s ar e s imilar to the method for cons truct ing

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)

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b imatr ix game s . ) A b irat ional game i s def ined by a quadrupl e (A , B ; C , D ) of r eal ) mxn matr ices tog ether with the Carte s ian produc t XxY of a l l

P a r t o f t h i s r e s e a r c h wa s s up p o r t e d b y C ON l C E T a n d UN S L , S an L u i s , Ar g en t i n a .

) ) ) ) ) )

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m -d imens ional probab i l ity vector s X and a l l n - d imens ional pro ­bab i l ity vectors Y . When B=D=Jm , n ' wher e Jm , n i s the mxn matrix with every el ement equal to 1 , a b irat ional game reduc es to an ord inary b imatr ix game .

I f (x , y ) E XxY , the payo ffs of the game (A , B ; C , D ) for pl ayer i, i = 1 , 2 , i s defined by E (x , y ) = xAy/xBy and F (x , y) = xCy/xDy , re spec t ivel y . E (x , y) ( F (x , y) ) i s def ined ( though po s s ibly equal to +00) prov ided i t s numerator and denominator are no t s imul ta ­neou s ly equa l to O .

A po int (x , y) in XxY i s an equi l ibr ium po int o f the game (A , B ; C , D ) if xAy/xBy � �Ay/�By for a l l � E X and xCy/xDy �

� xCn /xDn for a l l n E Y . Marchi [ 1 9 7 6 ] proved that every rat io nal game (A , B ; C , D ) with B > 0 and D > 0 ( i . e . , every el ement b . . , d . . of B , D are po s i t ive real ) ha s an equi l ibr ium po int . l.J l. J Von Neumann [ 1 9 3 7 ] in the cour s e o f analy z ing a model of econo mic growth , has been the f ir s t in cons ider ing a two - person z ero - sum game with nonl inear payo ff funct ion . Subsequent devel£. pment of the model ha s been synthes i z ed by Morgenst ern and Thomp son [ 1 9 7 6 ] . The same payoff func t ion appears in a spec ial ca s e o f a stochast ic game propo s ed by Shapl ey [ 1 9 5 3 ] . Marchi [ 1 9 7 6 ] ext ended and g eneral i z ed the equ i l ibr ium po int s of a r� t ional game to a . n� person game with a rat iOnal payoff func ­t ion . Marchi [ 1 9 7 9 ] and Marchi , Tara z aga , E lor z a [ 1 9 8 3 - 4 ] ap ­pl i ed such resul t s to obta in a new approach to expand ing eco ­nomie s .

I I . C O N S T R U C T I N G A G A M E W I T H G I V E N E Q U I L I B R I U M P O I N T S

Let (x , y) be an equi l ibr ium po int , w i th a = xAy/xBy and S = xCy/xDy the corr e spond ing payoffs . Deno t e by S ex ) , S (y ) , M (x ) and M (y) the fol l owing s et s :

S ex ) { i : M (y ) { i :

wher e A . ( B . ) l. . l. . i s lumn of C (D ) .

x . > O } l. A . y = l. . aB o y} l. .

the ith row

S (y)

and M (x)

of A (B ) and

0 : Yj > O }

0 : xC . = . J

C . ( D . ) t he . J . J

SXD .j }

j th co -

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THEOREM 2 . 1 . L e t (x , y ) r e pr e s e n t a p a i r of pro b ab i l i t y v e a t o r s

in XxY, l e t a , S b e r e a l num b e r s a n d l e t B , D > 0 b e mxn re a l

matr i a e s . T h e r e e xi s t two ma tr i a e s A and C s u a h t ha t t he b i r a ­

t i o na l g a m e (A , B ; C , D ) has (x , y ) a s an e q u i l i b r i um p o i n t a n d

a , S a s t h e aorre sponding payoff t o t h e p l a y e r s .

Pro o f . Without lo s s of g eneral ity we as sume that a > 0 and S > 0 (becaus e if xAy/xBy < 0 there exi s t s c > 0 such t hat x (A+cB ) y/xBy > 0 ; a s im i lar re sul t i s val id for S ) . Regard x

and y a s l inear transformat ions from Em and En to E 1 , r e spect i ­vely . L e t c 1 , . . . , cm _ 1 and a 1 , . . . , an _ 1 b e ba s e s re spect ively , for the nul l spac es o f

n - 1 A . = aB o + r A · ka k , � . �. k= 1 �

for j E S ly) , l e t C . = .

