UBC Theses and Dissertations

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UBC Theses and Dissertations

Bargaining solutions to the problem of exchange of uncertain ventures Weerahandi, Samaradasa 1976

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B A R G A I N I N G S O L U T I O N S T O T H E P R O B L E M O F E X C H A N G E O F U N C E R T A I N V E N T U R E S b y ( j f E E R A H A N D L S A M A R A D A S A B . S c , , V i d y o d a y a U n i v e r s i t y o f S r i L a n k a , 1 9 7 0 A T H E S I S ' S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E D E P A R T M E N T O F M A T H E M A T I C S ( I N S T I T U T E O F A P P L I E D M A T H E M A T I C S A N D S T A T I S T I C S ) W e a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A O C T O B E R , 1 9 7 6 ( c ) S a m a r a d a s a W e e r a h a n d i , 1 9 7 6 In presenting th is thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t i on of this thesis for f i nanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. M a t h e m a t i c s Department of • The Univers i ty of B r i t i s h Columbia 207S Wesbrook Place Vancouver, Canada V6T 1W5 Date i i Supervisors: Lawrence M. Clevenson and James V. Zidek ABSTRACT Consider the b e t t i n g problem where two i n d i v i d u a l s negotiate to determine the amount each w i l l bet. I t has already been established that when the two bettors both have concave u t i l i t y functions, there e x i s t mutually b e n e f i c i a l bets ( i . e . bets giving p o s i t i v e u t i l i t y to both players) merely i f players' subjective p r o b a b i l i t i e s on the b e t t i n g event d i f f e r . I t i s shown here that t h i s r e s u l t can be generalized to the case of more general u t i l i t y functions. The r e s u l t s are extended to a more general s i t u a t i o n , that of a stochastic exchange. It i s shown that the set of a l l f e a s i b l e solutions a v a i l a b l e for exchange for two r i s k averters i s a convex set with a known boundary. Aft e r defining a s o l u t i o n for the members of a class of exchange models i t i s shown i n the t h i r d chapter that the 'size' of the exchange prescribed by the s o l u t i o n tends to increase with the p a r t i c i p a n t s ' i n i t i a l wealth and with m u l t i p l i c a t i v e s h i f t s of the random var i a b l e characterizing the exchange. Furthermore the s i z e of the exchange may increase or decrease due to an additive s h i f t of t h i s random v a r i a b l e . In Chapter 4 i t i s shown by an axiomatic method that an i n d i v i d u a l engaged i n bargaining with incomplete information finds h i s ' f a i r ' demand (offer) by maximizing a generalized Nash function, GNF; t h i s GNF i s found to be the product of h i s u t i l i t y and a general mean of h i s opponent's uncertain u t i l i t y (from f i r s t i n d i v i d u a l ' s point of view). This general mean i s characterized by a parameter whose value may vary from person to person. Continuing the study on bargaining under incomplete information, a best i i i b a r g a i n i n g s t r a t e g y i s d e v e l o p e d i n t h e l a s t c h a p t e r u s i n g t h e t e c h n i q u e o f ' B a c k w a r d I n d u c t i o n ' . A c r i t e r i o n f o r c o m p a r i n g a v a i l a b l e b a r g a i n i n g s t r a t e g i e s i s a l s o e s t a b l i s h e d . i v T A B L E O F C O N T E N T S P a g e C H A P T E R 1 . I N T R O D U C T I O N 1 C H A P T E R 2 . E X C H A N G E P R O B L E M : E X I S T E N C E O F S O L U T I O N S 4 2 . 1 I n t r o d u c t i o n 4 2 . 2 M o d e l o f t h e B e t t i n g P r o b l e m 5 2 . 3 E x i s t e n c e o f M u t u a l l y F a v o r a b l e B e t s 8 2 . 4 A n E x t e n s i o n t o t h e E x c h a n g e P r o b l e m 1 4 2 . 5 G e o m e t r i c a l S t r u c t u r e o f t h e S e t o f A l l M u t u a l l y F a v o r a b l e E x c h a n g e s 2 8 C H A P T E R 3 . E X C H A N G E P R O B L E M : N A S H S O L U T I O N S 3 3 3 . 1 I n t r o d u c t i o n 3 3 3 . 2 E d g e w o r t h ' s C o n t r a c t C u r v e A n a l y s i s 3 4 3 . 2 . 1 E d g e w o r t h T h e o r y 3 4 3 . 2 . 2 P a r e t o O p t i m a l E x c h a n g e s 3 5 3 . 3 N a s h ' s S o l u t i o n s 3 9 3 . 3 . 1 T h e N a s h T h e o r y 4 0 3 . 3 . 2 F a i r E x c h a n g e s 4 1 3 . 3 . 3 R e s p o n s e o f t h e N a s h S o l u t i o n t o C h a n g e s i n C , P , a n d Y 4 6 C H A P T E R 4 . B A R G A I N I N G P R O B L E M : S O L U T I O N S U N D E R I N C O M P L E T E I N F O R M A T I O N 5 9 4 . 1 I n t r o d u c t i o n 5 9 4 . 1 . 1 B a c k g r o u n d 5 9 4 . 1 . 2 B a r g a i n i n g U n d e r I n c o m p l e t e I n f o r m a t i o n 6 1 4 . 2 T h e P r o b l e m 6 3 4 . 2 . 1 T h e N a t u r e o f t h e P r o b l e m 6 3 4 . 2 . 2 F o r m u l a t i o n o f t h e B a r g a i n i n g M o d e l 6 5 4 . 3 A n A x i o m a t i c A p p r o a c h t o t h e D e r i v a t i o n o f F a i r D e m a n d s 6 9 4 . 3 . 1 T h e A x i o m s 6 9 4 . 3 . 2 D e d u c i n g t h e F o r m o f G N F W h e n ft C o n t a i n s T w o E l e m e n t s 7 0 4 . 3 . 3 E x a m p l e 8 5 4 . 3 . 4 D e r i v i n g o f t h e G e n e r a l F o r m o f G N F 9 4 4 . 4 S u m m a r y a n d C o n c l u s i o n s 9 8 C H A P T E R 5 . B A R G A I N I N G P R O B L E M : A N O P T I M A L B A R G A I N I N G P R O C E S S 5 . 1 I n t r o d u c t i o n 5 . 2 O p t i m a l S t r a t e g y w h e n t h e O p p o n e n t ' s S t r a t e g y i s P r e p l a n n e d 5 . 2 . 1 N o t a t i o n a n d t h e M o d e l 5 . 2 . 2 C r i t e r i o n f o r C o m p a r i s o n o f B a r g a i n i n g S t r a t e g i e s 5 . 2 . 3 B a c k w a r d I n d u c t i o n P r o c e d u r e ( B I P ) 5 . 3 B a r g a i n i n g S t r a t e g i e s b y B a c k w a r d I n d u c t i o n P r o c e d u r e 5 . 3 . 1 I n t r o d u c t i o n 5 . 3 . 2 B a c k w a r d I n d u c t i o n P r o c e d u r e 1 0 0 1 0 2 1 0 2 1 0 4 1 0 7 1 2 0 1 2 0 1 2 1 V T a b l e o f C o n t e n t s ( C o n t ' d . ) P a g e B I B L I O G R A P H Y 1 2 7 A P P E N D I X A 1 3 0 A P P E N D I X B 1 3 3 A P P E N D I X C 1 3 5 v i L I S T O F F I G U R E S P a g e C H A P T E R 2 F i g . 2 . 5 . 1 3 1 F i g . 2 . 5 . 2 3 2 C H A P T E R 3 F i g . 3 . 2 . 1 3 6 F i g . 3 . 3 . 1 4 4 F i g . 3 . 3 . 2 5 2 F i g . 3 . 3 . 3 5 5 C H A P T E R 4 F i g . 4 . 3 . 1 , 7 1 F i g . 4 . 3 . 2 7 6 F i g . 4 . 3 . 3 7 7 F i g . 4 . 3 . 4 8 6 F i g . 4 . 3 . 5 8 8 F i g . 4 . 3 . 6 8 8 F i g . 4 . 3 . 7 9 0 F i g . 4 . 3 . 8 9 0 F i g . 4 . 3 . 9 9 1 F i g . 4 . 3 . 1 0 9 1 F i g . 4 . 3 . 1 1 9 3 F i g . 4 . 3 . 1 2 9 6 C H A P T E R 5 F i g . 5 . 1 1 0 7 A P P E N D I X C F i g . A . 4 . 1 1 3 6 v i i L i s t o f F i g u r e s ( C o n t ' d . ) P a g e F i g . A . 4 . 2 1 3 7 F i g . A . 4 . 3 1 4 0 v i i i ACKNOWLEDGEMENTS I am g r a t e f u l l y indebted to Professor Lawrence Clevenson for suggesting the problems treated i n chapters two and f i v e and for h i s guidance of my research work during nearly two years. I am also g r a t e f u l l y indebted to Professor Jim Zidek for suggesting the r e s t of the problems treated i n th i s thesis and for his guidance i n obtaining the re s u l t s i n chapters three and four and during the preparation of the thesis. I would also l i k e to express my gratitude to Professor S. W. Nash for acting as my advisor for more than four months during which time he devoted hi s time to help me to bring this d i s s e r t a t i o n into i t s present form. I would l i k e to extend my appreciation to Professors A. Marshall, K. Nagatani, S. W. Nash,£S. J. Press and W. T. Ziemba for t h e i r h e l p f u l suggestions and comments and for t h e i r c a r e f u l reading of the di s s e r t a t i o n . I would l i k e to express my gratitude to Professor F. H. Clarke for the proof of Lemma A.5.1. My sincere thanks also go to M. W. A l i , D. Fynn, K. Lee and K. Tsui for t h e i r constant willingness to enter into discussions. F i n a l l y , I would l i k e to thank Ms. L. D. Nelson for her ca r e f u l and patient typing and Mr. R. Brunn andiMfvjS. D.. D... Jayasingha for t h e i r expert d r a f t i n g . The f i n a n c i a l support of the Canadian Commenwealth Scholarship and Fellowship Association and of the University of B r i t i s h Columbia are g r a t e f u l l y acknowledged. C H A P T E R 1 I N T R O D U C T I O N T h e f i r s t p a r t o f t h i s t h e s i s , i n s p i r e d b y t h e w o r k o f H i l d r e t h [ 1 9 7 2 ] a n d H i l d r e t h a n d T e s f a s t s i o n [ 1 9 7 4 ] , d e a l s w i t h t h e s t o c h a s t i c m o d e l s a n d m a t h e m a t i c a l f o u n d a t i o n s o f t h e t h e o r y , i n t r o d u c e d b y H i l d r e t h [ 1 9 7 2 ] , o f ' b e t s ' a n d ' e x c h a n g e s ' . T h e b e t t i n g p r o b l e m i n v o l v e s t w o p l a y e r s d e t e r m i n i n g t h e a m o u n t e a c h w i l l . w a g e r . A n d t h e s t o c h a s t i c e x c h a n g e p r o b l e m i s a n e x t e n s i o n o f t h i s t o o t h e r t r a n s a c t i o n s w h i c h a r e c o n c e p t u a l l y o f t h e s a m e n a t u r e a s t h e b e t t i n g p r o b l e m e x c e p t t h a t t h e r a n d o m v a r i a b l e f r o m w h i c h t h e a d v e r s a r i e s ' u t i l i t i e s a r e e v a l u a t e d , m a y h a v e a m o r e g e n e r a l r a n g e o f p o s s i b l e v a l u e s . I n h i s w o r k , H i l d r e t h d e a l s w i t h t h e q u e s t i o n o f t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e ( t o b o t h p a r t i c i p a n t s ) e x c h a n g e s . H i s i n d i r e c t a p p r o a c h l e a d s h i m o n l y a s m a l l p a r t o f t h e w a y i n t o t h e l a r g e t e r r i t o r y o f m u t u a l l y f a v o r a b l e e x c h a n g e s , a n d t h e f i r s t o b j e c t i v e o f t h i s t h e s i s i s t o t r e a t , i n a s i m i l a r m a t h e m a t i c a l w a y , t h e u n e x p l o r e d t e r r i t o r i e s a n d t o e x t e n d t h e a n a l y s i s o f s t o c h a s t i c e x c h a n g e p r o b l e m t o a r e a s s u c h a s E d g e w o r t h c o n t r a c t c u r v e a n a l y s i s , s o l u t i o n s u n d e r b a r g a i n i n g , e t c . T h e l a t t e r p a r t o f t h i s t h e s i s c o n c e r n s w i t h t h e " b a r g a i n i n g p r o b l e m " a n d h a s p r a c t i c a l i m p l i c a t i o n s f o r e c o n o m i c a n a l y s i s , i n t h e c o n t e x t o f B i l a t e r a l M o n o p o l y . S o m e o f o u r w o r k i n t h i s a r e a w a s i n s p i r e d b y t h e w o r k s o f N a s h [ 1 9 5 0 ] a n d H a r s a n y i a n d S e l t e n [ 1 9 7 1 ] . I n o u r s e a r c h f o r a s o l u t i o n c r i t e r i o n f o r t h e e x c h a n g e p r o b l e m d e s c r i b e d i n t h e p r e v i o u s p a r a g r a p h s w e a r e n a t u r a l l y l e a d t o t h e b a r g a i n i n g p r o b l e m . H a v i n g d i s c o v e r e d t h a t t h e 2 s t o c h a s t i c e x c h a n g e p r o b l e m m u s t b e t r e a t e d a s ' b a r g a i n i n g u n d e r i n c o m p l e t e i n f o r m a t i o n ' i n t h e s p i r i t o f H a r s a n y i a n d S e l t e n [ 1 9 7 1 ] w e d e v o t e a s u b s t a n t i a l p a r t o f t h e t h e s i s t o t h i s p r o b l e m . T h i s p r o b l e m r e m a i n s l a r g e l y u n s o l v e d , h o w e v e r . O u r c o n t r i b u t i o n t o t h i s a r e a i s s u f f i c i e n t l y g e n e r a l s o a s t o a p p l y t o a n y b a r g a i n i n g s i t u a t i o n a n d t h e r e f o r e h a s p o s s i b l e a p p l i c a t i o n s i n M o n o p o l y - M o n o p s o n y a n d D u o p o l y s i t u a t i o n s i n E c o n o m i c s , r a t h e r t h a n t o o n l y t h e k i n d o f e x c h a n g e s i t u a t i o n s c o n s i d e r e d i n t h e e a r l y p a r t o f t h i s t h e s i s . T h e f o r m u l a t i o n a n d i m p l i c a t i o n s o f t h e m o d e l s a d o p t e d h e r e a r e m a i n l y b a s e d o n t h e i d e a t h a t , c h o i c e s a m o n g a l t e r n a t i v e s i n v o l v i n g d i f f e r e n t d e g r e e s o f r i s k c a n b e e x p l a i n e d b y m a x i m i z a t i o n o f e x p e c t e d u t i l i t y 1 . E x p e c t e d u t i l i t i e s a r e c a l c u l a t e d i n a c c o r d a n c e w i t h t h e V o n N e u m a n n - M o r g e n s t e r n t h e o r y o f u t i l i t y u s i n g p r o b a b i l i t y d i s t r i b u t i o n s o v e r r e l e v a n t e v e n t s . S o m e o f t h e o b j e c t i v e s o f t h e e a r l y c h a p t e r s a r e : ( 1 ) t h e e l a b o r a t i o n o f t h e f e a t u r e s o f t h e s t o c h a s t i c e x c h a n g e p r o b l e m a n d t h e v a l u e o f d e v e l -o p i n g a t h e o r y i n t h i s a r e a , ( 2 ) t h e d e v e l o p m e n t o f i n c e n t i v e s f o r s u c h e l a b o r a t i o n i n t e r m s o f s o m e i n i t i a l s t e p s f i r s t t a k e n b y H i l d r e t h i n h i s p i o n e e r i n g w o r k , ( 3 ) t h e e x p l a n a t i o n o f c e r t a i n p h e n o m e n a s u b s u m e d i n t h i s p r o b l e m b y a n a p p r o p i a t e s t o c h a s t i c m o d e l . T h e r e s u l t s w i l l a l s o h e l p t o e x p l a i n p h e n o m e n a w h i c h w e r e i n c o m p l e t e l y u n d e r s t o o d b y i n t u i t i o n . L a t e r c h a p t e r s d e v e l o p f u r t h e r b a r g a i n i n g t h e o r y , i n p a r t i c u l a r u n d e r i n c o m p l e t e i n f o r m a t i o n ; o u r c o n t r i b u t i o n s a r e s u r v e y e d b e l o w i n m o r e d e t a i l . 1 . S e e M . F r i e d m a n a n d L . J . S a v a g e [ 1 9 4 8 ] . 3 I n C h a p t e r 2 , w e d e v e l o p a s t o c h a s t i c m o d e l f o r a b e t t i n g p r o b l e m a n d s e a r c h f o r t h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s u n d e r w h i c h m u t u a l l y a g r e e a b l e b e t s e x i s t . T h e a p p r o a c h i s d i f f e r e n t f r o m t h a t o f H i l d r e t h [ 1 9 7 2 ] . T o f a c i l i t a t e p o s s i b l e f u r t h e r d e v e l o p m e n t o f t h i s t h e o r y w e a n a l y z e i n c o n s i d e r a b l e d e t a i l t h e s e t o f a l l f e a s i b l e e x c h a n g e s , f o r t h e s i t u a t i o n s u n d e r c o n s i d e r a t i o n . T h e p u r p o s e o f C h a p t e r 3 i s t o d o a n E d g e w o r t h c o n t r a c t c u r v e a n a l y s i s o n t h e f e a s i b l e e x c h a n g e s a n d t h e n c h a r a c t e r i z e s o l u t i o n s t o t h e p r o b l e m , i n t h e c o n t e x t o f b i l a t e r a l m o n o p o l y , u s i n g N a s h ' s [ 1 9 5 0 ] a n d B i s h o p ' s [ 1 9 6 4 ] s o l u t i o n c o n c e p t s , i n p a r t i c u l a r . A n i n t e r e s t i n g q u a l i t a t i v e c h a r a c t e r i z a -t i o n o f t h i s s o l u t i o n i s t h e n d o n e m a t h e m a t i c a l l y . T h e r e m a i n d e r o f t h i s t h e s i s i s d e v o t e d t o t h e b a r g a i n i n g p r o b l e m u n d e r i n c o m p l e t e i n f o r m a t i o n . I n C h a p t e r 4 , a c o n c e p t u a l d e c o m p o s i t i o n o f t h e p r o b l e m i s p r o p o s e d . D e a l i n g w i t h o n e o f t h e s e c o m p o n e n t s , a c l a s s o f g e n e r a l i z e d N a s h s o l u t i o n s i s d e r i v e d b y a n a x i o m a t i c a p p r o a c h . T h e s e w e p r o p o s e a s " b a s e l i n e s o l u t i o n s " f o r t h e o f f e r s e q u e n c e s m a d e i n t h e b a r g a i n i n g p r o c e s s . I n C h a p t e r 5 , a c r i t e r i o n i s i n t r o d u c e d t o c o m p a r e b a r g a i n i n g s t r a t e g i e s a n d t o f i n d w h i c h o n e o f t w o i s b e t t e r , i . e . w h i c h w o u l d y i e l d a h i g h e r e x p e c t e d u t i l i t y t o t h e b a r g a i n e r . A p r o c e d u r e i s t h e n p r o p o s e d t o f i n d a b e s t b a r g a i n i n g s t r a t e g y . A l t h o u g h p r a c t i c a l l y i m p o s s i b l e t o e v a l u a t e , o n e m a y u s e t h i s p r o c e d u r e t o c h e c k , f o r i n s t a n c e , w h e t h e r a n y p r o p o s e d s t r a t e g y , f o r e x a m p l e o n e d e r i v e d t a k i n g i n t o a c c o u n t p s y c h o l o g i c a l f a c t o r s , i s n e a r l y a s g o o d a s t h e b e s t o n e . 4 CHAPTER 2 EXCHANGE PROBLEM : EXISTENCE OF SOLUTIONS 2.1 INTRODUCTION The problem of "Exchanges and Bets" was defined i n a formal fashion by H i l d r e t h [1972] and [1974]. The b e t t i n g problem involves two people t r y i n g to determine the amount each williwage'r. aBy/aa;twager we mean the t y p i c a l " f r i e n d l y bet" between two people where what each person bets i s what he loses i n case he loses the bet. The exchanges problem i s a generalization of these ideas to a s i t u a t i o n i n v o l v i n g two decision makers i n which a c e r t a i n type of r i s k y transaction having a s p e c i a l feature i s involved. The feature i s that the e f f e c t of the transaction on the two decision makers' prospects can be represented by an exchange of a venture Y for -Y, where Y i s not an amount but rather a random v a r i a b l e on the sample space 0. . The e f f e c t of the transaction i s that one person pays the other Y(w) i f wefi i s r e a l i z e d . The b e t t i n g problem i s a p a r t i c u l a r case of t h i s s i t u a t i o n i n which the random v a r i a b l e takes only two values, a p o s i t i v e number and a negative number (the amount bet by each p l a y e r ) . Examples of bets and exchanges are given i n sections 2.2 and 2.4 r e s p e c t i v e l y . The major objective of t h i s chapter i s the study of the existence of mutually favorable bets and exchanges, when the decision makers have possibly d i f f e r e n t b e l i e f s about the random events involved i n the s i t u a t i o n , under various conditions on the u t i l i t y functions. Considering t h i s problem under some pr e s p e c i f i e d conditions on u t i l i t y functions 5 H i l d r e t h [1972] comes to the remarkable conclusion that, i f two r i s k averters have d i f f e r e n t subjective p r o b a b i l i t i e s for an event A, then there e x i s t mutually favorable bets based on A. He deduces t h i s r e s u l t from some of the characterizations he establishes for uncertain ventures. In t h i s chapter we consider the same problem more d i r e c t l y . Our approach d i f f e r s from that of Hildreth's and enables us to conclude more generally, without assuming any concavity property of the u t i l i t y functions, that decision makers having d i f f e r e n t subjective p r o b a b i l i t i e s can always f i n d mutually favorable bets. Studying the existence of mutually favorable exchanges, H i l d r e t h [1974] considers one family of ventures. In section 2.4 we consider a d i f f e r e n t family of ventures covering a range of i n t e r e s t i n g r e a l world s i t u a t i o n s that cannot be handled by Hildreth's approach. H i l d r e t h does not attempt (and h i s i n d i r e c t approach i s incapable of handling) to determine the set of a l l possible mutually favorable bets and exchanges, i . e . the set of a l l f e a s i b l e points. Attacking the problem d i r e c t l y we w i l l be able to do t h i s i n section 2.4. Such a study of the structure of the set of a l l mutually b e n e f i c i a l exchanges w i l l , f o r instance, enable us to do further analysis such as Edgeworth's contract curve analysis of the exchanges problem. 2.2 MODEL OF THE BETTING PROBLEM Consider two players, who w i l l be c a l l e d K and L throughout t h i s chapter b e t t i n g on the occurrence of events made up of possible outcomes of a random v a r i a b l e from which a B e r n o u l l i random v a r i a b l e can be defined by d i s t i n g u i s h i n g the events the two players bet i n favor of. An experiment 6 c o n s i s t i n g o f f l i p p i n g a c o i n w i t h u n k n o w n p r o b a b i l i t y o f " H e a d c o m m i n g u p " , t h e o c c u r r e n c e o f c e r t a i n p a r t i c u l a r s i d e s o f a d i e t o b e r o l l e d , a h o c k e y g a m e , a r e a l l e x a m p l e s o f g a m e s p r o v i d i n g a l t e r n a t i v e e v e n t s f o r t h e p l a y e r s t o b e t o n . I n s u r a n c e o f a p a r c e l , t o b e s h i p p e d , f o r l o s s a l s o p r e s e n t s a b e t t i n g p r o b l e m i n w h i c h t h e i n s u r a n c e c o m p a n y a n d t h e s e n d e r a r e a d v e r s a r i e s . T h e e v e n t o n w h i c h K b e t s s h a l l b e d e n o t e d b y A , a p r o p e r s u b s e t o f t h e s a m p l e s p a c e Q . T h u s K w i n s i f A o c c u r s a s t h e r e s u l t o f t h e r e s u l t o f t h e e x p e r i m e n t w h i l e L w i n s i f A d o e s n o t o c c u r , i . e . i f A ' , t h e c o m p l e m e n t o f A o c c u r s . We m i g h t s a y A ' i s t h e e v e n t t h a t L b e t s i n f a v o r o f . L e t u s s u p p o s e t h a t t h e o u t c o m e s o f t h i s e x p e r i m e n t , o r m o r e r i g o r o u s l y t h e e v e n t s i n a a - f i e l d F o f s u b s e t s o f 0, d o n o t h a v e a s t a n d a r d p r o b a b i l i t y a s s i g n m e n t t h a t m a y b e b a s e d o n p a s t e x p e r i e n c e , t r a d i -t i o n o r a n y o t h e r m e a n s . L e t u s a l s o s u p p o s e t h a t t h e p l a y e r s h a v e t h e i r o w n b e l i e f s o n s u c h e v e n t s r e f l e c t i n g t h e i r o w n i n f o r m a t i o n . T h e i r i n d i v i d u a l a s s i g n m e n t o f p r o b a b i l i t i e s i s r e f e r r e d t o i n t h e l i t e r a t u r e a s s u b j e c t i v e o r p e r s o n a l p r o b a b i l i t i e s . L e t p a n d q b e t h e r e s p e c t i v e s u b j e c t i v e p r o b a b i l i t i e s o f K a n d L t h a t A o c c u r s , i . e . p = P r o b . ( A o c c u r s a c c o r d i n g t o K ' s k n o w l e d g e ) , e t c . T h e y p r e d i c t a n y c o n s e q u e n c e s o f t h e b e t t h a t m a y r e s u l t w h e n t h e g a m e i s p e r f o r m e d b y m e a n s o f t h e i r p e r s o n a l p r o b a b i l i t i e s . T o a v o i d w a s t i n g t i m e o n u n i m p o r t a n t c a s e s t h a t m a y n e v e r a r i s e i n p r a c t i c e i t i s a s s u m e d t h a t 0 < p < 1 , 0 < q < 1 . T h e g a m e i s t o b e p e r f o r m e d o n c e a n d t h e p l a y e r s b e t p o s s i b l y d i f f e r e n t a m o u n t s . H o w e v e r t h e g a m e i s c a r r i e d o u t o n l y i f e a c h p l a y e r i s s a t i s f i e d w i t h t h e o f f e r o f t h e o t h e r . H a v i n g c o m e t o a c o m p r o m i s e a b o u t t h e a m o u n t s t o b e b e t p l a y e r s o b s e r v e t h e o u t c o m e o f t h e g a m e a n d L p a y s K w h a t e v e r h e b e t i f co e A i s r e a l i z e d a n d a l t e r n a t i v e l y K p a y s L i f to e A ' i s r e a l i z e d . A s i t w i l l lead to no confusion i n terminology, we s h a l l use the l e t t e r s K and L to denote the amounts bet by K and L, res p e c t i v e l y . A bet made up by players' o f f e r s i s said to be favorable to a c e r t a i n player i f h i s expected gain from h i s own point of view from that bet i s s t r i c t l y p o s i t i v e . A bet which i s found to be favorable to each player according to his own b e l i e f s w i l l be referred to as a mutually favorable bet^. We are assuming that neither player w i l l accept a given bet unless i t i s favorable to him. We are further assuming that there i s no cost involved i n p a r t i c i p a t i n g i n t h i s game and that no player w i l l be fine d or lose anything even i f he decides to quit the game once he has p a r t i c i p a t e d . These assumptions may not be completely r e a l i s t i c since at l e a s t personal inconveniences are not taken into account. By undertaking a bet (K, L ) , K's worth w i l l be changed e i t h e r from C to C+L or from C to C-K depending on which of A or A' occurs, 2 where C > 0 i s K's f i x e d i n i t i a l worth (c a p i t a l ) and 0 < K <_ C . This would r e s u l t i n a change of K's u t i l i t y from U (C) to U (C+L) K K or U(C-K) accordingly. Hence K's expected gain from a bet (K, L ) , K derived on the basis of h i s b e l i e f s , i s p lL(C+L) + (1-p) iL(C-K) - tX_(C) . K is. K Since a u t i l i t y function i s only determined up to a p o s i t i v e l i n e a r trans-formation one may conveniently use a normalized u t i l i t y function U , defined by IL,(x) = U (C+x), with U V(C) = 0 and rewrite t h i s formula as K K K p U (L) + (1-p) U (-K). Hence the requirement that a given bet (K, L) be K K favorable to K i s 1 A mutually favorable bet may not ne c e s s a r i l y mean that i n r e a l world s i t u a t i o n s people w i l l surely engage i n i t . For a discussion of the question as to why opportunities for m.f. bets are not completely exploited i n the r e a l world see H i l d r e t h [1972] and Hickman [1974]. 2 For random i n i t i a l prospects c f . H i l d r e t h and Tesfatsion [1974]. 8 p U R(L) + (1 - p) U K(-K) > 0 (2.2.1) S i m i l a r l y L would f i n d such a bet i s favorable to himself i f , (1 - q) U L(K) + q U L(-L) > 0 (2.2.2) with 0 < L <^  D and 0 < K <_ C, where 0 < p < 1 and 0 < q < 1 . The l.h.s. of the above in e q u a l i t y represents L's expected gain from (K, L) from h i s point of view, D > 0 being the i n i t i a l worth of L with 0 < L <^D. Any bet (K, L) s a t i s f y i n g both equations (2.2.1) and (2.2.2) i s mutually favorable. In t h i s chapter we w i l l e s t a b l i s h some necessary and s u f f i c i e n t conditions for the existence of such mutually favorable bets. In the previous section we said that a bet (K, L) formed by the of f e r s of the two players i s mutually favorable i f and only i f the equations (2.2.1) and (2.2.2) are both s a t i s f i e d . In t h i s section we s h a l l derive some necessary and s u f f i c i e n t conditions for these equations to have solutions. Throughout t h i s analysis we assume, unless otherwise mentioned, that any two r e a l numbers K and L s a t i s f y i n g 0 < K <_ C, 0 < L <_B are admissible as o f f e r s of the two players. From here on we s h a l l impose: 2.3 EXISTENCE OF MUTUALLY FAVORABLE BETS 9 Assumption 1: The u t i l i t y functions U R and U T are s t r i c t l y increasing Assumption 2: The u t i l i t y functions U and U are continuous everywhere on t h e i r domains of d e f i n i t i o n . Assumption 1 states that each player prefers more future wealth to l e s s . The s t r i c t n e s s of the monotonicity of U and U i s imposed to K Li ensure that no player's marginal u t i l i t y beicdmeshnegative.sThe second assumption was of course made for mathematical t r a c t a b i l i t y . I t should be pointed out that these assumptions ensure that U and U furnish inverse functions. The inverse function of U w i l l be denoted by D \ Of course these inverse functions are also s t r i c t l y increasing and continuous everywhere on the range of U. Since U(0) = 0 = U 1(0) the u t i l i t y functions and t h e i r inverses both have the property that they are p o s i t i v e on (0, °°) and are negative on (-°°, 0) . Recall that there e x i s t mutually b e n e f i c i a l bets i f and only i f the two i n e q u a l i t i e s (2.2.1) and (2.2.2) are simultaneously s a t i s f i e d by some (K, L ) . Rearranging the terms of these i n e q u a l i t i e s we f i n d that these requirements reduce to, - \ x [- i?p- V L ) ] > K > \ x [- A V " L ) ] ( 2 - 2 - 3 ) i . e . a necessary and s u f f i c i e n t condition"'" f o r the existence of mutually This i s given f o r the case p = q i n De Groot [1970] as an exercise and we s h a l l show i n t h i s section that i f U and V are both concave t h i s condition w i l l never be s a t i s f i e d when p = q. 10 f a v o r a b l e b e t s I s t h e e x i s t e n c e o f 0 < K <_ C a n d 0 < L <_ D s u c h t h a t ( 2 . 2 . 3 ) i s s a t i s f i e d . We s h a l l n o w o b t a i n m u c h s i m p l e r n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s w h e n t h e p l a y e r ' s u t i l i t y f u n c t i o n s p o s s e s s e x t r a p r o p e r t i e s . T h e o r e m 2 . 3 . 1 : S u p p o s e U a n d s a t i s f y A s s u m p t i o n s 1 a n d 2 , a n d p o s s e s s d e r i v a t i v e s a t t h e o r i g i n . L e t p a n d q. h e . t h e - s u b j e c t i v e p r o b a b i l i t i e s o f K a n d - L r e s p e c t i v e l y o f t h e e v e n t t h a t - K ^ w ' i l l w i n t h e - b e t . . T h e n p > q i s a a s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e b e t s f P r o o f : S u p p o s e p > q a n d l e t p = p / ( l - p ) , q = q / ( l - q ) . I t i s s e e n t h a t , i f - U ~ 1 ( - p U ( L ) ) > U ~ 1 ( - q U ( - L ) ) f o r s u f f i c i e n t l y s m a l l L > 0, s a y f o r L s u c h t h a t 0 < L < D a n d - U x ( - p U V ( L ) ) + U T ( - q U T ( - L ) ) <_ 2 C , t h e n t h e r e — K K L L e x i s t m u t u a l l y f a v o r a b l e b e t s . T o f i n d s u c h L d e f i n e h : R -> R b y h ( x ) = - U K X ( - p U R ( x ) ) - U L X ( - q U L ( - x ) ) T h u s , t o c o m p l e t e t h e p r o o f , i t i s s u f f i c i e n t t o s h o w t h a t n e a r t h e o r i g i n h ( x ) > 0 f o r x > 0. T o i n v e s t i g a t e t h e b e h a v i o r o f t h e f u n c t i o n h n e a r t h e o r i g i n d i f f e r e n t i a t e h w . r . t . x . T h e n w e h a v e , h ' ( x ) = p ( U ' . ( x ) ") " K ^ i : p u K ( x ) ) U T ' ( U T X ( - q U T ( - x ) - ) 1 . I n f a c t ] L Q g . £ . g i v e n L < L Q , o n e c a n f i n d K s . t . ( K , L ) i s m u t u a l l y f a v o r a b l e 11 for any x at which the functions I L . and U T are both d i f f e r e n t i a b l e . In p a r t i c u l a r since U~ x(-p U„(0)) = 0, lL7 X(-q U T(0)) = 0, is. K L L h'(0) p U^(0)/U^(0) q U[(0)/U^(0) p - q > 0, where the derivatives are nonzero, as the u t i l i t y functions are s t r i c t l y increasing. Hence h'(0) >' 0 and the d i r e c t s u b s t i t u t i o n of x = 0 i n the d e f i n i t i o n of h(x) gives h(0) = 0. Now since h i s d i f f e r e n t i a b l e at the o r i g i n and has a p o s i t i v e d e r i v a t i v e , _ h.(0)|<4W for a l l |x| < <5 with s u f f i c i e n t l y small <S . Hence 0 < h'(0)/2 < h(x)/x for a l l |x| < S with s u f f i c i e n t l y small 6 . This i n turn implies that h(x) > 0 for a l l 0 < x < 6 , thus completing the proof. I t should be pointed out that, i f we do not r e s t r i c t the event on which a given player may bet, t h i s theorem states that the two players can fin d mutually favorable bets whenever they have d i f f e r e n t subjective p r o b a b i l i t i e s on some £eveht",.cregard'less'. irfegfehey.^areifisfc"takers or r i s k averters. I t should be mentioned here that H i l d r e t h [1972] has come to the same conclusion, with the a d d i t i o n a l assumption of concave u t i l i t y functions, by a d i f f e r e n t approach.  1. I f the i n i t i a l prospects are ; u n c e r t a l n 5 t h i s event i s required to be independent of i n i t i a l prospects. 12 T h e o r e m 2 . 3 . 2 : S u p p o s e b o t h IL^ a n d U a r e c o n c a v e f u n c t i o n s a n d a r e  p o s i t i v e o n R + a n d n e g a t i v e o n R ~ . T h e n p > q i s a n e c e s s a r y c o n d i t i o n f o r  t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e b e t s , w h e r e p a n d q a r e a s i n T h e o r e m 2 . 3 . 1 . 2 P r o o f : S u p p o s e i t i s l m u t u a M y r : ^ ( i ) p U ( L ) + ' * ' K ( 1 - p ) IT ( - K ) > 0 a n d ( i i ) ( 1 - q ) U T ( K ) + q U T ( - L ) > 0 . T h e n f r o m K L i LI ( i ) i t f o l l o w s b y J e n s e n ' s i n e q u a l i t y t h a t , 0 < p U R ( L ) + ( 1 - p ) U R ( - K ) < U R ( p L - ( 1 - p ) K ) t h u s i m p l y i n g 0 < ( p L - ( l - p ) K ) . H e n c e K < p L / ( l - p ) a n d s i m i l a r l y f r o m ( i i ) K > q L / ( l - q ) . W r i t i n g t h e s e t w o i n e q u a l i t i e s t o g e t h e r w e h a v e , - p L > K > , q L f o r 0 < p , q < 1 . 1 - p 1 - q B u t t h i s c a n h o l d o n l y i f p / ( l - p ) > q / ( l - q ) o r e q u i v a l e n t l y i f p > q . T h i s c o m p l e t e s t h e p r o o f . T h i s t h e o r e m s t a t e s t h a t t w o r i s k a v e r t e r s c a n f i n d m u t u a l l y b e n e f i c i a l b e t s o n l y i f a g i v e n p l a y e r ' s s u b j e c t i v e p r o b a b i l i t y o f t h e e v e n t h e b e t s o n i s h i g h e r t h a n t h a t o f h i s o p p o n e n t - . O f c o u r s e i f t h e i n e q u a l i t y i s r e v e r s e d i t m a y b e p o s s i b l e t o s w i t c h t h e e v e n t s t h e y a r e b e t t i n g i n f a v o u r o f t o a c h i e v e a m u t u a l l y a g r e e a b l e w a g e r . I t i s n o w e v i d e n t t h a t r i s k a v e r t e r s c a n n e v e r f i n d m u t u a l l y f a v o r a b l e b e t s i f t h e e v e n t t o b e b e t o n h a s a n 1 . H e r e s t r i c t i n c r e a s i n g n e s s o f U i s n o t n e c e s s a r y . 2 . S o L b e t s o n A ° . 13 objective p r o b a b i l i t y which i s known to both p a r t i c i p a n t s . The following proposition i s also of some i n t e r e s t . Proposition 2.3.3: Suppose U R and U are concave and s a t i s f y assumptions 1 and 2 If (K Q, L q ) i s a mutually favorable wager, then given any 0 < L <^  L £ there e x i s t s 0 < K < K such that (K, L) also forms a mutually favorable — o wager. Proof: Since ( K D> ^ ) i s mutually favorable, we have U L 1 ( " ^ V - L o » < K o < " UK 1 (-P UK< Lo» with 0 < K <C, 0 < L <D. Since XL, i s concave, we have f or a > 1, o — o — K — TL_(a L) < a U (L) (take a X convex combination of 0 and x and then put K K a = 1/A. and L = A.x) . Hence, -U^C-p U R(a L ) ) 1 "U/C-P a U R ( D ) <_ -a U K 1(-p U R(L)) as a r e s u l t of convexity of U 1 . In a s i m i l a r fashion one derives that a U7 1 ( - q U T(-L)) <_ u!" 1(-q U (-a L ) ) . Now w r i t i n g L = (L /L)L and s e t t i n g XJ XJ L J_I O O 14 a = L /L > 1, we obtain o — a U - V q V - D ) < U - V q U L ( - L o ) ) < K Q < - u " 1 ^ U K ( L o ) < -a u ' V p U R ( L ) ) . % K - l a , Hence f or any 0 < L < L we have UT (-q U T(-L)) < — < -IL, (-p IL.(L)). O L i L Ot K K K - - _ Now s e t t i n g K = min{^-, y t u " (-q U L(-L)) - U~ (-p U L(L))]} we f i n d that (K, L) i s mutually favorable, because 0 < K < C, 0 < L <_D and u ' V q ^ ( - L ) ) < K < -U'Vp U R ( L ) ) . 2.4 AN EXTENSION TO THE EXCHANGE-: PROBLEM The stochastic exchanges problem introduced by H i l d r e t h [1974] i s as follows: Consider two decision makers K and L contemplating the exchange of a random venture Y (money value) for - Y; a venture i s a possible undertaking which would modify the decision makers' i n i t i a l prospects. The value of Y i s determined on the basis of aarandomavarlable taking values i n a set ft, weft being a s p e c i f i c sequence of developments i n the decision makers' environment. The decision makers agree that L w i l l pay K, Y(u)) i f weft i s r e a l i z e d . Let P be K's subjective p r o b a b i l i t y d i s t r i b u t i o n r e f l e c t i n g h i s own b e l i e f s on the events i n a a - f i e l d F of subsets of ft. And Q w i l l denote the p r o b a b i l i t y assignment of L on (ft, F). Hence Y i s a random v a r i a b l e with respect to e i t h e r decision maker. 15 I n o r d e r t o i m p o s e t h e c o n d i t i o n t h a t n e i t h e r d e c i s i o n m a k e r w o u l d ( u n p r o f i t a b l y ) o f f e r t h e o t h e r a s u r e t h i n g , i t i s a s s u m e d t h a t Y i s a r a n d o m v a r i a b l e t a k i n g b o t h p o s i t i v e a n d n e g a t i v e v a l u e s o n a s e t o f p o s i t i v e p r o b a b i l i t y w i t h r e s p e c t t o b o t h P a n d Q . F u r t h e r m o r e i t i s a s s u m e d t h a t p r o b a b i l i t y m e a s u r e s P a n d Q a r e ^ a b s b l u t e l y ^ . c o n t i n u o u s " w i t h r e s p e c t t o e a c h o t h e r , i . e . n u l l s e t s o f F a r e t h e s a m e u n d e r b o t h P a n d Q . We c o n c e i v e t h i s p r o b l e m a s o n e w h e r e d e c i s i o n m a k e r s h a v e f i x e d i n i t i a l w e a l t h s , a s b e f o r e . T h i s e n a b l e s u s t o w r i t e K a n d L ' s u t i l i t y f r o m t h e e x c h a n g e , u s i n g n o r m a l i z e d u t i l i t y f u n c t i o n s U a n d U s u c h t h a t U ( 0 ) = U ( 0 ) = 0 , a s F ^ U R ( Y ) = / U K ( i ) d P a n d E L U L ( _ Y ) = / U L ( _ Y ) d Q l S u c h a n e x c h a n g e w i l l b e m u t u a l l y f a v o r a b l e i f f , E „ U _ ( Y ) > 0 a n d E T U ( - Y ) > 0 . ( 2 . 4 . 1 ) W e w i s h t o e s t a b l i s h n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e e x c h a n g e s . B e f o r e a t t e m p t i n g t o d o t h i s w e o u t l i n e a h y p o t h e t i c a l , b u t f a i r l y s u g g e s t i v e ( o f a r e a l w o r l d s i t u a t i o n ) e x a m p l e i n w h i c h a n e x c h a n g e o f a v e n t u r e i s i n v o l v e d . M o r e i n t e r e s t i n g e x a m p l e s w i l l b e d i s c u s s e d i n t h e l a t e r p a r t o f t h i s s e c t i o n . 16 EXAMPLE 2.1: A decision maker, say K, wishes to s e l l an a r t i c l e belonging to him to anyone making a reasonable offer. Suppose his u t i l i t y for the a r t i c l e i s completely based on the time T at which the a r t i c l e w i l l go out of order for the f i r s t time (or the l i f e time). K has a p r i o r d i s -t r i b u t i o n function n (t) on the random variable T based on his past experience, the performance of the item, the way that he has been handling i t , etc. Let F (t) be the value of this a r t i c l e from K's point of view. K Suppose another decision maker, say L, i s interested i n purchasing this a r t i c l e for a reasonable price, say $R. His u t i l i t y on this a r t i c l e i s also completely based on the random variable T. Let II (t) be L's pr i o r d i s t r i b u t i o n function (subjective) on T based on the current condition of the item, i t s age, i t s current performance, the reputation of the brand, etc. Let F (t) be the value of this a r t i c l e from L's point of view. JL Now define a venture Y by Y = R - X, where X i s a random variable. From K's point of view X = F (T) and the prior d i s t r i b u t i o n of X i s determined by II , whereas from L's point of view X = F (T) K L and the p r i o r d i s t r i b u t i o n of X i s induced by II . This i s an example where J _ i the decision makers K and L exchange the venture Y for -Y. I t i s clear that this exchange i s mutually favorable i f f E U (R - X) > 0 and K K E U (X - R) > 0. In the par t i c u l a r case where U^(y) = U T(y) = y, L L K L F (t) = F (t) = k t , t r i v i a l l y , there ex i s t mutually favorable s e l l i n g K L prices (values for R) i f f E T(T) > EV(T), i . e . i f f the buyer feels that Li K. the a r t i c l e w i l l l a s t longer than how the s e l l e r feels. Our goal i s to get sim i l a r results i n more general cases. 17 W h e n t h e u t i l i t y f u n c t i o n o f K a n d L a r e b o t h c o n c a v e w e c a n e a s i l y o b t a i n a s i m p l e n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e e x c h a n g e s . P r o p o s i t i o n 2 . 4 . 1 : S u p p o s e b o t h U a n d U a r e b o t h c o n c a v e f u n c t i o n s K L i a n d a r e n o n z e r o e x c e p t a t t h e o r i g i n . T h e n t h e e x i s t e n c e o f a Y s u c h  t h a t E ^ ( Y ) > 0 > E ^ ( Y ) i s a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e o f  m u t u a l l y f a v o r a b l e e x c h a n g e s . P r o o f : I f Y i s m u t u a l l y f a v o r a b l e , t h e n Y s a t i s f i e s b o t h E ^ U ^ C Y ) > 0 K K a n d ' E U ( - Y ) > 0 . T h e n b y J e n s e n ' s i n e q u a l i t y J _ i .Li U K ( E R ( Y ) ) > 0 a n d U L ( E L ( - Y ) ) > 0 t h u s i m p l y i n g t h e r e s u l t . C o n s i d e r n o w a f a m i l y , [y/\ o f v e n t u r e s s u c h t h a t z e [ y ] = * z = a y f o r s o m e - °° < a < ° ° . C o n f i n i n g h i s a n a l y s i s t o f a m i l i e s o f v e n t u r e s h a v i n g t h i s p r o p e r t y , H i l d r e t h [ 1 9 7 4 ] h a s s h o w n , a s s u m i n g c o n c a v e u t i l i t y f u n c t i o n s " ' ' , t h a t t h e r e e x i s t m u t u a l l y f a v o r a b l e e x c h a n g e s i f f E ( Y ) a n d E ( Y ) a r e o f K L o p p o s i t e s i g n . U n f o r t u n a t e l y s o m e - o f t h e v e n t u r e s f o u n d i n a p p l i c a t i o n s d o n o t h a v e t h e a b o v e p r o p e r t y . F o r a v a r i e t y o f v e n t u r e s , i t i s t r u e t h a t w h e n Y i s a v a i l a b l e a s a n e x c h a n g e a Y i s a l s o a v a i l a b l e ; h o w e v e r t h i s d o e s n o t n e c e s s a r i l y m e a n t h a t a l l p o s s i b l e v e n t u r e s a r e o f t h e f o r m a Y f o r s o m e a . I t s h o u l d , t h e r e f o r e , b e n o t e d t h a t t h e a b o v e m e n t i o n e d n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s m a y n o t h o l d i n t h i s m o r e g e n e r a l c a s e a n d s o f u r t h e r a n a l y s i s i s n e c e s s a r y . 