• J

ry r eal number s . For i

x and y . Fo r i E S ex ) l et where A i k ar e arb i trary r ea l number s and

m - 1 SD . + I ykJ· c k , where ykJ. ar e arb i tra ­

. J k= 1 fI. S ex ) , l e t A . = ( 1 / 2 ) aB . . For j rt. S (y ) , . � . . � .

l et B . = ( 1 / 2 ) SB . . Then the b irat ional game (A , B ; C , D ) ha s an • J • J

equ i l ibt ium pD int (x ,y) .

A po int (x , y) o f XxY i s sa id to be a compl etely mixed po int if

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.J ) )

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X . > 0 , i = 1 , . . . , m , and. y . > 0 , j 1 , . . . , n . ) 1. J

THEOREM 2 . 2 . L e t (x , y) b e a aomp l e te l y mixed p o i n t and B ,D > O.

A n e a e s sary and suffi a i e n t a o ndi t i o n for t h e exi s t e n a e of a bira t i o na l game (A , B ; C , D ) t h a t ha s (x , y) a s i t s u n i q u e aomp l e -

t e l y -m i x e d e qu i l i b r i um p o i n t i s t h a t m n .

Pro o f . As sume m = n . Choo s e , a s before , a bas i s a 1 , . . . , an _ 1

for the nul l spac e of y . Let c 1 , . . . , cn_ 1 b e a bas i s for the

nul l spac e of x . Let A be the nxn matr ix who s e i.- t h row i s A . = aB o + a ; , i = 1 , . . . , n - 1 . Let A = aB Let C be the � . � . � n . n . nxn matr ix who s e j - th column i s C . SD . + C . j = 1 , . . . ,n- 1 .

• J • J J

Let C = BC • Then (A , B ; C , D ) ha s an equil ibr ium po int (x , y ) . . n . n Let (x * , y * ) be ano ther compl etely -miXed equil ibr ium po int with

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payoffs a * , S * . Then Ay* = a *By* and x *C = S *x *D , in part icu ­l ar A y* = a *B y * and x*C = S *x*D . I t fo l l ows from the n . n . . n . n definit ions of A and B that a * = a and S * = S ; but the

. n . n

sys t ems (A - aB ) z = 0 and ( C - SD) w = 0 have rank n - 1 , imp ly ing x = x* and y = y* .

The proof of the nec e s s ity i s s imilar to the proof g iven by Mil lham [ 1 9 7 3 , Theorem 2 ] .

We general i z e Theor em 2 . 2 for arb itrary probab il ity vector s .

THEOREM 2 . 3 . L e t (x , y) b e a p a i r of probab i l i ty v e e t o r s and

B , D > O . A n e c e s sary a n d suffi c i e n t eondi t i o n fo r the exi s t en ­

e e o f a b i r a t i o n a l game t ha t has (x , y ) a s i t s u n i q u e e qu i l i ­

brium p o i n t i s t ha t I S (x ) 1 = I S (y ) I .

We omit the proof s inc e it i s s imilar to that of Kre ep s [ 1 974] .

In our cas e we have t o replace systems ( 1 ) and ( 2 ) in Kr eep s ' paper by the fo l lowing

m L x . (c . . - c · 1 - S (d . . - d · 1 ) ) = 0

i= 1 1 1J 1 1 J 1 m L x i (c ij - C i 1 - S (d ij - d i 1 ) ) � 0

i= l

n L YJ. ( a iJ· .,a 1 j - S ( b iJ· -b 1 j )) = 0

j = l n L y . (a . . -a l j - S ( b . . - b 1 j ) ) � 0 j = 1 J 1J 1 J

j = 2 , ; . . , k .

)

2 ( 1 )

j = � +l , . . . ,n .

i = 2 , . . . , k1 . r (2)

j = kl + 1 , . . . ,m. J

Under the cond it ions g iven by Mi l lhan [ 1 9 7 3 , Theor em 4 ] two equi l ibr ium po int s in b imatr ix games are int erchangeabl e . Thi s r e sult i s found t o ho l d true a l so for b irat ional game s .

1 1 1 1 2 2 2 2 . . THEOREM 2 . 4 . L e t (x , y , a , S ) , (x , y , a , S ) b e two equ� l � -

b r i um p o i n t s and p a y o ffs for a b i r a t i ona l game (A , B ; C , D) . A 1 . 1 1 1 n e e e s sary and s uffi c i e n t e o ndi t i on fo .!' (x , y , a , S ) ,

(x 2 , y2 , a2 , S 2 ) t o b e i n t e r c hang e a b l e i s S (x 1 ) � M (y2 ) , S (x2 ) � M (y 1 ) , S (y 1 ) � M ex2 ) , S (/ ) � M (x 1 ) .