1 . I n d e p e n d e n c e o f v e n t u r e a n d i n i t i a l p r o s p e c t s ±s a l s o a s s u m e d 18 As indicated e a r l i e r , our goal i s a necessary and s u f f i c i e n t condition for the existence of mutually favorable ventures for exchange, t h i s condition being dependent merely on the nature of the venture and not on the u t i l i t y functions. C l e a r l y the generality of t h i s problem must be somewhat r e s t r i c t e d ; here t h i s i s done by r e s t r i c t i n g the f a m i l i e s of ventures to those having c e r t a i n s p e c i f i e d properties. We admit, i n p a r t i c u l a r , one important family, having a v a r i e t y of applications, whose members have features analogous to those of a bet. In most of the exchanges we encounter i n the r e a l world the venture a v a i l a b l e f or exchange i s a function of several f l e x i b l e parameters, the values of which are to be f i x e d by negotiation. The following are two such hypothetical examples where the venture can be determined by two parameters. We w i l l discuss Hildreth's method i n r e l a t i o n to these examples. Example 2.2: S r i Lanka (say K ) imports r i c e from China (say L ) while i t exports tea to China. Suppose the two countries have decided on a contract of the following form which guards against the p o s s i b i l i t y of future p r i c e i n f l a t i o n s : S r i Lanka i s to supply a c e r t a i n amount of tea, say a units to China and China i s to supply a c e r t a i n amount of r i c e , say 3 units to S r i Lanka, during each time period, both at current market prices (say $P and $Q, respectively) for the coming T time periods, a and 3 are to be fixed by negotiation. The prices of tea and r i c e i n the world market are expected to go up at constant but unknown addit i v e rates, say at rates t and r per unit time r e s p e c t i v e l y . The two decision makers K and L have t h e i r own subjective p r o b a b i l i t y d i s t r i -butions on t and r . It i s evident that the venture involved here can be written as 19 Y ( a , 3 ) = [ a ( Q + r T ) T - a Q T ] - [ g ( P + t T ) T - 3PT] 2 2 o r e q u i v a l e n t l y , T 2 Y ( a , 3 ) = j - ( a r - 3 t ) w h e r e r a n d t a r e r a n d o m v a r i a b l e s f r o m e i t h e r d e c i s i o n m a k e r ' s p o i n t o f v i e w . F o r t h i s p r o b l e m H i l d r e t h ' s r e s u l t i s a p p l i c a b l e o n l y i f a = 3 > a n d t h e r e i s n o r e a s o n w h y t h i s r e s t r i c t i o n s h o u l d b e i m p o s e d . I n f a c t i f w e m o d i f y t h i s e x a m p l e b y r e q u i r i n g t h a t " C h i n a w a n t s t o b u y e x a c t l y c ( f i x e d ) u n i t s o f r u b b e r a l s o f r o m S r i L a n k a u n d e r t h i s c o n t r a c t s o t h a t t h e v e n t u r e w i l l h a v e t h e f o r m T 2 Y ( a , 3 ) = ( a r - 3 t - c R ) w h e r e R i s t h e c o n s t a n t i n f l a t i o n r a t e f o r r u b b e r , t h e n H i l d r e t h ' s r e s u l t w i l l e v e n m o r e w i d e l y m i s s t h e m a r k . E x a m p l e 2 . 3 : An i n s u r a n c e c o m p a n y , s a y K , i n s u r e s a i r t r a v e l l e r s a g a i n s t l o s s o f l i f e o r i n j u r i e s . S u p p o s e t h e d i f f e r e n t k i n d o f p o s s i b l e i n j u r i e s a r e d i v i d e d i n t o n c a t e g o r i e s . K s p e c i f i e s t h a t $ L o f i n s u r a n c e c o v e r a g e o n a c e r t a i n p a s s e n g e r y i e l d s t h e b u y e r $ R i f t h e a i r c r a f t 2 0 c r a s h e s c a u s i n g t h i s p a r t i c u l a r p a s s e n g e r ' s d e a t h , a n d i f s u c h a t h c r a s h c a u s e s h i m a n i n j u r y o f i c a t e g o r y . I t i s r e a l i s t i c t o a s s u m e t h a t L < R , L < R^ f o r e a c h i , i = l , 2 , . . . , n . L e t X b e t h e a m o u n t t h a t K w i l l p a y t h e b u y e r a n d a s s u m e K h a s s u b j e c t i v e p r o b a b i l i t i e s p = P ( X = R ) , = P ( X = R ^ ) i = 1 , 2 , n . S u p p o s e a b u y e r , s a y L , w h o h a s t h e s u b j e c t i v e p r o b a b i l i t i e s q = P ( X = R ) , q_^ = P ( X = R ^ ) , i = 1 , 2 , n w i s h e s t o b u y a u n i t s ( a >^  0 ) o f t h e s p e c i f i e d p o l i c y a t $ a L . S u p p o s e K w i l l p a y L $ g X f o r a n i n s u r a n c e p o l i c y o f $ a L . N o w d e f i n e a v e n t u r e Y ( a , 3) = a L - g X . T h i s i s a n e x a m p l e w h e r e K a n d L e x c h a n g e a v e n t u r e Y f o r - Y w h i c h i s f l e x i b l e u p t o a c h a n g e i n a a n d B . T h i s e x c h a n g e i s m u t u a l l y f a v o r a b l e i f f b o t h E K U K ( Y ) = ( 1 " PK " E p i c ^ V a L ) + P V " e R ) + ^ i V - 0 * ! * > 0 . a n d E T U ( - Y ) = ( 1 - q - E q . ) I L ( - a L ) + q U (BR) + E q , U _ ( B R . ) > 0 L L Z L L i L i 1 LI 1 a r e s a t i s f i e d . I n t h i s e x a m p l e i t m a y r e a l l y b e t h e c a s e t h a t a = B • T h e n H i l d r e t h ' s r e s u l t p r o v i d e s t h e a n s w e r t o o u r q u e s t i o n . I f a 4 B, H i l d r e t h ' s r e s u l t f a i l s t o a p p l y . C o n s i d e r t h e p r o b l e m o f e x c h a n g e i n w h i c h t h e v e n t u r e Y i s a f u n c t i o n o f t w o b o u n d e d a n d p o s i t i v e f l e x i b l e p a r a m e t e r s , s a y a a n d B-T h e p r o b l e m w h e r e Y i s f u n c t i o n o f m o r e t h a n t w o f l e x i b l e p a r a m e t e r s i s 2 1 l e f t f o r p o s s i b l e f u t u r e s t u d i e s . A m e m b e r o f t h e f a m i l y o f v e n t u r e s u n d e r c o n s i d e r a t i o n i s d e n o t e d b y Y ( a , g ) , w h e r e a a n d g a r e t o b e c h o s e n b y n e g o t i a t i o n . A s s u m e E U ( Y ) a n d E U ( - Y ) a r e b o t h f i n i t e f o r a l l K K L L i f e a s i b l e a a n d g . L e m m a 2 . 4 . 2 : Y ( a , B) i s a r a n d o m v a r i a b l e d e f i n e d o n a p r o b a b i l i t y s p a c e 2 (Q, F , P ) . G i v e n CJ s u p p o s e Y ( a , B) : R R" i s c o n t i n u o u s , i n c r e a s e s ( s t r i c t l y ) w i t h .a a n d d e c r e a s e s ( s t r i c t l y ) w i t h B . L e t U b e a c o n t i n u o u s a n d ( s t r i c t l y ) i n c r e a s i n g u t i l i t y f u n c t i o n n o r m a l i z e d s u c h t h a t U ( 0 ) = 0 . S u p p o s e L i m P ( Y > 0 ) = 1 a n d L i m P ( Y > 0 ) = 0 , a -> co ot - 0 0 L i m P ( Y < 0 ) = 1 , a n d L i m P ( Y < 0 ) = 0 , a n d E U ( Y ( a , B ) ) < °° f o r f e a s i b l e B -> c o g -> — o o a , g . T h e n E U ( Y ( a , B)) = 0 c a n b e s o l v e d f o r a i n t e r m s o f B as_ a = g(B), g b e i n g a n i n c r e a s i n g f u n c t i o n d e f i n e d f r o m R o n t o R , i f  a n d o n l y i f P ( Y > 0 ) > 0 a n d P ( Y < 0 ) > 0 f o r e a c h a , g . P r o o f : C l e a r l y i f o n e o f P ( Y > 0 ) a n d P ( Y < 0 ) i s z e r o w h i l e t h e o t h e r i s p o s i t i v e E U ( Y ( a , B)) w i l l b e , r e s p e c t i v e l y , p o s i t i v e o r n e g a t i v e ; t h u s f o l l o w s t h e n e c e s s i t y p a r t o f " t h e f l e m m a : . c - . T o i a p r o v e t h e s u f f i c i e n c y p a r t n o w s u p p o s e P ( Y > 0 ) a n d P ( Y < 0 ) a r e b o t h p o s i t i v e . G i v e n g = B , w r i t e , o E U ( Y ( a , g Q ) ) = U ( Y ( a , B o)) - U ( - Y ( a , g Q ) ) Y > 0 Y < 0 = f ( a ) - h ( a ) 2 2 w h e r e f ( a ) i s p o s i t i v e a n d i n c r e a s i n g w h i l e h ( a ) i s p o s i t i v e a n d d e c r e a s i n g . S u p p o s e L i m P ( Y > 0 ) = 1. T h e n s i n c e L i m P ( Y < 0 ) = 0 a 0 0 a 0 0 i t i m m e d i a t e l y f o l l o w s , t h a t h ( a ) 0 a s a -> 0 0 . N o w s i n c e f ( a ) i s i n c r e a s i n g h ( a ) < f ( a ) f o r s u f f i c i e n t l y l a r g e a . S i m i l a r l y i t c a n b e s h o w n t h a t h ( a ) > f ( a ) f o r s u f f i c i e n t l y s m a l l a . H e n c e E U ( Y ( a , $ 0 ) ) i s p o s i t i v e f o r l a r g e a a n d i s n e g a t i v e f o r s m a l l a . N o w i t i s e v i d e n t t h a t t h e r e e x i s t s u n i q u e a = g ( 3 ) s u c h t h a t E U ( Y ( a , 3 ) ) = 0 o o o b e c a u s e E U ( Y ( a , 3 Q ) ) i s s t r i c t l y i n c r e a s i n g a n d c o n t i n u o u s i n a . T h a t g i s i n c r e a s i n g i s t r i v i a l l y s h o w n b y c o n t r a d i c t i o n . F o r , s u p p o s e t h e r e e x i s t s a., = g ( 3 - , ) < a 0 = g ( 3 0 ) w i t h 3 0 < 3 . , • T h e n 0 = E U ( Y ( a l 5 < E U ( Y ( a 2 , 3 - ^ ) b e c a u s e a± < a £ < E U ( Y ( a 2 , 3 2 ) ) b e c a u s e 3 X < 3 2 B u t a 2 = g ( 3 2 ) i m p l i e s E U ( Y ( a 2 > 3 2 ) ) = 0 t h u s l e a d i n g t o a c o n t r a d i c t i o n . T h e m a p p i n g g : R -> R i s o n t o b e c a u s e E U ( Y ( a , 3 ) ) = 0 c a n e q u i v a l e n t l y b e e x p r e s s e d f o r 3 i n t e r m s o f a . T h e o r e m 2 . 4 . 3 : S u p p o s e t h e v e n t u r e Y ( a , 3 ) a n d e a c h u t i l i t y f u n c t i o n , U a n d U , h a s t h e p r o p e r t i e s h y p o t h e s i z e d i n L e m m a 2 . 4 . 2 . A s s u m e IN. Li t h a t Y , U a n d U a r e a l l c o n t i n u o u s l y d i f f e r e n t i a b l e . A l s o s u p p o s e JN. J_I t h a t Y ( a , b ) = 0 , o n l y a >_ a , 3 ^_ b ( a t l e a s t i n a n e i g h b o r h o o d o f 2 3 ( a , b ) ) a r e a d m i s s i b l e a n d t h a t t h e b o t h p r o b a b i l i t y m e a s u r e s P a n d Q a s s i g n p r o b a b i l i t y 1 t o a b o u n d e d s u b s e t A o _ f _ . I f r 3 Y l J A f e \ 3 Y J _9a_ d Q a = a 3 = b d Q a = a 3 = b 9 Y d P a = a 3 = b 9 Y 9 a ^ J ( 2 . 4 . 1 ) d P a = a 3 = b t h e n , t h e r e e x i s t s 3 > b s u c h t h a t , g i v e n b < 3 3 t h e d e c i s i o n m a k e r s  c a n f i n d a > a m a k i n g Y ( a , 3 ) m u t u a l l y b e n e f i c i a l . P r o o f : I t f o l l o w s f r o m t h e p r e v i o u s l e m m a t h a t E U ( Y ( a , 3 ) ) = 0 a n d E U ( - Y ( a , 3 ) ) = 0 c a n b e e q u i v a l e n t l y e x p r e s s e d a s a = g ( 3 ) a n d a = h ( 3 ) r e s p e c t i v e l y , w h e r e g a n d h a r e b o t h c o n t i n u o u s l y d i f f e r e n t i a b l e a n d i n c r e a s i n g . B y e x a m i n i n g t h e r e s p o n s e s o f Y t o c h a n g e s i n a a n d 3 w e i m m e d i a t e l y n o t i c e t h a t E U ( Y ) > 0 i f f a > g ( 3 ) a n d t h a t E U ( - Y ) > 0 i f f a < h ( 3 ) . I t i s o b v i o u s t h a t t o f i n i s h t h e p r o o f w e a r e Li J-J r e q u i r e d o n l y t o p r o v e t h a t h ( 3 ) > g ( 3 ) f o r a l l p o i n t s i n a n e i g h b o r h o o d o f ( a , b ) . T o d o t h i s c o n s i d e r t h e f u n c t i o n f ( 3 ) d e f i n e d b y f ( 3 ) f ( 3 ) = h ( 3 ) - g ( 3 ) . O n d i f f e r e n t i a t i o n o f f ( 3 > w . r . t . 3 w e h a v e I t i s o f t e n a s s u m e d t o a v o i d p a r a d o x ( c f . A r r o w [ 1 9 7 1 ] ) . t h a t t h e u t i l i t y f u n c t i o n s a r e b o u n d e d . I f t h i s i s t h e c a s e , t h e r e q u i r e d c o n d i t i o n t h a t P ( A ) = Q ( A ) = 1 , A b e i n g b o u n d e d , i s u n n e c e s s a r y a n d T h e o r e m 2 . 4 . 3 h o l d s t r u e . 2 4 f ' ( 3 ) = h ' ( 3 ) - g ' ( 3 ) f 3 E L U L ( " Y ) + a = h ( 3 ) ! B E K U K ( Y ) 3 a K K ( 2 . 4 . 2 ) a = g ( 3 ) b e c a u s e , f o r i n s t a n c e , E ^ U ^ ( - Y ( a , 3 ) = 0 i m p l i e s — E ^ U ^ ( - Y ) + 4 ^ - • T— E T U ( - Y ) = 0 . S i n c e t h e p a r t i a l d e r i v a t i v e s o f U T ( - Y ) a n d d 3 9oi L L t, U „ ( Y ) w . r . t . a a n d 3 a r e a l l c o n t i n o u s w e c a n u s e t h e t e c h n i q u e o f K ' D i f f e r e n t i a t i o n U n d e r t h e I n t e g r a l S i g n ' t o r e d u c e ( 2 . 4 . 2 ) a s f ' ( S ) = — U ' ( Y ) d P | 3 B V l ; a r £ U K ( Y ^ D P a = g ( 3 ) f i f U L ( " Y ) d Q feUL('Y)dQ a = h ( B ) ( 2 . 4 . 3 ) C o n s i d e r n o w a d e c r e a s i n g s e q u e n c e o f a d m i s s i b l e ^ ° . T h e n s i n c e 8Y a n d U ' a r e b o t h c o n t i n u o u s w e h a v e , K -> [ — U ' ( Y ) 1 L 8 3 V Jia=a B=b a s 8Y n ->• °° . S i n c e — U ' ( Y ) i s b o u n d e d o n A f o r a n y ( a , 3 ) , w e c a n op K f i n d a c o n s t a n t M s u c h t h a t F < M f o r a l l n a n d a l l u e f~. n — 2 5 H e n c e t h e a p p l i c a t i o n o f t h e B o u n d e d C o n v e r g e n c e T h e o r e m y i e l d s , If « K « d P a = g ( g ) < ! j f > a - a U K ( Y ^ ' b » d P g=b = DJCO) v 3 3 y a = a 6=b d P O b t a i n i n g s i m i l a r r e s u l t s f o r t h e r e s t o f t h e t e r m s w h i c h a p p e a r i n ( 2 . 4 . 3 ) , w e d e d u c e f r o m t h e c o n t i n u i t y o f f t h a t f ' ( b ) = > v 9 3 y a = a . fi=k . & : - d P ^ 3 a a = a e ^ b & 3=b 0 , t h e p o s i t i v i t y f o l l o w i n g f r o m t h e h y p o t h e s e s ( s e e e q u a t i o n ( 2 . 4 . 1 ) ) . F u r t h e r m o r e f ( b ) = a - a = 0 . T h e s e t w o c o n d i t i o n s a r e s u f f i c i e n t t o c o n c l u d e a s i n T h e o r e m 2 . 3 . 1 , t h a t f ( B ) > 0 f o r a l l p o i n t s i n a n a p p r o p i a t e l y c h o s e n n e i g h b o r h o o d o f ( a , b ) . I t s h o u l d b e n o t e d t h a t i f a n d o n l y i f a <_ a a n d $ <_ b a r e a d m i s s i b l e , T h e o r e m 2 . 4 . 3 h o l d s v a l i d w i t h t h e i n e q u a l i t y ( 2 . 4 . 1 ) r e v e r s e d . I n i t s p r e s e n t f o r m c o n d i t i o n ( 2 . 4 . 1 ) i s n o t v e r y i n t u i t i v e . H o w e v e r i t s a p p l i c a t i o n t o p a r t i c u l a r c l a s s e s o f v e n t u r e s w i l l r e d u c e i t t o a m u c h s i m p l e r , p l a u s i b l e f o r m . T h e f o l l o w i n g c o r o l l a r y i s o n e s u c h c a s e . 2 6 C o r o l l a r y 2 . 4 . 4 ; L e t Y ( a , g ) = a S - g T , S a n d T b e i n g p o s i t i v e r a n d o m  v a r i a b l e s . S u p p o s e t h e u t i l i t y f u n c t i o n s , s u b j e c t i v e p r o b a b i l i t i e s a n d Y a l l s a t i s f y t h e r e q u i r e m e n t s o f T h e o r e m 2 . 4 . 3 . S u p p o s e o n l y n o n n e g a t i v e a a n d g a r e a d m i s s i b l e . T h e n , a s u f f i c i e n t c o n d i t i o n 1 f o r t h e e x i s t e n c e o f ( a , g ) l e a d i n g t o m u t u a l l y f a v o r a b l e v e n t u r e s i s t h a t E(T) E ( T ) I f t h e p l a y e r s ' u t i l i t y f u n c t i o n s a r e b o t h c o n c a v e , t h e n t h e a b o v e c o n d i t i o n  i s n e c e s s a r y t o o , f o r e x i s t e n c e o f m u t u a l l y b e n e f i c i a l v e n t u r e s . 8Y P r o o f : S e t ( a , b ) = ( 0 , 0 ) i n T h e o r e m 2 . 4 . 3 . T h e n s i n c e — = S a n d dCt 3Y — = - T , ( 2 . 4 . 4 ) t r i v i a l l y f o l l o w s f r o m ( 2 . 4 . 1 ) . T o p r o v e n o w t h e 3g n e c e s s i t y o f ( 2 . 4 . 4 ) w h e n t h e u t i l i t y f u n c t i o n s a r e c o n c a v e , l e t u s a p p l y P r o p o s i t i o n 2 . 4 . 1 t o Y = a S - g T . T h i s i m p l i e s t h a t , a E ^ ( S ) - g E f T ) > 0 > otE ( S ) - g E ( T ) , o r e q u i v a l e n t l y , E K ( S ) ^ ^ ^ E L ( S ) E R ( T ) a E L ( T ) a s a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e v e n t u r e s . T h e a b o v e i n e q u a l i t i e s j o i n t l y i m p l y ( 2 . 4 . 4 ) t h u s c o m p l e t i n g t h e p r o o f . 1. I t i s c l a i m e d b y C . H i l d r e t h t h a t t h i s r e s u l t h o l d s v a l i d e v e n i f t h e a s s u m p t i o n o f f i x e d i n i t i a l w e a l t h i s r e p l a c e d b y t h a t o f u n c r e t a i n i n i t i a l p r o s p e c t s o f b o t h K a n d L a r e s t o c h a s t i c a l l y i n d e p e n d e n t o f t h e v e n t u r e s . 2 7 T h e p l a u s i b i l i t y o f C o r o l l a r y 2 . 4 . 4 i s a s f o l l o w s . E n l a r g i n g T o r d i m i n i s h i n g S i n c r e a s e s d e c i s i o n m a k e r L ' s u t i l i t y a n d d e c r e a s e s K ' s u t i l i t y . H e n c e t h e d e c i s i o n m a k e r s t e n d t o b e a b l e t o f i n d m u t u a l l y a g r e e a b l e e x c h a n g e s w h e n t h e i r s u b j e c t i v e b e l i e f s m a k e E ( T ) / E ( S ) a d m i s s i b l e , c l e a r l y , t h e c o n c l u s i o n o f t h i s c o r o l l a r y m u s t b e m o d i f i e d a s f o l l o w s : " t h e r e e x i s t m u t u a l l y f a v o r a b l e e x c h a n g e s i f f t h e r a t i o o f t h e e x p e c t a t i o n s o n t h e r a n d o m v a r i a b l e s T a n d S , n a m e l y E ( T ) / E ( S ) , f r o m t h e t w o d e c i s i o n m a k e r s v i e w p o i n t s d i f f e r . " I t i s a l s o i n t e r e s t i n g t o n o t e h e r e t h a t , i n e i t h e r o f t h e s e c a s e s , i f t w o r i s k a v e r t e r s d o t h e i r e v a l u a t i o n s b y k n o w n o b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s o n t h e u n c e r t a i n -t i t i e s o f t h e e x c h a n g e s i t u a t i o n , t h e y w i l l n e v e r b e a b l e t o f i n d m u t u a l l y a g r e e a b l e v e n t u r e s , n o m a t t e r h o w d i f f e r e n t t h e i r u t i l i t y f u n c t i o n s a r e . A s a n a p p l i c a t i o n o f C o r o l l a r y 2 . 4 . 4 s e t , i n E x a m p l e 2 . 2 , S = L , t h e c o s t o f i n s u r a n c e c o v e r a g e a n d T = X , t h e a m o u n t t h a t t h e i n s u r a n c e c o m p a n y w o u l d p a y f o r i n j u r i e s o f t h e p a s s e n g e r ( a r a n d o m v a r i a b l e ) . S i n c e L i s a c o n s t a n t ( 2 . 4 . 4 ) i m p l i e s t h a t t h e r e e x i s t m u t u a l l y f a v o r a b l e e x c h a n g e s i n t h e a i r t r a v e l l e r ' s i n s u r a n c e s i t u a t i o n i f a n d o n l y i f E ( X ) < E ( X ) , i . e . i f f t h e p o l i c y b u y e r ' s e x p e c t a t i o n o f t h e a m o u n t t h e i n s u r a n c e c o m p a n y w o u l d h a v e t o p a y ( i n c a s e o f d i s a s t e r ) i s h i g h e r t h a n t h e c o m p a n y ' s e x p e c t a t i o n d e r i v e d b y e x p e r i e n c e . I n p a r t i c u l a r t h i s h a p p e n s w h e n p < q a n d p ^ < q ^ f o r a l l i , i . e . t h e p r o b a b i l i t y o f a l l c o n c e i v a b l e l e v e l s o f a d i s a s t e r i s h i g h e r f r o m b u y e r ' s p o i n t o f v i e w c o m p a r e d t o c o m p a n y ' s . h i g h a n d l o w . I f n e g a t i v e v a l u e s f o r a a n d g a r e a l s o 2 8 2 . 5 G E O M E T R I C A L S T R U C T U R E O F T H E S E T O F A L L M U T U A L L Y F A V O R A B L E E X C H A N G E S I n t h i s s e c t i o n w e s t u d y t h e g e o m e t r i c a l s t r u c t u r e o f t h e s e t o f a l l a d m i s s i b l e , m u t u a l l y f a v o r a b l e v e n t u r e s i n t h e s t o c h a s t i c e x c h a n g e s s i t u a t i o n i n t r o d u c e d i n t h e p r e v i o u s s e c t i o n , a n d s t u d y t h e t o p o l o g i c a l p r o p e r t i e s o f i t . S u c h a s t u d y w i l l n o t o n l y i l l u s t r a t e t h e a s p e c t s o f t h e e x c h a n g e s p r o b l e m w e h a v e l o o k e d a t b u t w i l l a l s o e n a b l e u s t o d e v e l o p t h e t h e o r y f u r t h e r . I n t h i s t r e a t m e n t , w e c o n c e i v e t h a t t h e c l a s s e s o f v e n t u r e s h a v i n g m e m b e r s o f t h e f o r m Y ( a , g ) = a S - g T , a a n d g b e i n g f l e x i b l e r e a l n u m b e r s . S u p p o s e o n l y n o n n e g a t i v e a , g a r e a d m i s s i b l e . 2 D e f i n e t h e s e t M i n R a s f o l l o w s : M = { ( g , a ) : g > 0 , a >_ 0 , E U ( - Y ( a , g ) ) > 0 , L L E U ( Y ( a , g ) ) > 0 } T o s t u d y M , s u p p o s e Y a n d e a c h u t i l i t y f u n c t i o n p o s s e s s t h e p r o p e r t i e s s t a t e d i n L e m m a 2 . 4 . 2 . T h i s a l l o w s u s t o e x p r e s s M i n a s i m p l e r f o r m ( s e e t h e p r o o f o f T h e o r e m 2 . 4 . 3 ) a s , M = { ( g , a): g > 0 , a >_ 0 , h ( g ) > a > g ( g ) } w h e r e a = h ( g ) i s t h e s o l u t i o n o f E U ( - Y ) = 0 a n d a = g ( g ) i s t h e i-i JL s o l u t i o n o f E U ( Y ) = 0 , f o r a . T h e f u n c t i o n s h a n d g a r e b o t h K K 2 9 c o n t i n u o u s a n d i n c r e a s i n g . S i n c e Y ( 0 , 0 ) a 0 , w e f i n d t h a t h a n d g a l s o g o t h r o u g h t h e o r i g i n . A d d i t i o n a l p r o p e r t i e s o f t h e u t i l i t y f u n c t i o n s w i l l i n d u c e s i m i l a r p r o p e r t i e s o n h a n d g . P r o p o s i t i o n 2 . 4 . 5 : ( i ) I f t h e u t i l i t y f u n c t i o n U i s c o n c a v e ( c o n v e x ) , t h e n K g i s c o n v e x ( c o n c a v e ) . ( i i ) I f t h e u t i l i t y f u n c t i o n U i s c o n c a v e ( c o n v e x ) , t h e n i _ i 1 h i s a l s o c o n c a v e ( c o n v e x ) . P r o o f : S u p p o s e g i s n o t c o n v e x . T h e n t h e r e e x i s t t w o p o i n t s 3 - p and 0 < A < 1 s u c h t h a t g ( A 3 i + ( 1 - A ) 3 2 ) >. * g ( 6 i ) + ( 1 - A ) g ( 3 2 ) . L e t g ( A 3 i + ( l - A ) 3 2 ) , a , = g ( $ i ) a n d a ? g ( 3 2 ) s o t h a t a > A o t i + ( l - A ) a 2 . H e n c e , E K U K Y ( a , X 3 i + C 1 - A ) 3 2 ) = E K U K { a S - ( A 3 i + ) 3 2 ) T } > E K U K { ( A a i + U - A ) a 2 ) S - ( A 3 i + ( 1 - A ) 3 2 ) T } s i n c e U i s i n c r e a s i n g . K = E K U K { A C a i S - 3 i T ) + C l - A ) ( a 2 S - 3 2 T ) } B u t s i n c e a = g C A 3 i + ( 1 - A ) 3 2 ) w e h a v e E L D Y ( a . A B i + ) S 2 ) = 0 a n d t h u s w e h a v e , 0 > E IT { A ( a i S - 3 i T ) + ( i - A ) ( a 2 S - 3 2 T ) } ( 2 . 5 . 1 ) O n t h e o t h e r h a n d t h e c o n c a v i t y o f U i m p l i e s , E IT {A ( a i S - 3 i T ) + ( l - A ) ( a 2 S - 3 2 T ) } 3 0 - V A U K ( a i S ~ S l T ) + C 1 - A ) U K C a 2 S - B 2 T ) } = X E R U K C a i S - B i T ) + ( 1 - A ) E K U K ( a 2 S - B 2 T ) 0 b e c a u s e a i = g ( B i ) a n d a 2 = g ( 8 2 ) , m a k i n g e a c h o f t h e a b o v e e x p e c t a t i o n s z e r o . T h i s c o n t r a d i c t s ( 2 . 5 . 1 ) t h u s p r o v i n g t h a t g i s c o n c a v e . I f U i s c o n v e x a l l t h e f o r e g o i n g i n e q u a l i t i e s r e v e r s e t o p r o v e g i s c o n c a v e . T h e p r o o f o f p a r t ( i i ) o f t h e p r o p o s i t i o n w i l l f o l l o w a l o n g s i m i l a r l i n e s . T h e f o r e g o i n g r e s u l t s y i e l d a f a i r l y d e t a i l e d m a p o f M u n d e r c o n v e x i t y p r o p e r t i e s o f t h e u t i l i t y f u n c t i o n s . F i r s t o f a l l , i f b o t h u t i l i t y f u n c t i o n s a r e c o n c a v e M w i l l b e a s t h e s h a d e d a r e a s h o w n i n F i g u r e 2 . 5 . 1 b o u n d e d b y t h e t w o c u r v e s a = h v B ) a n d a = g ( .B ) . We h a v e s e e n t h a t i f E ^ C O / E ^ S ) > E L ( T ) / E L ( S ) , g w i l l l i e c o m p l e t e l y a b o v e h o n t h e f i r s t q u a d r a n t m a k i n g M e m p t y . O t h e r w i s e i t w i l l b e a b o u n d e d o r u n b o u n d e d c o n v e x s e t . I f a l l n o n n e g a t i v e ( o t , B ) a r e a d m i s s i b l e , M w i l l b e t h e s e t o f a l l ( a , B ) l e a d i n g t o m u t u a l l y f a v o r a b l e + 2 v e n t u r e s . A n d i f A , a s t i b j s e t o f R , i s t h e s e t o f a l l a d m i s s i b l e ( a , B ) ' s , o n l y p o i n t s i n A A M w i l l l e a d t o m u t u a l l y f a v o r a b l e v e n t u r e s . T h e s e t M w h e n o n e o r b o t h u t i l i t y f u n c t i o n s a r e c o n v e x i s s h o w n i n F i g u r e 2 . 5 . 2 ( t h e i n t e r s e c t i o n o f t h e a r e a b e l o w h a n d t h e a r e a a b o v e g ) . F i g u r e 2 . 5 . 2 ( a ) i l l u s t r a t e s t h a t w h e n t h e u t i l i t y f u n c t i o n s a r e n o t c o n c a v e , t h e c o n d i t i o n E ( T ) / E ( S ) < E ( T ) / E ( S ) i s n o t n e c e s s a r y , t h o u g h i t i s s u f f i c i e n t , f o r t h e K K L i LI e x i s t e n c e o f m u t u a l l y f a v o r a b l e e x c h a n g e s . I t i s a l s o o f i n t e r e s t t o o b s e r v e t h a t i f b o t h u t i l i t y f u n c t i o n s a r e c o n v e x a n d i f t h i s c o n d i t i o n i s s a t i s f i e d g i v e n a n y a > 0 , t h e r e a r e 8.' s m a k i n g Y ( a , f t ) a m u t u a l l y b e n e f i c i a l e x c h a n g e . 31 ( a ) E k ( T ) / E k ( S ) < E t ( T ) / £ { ( S) P ( b ) E k ( T ) / E k ( S ) > E{[1) / E i ( S ) I3 F IG.2 .5 ,1 : Set M w h e n U k a n d a re bo th c o n c a v e 3 3 C H A P T E R 3 E X C H A N G E P R O B L E M : N A S H S O L U T I O N S 3 . 1 I N T R O D U C T I O N T h e p r e v i o u s c h a p t e r i s a n i n v e s t i g a t i o n o f t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e e x c h a n g e s a n d a c h a r a c t e r i z a t i o n o f t h e s e t m a d e u p b y t h e m . W h e n n u m e r o u s e x c h a n g e s a r e p o s s i b l e , w e h a v e n o t y e t s p e c i f i e d w h i c h p o i n t t h e d e c i s i o n m a k e r s w o u l d a g r e e u p o n f o r t h e t r a n s a c t i o n . H o w e v e r i t i s n o t o u r a i m h e r e t o p r o p o s e a n e w t h e o r y o f h o w t h e s o l u t i o n i s a c h i e v e d . T h e o b j e c t i v e o f t h i s c h a p t e r i s t o u t i l i z e t h e a l r e a d y d e v e l o p e d t h e o r y i n a p a r t i c u l a r c o n t e x t a n d t o o b t a i n a q u a l i t a t i v e c h a r a c t e r i z a t i o n o f s t o c h a s t i c e x c h a n g e s . S u c h a t r e a t m e n t w i l l n o t o n l y j u s t i f y t h e i n t u i t i v e a n d u n i n t u i t i v e f e a t u r e s o f a n e x c h a n g e s s i t u a t i o n b u t w i l l a l s o g i v e i n s i g h t s t h a t a r e p e c u l i a r t o s u c h s i t u a t i o n s a s t h e o n e w e a r e d e a l i n g w i t h . I n t h e f i r s t p a r t o f t h i s c h a p t e r w e s h a l l d o E d g e w o r t h ' s c o n t r a c t c u r v e a n a l y s i s w h i c h w i l l p r o v i d e a m u l t i p l i c i t y o f i n c o m p a r a b l e s o l u t i o n s ( i . e . t h e P a r e t o o p t i m a l s e t ) . T h e n d e v i a t i n g a b i t f r o m t h e t r a d i t i o n a l e c o n o m i c a n a l y s i s , w e u s e N a s h ' s a p p r o a c h u n d e r t h e r e q u i r e d c o n d i t i o n s , a s s u m e d v a l i d h e r e , t o f i n d t h e s o l u t i o n ( p o i n t o f a g r e e m e n t ) o f t h e e x c h a n g e p r o b l e m t o b e c a l l e d " t h e f a i r e x c h a n g e " s o l u t i o n u n d e r c o m p l e t e i n f o r m a t i o n . T h e n w e w i l l c o n s i d e r t h e e f f e c t s o f v a r i o u s f a c t o r s , s u c h a s t h e i n i t i a l w e a l t h s a n d t h e s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s o f t h e d e c i s i o n m a k e r s , o n t h e N a s h s o l u t i o n o f t h e e x c h a n g e p r o b l e m . 3 4 I n t h i s c h a p t e r t o o , w e c o n s i d e r o n l y e x c h a n g e s i t u a t i o n s i n w h i c h t h e v e n t u r e t o b e e x c h a n g e d f r o m t h e d e c i s i o n m a k e r K t o t h e d e c i s i o n m a k e r L i s a f u n c t i o n o f t w o a r b i t r a r y p a r a m e t e r s a a n d g , w h e r e a e R a n d g e R. U n l e s s o t h e r w i s e m e n t i o n e d , w e a s s u m e t h a t a l l n o n n e g a t i v e v a l u e s a r e a d m i s -s i b l e f o r a a n d g . T h e d e c i s i o n m a k e r s n e g o t i a t e t o f i x s u i t a b l e v a l u e s f o r a a n d g . W h e n a = a a n d g = g a r e f i x e d , K g i v e s Y ( a , g) t o L , w h e r e Y i s a r a n d o m v a r i a b l e t a k i n g n e g a t i v e a s w e l l a s p o s i t i v e r e a l v a l u e s f r o m e i t h e r d e c i s i o n m a k e r s p o i n t o f v i e w . M o r e o v e r i n t h i s c h a p t e r w e s h a l l c o n f i n e o u r a t t e n t i o n t o r i s k a v e r t e r s , i . e . d e c i s i o n m a k e r s w h o s e u t i l i t y f u n c t i o n s a r e c o n c a v e . A s s u m e t h a t t h e s e u t i l i t y f u n c t i o n s a r e c o n t i n u o u s l y d i f f e r e n t i a b l e . 3 . 2 E D G E W O R T H ' S C O N T R A C T C U R V E A N A L Y S I S T h e a n a l y s i s i n t h i s s e c t i o n i s p r i m a r i l y b a s e d o n t h e t h e o r y p r e s e n t e d b y E d g e w o r t h [ 1 8 8 1 ] i n " M a t h e m a t i c a l P s y c h i c s " . A s s u c h , w e b r i e f l y o u t l i n e h e r e t h e r e l e v a n t p o r t i o n o f h i s v a s t a n d r e m a r k a b l e t h e o r y . 3 . 2 . 1 E D G E W O R T H T H E O R Y T h e p r o b l e m i s c o n c e r n e d w i t h t w o i n d i v i d u a l s , K a n d L , e a c h p o s s e s s i n g q u a n t i t i e s o f o n e o f t w o c o m m o d i t i e s t o b e e x c h a n g e d w i t h e a c h o t h e r . L e t x a n d y b e t h e r e s p e c t i v e a m o u n t s o f t h e c o m m o d i t i e s i n t e r -c h a n g e d . C o n c e n t r a t i n g o n t h e e x c h a n g e s w h i c h a r e P a r e t o o p t i m a l , i . e . t h o s e w h i c h c a n n o t y i e l d g r e a t e r u t i l i t y t o o n e t r a d e r w i t h o u t d e c r e a s i n g t h a t o f t h e o t h e r , E d g e w o r t h s h o w s t h a t s u c h c o n t r a c t s w i l l s a t i s f y t h e c o n d i t i o n , 35 3x 3y 3y 3x U . ^ J J where (f> and IT are the respective u t i l i t i e s of K and L . The equation (3.2.1) y i e l d s a curve on the x-y plane, c a l l e d the "contract curve". The admissible "contracts" are those that are at l e a s t as desirable to both traders as t h e i r i n i t i a l holdings. That part of the curve given by (3.2.1) con s i s t i n g of admissible contracts i s c a l l e d the "range of p r a c t i c a b l e bargains". 3.2.2 PARETO OPTIMAL EXCHANGES In t h i s section we consider only the class of ventures where Y ( a , B ) = aS - 3T, S and T being nonnegative random va r i a b l e s . In Chapter 2 we have seen that the set of a l l ( a , B ) leading to mutually favorable ventures i s the convex set bounded by the two curves, a = g ( B ) and a = h ( B ) . Assume E^(T) / E^(S) < E (T) / E T(S) so that t h i s set i s nonempty. In the K K L L graph of a vs. B > on the curve determined by g , decision maker K's u t i l i t y i s zero and when we move above t h i s curve h i s u t i l i t y increases; where as on the curve determined by h L's u t i l i t y i s zero and,when we move below t h i s curve i t increases. The locus of the points on the graph of a vs. B giving a decision maker the same u t i l i t y i s c a l l e d an in d i f f e r e n c e curve. The two sets of i n d i f f e r e n c e curves of K and L are indicated i n Figure 3.2.1 . The i n d i f f e r e n c e curve of K corresponding to a u t i l i t y v i s the solu t i o n of E U ( Y ( a , B ) ) - v = 0 say a = g ( B ) . I t can be K K V shown as i n Proposition 2.4.2 that the in d i f f e r e n c e curves of K are a l l convex. S i m i l a r l y L's i n d i f f e r e n c e curves are a l l concave. The equations 3 6 o f t h e l a t e r f a m i l y i s g i v e n b y E U ( - Y ( a , B ) ) - u = 0 o r e q u i v a l e n t l y , s a y b y a = h ( 3 ) > u b e i n g a p a r a m e t e r . I t i s e a s i l y s h o w n t h a t t h e f a m i l y o f s t r i c t l y c o n c a v e i n d i f f e r e n c e c u r v e s a n d t h e f a m i l y o f s t r i c t l y c o n v e x i n d i f f e r e n c e c u r v e s t h u s g e n e r a t e d w i l l t o u c h e a c h o t h e r , i f e v e r , a l o n g p o i n t s o n a d e c r e a s i n g c u r v e , b e c a u s e n o d e c i s i o n m a k e r ' s i n d i f f e r e n c e c u r v e s i n t e r s e c t . I t i s i n t u i t i v e t h a t s u c h a c u r v e e x i s t s i f t h e t w o c u r v e s h ^ a n d g ^ i n t e r s e c t s e a c h o t h e r a t t w o p o i n t s o n t h e f i r s t q u a d r a n t . T h e c u r v e A B i n F i g u r e 3 . 2 . 1 r e p r e s e n t s t h e l o c u s o f t h e p o i n t s a t w h i c h t h e t w o s e t s o f i n d i f f e r e n c e c u r v e s t o u c h t a n g e n t i a l l y . N o t i c e t h a t a t a n y p o i n t o n t h i s c u r v e , t h e d e c i s i o n m a k e r s c a n n o t m o v e i n a n y d i r e c t i o n w i t h m u t u a l c o n s e n t . I n c o n t r a s t a t a n y p o i n t i n M o t h e r t h a n t h e p o i n t s o n t h a t c u r v e t h e y c a n m o v e t o g e t h e r , w i t h m u t u a l c o n s e n t , i n a n y d i r e c t i o n b e t w e e n t h e i r r e s p e c t i v e c u r v e s o f i n d i f f e r e n c e . H e n c e t h e p o i n t s o n t h e c u r v e A B a n d o n l y t h e s e r e p r e s e n t P a r e t o o p t i m a l p o i n t s . I t i s c l e a r t h a t t h e c o r e o f t h i s e x c h a n g e p r o b l e m i s n o n e m p t y , i . e . t h e r e e x i s t m u t u a l l y f a v o r a b l e ( a , B ) l e a d i n g t o P a r e t o o p t i m a l e x c h a n g e s , i f t h e r e e x i s t v, u s u c h t h a t t h e c u r v e s h a n d g i n t e r s e c t e a c h o t h e r a t t w o u v p o i n t s o n t h e f i r s t q u a d r a n t . H e n c e t h e c o r e i s n o n e m p t y i f f t h e r e e x i s t ( a , B ) s u c h t h a t , 9 E K U K ( Y ( a , B ) ) 8 B 3 E K U R ( Y ( a , B ) ) 8 a 8 E L U L ( Y ( a , B ) ) 9B 3 E L U L ( Y ( a , B ) ) 8 a 37 a F I G . 3 . 2 . h I N D I F F E R E N C E C U R V E S A N D T H E C O N T R A C T C U R V E . 3 8 b e c a u s e e a c h s i d e o f t h e a b o v e e q u a t i o n r e p r e s e n t s t h e s l o p e o f t h e f a m i l y o f i n d i f f e r e n c e c u r v e s o f t h e c o r r e s p o n d i n g d e c i s i o n m a k e r . I t i s a l s o o f i n t e r e s t t o n o t e t h a t , i f t h e t w o c u r v e s h a n d g i n t e r s e c t e a c h o t h e r a t a p o i n t o n t h e f i r s t q u a d r a n t o t h e r t h a n ( 0 , 0 ) , t h e n t h e c o r e o f t h e p r o b l e m i s b o u n d e d . T h e p a r t o f t h e c u r v e A B w i t h i n M i s t h e r a n g e o f p r a c t i c a b l e b a r g a i n s . We h a v e a l r e a d y a s s u m e d t h a t t h e u t i l i t y f u n c t i o n s a r e c o n t i n u o u s l y d i f f e r e n t i a b l e a n d e i t h e r t h a t t h e u t i l i t y f u n c t i o n s a r e b o u n d e d o r t h a t P a n d Q a s s i g n p r o b a b i l i t y 1 t o a b o u n d e d s u b s e t o f ft . H e n c e f r o m E d g e w o r t h ' s t h e o r y o r d i r e c t l y f r o m t h e e q u a t i o n g ^ ( 3 ) = h / ( 3 ) , t h e p o i n t s ( a , 3 ) l e a d i n g t o P a r e t o o p t i m a l v e n t u r e s s a t i s f y , 3 E K U K ( Y ( a , 3 ) ) 3 E L U L ( - Y ( a , 3 ) ) 3 E ^ U ^ ( Y ( a , 3 ) ) 3 E L U L ( - Y ( a , 3 ) ) 3a" " 9 3 ~ 3 3 " 3 a ( 3 . 2 . 2 ) S i n c e Y ( a , 3 ) = a S - 3 T , d i f f e r e n t i a t i o n u n d e r i n t e g r a l s i g n i m p l i e s , E R ( U ^ ( Y ) S ) E L ( U ^ ( - Y ) T ) - E ^ U ^ Y ) ! ) E L ( U ^ ( - Y ) S ) = 0 ( 3 . 2 . 3 ) a s t h e c o n d i t i o n s a t i s f i e d b y P a r e t o o p t i m a l p o i n t s , w h e r e Y = a S - 3 T . T h e p a r t o f t h e c o n t r a c t c u r v e w i t h i n M i s t h e s o l u t i o n g i v e n > t r a d i t i o n a l l y i n e c o n o m i c s , t o t h e e x c h a n g e p r o b l e m . T h u s , b y n e g o t i a t i o n , K a n d L w i l l c o m e t o a n a g r e e m e n t a t a p o i n t o n t h i s c u r v e . A t t e m p t s w i l l b e m a d e i n l a t e r s e c t i o n s t o e s t i m a t e t h e e x a c t p o i n t . 3 9 T o d e t e r m i n e t h e u t i l i t y p o s s i b i l i t y s e t f o r a g i v e n e x c h a n g e p r o b l e m o n e c a n u s e e q u a t i o n ( 3 . 2 . 3 ) . T h e k e y e l e m e n t i n v o l v e d i n t h i s p r o c e d u r e i s t h e e q u a t i o n o f t h e u t i l i t y f r o n t i e r f o r m e d b y P a r e t o o p t i m a l p o i n t s . I t c a n b e d e t e r m i n e d a s f o l l o w s . L e t W = E U ( Y ( a , B ) ) a n d K K K W = E U ( - Y ( a , B ) ) d e n o t e t h e c o r r e s p o n d i n g u t i l i t i e s . N o w i f w e r e l a t e W a n d W u s i n g e q u a t i o n ( 3 . 2 . 2 ) t o r e p l a c e a a n d B , w e o b t a i n t h e K L e q u a t i o n o f t h e u t i l i t y f r o n t i e r . 3 . 3 N A S H ' S S O L U T I O N S I n t h i s s e c t i o n w e s h a l l d i s c u s s t h e s o l u t i o n s o f t h e e x c h a n g e p r o b l e m , w h e n i t i s a b i l a t e r a l m o n o p o l y s i t u a t i o n . H e n c e t h e t w o c o u n t r y n e g o t i a t i o n m o d e l a p p e a r i n g i n E x a m p l e 2 . 2 , i s c a p t u r e d h e r e w h i l e t h e a i r t r a v e l l e r s i n s u r a n c e p r o b l e m d e s c r i b e d i n E x a m p l e 2 . 3 i s n o t . T h e p r e v i o u s s e c t i o n m a d e p l a u s i b l e t h e h y p o t h e s i s t h a t a n y a g r e e m e n t b e t w e e n r a t i o n a l d e c i s i o n m a k e r s t e n d s t o f a l l o n t h e E d g e w o r t h c o n t r a c t c u r v e . M o r e o v e r , a n y o p t i m a l a g r e e m e n t , a n d w e s t i l l l a c k o f e v i d e n c e t h a t s u c h a n a g r e e m e n t e x i s t s , w i l l l i e o n t h i s c o n t r a c t c u r v e a n d i n s i d e M . E n o r m o u s b a r g a i n i n g t h e o r i e s h a v e b e e n d e v e l o p e d t o d e t e r m i n e t h e a g r e e m e n t p o i n t f r o m t h e m u l t i p l i c i t y o f i n c o m p a r a b l e s o l u t i o n s ( P a r e t o o p t i m a l s e t ) . A d e t a i l e d d i s c u s s i o n o f t h e s e b a r g a i n i n g t h e o r i e s w i l l b e u n d e r t a k e n i n C h a p t e r 4 . I n t h i s s e c t i o n w e b r i e f l y o u t l i n e t h e s o l u t i o n t o t h e b a r g a i n i n g p r o b l e m s u g g e s t e d b y N a s h [ 1 9 5 0 ] a n d u s e i t t o f i n d t h e s o l u t i o n t o t h e e x c h a n g e p r o b l e m . T h e s o l u t i o n g i v e n b y N a s h ' s p r o c e d u r e i s k n o w n a s t h e " f a i r s o l u t i o n " , m e a n i n g t h a t t h i s p r o c e d u r e g i v e s t h e u t i l i t i e s t w o i n t e l -l i g e n t , r a t i o n a l i n d i v i d u a l s w o u l d r e c e i v e f r o m t h e e x c h a n g e , i f t h e y 4 0 c o l l a b o r a t e f o r m u t u a l b e n e f i t . T h i s i s a s i t u a t i o n w h e r e b o t h i n d i v i d u a l s a r e s u p p o s e d t o h a v e c o m p l e t e i n f o r m a t i o n a b o u t t h e b a r g a i n i n g s i t u a t i o n ; s o t h e s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n a n d u t i l i t y f u n c t i o n o f e a c h i n d i v i d u a l i s k n o w n t o t h e o t h e r . We s h a l l o b t a i n s o l u t i o n s u n d e r m o r e g e n e r a l c o n d i t i o n s i n l a t e r c h a p t e r s . 3 . 3 . 1 T H E N A S H T H E O R Y N a s h p o s t u l a t e s t w o r a t i o n a l p l a y e r s b a r g a i n i n g i n a s i t u a t i o n w h e r e e a c h h a s f u l l k n o w l e d g e a b o u t t h e s i t u a t i o n . C o n c e i v i n g t h e p r o b l e m p u r e l y i n t e r m s o f t h e p l a y e r s ' u t i l i t y i n c r e m e n t s w i t h r e s p e c t t o t h e c o n f l i c t p o i n t , h e a x i o m a t i c a l l y a s s u m e s t h a t , a s o l u t i o n t o t h e b a r g a i n i n g p r o b l e m s h o u l d p o s s e s s : 1 . P a r a t o O p t i m a l i t y - t h e p o i n t o f a g r e e m e n t ( s o l u t i o n ) m u s t l i e o n t h e n o r t h e a s t b o u n d a r y o f t h e j o i n t u t i l i t y - p o s s i b i l i t y s e t ( u t i l i t y f r o n t i e r . ) . 2. S y m m e t r y - i f t h e u t i l i t y f r o n t i e r i s s y m m e t r i c a l t h e s o l u t i o n y i e l d s e q u a l u t i l i t i e s t o t h e t w o p l a y e r s . 3 . T r a n s f o r m a t i o n I n v a r i a n c e - t h e s o l u t i o n i s n o t a l t e r e d b y a l i n e a r , o r d e r - p r e s e r v i n g t r a n s f o r m a t i o n o f t h e u t i l i t y f u n c t i o n o f e i t h e r p a r t y . 4 . I n d e p e n d e n c e o f I r r e l e v a n t A l t e r n a t i v e s - g i v e n t w o b a r g a i n i n g s i t u a t i o n s s u c h t h a t t h e u t i l i t y - p o s s i b i l i t y s e t ( r e s p e c t i v e l y , s o l u t i o n p o i n t ) f o r t h e s e c o n d o n e c o n t a i n s ( i s c o n t a i n e d i n ) t h a t o f t h e f i r s t t h e n t h e t w o b a r g a i n i n g s i t u a t i o n s h a v e t h e s a m e s o l u t i o n . N a s h t h e n d e d u c e s t h e r e m a r k a b l e r e s u l t t h a t t h e o n l y s o l u t i o n s a t i s f y i n g a x i o m s 1 - 4 i s f o u n d b y m a x i m i z i n g t h e p r o d u c t o f t w o u t i l i t i e s . 41 V a r i o u s a u t h o r s ( c f . B i s h o p [1963], a n d L u c e a n d R a i f f a [1957] h a v e c r i t i c i z e d N a s h ' s a x i o m s , p a r t i c u l a r l y 3 a n d 4 . A l t e r n a t i v e s o l u t i o n s t o t h e b a r g a i n i n g p r o b l e m h a v e b e e n d e v e l o p e d b y B i s h o p [1963], K a l a i a n d S m o r o d i n s k y [1975], a n d R a i f f a [1953] b y r e p l a c i n g o n e o r b o t h o f t h e s e t w o a x i o m s . E x p e r i m e n t a l t e s t s c o n d u c t e d b y N y d e g g e r a n d O w e n [1975] i n d i c a t e t h a t t h e i r s u b j e c t s r e s p o n s e s c o n f o r m e d t o b o t h s e c o n d a n d f o u r t h a x i o m s w h i l e t h e y v i o l a t e d t h e t h i r d . 