Pro o f ·

x : > 0 1 x � > 0 1

Suppo se

imp l i e s impl i e s

(x l , y 2 ) A . y 2

1 . A . y 1 1 .

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and (x 2 , y l ) 2 2 a, B . y 1 . 1 2 a, B . Y 1 •

or or

are equi l ibr ium po int s . Then

S (x l ) 2 £ M ey ) , and s imilarly S (x 2 ) .£:; 1 M (y ) . The rema inder

of the nec e s s ity aspec t s of the proo f are c l ear .

Suppo s e , on the other hand , that the g iven cond it ions holds . The 1 2 2 2 2 cond it ion S ex ) .£:; M ey ) impl ie s that if Ai . y < a, B i . y then

x� = 0 , and the cond it ion S (y2 ) £ M (x l ) impl i e s that if x l c . B I x I n . then y� 0 , from which if fo l l ows that (x l , y 2 )

• J • J J i s an equi l ibr ium po int with payo ffs a, 2 and B l . The re s t o f the proof is ident ical in natur e to that s tated .

REFEREN C E S

[ 1 ] KRE P S , V . L . [ 1 9 7 4 ] , B..i.matJr...i. x g am e.6 w..i.t h un..i.q u e eq u..i.i..i. bJr...i.um po ..i.nt.6 , In t J o u rn a l o f G am e T h eo r y , vo l . 3 , I s s u e 2 : 1 1 5- 1 1 8 .

[ 2 ] MAR C H I , E . [ 1 9 7 6 ] , E q u..i.i..i. bJr...i. um p o ..i. nt.6 0 6 Jr.at..i.o n ai N - p eJr..6 o n g am e.6 , J o u r n a l Ma t h . An a l . a n d A p p l i c . 5 4 , 1 : 1 - 4 .

[ 3 ] MARC H I , E . [ 1 9 7 9 ] , Ei m o d e io d e c.Jr. e c...i.m..i. e nto d e v o n N eum an n p aJr.a un num eJr.o aJr. b..i.tJr.aJr...i.o d e pai.6 e.6 , R ev . Un i o n Ma t . A r ­g�nt i n a . 2 9 : 8 5 - 9 5 .

[ 4 ] MARCHI , E , TARAZAGA , P . a n d E L O R Z A , E . [ 1 9 8 3 - 4 ] , F UJr.t h eJr. t o ­p..i.c..6 ..i.n v o n N euma n n gJr.o wth mo d ei , P o r t u g a l i e Ma t h ema t i c a 4 2 ( 3 ) : 2 5 5 - 2 6 4 .

[ 5 ] MILLHAN , C . B . [ 1 9 7 3 ] , Co n.6 :tJr. uc.t..i.ng bim atJr.i x g am e..6 w..i.t h .6 p e c.iai pJr.o p eJr.ti e.6 , N a v . Re s . L o g . Q u a r t . 1 9 , N ° 4 . 7 0 9 - 7 1 4 .

[ 6 ] MORGENT E RN , O . a n d THOMP S ON , G . L . [ 1 9 7 6 ] , M at h emati c.ai T h e.o Jr. y 0 6 e x p a n ding a n d c. o ntJr.a c.ti ng ec.o no mie.6 , D . C . H e a t h ,

-

Lex ing to n � Ma s s �

) ) ) ) ) ) ) )

) ) ) ') :I

) )

\ )

)

}

)

) )

167

[ 7 ] S HAPt EY , L . S . [ 1 9 5 6 ] , Sto c h�� tlc g �m e� , P r o c . N a t . A c a d . S c i . U . S . A . 3 9 : 1 0 9 5 - 1 1 0 0 .

[ 8 ] V o n NEUMANN , J . [ 1 9 3 7 ] , � b e� eln � ko no ml� c h e� Gt elc hung� ­� y� t em und eln e V e��ttg em eln e�ung d e� B� o uwe�� c h en Flx ­p u n kt� �t z e� , E r g e bn i s s e e in e s Ma t h ema t i s c h e n Ko l l o qu ium s 8 , 7 3 - 8 3 . T r an s l a t e d in Rev i ew o f E c o n om i c s S t u d i e s 1 9 4 5 - 6 .

I n s t i t u t o d e Ma t �mi t i c a Ap l i c a d a S an L u i s

Un iv e r s i d a d Na c io n a l d e S a n L u i s

Ej e r c i t o d e l o s An d e s 9 5 0 5 7 0 0 - S an L u i s

Rec ib ido en mar z o de 1 9 9 0 .