3.3.2 F A I R E X C H A N G E S We c o n s i d e r o n l y t h e p r o b l e m w h e r e t h e c o r e i s n o n e m p t y a n d b o u n d e d . W h e n Y(a, B ) = a S - B T , c o n d i t i o n s u n d e r w h i c h t h i s i s t r u e h a v e b e e n g i v e n i n S e c t i o n 3.2.2 . W h e n d e c i s i o n m a k e r s K a n d L c h o o s e a = a Q, B = B Q a s t h e p a r a m e t e r s o f t h e a g r e e m e n t t h e r e s p e c t i v e u t i l i t i e s t h e y w o u l d g a i n a r e d e n o t e d b y W^(a^, B Q ) = E U (Y(a o, B q ) ) a n d W T ( a , B ) = E U (-Y(a , B )) • L e t A b e t h e a d m i s s i b l e s e t o f (a, B ) L o o L L o o ( t h e s e t o f n o n r a n d o m i z e d s t r a t e g i e s ) . T h e n a c c o r d i n g t o t h e N a s h t h e o r y K a n d L , w h o a r e a s s u m e d t o b e i n t h e f a i r b a r g a i n i n g s i t u a t i o n , w o u l d d e c i d e t o c h o o s e ( a , B ) t o m a x i m i z e o v e r A , WR(a, B ) W L (a, B ) A W(a, B ) (3.3.1) I f A i s c l o s e d t h i s m a x i m u m i s a t t a i n e d a t a n a d m i s s i b l e p o i n t . N o w d i f f e r e n t i a t i o n o f W d e f i n e d i n e q u a t i o n (3.3.1) w . r . t . a a n d B r e s p e c t i v e l y a n d s e t t i n g t h e r e s u l t e q u a l t o 0 y i e l d s , 42 - e a ( Y ) U L ( - Y > } + E L V - y ) vU U K ( Y ) } = 0 ( 3 . 3 . 2 ) ( 3 . 3 . 3 ) a t m a x i m a a n d m i n i m a o f ( 3 . 3 . 1 ) , p r o v i d e d ( a , 3 ) i s n o t c o n s t r a i n e d . H e n c e t h e ( a , 3 ) m a x i m i z i n g W ( a , 3 ) a r e s o l u t i o n s o f e q u a t i o n s ( 3 . 3 . 2 ) a n d ( 3 . 3 . 3 ) w h e n e v e r t h e y e x i s t . O f c o u r s e t h e s e p o i n t s l i e o n t h e c o n t r a c t c u r v e g i v e n b y ( 3 . 2 . 2 ) . I f t h i s ( a , 3 ) i s a d m i s s i b l e , t h e n i t i s t h e s o l u t i o n g i v e n b y N a s h ' s f a i r b a r g a i n c r i t e r i o n . I n p a r t i c u l a r t h i s i s t h e c a s e i f a l l n o n - n e g a t i v e v a l u e s o f a a n d 3 a r e a d m i s s i b l e . I n p r e p a r a t i o n f o r t h e n e x t s e c t i o n , l e t u s n o w c o n s i d e r t h e c l a s s o f v e n t u r e s w h o s e e l e m e n t s a r e o f t h e f o r m Y ( a ) = a X w h e r e a >_ 0 i s a n a r b i t r a r y c o n s t a n t a n d X i s a r a n d o m v a r i a b l e t a k i n g p o s i t i v e a n d n e g a t i v e v a l u e s . A b e t t i n g p r o b l e m w h e r e t h e l o s e r p a y s t h e w i n n e r $ a i s a n e x a m p l e i n w h i c h X = 1 i f K w i n s a n d X = - 1 o t h e r w i s e . H i l d r e t h [ 1 9 7 4 ] h a s s h o w n t h a t t h e r i s k a v e r t e r s K a n d L c a n f i n d m u t u a l l y b e n e f i c i a l v e n t u r e s f r o m t h i s c l a s s i f f E ^ ( X ) > 0 > E T ( X ) . L e t = E ^ T j ( a X ) , K L K K K V T = E U ( - a X ) a n d V = V * V . D i f f e r e n t i a t i n g V w . r . t . a w e g e t , i - i L i LI K. LI dV_ d a = - E K U R ( a X ) E L ( X U ^ ( - a X ) ) + E ^ U ^ ( - a X ) E ^ ( X U ^ ( a X ) ) = f ( a ) a n d , 4 3 d a E T U ( - a X ) E ( X 2 U " ( a X ) } a n d t h e r e f o r e , i n p a r t i c u l a r < 0 f ( a ) = 0 I f t h e N a s h s o l u t i o n t o t h i s p r o b l e m e x i s t s a n d a l l a >_ 0 a r e a d m i s s i b l e , t h e n i t i s g i v e n b y , O f c o u r s e w h e n a i s c o n s t r a i n e d , t h e N a s h s o l u t i o n m a y o c c u r a t a d i f f e r e n t p o i n t . H o w e v e r w e c o n t i n u e o u r a n a l y s i s o n l y f o r t h e c a s e w h e r e i t i s g i v e n b y ( 3 . 3 . 4 ) . E X A M P L E 3 . 2 . 1 : C o n s i d e r t h e p a r t i c u l a r b e t t i n g p r o b l e m w h e r e K ' s u t i l i t y f u n c t i o n i s , f ( d ) = 0 ( 3 . 3 . 4 ) t x i f x > 0 , l > t > 0 V x ) = X i f x < 0 a n d L ' s u t i l i t y f u n c t i o n i s , Vx+T 1 i f x > 0 U L ( x ) = i f x < 0 x 4 4 B o t h u t i l i t y f u n c t i o n s a r e c o n t i n u o u s , c o n c a v e a n d i n c r e a s i n g . C o n s i d e r t h e b e t i n d e x e d b y (a, g ) . T h e n K ' s u t i l i t y i s W (a, g ) = p t a - ( 1 - p ) g a n d L ' s u t i l i t y i s W ( a , g ) = ( 1 - q ) ( / g + 1 - 1 ) - q a . L e t u s f i r s t d e t e r m i n e L t h e s e t o f a l l m u t u a l l y f a v o r a b l e b e t s . S i n c e U a n d U a r e n o t d i f -f e r e n t i a b l e a t t h e o r i g i n T h e o r e m 2 . 3 . 1 d o e s n o t a p p l y a n d s o t h e c o n d i t i o n p > q d o e s n o t g u a r a n t e e t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e b e t s . I t i s c l e a r , h o w e v e r , t h a t t h e r e e x i s t s u c h b e t s i f a n d o n l y i f t h e r e e x i s t n o n -n e g a t i v e a a n d g ( s m a l l e r t h a n t h e i r c a p i t a l s ) s u c h t h a t , 1-2. { P J < a < •1=3. (/g+T - i) I t i s n o w e v i d e n t t h a t t h e r e e x i s t s u c h b e t s i f f t h e s l o p e o f t h e g r a p h o f a = ( ( l - q ) / q ) ( / g + 1 - 1 ) i s g r e a t e r t h a n t h a t o f a = ( ( l - p ) / p ) ( g / t ) a t t h e o r i g i n ( s e e F i g u r e 3 . 3 . 1 ) . H e n c e a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f m u t u a l l y f a v o r a b l e b e t s i s t h a t ( l - p ) / p t < ( l - q ) / 2 q . N o t i c e t h a t p > q i s n e c e s s a r y b u t n o t s u f f i c i e n t f o r e x i s t e n c e o f s u c h b e t s . L e t u s s u p p o s e , t h e r e f o r e , t h a t t < p ( l - q ) / 2 q ( l - p ) . S i n c e t h e u t i l i t y f u n c t i o n s a r e d i f f e r e n t i a b l e a t a n y p o i n t o t h e r t h a n a t t h e o r i g i n w e c a n f i n d t h e e q u a t i o n o f t h e c o n t r a c t c u r v e t o t h i s p r o b l e m b y f i n d i n g t h e p o i n t s (a, g ) • w h e r e t h e s l o p e s o f t h e i n d i f f e r e n c e c u r v e s a r e e q u a l , i . e . w h e r e , p t O z £ i = ( i _ p ) q 2/g+T 45 P F I G . 3 .3 .1 : C o n t r a c t c u r v e a n d t h e N a s h s o l u t i o n of t h e b e t t i n g p r o -b l e m i n e x a m p l e 3.2,1. 4 6 a n d a i s a r b i t r a r y s o l o n g a s ( a , (3) r e m a i n s f e a s i b l e . T h u s , 2 _ - 1 = 3 ( s a y ) f o r a n y a g r e e m e n t a n d t h i s i s w h a t K m u s t o f f e r i n a n y o p t i m a l b a r g a i n i n g s i t u a t i o n . T o f i n d t h e N a s h s o l u t i o n t o t h i s p r o b l e m w e d i f f e r e n t i a t e W = W„W = [ p t a - ( l - p ) B ] [ ( 1 - q ) ( / B + l - 1 ) - q a ] a n d e q u a t e t h e d e r i v a t i v e K L t o 0 . T h u s w e h a v e , ( ( 1 - q ) ( / e + T - 1 ) - q a ) p t - ( p t a - ( l - p ) B ) q act = - 2 q p t a + ( p ( l - q ) t ( / B + T - 1 ) + q ( l - p ) B ) = 0 S i n c e w e k n o w t h a t t h e s o l u t i o n w i l l l i e o n t h e c o n t r a c t c u r v e , t h e a b o v e e q u a t i o n i m p l i e s t h e N a s h s o l u t i o n f o r ( a , B ) , [ ( l - p ) B / 2 p t + ( 1 - q ) (/B" + 1 - D / 2 q , B ] . 3.3.3 R E S P O N S E O F T H E N A S H S O L U T I O N T O C H A N G E S I N C , P , A N D Y I n t h i s s e c t i o n w e s t u d y t h e r e l a t i o n b e t w e e n t h e N a s h s o l u t i o n a n d t h e p r e v a i l i n g c i r c u m s t a n c e s o f a n e x c h a n g e s i t u a t i o n . We s h o u l d b e i n t e r e s t e d i n r e s p o n s e s o f t h e N a s h s o l u t i o n t o v a r i o u s c h a n g e s i n Y , p l a y e r s ' s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s a n d t h e i r i n i t i a l w e a l t h s . EllzSl t / 2 I q ( l - P ) ' 4 7 H e r e w e s h a l l d e a l o n l y w i t h t h e c l a s s o f v e n t u r e s { a X } d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . I n t h e f o l l o w i n g a n a l y s i s w e s h a l l u s e t h e e q u a t i o n ( 3 . 3 . 4 ) d e v e l o p e d f o r N a s h ' s f a i r b a r g a i n i n g . H o w e v e r i t s h o u l d n o t b e o v e r l o o k e d t h a t o u r c o n c l u s i o n s a r e v a l i d o n l y w h e n N a s h ' s u t i l i t y p r o d u c t i s m a x i m i z e d . F o r , i n a b a r g a i n i n g s i t u a t i o n , s u p p o s e t h e a g r e e m e n t i s s r r e a c h e d b y m a x i m i z i n g t h e g e n e r a l i z e d u t i l i t y p r o d u c t V V , a s B i s h o p K t i [ 1 9 6 4 ] c o n c l u d e s b y t a k i n g t h e t i m e , f o r w h i c h t h e b a r g a i n i n g i s c a r r i e d o u t , i n t o a c c o u n t , w h e r e r a n d s a r e t h e r a t e s o f t i m e d i s c o u n t s o f K a n d L r e s p e c t i v e l y . I f t h i s i s t h e c a s e t h e s o l u t i o n a i s f o u n d b y , - J E K U K ( a X ) E L ( X U ^ ( - a X ) ) + E ^ - a X ) • E K ( X U ^ ( a X ) ) = 0 I t w i l l b e c o m e c l e a r t h a t t h e f o l l o w i n g q u a l i t a t i v e p r o p e r t i e s a r e n o t a f f e c t e d b y t h e p r e s e n c e o f t h e c o n s t a n t s / r . S H I F T S I N I N I T I A L W E A L T H S C A N D D L e t u s s t a r t b y i n v e s t i g a t i n g t h e e f f e c t s o f C a n d D , t h e i n i t i a l w e a l t h s , o n a . S u c h e f f e c t s a r e c a l l e d w e a l t h e f f e c t s . B a r g a i n e r K ' s w e a l t h m i g h t c h a n g e , f o r i n s t a n c e , w h e n h e r e o p e n s a p r e v i o u s l y c l o s e d d e a l w i t h L ( e . g . a s w h e n y e a r l y , K b e t s w i t h L o n t h e h o c k y f i n a l s w i t h a n a l t e r e d w e a l t h , a n d s a m e s u b j e c t i v e b e l i e f s , t a s t e s a n d p r e f e r e n c e s a s b e f o r e ) . We a r e i n t e r e s t e d i n d e t e r m i n i n g h o w t h e p o i n t o f a g r e e m e n t , a r r i v e d a t b y b a r g a i n i n g , w o u l d c h a n g e u n d e r t h e s e a l t e r e d c i r c u m s t a n c e s . T o d o t h i s K ' s u t i l i t y f u n c t i o n i s r e l a b e l e d a s U ( C + Z ) - U ( C ) = U ( Z ) , 4 8 w h e r e C i s K ' s i n i t i a l w e a l t h . R e c a l l t h a t w e a r e a c c e p t i n g a s v a l i d , a s s u m p t i o n s w h i c h i m p l y t h a t t h e s o l u t i o n t o t h i s e x c h a n g e p r o b l e m i s g i v e n b y f ( a ) = 0 w h e r e f ( a ) = E L U L ( - a X ) • E K ( X U ^ . ( C + a X ) ) E L ( X U ^ ( - a X ) ) [ E K U K ( C + a X ) - U K ( C ) ] T h e n w e h a v e , d a d C e \ 3f_ 3G 9 f 3 a a = a B u t 3 a a = a i s n e g a t i v e a n d t h e r e f o r e ^ h a s t h e s a m e s i g n a s , i . e . , u C 3 C d a >_ d C < 0 i f f 3 f ( a ) 3C a = a D i f f e r e n t i a t i n g f ( a ) w . r . t . C w e h a v e , 3C 3 U K ( C ) a = a - = E L U L ( _ a X ) E K { X " K ( C + a X ) } " V X U L ( - a x ) } t E K ^ ( C + a X ) - ' ^ ] 4 9 [ { E K U K ( C + a X ) U K ( C ) } E K ( X U j ; ( C + a X ) } - E K { X U ^ . ( C + a X ) } { E K U ^ ( C + a X ) - U ^ ( C ) } ] b e c a u s e f ( a ) = 0 [ E R U K ( a X ) E R ( X U ^ ( a X ) ) - E ^ X U ^ a X ) ) { E ^ C a X ) - 1 ^ . ( 0 ) } ] ( 3 . 3 . 5 ) { E K U K ( a X ) > 2 df" E R U ^ . ( a X ) - 11^.(0) E K U K ( a X ) a = a w h e r e 8 = E U ( - a X ) / E ^ U ( a X ) > 0 a s a i s m u t u a l l y f a v o r a b l e . H e n c e a L L K K w i l l i n c r e a s e o r d e c r e a s e w i t h C a c c o r d i n g a s { E U ' ( a X ) - U ' ( 0 ) } / E U ( a X ) JN. K Is. K. K i n c r e a s e s o r d e c r e a s e s i n a a t a . I n f a c t i t s e e m s t h a t f o l l o w i n g i s a r e a s o n a b l e a s s u m p t i o n u n d e r w h i c h w e c o u l d s t a t e a d e f i n i t e c o n c l u s i o n . A s s u m p t i o n 3 . 1 : A s s u m e f o r U = a n d U = U T t h a t E ( U ' ( a X ) - U ' ( 0 ) ) / : ~- K L E U ( a X ) i s a n i n c r e a s i n g f u n c t i o n o f a f o r w h i c h E U ( a X ) > 0 . I n o r d e r t o s h o w t h i s a s s u m p t i o n i s p l a u s i b l e , f i r s t l y w e s h a l l p r o v e t h a t a s s u m p t i o n 3 . 1 i s t r u e f o r c o n s t a n t r a n d o m v a r i a b l e s X , u n d e r v e r y m i l d c o n d i t i o n s o n U , s e c o n d l y w e s h a l l g i v e a n e x a m p l e w h e r e a s s u m p t i o n 3 . 1 i s s h o w n b y s i m u l a t i o n t o b e t r u e e v e n w h e n X i s n o t a c o n s t a n t r a n d o m v a r i a b l e . P r a t t [ 1 9 6 4 ] a n d A r r o w [ 1 9 7 1 ] h a v e s h o w n t h a t t h e a b s o l u t e r i s k a v e r s i o n , d e f i n e d b y r ( x ) = - U " ( x ) / U ' ( x ) i s a n e c e s s a r y c o n c e p t f o r c h a r a c t e r i z i n g b e h a v i o r u n d e r u n c e r t a i n t y a n d t y p i c a l l y a s s u m e t h a t r 5 0 d e c r e a s e s w i t h x . T h e f o l l o w i n g p r o p o s i t i o n h o l d s w h e n t h i s r e q u i r e m e n t i s f u l f i l l e d . P r o p o s i t i o n 3 . 3 . 1 : L e t U ( x ) b e a n i n c r e a s i n g a n d c o n c a v e u t i l i t y f u n c t i o n  s u c h t h a t U ( 0 ) = 0 . I f U " ( x ) / U ' ( x ) i n c r e a s e s w i t h x , t h e n U ' ( x ) - U ' ( 0 ) U ( x ) a l s o i n c r e a s e s w i t h x 4 0 , p r o v i d e d t h a t U ' ( x ) i s t w i c e d i f f e r e n t i a b l e . P r o o f : L e t t i n g g ( x ) = ( U * ( x ) - U ' ( 0 ) ) > / U ( x ) , w e h a v e f o r x 4 0 , , ( x ) = U ( x ) U " ( x ) - ( U ' ( x ) - U ' ( 0 ) ) U ' ( x ) U ( x ) 2 = ^ ~ ^ [ U ( x ) ^ $ r + ( U ' ( 0 ) - U ' ( x ) ) ] U ( x ) U W . U ' ( x ) , , . A *-f h ( x ) U ( x ) Z N o t i c e t h a t h i s a d i f f e r e n t i a b l e f u n c t i o n s u c h t h a t h ( 0 ) = 0 a n d h ' ( x ) = U ( x ) ( U " ( x ) / U ' ( x ) ) . H e n c e h i s i n c r e a s i n g f o r a l l x > 0 a n d i s d e c r e a s i n g f o r a l l x < 0 t h u s i m p l y i n g t h a t h ( x ) >_ 0 f o r a l l x C o n s e q u e n t l y g ' ( x ) >_ 0 , t h u s p r o v i n g t h e d e s i r e d r e s u l t . 5 1 E X A M P L E 3 . 3 . 1 : L e t U ( x ) = 7 ( 1 - e X ) + 8 ( 1 - e ~ X ) . T h e n U ( x ) i s i n c r e a s i n g , c o n c a v e a n d U " ( x ) / U ' ( x ) i s i n c r e a s i n g . S u p p o s e t h e r a n d o m v a r i a b l e X t a k e s t h e v a l u e s 3 a n d - 4 w i t h p r o b a b i l i t i e s . 8 a n d . 2 r e s p e c t i v e l y . T h e n a M o n t e C a r l o c a l c u l a t i o n s h o w s t h a t E ( U ' ( a X ) - U ' ( 0 ) ) / E U ( a X ) i s i n c r e a s i n g f o r a l l a e x c e p t t h o s e a f o r w h i c h E U ( a X ) = 0 ( s e e A p p e n d i x A ) . T h e g r a p h o f t h i s f u n c t i o n i s s h o w n i n F i g u r e 3 . 3 : 2 f o r E U ( a X ) > 0 . T h u s w h a t w e h a v e o b s e r v e d i s t h a t : P r o p o s i t i o n 3 . 3 . 1 : W h e n t h e w e a l t h o f a b a r g a i n e r i s i n c r e a s e d , u n d e r  A s s u m p t i o n 3 . 1 , t h e t w o b a r g a i n e r s w i l l a g r e e a t a h i g h e r a . R e c a l l t h a t a r e p r e s e n t s t h e " s i z e " o f t h e e x c h a n g e t o b e p e r f o r m e d . I n p a r t i c u l a r , w h e n t h e v e n t u r e i s a b e t o f t h e t y p e d e s c r i b e d i n s e c t i o n 3 . 3 . 2 , a r e p r e s e n t s t h e a m o u n t o f m o n e y a b a r g a i n e r i s w i l l i n g t o p a y t h e o p p o n e n t , i f h e l o s e s . S i n c e , u n d e r A s s u m p t i o n 3 . 1 , a i n c r e a s e s w i t h C w e r e a c h t h e i n t u i t i v e l y c o r r e c t c o n c l u s i o n t h a t a r i s k a v e r t e r t y p i c a l l y b e t s m o r e w h e n h e b e c o m e s w e a l t h i e r . S H I F T S I N P , Q : We n o w a s s u m e f i x e d i n i t i a l w e a l t h s . A s h i f t i n P o c c u r s i f K h a s r e o p e n e d a d e a l w i t h L w i t h m o d i f i e d b e l i e f s . S u c h a s h i f t , f o r K s a y , c a n b e s t u d i e d b y c o n s i d e r i n g a t r a n s f o r m a t i o n o f o r i g i n a l r a n d o m v a r i a b l e X . F o l l o w i n g A r r o w [ 1 9 7 1 ] w e c o n c e i v e o f a f a m i l y o f s u c h t r a n s f o r m a t i o n s , e a c h m e m b e r b e i n g c h a r a c t e r i z e d b y t h e v a l u e o f a s h i f t p a r a m e t e r h . W e s h a l l b e i n t e r e s t e d i n ( i ) a s i m p l e m o v e m e n t o f t h e d i s t r i b u t i o n t o t h e r i g h t o r l e f t , i . e . X ( h ) = X + h , 03 CD 1V> ALPHA 5 3 a n d i n ( i i ) a s i m p l e e x p a n s i o n o f t h e d i s t r i b u t i o n a b o u t a c e n t r e , X , i . e . X ( h ) = X + ( 1 + h ) ( X - X ) . R e m e m b e r i n g t h a t X i s s h i f t e d o n l y f r o m K ' s s t a n d p o i n t w e r e w r i t e t h e e q u a t i o n ( 3 . 3 . 4 ) a s , f ( a ) = E L U L ( - a X ) E K { X ( h ) U ^ ( a X ( h ) ) } - E ^ X U ^ C - a X ^ E ^ U ^ a X C h ) ) } = 0 ( 3 . 3 . 6 ) u s i n g n o r m a l i z e d u t i l i t y f u n c t i o n s ( C a n d D a r e n o w c o n s t a n t ) . T o i n v e s t i g a t e t h e d i r e c t i o n i n w h i c h a i s d e p e n d e n t o n h w e n e e d t o e x a m i n e t h e s i g n o f r£ . A s i s s h o w n i n t h e p r e v i o u s s e c t i o n t h e s i g n o f 4 r i s a n d h d f t h e s a m e a s t h a t o f — -9 h - . D i f f e r e n t i a t i o n o f f ( a ) w . r . t . a y i e l d s , a = - = E L U L ( - a X ) E K { X ' ( h ) U ^ . ( a X ( h ) ) + a X ( h ) X ' ( h ) U £ ( a X ( h ) ) } E T { X U ' ( - a X ) } K j a X * ( h ) U ' ( a X ( h ) ) } L L K K [ E K U R ( a X ( h ) ) E K { X ' ( h ) U ^ ( a X ( h ) + a X ( h ) X ' ( h ) U £ ( a X ( h ) ) } ] u s i n g e q u a t i o n ( 3 . 3 . 6 ) , w h e r e 0 = E U ( - a X ) / E U ( c n X ( H ) ) > 0 . F o r L L K K s i m p l i c i t y h e r e w e c o n s i d e r o n l y t h e a d d i t i v e s h i f t X ( h ) = X + h . T h e o t h e r c a s e m a y b e c a r r i e d o u t i n a s i m i l a r w a y . H e n c e f o r a n a d d i t i v e s h i f t , 5 4 3 f 9 h a = - = 6 E K U K ( a X ( h ) E K U ^ ( a X ( h ) ) + 6 a [ E ^ U ^ . ( a X ( h ) ) E j , { X ( h ) U ^ ( a X ( h ) ) } - E R { X ( h ) U £ ( a X ( h ) ) } E K { U ^ . ( a X ( h ) ) } ] 6 { E U ( a X ( h ) ) } ' K- K E K U ^ ( a X ( h ) ) E K U K ( a X ( h ) ) - d E ^ U ^ . ( a X ( h ) ) = 6 { E K U K ( a X ( h ) ) } 2 ^ r _ E K U ^ . ( a X ( h ) ) > , a r E K U K ( a X ( h ) ) J a = -T h u s , a n a d d i t i v e s h i f t o f K ' s p r o b a b i l i t y d i s t r i b u t i o n i n c r e a s e s o r d e c r e a s e s a , t h e s i z e o f t h e e x c h a n g e , a c c o r d i n g a s a E ( U ' ( a X ( h ) ) ) / E ( U ( a X ( h ) ) ) i n c r e a s e s o r d e c r e a s e s a t a . A s b e f o r e w e K K K K c a n i n f a c t c o m e t o a d e f i n i t e c o n c l u s i o n u n d e r t h e f o l l o w i n g r e a s o n a b l e a s s u m p t i o n : A s s u m p t i o n 3 . 2 : A s s u m e f o r U = U R a n d U = U L t h a t a E U ' ( a X ) / E U ( a X ) i s a n i n c r e a s i n g f u n c t i o n o f a , a t a n y a f o r w h i c h E U ( a X ) > 0 , w h e r e X = X ( h ) . T h e p l a u s i b i l i t y o f t h i s a s s u m p t i o n i s a g a i n e s t a b l i s h e d b y a n e x a m p l e . F u r t h e r m o r e i t i s o b v i o u s t h e a s s u m p t i o n t e n d s t o b e t r u e w h e n U i s n e a r l y l i n e a r a t a . E X A M P L E 3 . 3 . 2 : C o n s i d e r t h e s a m e u t i l i t y f u n c t i o n U ( x ) = 7 ( l - e _ X ) - 2 x ^ + 8 ( l - e ) a n d t h e r a n d o m v a r i a b l e X = X i n E x a m p l e 3 . 3 . 1 . T h e n a M o n t e C a r l o c a l c u l a t i o n s h o w s t h a t a E U ( a X ) / E U ( a X ) i s m o n o t o n i c i n c r e a s i n g CO r o F I G U R E 3 . 3 . 3 G R A P H O F a E U ' ( a X ) / E U ( a X ) *» G A M A V S I N E X A M P L E 3 . 3 . 2 a = A L P H A O Q -C D a CD a a CD a U1 0.0 0.02 0.04 0.06 0.08 0.1 ALPHA 0.12 0.14 0.16 0.18 5 6 f o r a l l a s u c h t h a t E U ( a X ) > 0 ( s e e A p p e n d i x B ) . T h e g r a p h o f t h i s f u n c t i o n f o r o u r e x a m p l e i s s h o w n i n F i g u r e 3 . 3 . 3 . H e n c e w e c a n c o n c l u d e t h a t : P r o p o s i t i o n 3 . 3 . 2 : U n d e r a s s u m p t i o n 3 . 2 , t h e t w o b a r g a i n e r s w i l l a g r e e  a t a h i g h e r a a s a r e s u l t o f a s i m p l e s h i f t o f ( i ) K ' s s u b j e c t i v e 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 X t o t h e r i g h t , o r , ( i i ) L ' s s u b j e c t i v e 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 X t o t h e l e f t . S H I F T S I N X : T h e s h i f t s i n X o c c u r f o r e x a m p l e w h e n t h e v a l u e o f a p a r a m e t e r , s a y t h e p r i c e o f a n a r t i c l e i n v o l v e d i n t h e t r a n s a c t i o n , i s c h a n g e d , t h u s a f f e c t i n g t h e t w o s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s i n a s a m e f a s h i o n . T w o p a r t i c u l a r t y p e s o f s h i f t s a r e o f o u r i n t e r e s t : ( i ) a s i m p l e m o v e m e n t o f t h e f o r m X ( h ) = X + h a n d ( i i ) a s i m p l e e x p a n s i o n o f X a b o u t a c e n t r a l v a l u e X * , X ( h ) = X * + ( 1 + h ) ( X - X * ) . F i r s t c o n s i d e r t h e s h i f t X ( h ) = X + h . I n s o m e c o n t e x t s s u c h r e s p o n s e s o f a d u e t o c h a n g e s i n h a r e c a l l e d " p r i c e e f f e c t s " . We h a v e a l r e a d y s e e n i n t h e p r e v i o u s s e c t i o n t h a t w h e n X i s s h i f t e d t h i s w a y f r o m o n l y K ' s s t a n d p o i n t , a t y p i c a l l y i n c r e a s e s . B y s y m m e t r y a w i l l t y p i c a l l y t e n d t o d e c r e a s e i f s u c h a s h i f t t a k e s p l a c e f r o m o n l y L ' s p o i n t o f v i e w ( w h e n h i s p o s i t i v e ) . N o w i t i s e v i d e n t t h a t s u c h a s i m p l e t r a n s l a t i o n o f X v i s i b l e t o b o t h K a n d L c a n c a u s e a t o i n c r e a s e o r d e c r e a s e b e c a u s e o f t h i s " p u s h - p u l l " a c t i o n o f t h e t w o b a r g a i n e r s . T h u s t h e r e i s n o d e f i n i t e d i r e c t i o n i n w h i c h a m o v e s b y a s i m p l e t r a n s l a t i o n o f X . T o g u e s s h o w e v e r t h a t t h e e f f e c t s o f a n y s o r t o f m o v e m e n t o f X v i s i b l e t o b o t h K a n d L h a s n o p r e d i c t a b l e e f f e c t o n a i s n o t j u s t i f i e d 5 7 e i t h e r . S o m e s i m p l e e x p a n s i o n s o f X d o g i v e r i s e t o a p r e d i c t a b l e m o v e m e n t i n a . F i r s t c o n s i d e r a m u l t i p l i c a t i v e s h i f t o f X a r o u n d t h e o r i g i n , i . e . X ( h ) = ( l + h ) X . T o s t u d y t h e r e s p o n s e s o f a d u e t o s u c h a m a g n i f i c a t i o n ( a l t e r a t i o n ) w e u s e a m e t h o d s i m i l a r t o w h a t T o b i n [ 1 9 5 7 ] u s e d i n a d i f f e r e n t c o n t e x t . L e t a ( h ) b e t h e b a r g a i n i n g s o l u t i o n t o t h e e x c h a n g e p r o b l e m w h e n t h e r a n d o m v a r i a b l e , b y w h i c h t h e v e n t u r e s a r e g e n e r a t e d , i s X ( h ) . T h e n , E R U K ( a ( h ) X ( h ) ) E L { X ( h ) U | ( - a ( h ) X ( h ) ) } + E ^ U ^ ( - a ( h ) X ( h ) ) E v { X ( h ) U l ( a ( h ) X ( h ) ) } = 0 ( 3 . 3 . 8 ) a n d E U ( a X ) E { X U * (—otX) } + E U ( - a X ) E 1 7 { X U ' ( a X ) } = 0 J s - J s . L L L L K K ( 3 . 3 . 9 ) w h e r e X ( h ) = ( l + h ) X . M u l t i p l y i n g ( 3 . 3 . 9 ) b y ( 1 + h ) w e h a v e , W l + h ( 1 + h ) X ) E j X d ^ U ^ - f ^ - d ^ X ) } + a ( l + H X ) E ^ d C l + h J X u ' O ( l + h ) X - , ) } = 0 1 + h S i n c e X ( h ) = X l + h ) X i n ( 3 . 3 . 8 ) n o w i t i s c l e a r t h a t a ( h ) = a / ( l + h ) s a t i s f i e s t h e e q u a t i o n ( 3 . 3 . 9 ) . H e n c e , 5 8 P r o p o s i t i o n 3 . 3 . 4 : I f_ a i s t h e b a r g a i n i n g s o l u t i o n w h e n t h e r a n d o m  v a r i a b l e g e n e r a t i n g t h e v e n t u r e s i s X , t h e n a/(1+h) i s t h e b a r g a i n i n g  s o l u t i o n w h e n t h e r a n d o m v a r i a b l e i s X ( l + h ) . N o w w e a r e i n a p o s i t i o n t o c o n s i d e r a m u l t i p l i c a t i v e s h i f t a b o u t a n a r b i t r a r y c e n t e r X * . T h e n X ( h ) c a n b e p a r t i t i o n e d a s , X ( h ) = X * + ( 1 + h ) ( X - X * ) = ( l + h ) X - h X * H e n c e t h i s m u l t i p l i c a t i v e s h i f t i s a s u m o f a m u l t i p l i c a t i v e s h i f t a b o u t t h e o r i g i n a n d a d o w n w a r d a d d i t i v e s h i f t h X * . W e - h a v e s e e n t h a t t h e f i r s t m e m b e r o f t h i s p a r t i t i o n w i l l i n d u c e a d e f i n i t e d i r e c t i o n o f m o v e m e n t i n a a n d t h e s e c o n d m e m b e r w i l l i n d u c e a n i n d e f i n i t e d i r e c t i o n o f m o v e m e n t i n i t . O n e c a n n o t , t h e r e f o r e , c o m e t o a d e f i n i t e g e n e r a l c o n c l u s i o n . A l l w e c a n s a y i s t h a t i f t h e a d d i t i v e s h i f t h X * h a s a p o s i t i v e e f f e c t i n a , t h e n a w i l l d e c r e a s e a t a g r e a t e r p r o p o r t i o n t h a n t h e p r o p o r t i o n a t w h i c h X i n c r e a s e s , a n d o t h e r w i s e i t r e i n f o r c e s t h e e f f e c t o f t h e m u l t i p l i c a t i v e s h i f t a b o u t t h e o r i g i n . 5 9 CHAPTER 4 BARGAINING PROBLEM : SOLUTIONS UNDER INCOMPLETE INFORMATION 4.1 INTRODUCTION 4.1.1 BACKGROUND "Bargaining" l a b e l s a v a r i e t y of processes in v o l v i n g 2 or more (here 2) pa r t i e s c a l l e d bargainers which seek to reach agreement about the exchange of quantities of goods, union-management wage negotiation being a t y p i c a l example. U n t i l the 1930s, only the range on the Edgeworth contract curve i n which the settlement takes place had been s p e c i f i e d . Then at the beginning of 1930s attempts were made by Zeuthen [1930] and Hicks [1932] to model union-management negotiations and to specify the points of agreement i n terms of the parameters of the model. Since then a great deal of theory based on a v a r i e t y of models has been developed, inc l u d i n g the work of Nash [1950] and [1953], Pen [1952], Bishop [1963] and [1964], Folds [1964], Cross [1965] and [1969]. This theory o f f e r s a v a r i e t y of determinate solutions for the bargaining problem, and embraces, as p a r t i c u l a r cases, such basic economic problems as b i l a t e r a l monopoly, duopoly, etc. (For applications of the bargaining problem c f . de Menil [1971]). E s s e n t i a l l y there have been three approaches to the bargaining problem. F i r s t l y , there are theories based on von Neuman and Morgenstern's theory of games and include those of Bishop [ 1 9 6 3 ] , K a l a i and Smorodinsky [1975], Nash [1950], and R a i f f a [1953] (and Luce and R a i f f a [1957]). Secondly there 6 0 i s t h e w o r k o f a u t h o r s s u c h a s P e n [ 1 9 5 2 ] , a n d S a r a y d a r [ 1 9 6 5 ] w h o h a v e a p p r o a c h e d t h e b a r g a i n i n g p r o b l e m u s i n g m o d e l s b a s e d o n t h e m o d e l f o r e m p l o y e r - u n i o n w a g e n e g o t i a t i o n s s u g g e s t e d b y Z e u t h e n [ 1 9 3 0 ] . F i n a l l y i n t h e t h i r d c l a s s o f b a r g a i n i n g m o d e l s , a r e t h o s e o f B i s h o p [ 1 9 6 4 ] , C r o s s [ 1 9 6 5 ] , a n d F o l d e s [ 1 9 6 4 ] , w h i c h h a v e b e e n d e v e l o p e d f r o m t h e e l e m e n t s s k e t c h e d r o u g h l y b y H i c k s [ 1 9 3 2 ] i n w h i c h t h e " t i m e " f a c t o r ( d u r a t i o n o f b a r g a i n i n g ) i s t a k e n i n t o a c c o u n t . O f t h e s e t h e o r i e s p e r h a p s t h a t o f N a s h i s t h e m o s t p r e c i s e . H o w e v e r , i t i s m o r e d e s c r i p t i v e t h a n i t i s a p p l i c a b l e , s i n c e l i t e r a l l y i t a p p l i e s o n l y t o a b a r g a i n i n g s i t u a t i o n i n v o l v i n g t w o r a t i o n a l b a r g a i n e r s w i t h p e r f e c t k n o w l e d g e a b o u t t h e b a r g a i n i n g s i t u a t i o n . F u r t h e r m o r e , h i s t h e o r y p r o v i d e s o n l y a c h a r a c t e r i z a t i o n o f t h e p o i n t o f s e t t l e m e n t ( a s o l u t i o n c o n c e p t ) i n a t w o - p l a y e r b a r g a i n i n g s i t u a t i o n w i t h o u t a n y i n s i g h t s a b o u t t h e b a r g a i n i n g p r o c e s s i t s e l f . I n t e r e s t i n g l y e n o u g h , H a r s a n y i [ 1 9 5 6 ] h a s s h o w n t h a t Z e u t h e n ' s b a r g a i n i n g t h e o r y ( H a r s a n y ' s v e r s i o n b e i n g i n t e r m s o f m o d e r n u t i l i t y t h e o r y ) i m p l i e s t h e s a m e o u t c o m e a s N a s h ' s t h e o r y , e v e n t h o u g h t h e i r a p p r o a c h e s a r e d i f f e r e n t . I t s h o u l d a l s o b e p o i n t e d o u t t h a t m o s t o f t h e o t h e r t h e o r i e s i n c l u d i n g t h o s e o f C r o s s [ 1 9 6 5 ] , B i s h o p [ 1 9 6 4 ] , a n d F o l d e s [ 1 9 6 4 ] i m p l y s o l u t i o n s t h a t a r e e i t h e r i d e n t i c a l w i t h N a s h ' s s o l u t i o n o r a g r e e w i t h N a s h ' s s o l u t i o n i n s p e c i a l c a s e s . I n p a r t i c u l a r m o s t o f t h e t h e o r i e s t h a t t a k e t i m e a s p e c t s i n t o a c c o u n t c o n c l u d e ( a t l e a s t i n s p e c i a l c a s e s ) t h a t t h e r r 2 1 a g r e e m e n t i s r e a c h e d w h e n i s a m a x i m u m w h e r e a s N a s h ' s t h e o r y i m p l i e s t h a t t h i s o c c u r s i n a s i t u a t i o n w i t h t w o s i m i l i a r p l a y e r s w h e n i - s a m a x i m u m , d e n o t i n g u t i l i t i e s a n d r ^ t h e d i s c o u n t r a t e ( c o n s t a n t ) p e r u n i t t i m e o f t h e p l a y e r i . 61 O u r a n a l y s i s o f t w o - p e r s o n b a r g a i n i n g u n d e r i n c o m p l e t e i n f o r m a t i o n r e t a i n s s o m e o f t h e s p i r i t o f N a s h ' s t h e o r y w h i c h w e h a v e b r i e f l y o u t l i n e d i n S e c t i o n 3 . 3 . 1 . 4 . 1 . 2 B A R G A I N I N G U N D E R I N C O M P L E T E I N F O R M A T I O N M o s t o f t h e w o r k o n b a r g a i n i n g d e a l s w i t h s i t u a t i o n s w h e r e t h e p l a y e r s h a v e f u l l k n o w l e d g e a b o u t p a y o f f s o f t h e g a m e , e a c h o t h e r s ' u t i l i t i e s , e t c . o r b e c a u s e o f j u d i c i o u s a s s u m p t i o n s ( c f . C r o s s [ 1 9 6 5 ] ) t h e p l a y e r s ' k n o w l e d g e a b o u t s u c h f a c t o r s b e c o m e s i r r e l e v a n t . H a r s a n y i a n d S e l t e n [ 1 9 7 2 ] d e f i n e a n d s o l v e t h e b a r g a i n i n g p r o b l e m w h e n t h e p l a y e r s ' i n f o r m a t i o n i s i n c o m p l e t e a n d t h e i r w o r k t h e r e f o r e f a l l s o u t s i d e o f t h e c a t e g o r i e s j u s t d e s c r i b e d . H a r s a n y i a n d S e l t e n [ 1 9 7 2 ] f o r m u l a t e a t w o - p e r s o n b a r g a i n i n g m o d e l u n d e r i n c o m p l e t e i n f o r m a t i o n , w h i c h t a k e s i n t o a c c o u n t , r a t h e r t h a n m a k e s i r r e l e v a n t i m p o r t a n t a s p e c t s o f s u c h a s i t u a t i o n . T h e y c o n s i d e r t w o p l a y e r s e a c h o f w h o m i s o n e o f a f i n i t e n u m b e r o f p o s s i b l e s p e c i f i e d t y p e s ; t h e p l a y e r ' s a c t u a l t y p e d e p e n d s o n t h e p a y o f f s t r u c t u r e o f t h e g a m e , h i s u t i l i t y , e t c . a n d i s u n k n o w n t o h i s a d v e r s a r y . I t i s a s s u m e d t h a t e a c h p l a y e r h a s a s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n o v e r h i s o p p o n e n t ' s p o s s i b l e t y p e s . T h e e x i s t e n c e o f a p r o b a b i l i t y m a t r i x t o r e p r e s e n t t h e p l a y e r s ' j o i n t s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s i s p o s t u l a t e d a n d a s s u m e d k n o w n t o b o t h p l a y e r s . H a r s a n y i a n d l S e l t e r i . ' r a n a l y z e . . t h e ^ b a r g a i n i n g m o d e l ^ ' " b y g e x p l o i t i n g a n N - p l a y e r g a m e a n a l o g y , t h e t y p e s f o r m a l l y r e p r e s e n t i n g s u b p l a y e r s i n t h e a n a l o g u e . O u t o f t h e s e t o f a l l p o s s i b l e N - t u p l e s o f e x p e c t e d p a y o f f s g e n e r a t e d b y v a r y i n g t h e b a r g a i n i n g s t r a t e g i e s o f t h e p l a y e r s , " s t r i c t e q u i l i b r i u m " p o i n t s a r e c h o s e n a n d f r o m t h e s e t h e " e q u i l i b r i u m s e t " i s g e n e r a t e d b y t a k i n g m i x t u r e s o f s t r i c t e q u i l i b r i u m p o i n t s . F i n a l l y a n a x i o m a t i c t h e o r y i s 6 2 d e v e l o p e d t o s e l e c t a u n i q u e e q u i l i b r i u m p o i n t , t h e " g e n e r a l i z e d N a s h s o l u t i o n " . T h i s p o i n t i s f o u n d b y m a x i m i z i n g a f u n c t i o n a l c a l l e d t h e g e n e r a l i z e d N a s h p r o d u c t . H a r s a n y i a n d S e l t e n ' s w o r k i s w i t h o u t d o u b t , a p r o f o u n d a n d h i g h l y o r i g i n a l c o n t r i b u t i o n t o t h e m a t h e m a t i c a l t h e o r y o f b a r g a i n i n g . T h e i r a x i o m s w i l l n o t a l l b e s a t i s f i e d t y p i c a l l y a n d w e s h a l l n o w i n d i c a t e w h y a n d l a t e r a d d r e s s o u r s e l v e s m o r e p o s i t i v e l y t o t h e s e d i f f i c u l t i e s . F i r s t o f a l l H a r s a n y i a n d S e l t e n p u r s u e t o o f a r t h e N - p l a y e r a n a l o g y o f t h e t w o - p e r s o n b a r g a i n i n g g a m e . I t i s m i s l e a d i n g t o a d a p t f o r t h e t w o - p e r s o n b a r g a i n i n g g a m e t h e a x i o m s t h a t a r e m e a n t f o r t h e N - p l a y e r a n a l o g y . F o r i n s t a n c e , f o r t h e t w o - p e r s o n b a r g a i n i n g g a m e ( i ) a f f i n e i n v a r i a n c e ( H a r s a n y i a n d S e l t e n ' s A x i o m 5 ) d o e s n o t h o l d ; ( i i ) t h e s e t s E V + J U , < 2 a r e n o t a t t a i n a b l e , V b e i n g p l a y e r l ' s u t i l i t y w h e n h e i s t y p e i a n d V_. b e i n g p l a y e r 2 ' s u t i l i t y w h e n h e i s t y p e j . F u r t h e r m o r e i t i s u n r e a s o n a b l e t o a s s u m e t h e p l a y e r s ' j o i n t s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n i s k n o w n t o b o t h p l a y e r s . A l t h o u g h H a r s a n y i a n d S e l t e n d o n o t m a k e i t c l e a r , t h i s a s s u m p t i o n i s n e c e s s a r y t o g i v e m e a n i n g t o t h e i r e q u i l i b r i u m s o l u t i o n . A p p a r e n t l y t h e y a l s o a s s u m e ? e i t h e r t h a t t h e p l a y e r s ' j o i n t s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n i s c o n s t a n t d u r i n g t h e b a r g a i n i n g , o r t h a t i t i s t h e l i m i t i n g d i s t r i b u t i o n a t t h e t e r m i n a t i o n o f b a r g a i n i n g . T h e l a t e r i s t h e m o r e r e a l i s t i c a l t e r n a t i v e . E v e n s o , d i f f i c u l t i e s a r i s e b e c a u s e t h i s l i m i t i n g d i s t r i b u t i o n w o u l d o b v i o u s l y d e p e n d o n t h e b a r g a i n i n g p r o c e s s s o c o u l d n o t b e u s e d a s H a r s a n y i a n d S e l t e n d o i n t h e i r g e n e r a l i z e d N a s h p r o d u c t t o d e t e r m i n e a n e q u i l i b r i u m s t r a t e g y , a p r i o r w i t h w h i c h t o c a r r y o u t t h e n e g o t i a t i o n s . 6 3 B y t h e s e c o m m e n t s w e a r e e s s e n t i a l l y d e n y i n g t h e v a l i d i t y o f t h e g a m e t h e o r e t i c a l a p p r o a c h t o t h e s o l u t i o n o f t h e b a r g a i n i n g p r o b l e m a t a p r a c t i c a l l e v e l . W e a r e n o t , o f c o u r s e , d e n y i n g t h i s a p p r o a c h h a s g r e a t d e s c r i p t i v e v a l u e a n d m a y w e l l a p p l y i n c e r t a i n p a r t i c u l a r l y s i m p l e c a s e s . I t i s p l a u s i b l e t h a t t h e s o l u t i o n t o t h e b a r g a i n i n g p r o b l e m w i t h i n c o m p l e t e i n f o r m a t i o n i s n o t a s i n g l e p o i n t a s t h e g a m e t h e o r e t i c a l a p p r o a c h s u g g e s t s , b u t r a t h e r , 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 v e r t h e s p a c e o f p o s s i b l e s o l u t i o n s . I n t h e s e q u e l , a c o n c e p t u a l d e c o m p o s i t i o n o f t h e b a r g a i n i n g p r o b l e m u n d e r i n c o m p l e t e i n f o r m a t i o n i n t o c o m p o n e n t p r o b l e m s i s s u g g e s t e d . A n a t t e m p t w i l l t h e n b e m a d e t o a n a l y z e o n e o f t h e c o m p o n e n t s o f t h e p r o b l e m , l e a v i n g t h e r e s t f o r p o s s i b l e f u t u r e s t u d i e s . 4 . 2 T H E P R O B L E M 4 . 2 . 1 T H E N A T U R E O F T H E P R O B L E M C o n s i d e r a s i t u a t i o n w h e r e t w o b a r g a i n e r s e n g a g e i n b a r g a i n i n g w i t h i n c o m p l e t e i n f o r m a t i o n a b o u t p a y o f f s , e a c h o t h e r s u t i l i t i e s , e t c . S u c h a s i t u a t i o n i s d e s c r i b e d b y H a r s a n y i a n d S e l t e n [ 1 9 7 2 ] . T h e u n c e r t a i n t i e s i n v o l v e d m a y b e c l a s s i f i e d i n t o t w o b r o a d c a t e g o r i e s . T h e f i r s t c a t e g o r y i n c l u d e s s t o c h a s t i c u n c e r t a i n t i e s q u a n t i f i a b l e i n t e r m s o f b a r g a i n e r s ' s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s a b o u t p a s t , p r e s e n t , o r f u t u r e s t a t e s o f n a t u r e . A c o n t r i v e d e x a m p l e m a y c l a r i f y t h e n a t u r e o f t h e d i s t r i b u t i o n i n q u e s t i o n . T w o c o u n t r i e s , A a n d B , n e g o t i a t e a " d e a l " g i v i n g B a n a m o u n t o f o i l , t o b e w o r k e d o u t , i n r e t u r n f o r a c e r t a i n n u m b e r o f a u t o -m o b i l e s a t a f i x e d p r i c e , o v e r a p e r i o d o f 1 0 y e a r s , t h e n u m b e r a n d p r i c e a l s o t o b e w o r k e d o u t . H e r e t h e u t i l i t y o f a n y a g r e e m e n t w i l l b e d e t e r m i n e d 6 4 b y f u t u r e i n f l a t i o n a r y r a t e s a n d t h e t w o b a r g a i n e r s A a n d B h a v e s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s o n t h e s e . T h e s e c o n d c a t e g o r y i n c l u d e s t h e u n c e r t a i n t i e s a b o u t t h e o p p o n e n t ' s t a s t e s a n d p r e f e r e n c e s o r a n y o t h e r f a c t o r s r e l a t e d t o h i s u t i l i t y p a y o f f s . T h e b a r g a i n i n g p r o c e s s c o n c e i v e d h e r e i s a s f o l l o w s : E a c h b a r g a i n e r m a k e s a d e m a n d ( o r o f f e r ) , i n t e r m s o f u t i l i t y u n i t s o r p h y s i c a l p a y o f f u n i t s , s i m u l t a n e o u s l y o r o n e a f t e r t h e o t h e r . I f t h e r e a r e f e a s i b l e s o l u t i o n s s a t i s f y i n g b o t h b a r g a i n e r s ' d e m a n d s t h e y c o m e t o a n a g r e e m e n t a t o n e o f t h e s e . O t h e r w i s e t h e y m a y c o n c e d e a n d m a k e n e w d e m a n d s , p e r h a p s w i t h t h e i r b e l i e f s a b o u t u n c e r t a i n t i e s o f t h e b a r g a i n i n g s i t u a t i o n u p d a t e d . S h o u l d n o t a t l e a s t o n e o f t h e m c o n c e d e , c o n f l i c t r e s u l t s a n d p a y o f f s i n t h i s s i t u a t i o n ( g i v e n b y t h r e a t s , n o d e a l , e t c . ) w i l l b e e n f o r c e d . R a t i o n a l b a r g a i n e r s m a k e d e m a n d s a t e v e r y s t a g e o f t h e b a r g a i n i n g p r o c e s s b a s e d o n s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s ( p r i o r s ) o n e a c h o t h e r ' s u t i l i t y p a y o f f s o r o v e r t h e r a n g e o f p o s s i b i l i t i e s f o r u n k n o w n p a r a -m e t e r s t h a t d e t e r m i n e t h e s e p a y o f f s . I n g e n e r a l , b a r g a i n e r s w i l l m a k e u s e o f t h e i n f o r m a t i o n g a i n e d f r o m t h e b a r g a i n i n g p r o c e s s t o s e q u e n t i a l l y u p d a t e t h e s e p r i o r s . A c o m p l e t e b a r g a i n i n g s c h e m e w o u l d b a s i c a l l y c o n s i s t o f t h e f o l l o w i n g c o m p o n e n t s : ( i ) h o w t h e s u b j e c t i v e p r o b a b i l i t i e s c h a n g e f r o m s t a g e n t o n + 1 ( i i ) c r i t e r i a f o r f i n d i n g o p t i m a l o f f e r s a t s t a g e n + 1 ( i i i ) c h a r a c t e r i z i n g t h e s a t i s f a c t o r y a g r e e m e n t s a t s t a g e n + 1 . I n t h i s s t u d y w e w i l l n o t s o l v e p r o b l e m s ( i ) a n d ( i i ) a n d i n p a r t i c u l a r n o t i n t r o d u c e a m e c h a n i s m f o r u p d a t i n g t h e p r i o r s ; t h e m a i n o b j e c t i v e o f o u r s t u d y i s t o i n v e s t i g a t e t h e c r i t e r i o n u s e d b y a r a t i o n a l b a r g a i n e r t o f i n d a " f a i r d e m a n d " ( i n t h e s e n s e o f N a s h , a p p r o p r i a t e l y 6 5 g e n e r a l i z e d ) , a s a b a s e - l i n e o r r e f e r e n c e o n w h i c h t o b a s e h i s a c t u a l d e m a n d a t s t a g e n + 1 . I f t h e b a r g a i n e r i n q u e s t i o n m u s t d e c i d e w h e t h e r t o a c c e p t a n o f f e r m a d e b y t h e o p p o n e n t , h e m a y d o s o i f i t g i v e s h i m a t l e a s t h i s f a i r d e m a n d . O n t h e o t h e r h a n d i f a b a r g a i n e r i s t o m a k e a d e m a n d h e w o u l d d e m a n d a n a m o u n t a t l e a s t a s d e s i r a b l e a s h i s f a i r d e m a n d . T h e m e a n i n g o f ' f a i r d e m a n d ' w i l l b e m a d e p r e c i s e b e l o w . A n d a l t h o u g h i t i s c o n c e i v e d o f i n t h e t e r m s g i v e n a b o v e , t h e d e f i n i t i o n b e l o w w i l l r e v e a l a g a p i n o u r a n a l y s i s , v i z . t h e r e i s n o m o r e t h a n a w e a k l i n k b e t w e e n t h e c o n c e p t u a l a n d t e c h n i c a l m e a n i n g s o f ' f a i r d e m a n d ' g i v e n h e r e . N e v e r t h e -l e s s , t h e r e s u l t s a r e v e r y s u g g e s t i v e a n d i t i s h o p e d t h a t a c c e p t a b l e a x i o m s w i l l b e f o u n d t o s t r e n g t h e n t h e l i n k i n q u e s t i o n . 4 . 2 . 2 F O R M U L A T I O N O F T H E B A R G A I N I N G M O D E L S u p p o s e e a c h b a r g a i n e r ' s u t i l i t y i s a f u n c t i o n o f p h y s i c a l u n i t s s o m e o f w h i c h a r e v a r i a b l e s a y ) ; t h e v a l u e o f x . i s t o «. 1 n l b e f i x e d b y n e g o t i a t i o n s . T h e u t i l i t i e s o f B a r g a i n e r 1 a n d 2 e v a l u a t e d a t a g i v e n x w i l l b e d e n o t e d b y U ( x ) a n d V ( x ) , r e s p e c t i v e l y . T h e p a y o f f s , t h e t w o b a r g a i n e r s w o u l d r e c e i v e i n c a s e o f c o n f l i c t a r e d e n o t e d b y c a n d d , r e s p e c t i v e l y . F o r a g i v e n x B a r g a i n e r 1 h a s a p r i o r d i s t r i b u t i o n T T ^ d e r i v e d f r o m h i s s u b j e c t i v e b e l i e f s , o n t h e r a n g e o f ( V ( x ) , d ) a n d B a r g a i n e r 2 h a s a p r i o r d i s t r i b u t i o n T T ^ o n t h e r a n g e o f ( U ( x ) , c ) . T h e s e p r i o r d i s t r i b u t i o n s m a y h a v e b e e n i n d u c e d b y t h e s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s o n t h e u n k n o w n p a r a m e t e r s l a b e l l i n g t h e u t i l i t y p a y o f f s . 6 6 U s u a l l y b a r g a i n e r s d o n o t b a r g a i n o v e r u t i l i t i e s ; r a t h e r t h e y b a r g a i n i n t e r m s o f x . E a c h b a r g a i n e r m a y d e m a n d o n e x a t a t i m e o r o n e o u t o f s e v e r a l x ' s y i e l d i n g h i m t h e s a m e u t i l i t y . T h e p r o c e s s o f s e a r c h i n g f o r a n x s a t i s f y i n g b o t h b a r g a i n e r s ' d e m a n d s i n v o l v e s a m e c h a n i s m w h e r e b y b a r g a i n i n g i s c a r r i e d o u t . T h e f o l l o w i n g a r e e x a m p l e s o f t w o s u c h m e c h a n i s m s : ( 1 ) E a c h b a r g a i n e r s u b m i t s t o a n a r b i t r a t o r , a l i s t o f e q u i v a l e n t d e m a n d s , { x } . T h e a r b i t r a t o r s e a r c h s t h e s e l i s t s f o r a c o m m o n v a l u e x . I f s u c h a n x i s f o u n d h e a n n o u n c e s i t a s t h e s o l u t i o n . O t h e r -w i s e b a r g a i n e r s m a k e u p n e w l i s t s o f e q u i v a l e n t d e m a n d s o r , a l t e r n a t i v e l y , e n t e r t h e c o n f l i c t s i t u a t i o n . A s e a r c h f o r a p e a c e a g r e e m e n t b e t w e e n t w o c o u n t r i e s w h e r e a n a r b i t r a t o r t a l k s t o d e l e g a t e s f r o m t h e t w o c o u n t r i e s s e p a r a t e l y i s o f t h i s k i n d . ( 2 ) B a r g a i n e r 1 f i r s t a n n o u n c e s a l l x ' s a c c e p t a b l e t o h i m a t t h a t m o m e n t a n d B a r g a i n e r 2 d e t e r m i n e s i f a n y o f t h e s e a t l e a s t f u l f i l s h i s e x p e c t a t i o n s . I f s o , h e a c c e p t s t h a t p a r t i c u l a r o f f e r x . O t h e r w i s e h e m a y c o u n t e r b y a n n o u n c i n g o f f e r s a c c e p t a b l e t o h i m s e l f o r a l t e r n a t i v e l y t h a t h e p r e f e r s c o n f l i c t l e a v i n g B a r g a i n e r 1 t o m a k e a c o n c e s s i o n o r e l s e p r o d u c e c o n f l i c t . T h e b a r g a i n i n g c o n t i n u e s s e q u e n t i a l l y i n t h i s w a y . E m p l o y e r - u n i o n w a g e n e g o t i a t i o n s a r e s o m e t i m e s c a r r i e d o u t t h i s w a y . A c r i t e r i o n i s r e q u i r e d b y w h i c h a b a r g a i n e r c a n d e t e r m i n e a l i s t o f f a i r d e m a n d s , i . e . a b a s e l i n e f o r h i s d e m a n d s . R e c a l l t h a t i n b a r g a i n i n g w i t h c o m p l e t e i n f o r m a t i o n a f a i r d e m a n d f o r e i t h e r b a r g a i n e r w a s a n y o n e w h i c h m a x i m i z e s t h e N a s h u t i l i t y p r o d u c t . I n o r d e r t o e s t a b l i s h s o m e t h i n g a n a l o g o u s t o t h i s i n c a s e o f i n c o m p l e t e i n f o r m a t i o n l e t u s v i e w t h e p r o b l e m f r o m B a r g a i n e r l ' s s t a n d p o i n t . F o r a n y g i v e n x , B a r g a i n e r 1 r e g a r d s h i s o p p o n e n t ' s u t i l i t y V ( x ) , o r s i m p l y V , a n d c o n f l i c t p a y o f f d , a s r a n d o m v a r i a b l e s , s a y V a n d d d e f i n e d o n t h e s a m e p r o b a b i l i t y s p a c e , s a y ( f t , F , P ) ; w e f t i s a p o s s i b l e v a l u e o f t h e p a r a m e t e r s ( u n k n o w n t o B a r g a i n e r 1 ) d e f i n i n g V , a n d P r e p r e s e n t s B a r g a i n e r l ' s s u b j e c t i v e 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 t h e e v e n t s i n a o - f i e l d F o f s u b s e t s o f ft. N o t e t h a t U , c , V a n d d a r e a l l d e t e r m i n e d b y t h e a p p r o p r i a t e b a r g a i n e r ' s u t i l i t y f u n c t i o n . G i v e n x , d e n o t e b y <j>(w; x ) t h e e q u i v a l e n c e - c l a s s o f u t i l i t i e s w i t h l a b e l w , g e n e r a t e d b y t h e e q u i v a l e n t u t i l i t y f u n c t i o n s o f B a r g a i n e r 2 , t w o u t i l i t y f u n c t i o n s b e i n g c a l l e d e q u i v a l e n t i f o n e i s a p o s i t i v e a f f i n e t r a n s f o r m a t i o n o f t h e o t h e r . A d i f f i c u l t y c o n f r o n t s t h e B a r g a i n e r 1 i n c h o o s i n g cf> ( f o r V ) o u t o f t h e a v a i l a b l e a l t e r n a t i v e s . T o a v o i d t h i s d i f f i c u l t y w e a s s u m e t h e " i n t e r p e r s o n a l c o m p a r a b i l i t y " " ' " o f t h e m e m b e r s o f {(|>((jj; x ) : w e f t } . T h i s r e q u i r e m e n t d o e s n o t s e e m u n r e a s o n a b l e a f t e r a l l , <j> i s B a r g a i n e r l ' s o w n c h o i c e a n d y i e l d s h i s l a b e l s f o r B a r g a i n e r 2 ' s p o s s i b l e u t i l i t y c l a s s e s t o s e r v e w e l l i n t h e i r r o l e a s l a b e l s t h e {<t>(w; x ) : w e f t } s h o u l d b e c o m p a r a b l e , s a y h a v e t h e s a m e m a x i m u m , f o r e x a m p l e . S o m a k i n g t h e m c o m p a r a b l e d o e s n o t s e e m n e a r l y a s d i f f i c u l t h e r e a s w o u l d b e t h e c a s e i n t h e s i t u a t i o n w h e r e t h e s e l a b e l s a c t u a l l y r e p r e s e n t e d t h e u t i l i t y f u n c t i o n s o f d i f f e r e n t i n d i v i d u a l s . W e n o w i n t r o d u c e a s i m p l i f y i n g a s s u m p t i o n , n a m e l y t h a t t h e v a l u e o f a d e m a n d x i s d e t e r m i n e d b y U ( x ) - c a n d V ( x ) - d , i n t h e u t i l i t y i n c r e m e n t s w i t h r e s p e c t t o t h e c o n f l i c t p o i n t s . T h e n i t f o l l o w s t h a t w i t h o u t l o s s o f g e n e r a l i t y w e m a y t a k e c = 0 = d ( w ) f o r a l l w e f t ( V w = ~ ^ a n d o n l y I s d e t e r m i n e d b y B a r g a i n e r 2 ' s c a r d i n a l u t i l i t y f u n c t i o n ) . 1 . F o r a g e n e r a l d i s c u s s i o n o f t h e " i n t e r p e r s o n a l c o m p a r a b i l i t y " o f u t i l i t i e s s e e L u c e a n d R i f f a [ 1 9 5 7 ] . 6 8 L e t TT d e n o t e t h e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n o f t h e r a n d o m v a r i a b l e V ( n e w ) i n d u c e d f r o m P . A s a n e x t e n s i o n o f t h e c a s e o f c o m p l e t e i n f o r m a t i o n t o t h a t o f i n c o m p l e t e i n f o r m a t i o n a s s u m e t h a t B a r g a i n e r 1 d e t e r m i n e s w h a t w e w i l l c a l l h i s " f a i r " d e m a n d s b y m a x i m i z i n g a c e r t a i n f u n c t i o n , t o b e c a l l e d t h e " G e n e r a l i z e d N a s h F u n c t i o n " ( G N F ) , o f t h e u t i l i t y o f t h e d e m a n d t o B a r g a i n e r 1 , t h e u t i l i t i e s o f t h e d e m a n d d e p e n d i n g o n h i s t y p e o f B a r g a i n e r 2 , a n d P . C o n s i d e r a p o i n t t h a t g i v e s B a r g a i n e r 2 a p a y o f f V ( c o ) i f co i s t r u e a n d g i v e s B a r g a i n e r 1 a p a y o f f U > 0 . B a r g a i n e r l ' s u t i l i t y o f t h i s p o i n t , t h e r e f o r e , i s U i f V ( c o ) > 0 a n d i s 0 o t h e r -w i s e b e c a u s e s u r e l y B a r g a i n e r 2 w o u l d p r e f e r t h e c o n f l i c t p o i n t t o t h i s p o i n t i f h i s u t i l i t y i s n e g a t i v e . H e n c e B a r g a i n e r l ' s u t i l i t y o f s u c h a p o i n t i s U P ( £ T ) + 0 . [ 1 - P ( £ T ) ] , w h e r e Q" = { c o/V ( co ) > 0 } . A l s o i f V ( c o ) i s n e g a t i v e , i t s m a g n i t u d e c a n n o t b e r e l e v a n t b e c a u s e a s e t t l e m e n t w i l l n e v e r t a k e p l a c e a t s u c h a p o i n t , a n d t h u s a l l n e g a t i v e V ( c o ) w i l l b e c a r d i n a l l y e q u i v a l e n t t o 0 . H e n c e w e a s s u m e G N F c a n b e w r i t t e n a s : T X ( U ; ( V ( u ) ) , co e Q}; P ) i f U > 0 , V ( u ) > 0 f o r a l l co G N F = < T 2 ( U P ; { V + ( c o ) , to e fi}; P ) i f U > 0 , 8 3 . 0 ' ^ o t h e r w i s e . w h e r e , i t s h o u l d b e r e m e m b e r e d , U a n d V ( c o ) f o r e a c h co a r e f u n c t i o n s o f d e m a n d . N o w n o t i n g t h a t t h e f u n c t i o n s a n d 1^ a r e d e f i n e d o n m u t u a l l y e x c l u s i v e d o m a i n s a n d l e t t i n g U = U P ( f t ' ) , G N F c a n s i m p l y b e w r i t t e n a s : 6 9 G N F = T ( U + ; { V + ( c o ) , a) e ft}; P ) w i t h T ( 0 ; { V + ( t o ) , u e ft};P) = T ( U + ; { V + ( t o ) H 0 f o r a l l u e ft}; P ) = 0 . T h e f u n c t i o n T t h u s d e f i n e d i s i n t e n d e d a s a n a n a l o g u e o f t h e N a s h p r o d u c t a n d a s s u c h i t m e a s u r e s t h e j o i n t p r e f e r e n c e o f t h e t w o b a r g a i n e r s . H e n c e T m u s t a s s i g n l a r g e r v a l u e s t o p o i n t s p r e f e r r e d b y b o t h b a r g a i n e r s . 4 . 3 A N A X I O M A T I C A P P R O A C H TO T H E D E R I V A T I O N  O F F A I R D E M A N D S We h a v e a s s u m e d t h a t i n o r d e r t o f i n d a " f a i r d e m a n d B a r g a i n e r 1 m a x i m i z e s t h e f u n c t i o n T ( U + ; { V + ( w ) , to e ft}; P ) . T h e t e r m " f a i r " i s u s e d h e r e m a i n l y b e c a u s e T w i l l s e r v e a s a g e n e r a l i z a t i o n o f t h e N a s h p r o d u c t d e v e l o p e d f o r f a i r b a r g a i n s u n d e r c o m p l e t e i n f o r m a t i o n . F u r t h e r j u s t i f i c a t i o n s f o r t h i s d e s i g n a t i o n w i l l b e g i v e n l a t e r i n t h i s s e c t i o n . 4 . 3 . 1 T H E A X I O M S We w i l l d e r i v e a p r e c i s e f o r m f o r T f r o m t h e f o l l o w i n g t w o a x i o m s p l u s a t h i r d t o b e s t a t e d l a t e r . A x i o m 1 : - A b a r g a i n e r ' s c l a s s o f f a i r d e m a n d s d o e s n o t v a r y u n d e r p o s i t i v e s c a l a r t r a n s f o r m a t i o n o f t h e o p o n e n t ' s o r h i s o w n u t i l i t y f u n c t i o n . A x i o m 2 : - I n b a r g a i n i n g w i t h c o m p l e t e i n f o r m a t i o n , f a i r d e m a n d s a r e f o u n d b y m a x i m i z i n g t h e N a s h p r o d u c t . 7 0 A x i o m 1 i s s i m p l y a g e n e r a l i z a t i o n o f N a s h ' s a x i o m o f l i n e a r i n v a r i a n c e a n d s t a t e s t h a t , i f T ( U + ; { V + ( t o ) , to e Q,}; P ) m a x i m i z e d b y U ^ 0 , t h e n T ( a U + ; { b V + ( t o ) , to e fi}; P ) i s a l s o m a x i m i z e d b y t h e s a m e U f o r a l l r e a l a a n d b . A x i o m 2 s a y s t h a t a b a r g a i n e r ' s f a i r d e m a n d i n a b a r g a i n i n g s i t u a t i o n w i t h c o m p l e t e i n f o r m a t i o n i s t h e a m o u n t h e w o u l d r e c e i v e a c c o r d i n g t o N a s h ' s p r o c e d u r e f o r d e t e r m i n i n g f a i r b a r g a i n s , i . e . i f V + ( t o ) = V + f o r a l l to e Q, t h e n h e w o u l d m a x i m i z e T = U + V + . We a l s o a s s u m e t h a t ( i ) I f i t i s p o s s i b l e f o r a b a r g a i n e r t o i n c r e a s e h i s u t i l i t y w i t h o u t l o w e r i n g h i s o p o n e n t ' s u t i l i t y , h e d o e s s o , i . e . t h e b a r g a i n i n g p r o c e s s i s e f f i c i e n t a n d t h a t ( i i ) E a c h b a r g a i n e r i s a w a r e t h a t h i s o p o n e n t i s a l s o e f f i c i e n t . S i n c e T a s s i g n s l a r g e r v a l u e s t o p o i n t s p r e f e r r e d b y b o t h b a r g a i n e r s , t h i s a s s u m p t i o n i m p l i e s t h a t T ( U + ; ( V + ( t o ) , to e fi}; P ) i s s t r i c t l y i n c r e a s i n g i n U > 0 , a n d V ( t o ) > 0 f o r a l l to e . 4 . 3 . 2 C P E D U C I ^ G ; ; : / T H E F O R M O F G N F W H E N Q C O N T A I N S TWO E L E M E N T S  P R E L I M I N A R I E S : L e t Q = (a , t o 2 ) , = V ( t o 1 ) , a n d = V ( t o 2 ) . I n t h i s p a r t i c u l a r c a s e P h a s t h e f o r m : T ( u + , V * , V * , p ) w i t h T ( 0 , V\J\ V * , p ) = T ( U + , 0 , 0 , p ) = 0 f o r a l l p w h e r e p = P ( t o ^ ) , t h e p r o b a b i l i t y t h a t B a r g a i n e r 2 ' s u t i l i t y i s ( I n t h i s c a s e w e s a y t h a t B a r g a i n e r 2 h a s t w o p o s s i b l e t y p e s f r o m B a r g a i n e r l ' s s t a n d p o i n t . ) We c o n s i d e r o n l y b a r g a i n i n g g a m e s w h e r e t h e s e t S g e n e r a t e d b y ( U , V ^ , V^) w h e n X r a n g e o v e r a l l f e a s i b l e v a l u e s h a s b o u n d e d p r o j e c t i o n s i n t h e U x V 1 a n d U x V ? p l a n e s . F u r t h e r m o r e 71 F I G . 4 3 . 1 : T H E S E T O F C O N C E I V A B L E A G R E E M E N T S A N D I T S P R O J E C T I O N S . 7 2 S i s r e q u i r e d t o b e c o n v e x ; i f t h i s i s n o t s o w i t h p u r e c o n t r a c t s ( c h o i c e s o f x ) i t m a y b e a c h i e v e d b y i n c l u d i n g r a n d o m i z e d c o n t r a c t s . G i v e n p , T i s t o b e m a x i m i z e d o v e r t h e s e t S . F o l l o w i n g N a s h , w e a s s u m e t h a t i n a n y s u c h b a r g a i n i n g g a m e B a r g a i n e r 1 h a s i n t e r m s o f u t i l i t y u n i t s a u n i q u e f a i r d e m a n d . T o d e r i v e t h e f o r m o f T w e m a k e u s e o f t h e f o l l o w i n g t h e o r e m s , t h e p r o o f s o f w h i c h a r e g i v e n i n A p p e n d i x r C . T h e o r e m 4 . 3 . 1 : L e t f : ( 0 , °°) x ( 0 , °°) -> ( 0 , °°) b e d i f f e r e n t i a b l e a n d  s t r i c t l y i n c r e a s i n g i n b o t h v a r i a b l e s . A s s u m e , ( i ) f i s m a x i m i z e d o v e r a n y c o n v e x c o m p a c t s e t S c o n t a i n i n g t h e  o r i g i n i n t h e x - y p l a n e , a t a u n i q u e p o i n t , s a y ( x ( S ) , y ( S ) ) . ( i i ) _ i f T c ( ° » °°) x ( 0 , °°) ( 0 , °°) x ( 0 , °°) i s d e f i n e d b y T c ( X , y ) ( c x , y ) , t h a t ( x ( T c ( S ) ) , y ( T c ( S ) ) ) = ( c x ( S ) , y ( S ) ) . T h e n f m u s t b e o f t h e f o r m : f ( x , y ) = F ( x B ( y ) ) f o r a l l x , y f o r s o m e f u n c t i o n F a n d B . C o r o l l a r y 4 . 3 . 2 . L e t f ( 0 , °°) -> ( 0 , °°) b e d i f f e r e n t i a b l e a n d s t r i c t l y  i n c r e a s i n g i n a l l i t s v a r i a b l e s a n d a s s u m e t h a t f , a s a f u n c t i o n o f 3 v a r i a b l e s h a s p r o p e r t i e s ( i ) a n d ( i i ) o f T h e o r e m 4 . 3 . 1 w i t h r e s p e c t t o  t h e x - y p l a n e a n d t h e x - z p l a n e s e p a r a t e l y . T h e n , f m u s t b e o f t h e f o r m : 7 3 f ( x , y , z ) = G ( x A ( y , z ) ) f o r s o m e f u n c t i o n s G a n d A . T h e o r e m 4 . 3 . 3 : L e t f : ( 0 , °°) ( 0 , «>) , d e f i n e d b y f ( x , y , z ) = x h ( y , z ) , b e d i f f e r e n t i a b l e a n d s t r i c t l y i n c r e a s i n g i n a l l  i t s v a r i a b l e s . A s s u m e , ( i ) f i s m a x i m i z e d o v e r Q x R , Q a n d R b e i n g a n y c o n v e x - c o m p a c t s u b s e t s o f t h e x - y a n d x - z p l a n e s r e s p e c t i v e l y , e a c h c o n t a i n i n g t h e r e s p e c t i v e o r i g i n , i s a t t a i n e d a t a u n i q u e p o i n t , s a y ( x ( Q x R ) , y ( Q x R ) ) . 3 3 ( i i ) i f _ X^: ( 0 , °°) ( 0 , °°) i s d e f i n e d b y ^ ( x , y , z ) -»-( x , c y , c z ) , t h a t ( X ( i c ( Q x R ) ) , y ( A c ( Q x R ) ) , z ( ^ C Q x R ) ) ) = ( x ( Q x R ) , c y ( Q x R ) , c z ( Q x R ) ) . T h e n h m u s t b e o f t h e f o r m h ( y , z ) = C ( z ) N ( | ) . F U N C T I O N A L A N A L Y S I S : We w i l l n o w s h o w t h a t a n y g e n e r a l i z e d N a s h f u n c t i o n s a t i s f y i n g o u r a x i o m s i s o f t h e f o r m , 1 T ( U + , V * , V * , P ) = U + [ p L + ( 1 - p ) V + L ] L , w h e r e L i s a c o n s t a n t . F o r a n y g i v e n p , 0 <_ p <_ 1 , T i s a f u n c t i o n d e f i n e d o n t h e f i r s t q u a d r a n t o f 3 - d i m e n s i o n a l E u c l e a d i a n s p a c e . C o n s i d e r f i r s t t h e 7 4 — + + + — + c a s e w h e r e U > 0 , > 0 a n d > 0 , U b e i n g B a r g a i n e r l ' s u t i l i t y e v a l u a t e d a t a n y f i x e d d e m a n d a n d a n d V 2 , t h e u t i l i t i e s f o r t h a t d e m a n d B a r g a i n e r 2 w o u l d g e t i f h i s t y p e w e r e 1 a n d 2 , r e s p e c t i v e l y . ' F r o m . a x i o m s 1 , 2 a n d C o r o l l a r y 4 . 3 . 2 w e d e d u c e , T ( U + , V + , V + ; p ) = T ( U , V r V 2 ; p ) = F ( U B ( V ] _ , V 2 ; p ) ; p ) p r o v i d e d T i s e v e r y w h e r e d i f f e r e n t i a b l e . I n p a r t i c u l a r i f w e s e t = V 2 = V > 0 , s i n c e B a r g a i n e r 2 t h e n r e c e i v e s t h e s a m e a m o u n t r e g a r d -l e s s o f h i s t y p e A x i o m 2 i m p l i e s t h a t , F ( U B ( V , V ; p ) ; p ) = U V f o r a l l U , V . i . e . F i s t h e u s u a l N a s h p r o d u c t f o r b a r g a i n i n g w i t h c o m p l e t e i n f o r m a t i o n . W h e n V = 1 w e g e t F ( x ; p ) = x A ( p ) , w h e r e x = U B ( 1 , 1 ; p ) a n d A ( p ) = 1 / B ( 1 , 1 ; p ) w i t h B ( l , 1 ; p ) 4 0 . H e n c e T r e d u c e s t o , T = U h ( V 1 , V 2 ; p ) f o r U > 0 , V± > 0 , V 2 > 0 ( 4 . 3 . 1 ) w h e r e M V ^ V ^ p ) = B(V±, V ^ p ) A ( p ) . N o t i c e t h a t t h e s e c o n d p a r t o f A x i o m 1 i m p l i e s t h e c o n d i t i o n ( i i ) o f T h e o r e m 4 . 3 . 3 w h e r e Q a n d R a r e t h e u t i l i t y - p o s s i b i l i t y s e t s w h e n B a r g a i n e r 2 i s o f t y p e 1 a n d 2 r e s p e c t i v e l y . S i n c e w e h a v e 7 5 a l r e a d y a s s u m e d A l l t h e c o n d i t i o n s o n T n e c e s s a r y f o r t h e a p p l i c a t i o n o f T h e o r e m 4 . 3 . 3 , w e n o w a p p l y t o r e d u c e T f u r t h e r a n d g e t , T = U C C V ^ p ) N V ' A g a i n w h e n = = V w e m a y a p p l y t h e b o u n d a r y c o n d i t i o n U C ( V ; p ) N ( l ; p ) = U V a n d t h u s o b t a i n , V T ( U + , V * , V + ; p ) = U V1 M ( ^ - ; p ) ( 4 . 3 . 2 ) f o r U > 0 , V 1 > 0 , V 2 > 0 a n d 0 <_ p <_ 1 , w h e r e M ( x ; p ) = N ( x ; p ) / N ( l ; p ) . T o c o m p l e t e o u r a n a l y s i s w e n o w c o n s i d e r s o m e o f t h e s t o c h a s t i c a s p e c t s o f b a r g a i n i n g w i t h i n c o m p l e t e i n f o r m a t i o n . S T O C H A S T I C A N A L Y S I S : F r o m B a r g a i n e r l ' s s t a n d p o i n t B a r g a i n e r 2 c a n a s s u m e e i t h e r o f t w o t y p e s ( i n w h i c h h e h a s d i f f e r e n t u t i l i t i e s ) w i t h c e r t a i n p r o b a b i l i t i e s . W e , t h e r e f o r e , c a n t h i n k o f t h i s s i t u a t i o n a s o n e i n w h i c h a c o i n ( p o s s i b l y b i a s e d ) i s t o s s e d t o d e t e r m i n e B a r g a i n e r 2 ' s t y p e , 1 o r 2 a c c o r d i n g a s H o r T i s o b t a i n e d . I n t h i s v e r s i o n o f t h e p r o b l e m B a r g a i n e r 1 k n o w s h o w t o f i n d a " f a i r " d e m a n d a f t e r t h e c o i n i s t o s s e d , n a m e l y h e m a x i m i z e s t h e N a s h p r o d u c t . H o w e v e r , B a r g a i n e r l ' s p r o b l e m i s t h a t o f f i n d i n g a " f a i r " d e m a n d b e f o r e t h e c o i n i s t o s s e d . W e h a v e s h o w n t h a t t o f i n d a f a i r d e m a n d , m e a n i n g h e r e o n e w h i c h m a x i m i z e s T , 7 6 V 2 h e m a y e q u i v a l e n t l y m a x i m i z e , U . [ V ^ M ( — ; p ) ] A_ U . [ V * ( p ) ] . C o m p a r i n g t h e f o r m o f T w i t h t h a t o f t h e u s u a l N a s h p r o d u c t w e s e e t h a t V * ( p ) m a y b e r e g a r d e d c o n c e p t u a l l y , a s b a r g a i n e r 2 ' s " e f f e c t i v e u t i l i t y " i n s p i t e o f t h e f a c t t h a t V * ( p ) n e e d n o t b e a u t i l i t y f u n c t i o n i n t h e u s u a l s e n s e i n t h e m i x e d g a m e . T h e c r i t e r i a u s e d b y B a r g a i n e r 1 i n d e t e r m i n i n g a f a i r d e m a n d b e f o r e a n d a f t e r t h e c o i n i s t o s s e d i s i l l u s t r a t e d i n F i g u r e 4 . 3 . 2 . F I G U R E 4 . 3 . 2 : C R I T E R I A P R O D U C I N G F A I R D E M A N D S B E F O R E A N D A F T E R T H E C O I N T O S S . 7 7 T o f i n d a r e a s o n a b l e f o r m f o r V * c o n s i d e r a m o r e c o m p l i c a t e d v e r s i o n o f t h i s c o i n t o s s i n g g a m e i n w h i c h t w o c o i n s a r e s e q u e n t i a l l y t o s s e d , t h e f i r s t o n e w i t h P ( H ) = a a n d t h e s e c o n d w i t h P ( H ) = p o r q a c c o r d i n g a s H o r T i s o b t a i n e d o n t o s s 1 . B a r g a i n e r 2 ' s t y p e i s 1 o r 2 a c c o r d i n g a s H o r T i s o b t a i n e d o n t o s s 2 . B a r g a i n e r 1 i s r e q u i r e d t o m a k e a " f a i r " d e m a n d b e f o r e t h e c o i n s a r e t o s s e d . W e h a v e a l r e a d y s p e c i f i e d , a s s h o w n i n F i g u r e 4 . 3 . 3 , w h a t h e w o u l d h a v e d o n e a f t e r t h e f i r s t c o i n t o s s a s w e l l a s a f t e r t h e s e c o n d c o i n t o s s . F I G U R E 4 . 3 . 3 : C R I T E R I A P R O D U C I N G F A I R D E M A N D S A F T E R T H E F I R S T C O I N T O S S A N D A F T E R T H E S E C O N D C O I N T O S S . 7 8 O n t h e o n e h a n d , s i n c e i n t h i s g a m e B a r g a i n e r 2 w i l l b e o f t y p e 1 w i t h p r o b a b i l i t y a p + ( 1 - a ) q a n d w i l l b e o f t y p e 2 w i t h p r o b a b i l i t y 1 - [ a p + ( 1 - a ) q ] , B a r g a i n e r 1 w o u l d m a x i m i z e , \N± M| V 2 r r - ; > a p + ( 1 - a ) q ( 4 . 3 . 3 ) O n t h e o t h e r h a n d , r e g a r d i n g V * a s B a r g a i n e r 2 ' s e f f e c t i v e u t i l i t y , h e m a y e q u i v a l e n t l y m a x i m i z e , U V * ( p ) M * V * ( q ) V * ( p ) ' ( 4 . 3 . 4 ) w h e r e V * ( p ) 4 0 b e c a u s e T i s a s t r i c t l y i n c r e a s i n g f u n c t i o n f o r U > 0 , > 0 a n d > 0 m a k i n g M p o s i t i v e e v e r y w h e r e . We w i l l s a y t h a t B a r g a i n e r l ' s c r i t e r i o n f o r d e t e r m i n i n g h i s " f a i r " d e m a n d i s " c o n s i s t e n t " i f t h e q u a n t i t i e s i n e q u a t i o n s ( 4 . 3 . 3 ) a n d ( 4 . 3 . 4 ) a r e i d e n t i c a l . A x i o m 3 : - T h e c r i t e r i o n d e t e r m i n i n g B a r g a i n e r ' s f a i r d e m a n d i s s t o c h a s t i c a l l y c o n s i s t e n t . U n d e r A x i o m 3 w e t h u s h a v e , U V X M a p + ( l - a ) q f \ = U V X M 2 V p •M* ) I ^ J i f M v 7 ; n M v7 } P ( 4 . 3 . 5 ) f o r a l l 0 <_ a , p , q <^  1 7 9 N o t i c e t h a t a c c o r d i n g t o A x i o m 1 , M 1 a n d M - ; 0 H e n c e s e t t i n g q = 0 , p = 1 a n d x = — i n ( 4 . 3 . 5 ) w e g e t 1 M ( x ; a ) = l . M * ( x ; a ) f o r a l l x , a . T h u s t h e e q u a t i o n ( 4 . 3 . 5 ) r e d u c e s t o , M ( x ; a p + ( 1 - a ) q ) = M ( x ; p ) M M ( x ; q ) M ( x ; p ) ' ( 4 . 3 . 6 ) I n p a r t i c u l a r l e t t i n g p = 1 w e f i n d t h a t t h e f u n c t i o n M s a t i s f i e s t h e e q u a t i o n , M ( x ; a + ( 1 - a ) q ) = M ( M ( x ; q ) ; a ) ( 4 . 3 . 7 ) f o r a l l x > 0 , 0 <_ q ^ 1 , 0 <_ a <_ 1 , b e c a u s e M ( x , 1 ) = 1 . D i f f e r e n t i a t i n g ( 4 . 3 . 7 ) w i t h r e s p e c t t o q a n d t h e n s e t t i n g q = 0 w e s e e t h a t M ( x , a ) s a t i s f i e s t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n , ( 1 ~ a ) 3 ^ = 3 x " E ( X ) ( 4 . 3 . 8 ) w h e r e E ( x ) = — M ( x , q ) q = 0 i s a f u n c t i o n o f x a l o n e . I t c a n b e s h o w n ( c f . S n e d d o n [ 1 9 5 7 ] ) t h a t t h e g e n e r a l s o l u t i o n o f ( 4 . 3 . 8 ) i s 8 0 M ( x , a ) = H ( ( l - a ) D ( x ) ) , w h e r e H i s a n a r b i t r a r y f u n c t i o n a n d 1 _ d x D ( x ) = e . B u t t h e b o u n d a r y c o n d i t i o n M ( x , 0 ) = x i m p l i e s t h a t H = D S i n c e T m u s t b e s y m m e t r i c i n a n o b v i o u s s e n s e i t f o l l o w s t h a t , M ( x , p ) = x M ( ^ , 1 - p ) b y i n t e r c h a n g i n g a n d V^, p a n d ( 1 - p ) . H e n c e , H ( ( l - p ) D ( x ) ) = x H ( p D ( i ) ) ( 4 . 3 . 9 ) w h e r e H = D \ H ( 0 ) = 1 ( p u t p = 1 i n 4 . 3 . 9 ) , a n d t h e r e f o r e D ( l ) D i f f e r e n t i a t i n g ( 4 . 3 . 9 ) w i t h r e s p e c t t o p y i e l d s , - D ( x ) H » ( ( l - p ) D ( x ) ) = x D (k H ' ( p D A ) . X X P u t t i n g p = 0 a n d n o t i n g t h a t D ' ( x ) H ' ( D ( x ) ) = 1 , w e g e t . k D ( x ) = x D ( ^ ) D ' ( x ) f o r a l l x > 0 w h e r e k = - 1 / H ' ( 0 ) 4 0 s i n c e H i s s t r i c t l y m o n o t o n i c . D e f i n i n g t h e f u n c t i o n <j> b y (|>(y) = D ( e y ) a n d s e t t i n g x = e y i n t h e a b o v e e q u a t i o n o b t a i n t h e d i f f e r e n c e - d i f f e r e n t i a l e q u a t i o n , 81 k <|><y) = * ( - y ) * ' ( y ) f o r a l l y e R ( 4 . 3 . 1 0 ) w h e r e k 4 0 . N o w w e u t i l i z e t h e f o l l o w i n g l e m m a t h e p r o o f o f w h i c h i s g i v e n i n t h e A p p e n d i x C . L e m m a 4 . 3 . 4 : T h e g e n e r a l s o l u t i o n o f e q u a t i o n ( 4 . 3 . 1 0 ) i s , + (y) = - [ 1 + e k m y ] f o r a l l k 4 0 a n d + y a s w e l l i f m — k = + 1 , w h e r e m i s a n o n z e r o a r b i t r a r y c o n s t a n t . 1 k m H e n c e , s i n c e D ( l ) = 0 , D ( x ) = — [ 1 - x ] : o r + £ n x a n d i n t u r n m — w e c o n c l u d e , s i n c e M s h o u l d b e p o s i t i v e , t h a t , M ( x ; p ) = [ p + ( 1 - p ) x L ] L o r x x P w i t h L = k m a r b i t r a r y a n d t h a t , T ( t T + , v+, V + ; P ) U [ p + ( 1 - p ) v ^ ] L u v x p v 2 o r 1 - p ( 4 . 3 . 1 1 ) f o r U + > 0 , > 0 , V 2 > 0 a n d 0 <_ p <_ 1 . N o t e t h a t f o r a l l L e R , 8 2 U [ p + ( 1 - p ) V ^ l ^ i s i n c r e a s i n g i n U , a n d V ^ , a s r e q u i r e d . N o w - + + + w e d e f i n e t h e v a l u e o f T a l o n g t h e a x e s U , , a s t h e p r o p e r l i m i t s o t h a t t h e c o n t i n u i t y i s p r e s e r v e d . I t i s a l s o o f i n t e r e s t t h a t , l i m [ p + ( 1 L - * 0 P ) v h 1 - v]->. T h i s l e a d s u s t o d e f i n e , [ p V ^ + ( 1 - p ) V^]L i f I | V ( P ) | | , = v p v 1 _ p 1 2 i f L = 0 H e n c e t h e g e n e r a l i z e d N a s h f u n c t i o n h a s t h e f o r m , G N F L = U + | | V ( p ) + | | L f o r a l l U , V , 0 <_ p < 1 w h e r e L e R R e m a r k s : N o t i c e t h a t t h e L - n o r m | | v + ( p ) | | L i s a g e n e r a l m e a n ( c f . N i l a n N o r r i s [ ' 7 6 ] ) o f B a r g a i n e r 2 ' s u t i l i t i e s . T h u s , w h a t w e h a v e s h o w n i s t h a t , i n o r d e r t o f i n d a ' f a i r ' d e m a n d , a b a r g a i n e r m a y m a x i m i z e t h e p r o d u c t o f h i s u t i l i t y a n d a g e n e r a l m e a n o f h i s a d v e r s a r y ' s p o s s i b l e u t i l i t i e s . T h i s i s 8 3 p r e c i s e l y t h e e x p e c t e d a n a l o g u e i n t h e c a s e o f i n c o m p l e t e i n f o r m a t i o n o f N a s h ' s p r o d u c t i n t h e c a s e o f c o m p l e t e i n f o r m a t i o n . I t i s o f i n t e r e s t t h a t , 1 ( i ) l i m [ p V + L + ( 1 - p ) V 2 L ] L = S u p V + L -> 0 0 i a n d 1 ( i i ) l i m [ p V + L + ( 1 - p ) V 2 L ] L = I n f V * L e t 0 . , 0 * , a n d 0 . b e t h e s o l u t i o n s g i v e n b y t h e c r i t e r i a , * l h + - + + + + U I n f V \ , U S u p V \ a n d U r e s p e c t i v e l y . T h e n i t i s e v i d e n t t h a t , U(01) < U ( 0 . ) < U ( 0 * ) f o r a l l i i . e . b a r g a i n e r u s i n g t h e c r i t e r i o n G N F _ r o i s d e m a n d i n g t h e l e a s t u t i l i t y h e w o u l d h a v e d e m a n d e d i n c o m p o n e n t b a r g a i n i n g g a m e s , w h e r e a s b a r g a i n e r u s i n g t h e c r i t e r i o n G N F ^ i s d e m a n d i n g t h e m o s t u t i l i t y h e w o u l d h a v e d e m a n d e d i n c o m p o n e n t b a r g a i n i n g g a m e s . I n f a c t , t h e h i g h e r t h e v a l u e o f L , t h e h i g h e r i s B a r g a i n e r l ' s f a i r d e m a n d . T h e r o l e o f L i s s o m e w h a t m y s t e r i o u s . W e i n t e r p r e t i t , z e i £ h e r s a s i e x p r e s s i n g w h a t P e n [ 1 9 5 2 ] h a s c a l l e d a n d e x p r e s s e d i n a d i f f e r e n t w a y , t h e b a r g a i n e r s ' ' n o n - a c t u a r i a l 8 4 m e n t a l i t y . ' H e r e L r e p r e s e n t s ( i n P e n ' s w o r d s ) " t h e s a t i s f a c t i o n ( o r d i s s a t i s f a c t i o n ) t h e s u b j e c t d e r i v e s f r o m t h e r i s k - t a k i n g a s a f a c t i n i t s e l f . . . " J o r a s a m e a s u r e o f b a r g a i n e r ' s a g g r e s s i o n . O u r j u s t i f i c a t i o n f o r t h i s i n t e r p r e t a t i o n o f t h e r o l e , o f L i s " f o u n d ' i n t h e p r o p e r t i e s , ( i ) G N F = U ( I n f V ) , " - - ° ° i ( i i ) G N F = U ( S u p V . ) a n d ( i i i ) G N F i s i n c r e a s i n g i n L . T h e s e p r o p e r t i e s s h o w t h a t a s L i n c r e a s e s B a r g a i n e r 1 i s s u b j e c t i n g h i m s e l f t o p r o g r e s s i v e l y g r e a t e r :.&S3. 3 s - . r g a i n e r i l-< • . ' " t c .: -T-. r i s k o f c o n f l i c t . o f ccntlict. W i t h t h i s s t r u c t u r e o f t h e G N F t h e t e r m i n o l o g y ' f a i r d e m a n d ' m a y b e f u r n i s h e d f u r t h e r j u s t i f i c a t i o n . 1 . I n p a r t i c u l a r i f L = 0 H a r s a n y i a n d S e l t e n ' s w o r k s h o w s e s s e n t i a l l y t h a t i f B a r g a i n e r 2 k n e w U , V ^ , V ^ , p ^ a n d p ^ h e w o u l d r e a l i z e t h a t B a r g a i n e r l ' s p o s i t i o n a n d t h e r e f o r e h i s o w n w a s u n a l t e r a b l e . H i s o n l y a l t e r n a t i v e w o u l d b e t o t r y t o c a u s e p ^ , p 2 t o c h a n g e . 2 . C o n s i d e r a s i t u a t i o n w h e r e t h e u t i l i t i e s t h a t B a r g a i n e r 1 w o u l d h a v e r e c e i v e d f r o m e i t h e r c o m p o n e n t b a r g a i n i n g g a m e ( w i t h c o m p l e t e i n f o r m a -t i o n ) b y N a s h ' s f a i r b a r g a i n p r o c e d u r e , a r e t h e s a m e . T h e n n o t i c e t h a t e v e n i n t h e b a r g a i n i n g s i t u a t i o n w i t h i n c o m p l e t e i n f o r m a t i o n , B a r g a i n e r l ' s f a i r d e m a n d g i v e n b y G N F i s t h e s a m e a s w h a t h e w o u l d h a v e r e c e i v e d f r o m c o m p o n e n t b a r g a i n i n g g a m e s , r e g a r d l e s s o f t h e m a g n i t u d e o f L . N o t i c e t h a t , w h e n L = l - G N F ^ r e d u c e s : t o t h e i i s u a l t - ( a r i t h m e t i c ) m e a n v a l u e o f JLi AJ V ^ a n d s v | g a T h e - d M i e e s - E s - l:o. a n d 0, , . r e s p e c t i v e l y , - l y i e l d ^ t h e c o r r e s -p S i i d M g - h a r m o n l c S a f i d " ' g S o f f l d f c r - d L C - - ' -me i ah s - I t s h o u l d b e p o i n t e d o u t t h a t w h e n L = 0 , o u r G N F i s e q u a l i n a n a p p r o p r i a t e s p e c i a l c a s e o f t h e g e n e r a l i z e d N a s h p r o d u c t o b t a i n e d b y H a r s a n y i a n d S e l t e n [ 1 9 7 2 ] i n t h e i r t r e a t m e n t o f t h e t w o p e r s o n b a r g a i n i n g g a m e , e x c e p t f o r t h e d e f i n i t i o n s o f v a r i a b l e s . a n d . t h e s e t o v e r w h i c h t h e m a x i m i z a t i o n i s c a r r i e d o u t . I t i s 1 . c f . L o e v e [ 1 9 6 3 ] . 8 5 e a s i l y s e e n t h a t i f A x i o m 1 i s r e p l a c e d b y t h e f o l l o w i n g o n e , w h i c h i s s t r o n g e r t h a n A x i o m 1 a n d s i m i l a r t o o n e o f H a r s a n y i a n d S e l t e n ' s a x i o m s ( t h e i r A x i o m o f L i n e a r I n v a r i a n c e ) , i n f a c t , o u r w h o l e c l a s s o f G N F ' s r e d u c e s t o t h e s i n g l e e l e m e n t D + v f v f P - u + || « + ( P ) | | = A x i o m : - A b a r g a i n e r ' s c l a s s o f f a i r d e m a n d s d o e s n o t v a r y u n d e r p o s i t i v e s c a l a r t r a n s f o r m a t i o n s o f h i s o w n u t i l i t y f u n c t i o n , B a r g a i n e r 2 ' s t y p e 1 u t i l i t y f u n c t i o n , o r B a r g a i n e r 2 ' s t y p e 2 u t i l i t y f u n c t i o n . A l t h o u g h H a r s a n y i a n d S e l t e h ' s a x i o m o f l i n e a r i n v a r i a n c e s e e m s r e a s o n a b l e i n h i s n - p l a y e r a n a l o g u e ( h e r e p l a y e r 1 h a s o n e t y p e a n d p l a y e r 2 h a s t w o t y p e s s o n = 3 ) , t h e a b o v e a x i o m d o e s n o t s e e m s u i t a b l e i n o u r c o n t e x t s i n c e B a r g a i n e r 2 h a s e x a c t l y 1 ( a l b e i t ) u n k n o w n u t i l i t y f u n c t i o n . I n f a c t , t h e c o n s e q u e n c e o f i m p o s i n g t h i s c r i t e r i o n n a m e l y t h a t f a i r d e m a n d s s h o u l d b e f o u n d b y m a x i m i z i n g G N F = U V P P i s u n r e a s o n a b l e f o r a w h o l e r a n g e o f c h o i c e s o f V_^, a n d p . C o n s i d e r t h e f o l l o w i n g e x a m p l e t o j u s t i f y t h i s f a c t a n d t o c l a r i f y o u r r e s u l t s . 4 . 3 . 3 E X A M P L E S r i L a n k a ( C e y l o n ) i s t h e o n l y b u y e r o f " U m b a l a k a d a " ( D r i e d M a l d i v e f i s h ) f r o m t h e M a l d i v e i s l a n d s a n d C e y l o n c a n n o t b u y t h i s p r o d u c t f r o m a n y -w h e r e e l s e . T h u s a r i s e s a b i l a t e r a l m o n o p o l y s i t u a t i o n . L e t u s f o r m u l a t e a s i m p l i f i e d m o d e l w i t h f a k e f i g u r e s t o s e e h o w t h e p r i c e a n d q u a n t i t y d e t e r m i n a t i o n i s r e a l i z e d i n t h i s s i t u a t i o n , i n t e r m s o f b a r g a i n i n g w i t h i n c o m p l e t e i n f o r m a t i o n . 8 6 A S r i L a n k i a n c o m p a n y S , s a y B a r g a i n e r 1 , i m p o r t s u m b a l a k a d a f r o m t h e M a l d i v e i s l a n d s ( M ) s a y B a r g a i n e r 2 , a n d s e l l s i t i n S r i L a n k a . S i s a m o n o p o l i s t " ^ i n S r i L a n k a a n d h e k n o w s t h a t t h e d e m a n d f u n c t i o n f o r 8 u m b a l a k a d a i n S r i L a n k a i s g i v e n b y p " = —JT - , q ^ 0 F I G U R E 4 . 3 . 4 : D E M A N D F U N C T I O N F O R U M B A L A K A D A I N S R I L A N K A T h u s o u r m o d e l d i f f e r s f r o m t h e t y p i c a l b i l a t e r a l m o n o p o l y m o d e l s d i s c u s s e d i n m i c r o - e c o n o m i c b o o k s s u c h a s H e n d e r s o n a n d Q u a n d t [ 1 9 7 1 ] w h e r e t h e m o n o p s o n i s t s e l l s h i s o u t p u t a n d m o n o p o l i s t b u y s h i s i n p u t i n p e r f e c t l y c o m p e t i t i v e m a r k e t s a t f i x e d p r i c e s . 8 7 w h e r e q i s t h e q u a n t i t y t h a t c a n b e s o l d w h e n t h e s e l l i n g p r i c e i s ( i n r u p e e s ) . F o r e a c h u n i t o f u m b a l a k a d a i m p o r t e d S h a s a p p r o x i m a t e l y a c o s t o f R s . 1 f o r t r a n s p o r t a t i o n , d i s t r i b u t i o n , m a i n t e n a n c e a n d o t h e r c o s t s i n c u r r e d i n t h i s b u s i n e s s . H e n c e i f S b u y s q u n i t s o f t h i s p r o d u c t f r o m M a t a p r i c e R s . ~ p i t s p r o f i t i s ( p ' ~ p ) q - 1 q. A s s u m i n g t h a t p r o f i t i s p a r a l l e l , o r r a t h e r e q u a l t o M ' s u t i l i t y . W e h a v e , S u p p o s e a s i n g l e M a l d i v i a n C o r p o r a t i o n M i s r e s p o n s i b l e f o r e x p o r t i n g u m b a l a k a d a t o S r i L a n k a . I n S ' s v i e w M h a s t w o p o s s i b l e p r o d u c t i o n c o s t s , e t c . a n d u n d e r t h e s u p p o s i t i o n t h a t M ' s u t i l i t y x i s p a r a l l e l , o r r a t h e r e q u a l t o t h e p r o f i t , S ' s s u b j e c t i v e b e l i e f i s t h a t M ' s u t i l i t y i n c r e m e n t w i t h r e s p e c t t o t h e c o n f l i c t p o i n t o f c o r p o r a t i o n c o l l a p s e , f o r s e l l i n g q u n i t s o f u m b a l a k a d a a t p r i c e RsVp- i s , = p q - q w i t h p r o b a b i l i t y . 9 ^ = p q - 5 q w i t h p r o b a b i l i t y . 1 w h e r e p ^ 0 a n d 0 <_ q <_ 3. I n o r d e r t o f i n d t h e u t i l i t y p o s s i b i l i t y s e t s f o r t h i s s i t u a t i o n n o t e t h a t , I n r e a l i t y t h i s s h o u l d r e f l e c t M ' s f o r e i g n e x c h a n g e r e q u i r e m e n t s , o p p o r t u n i t y o f e m p l o y m e n t e t c . t o o a s w e l l a s i t s p r o f i t . 1 F IG. 4 . 3 . 5 : Graph of g Vs. q . 2 3 q F IG . A . 3 . 6 ? U t i l i t y — p o s s i b i l i t y s e t .when bargai ne r 2 i s of type 1. 8 9 U + V = 8 a . - 2 q = g ( q ) . q + 1 S i n c e t h e m a x i m u m o f t h i s o c c u r s a t q = 1 w e h a v e 0 <^  U + <^  2 f o r a l l a d m i s s i b l e q . H e n c e t h e s e t o f a l l f e a s i b l e u t i l i t y c o u p l e s w i l l b e a s u b s e t o f t h e r e g i o n b o u n d e d b y t h e t w o s t r a i g h t l i n e s U + = 2 a n d U + = 0 . I n p a r t i c u l a r s i n c e w h e n q = 1 w e h a v e - 1 <^  = ( p - 1 ) < °° - oo < u = 4 - ( p + 1 ) <^  3 i t i s c e r t a i n t h a t t h e p o i n t s o n t h e l i n e U + = 2 i n t h e f i r s t q u a d r a n t o f t h e g r a p h a n d h e n c e a l l t h e p o i n t s i n t h e s h a d e d r e g i o n i n F i g u r e 4 . 3 . 6 a r e f e a s i b l e . M o r e o v e r , t h e s e a r e t h e o n l y f e a s i b l e s o l u t i o n s w i t h n o n n e g a t i v e u t i l i t i e s . S i m i l a r l y U + V 2 = j f ^ i - 6 q = h ( q ) a n d s i n c e t h e m a x i m u m a n d m i n i m u m o f h o c c u r s a t q = . 1 6 a n d q = 1 / 3 , r e s p e c t i v e l y i n t h e a d m i s s i b l e r e g i o n , i t f o l l o w s t h a t - 1 2 <_ U + . 1 4 . A s b e f o r e i t i s , n o w e a s i l y s e e n t h a t t h e f e a s i b l e s o l u t i o n s w i t h n o n n e g a t i v e u t i l i t i e s c o m p r i s e t h e r e g i o n u n d e r t h e s t r a i g h t l i n e s V + = . 1 4 i n t h e f i r s t q u a d r a n t o f t h e g r a p h U V s V ^ . N o w s u p p o s e S ' s p r o c e d u r e f o r f i n d i n g a ' f a i r ' d e m a n d i s t o f i n d ( p , q ) y i e l d i n g t h e a m o u n t o f u t i l i t y t h a t m a x i m i z e s U + || V ( p ) + | | ^ » i . e . m a x i m i z e s , U [ . 9 ( 2 - U ) + . 1 ( 1 4 - U ) ] i f 0 < U < . 1 4 G N F 1 = U + [ . 9 V * + . 1 V * ] = \ . 9 U ( 2 - U ) i f . 1 4 < U < 2 0 o t h e r w i s e . T h e g r a p h o f t h i s f u n c t i o n i s a s i n F i g u r e 4 . 3 . 9 . H e n c e S ' s ' f a i r ' FIG. 4 . 3 . 8 : U t i l i t y - p o s s i b i l i t y s e t w h e n b a r g a i n e r 2 is of t y p e 2 . 91 0.90 h 0.23 F IG. 4.3.9 : Graph of GNF Vs. U. P o F IG. 4 - . 3 . 1 0 : 5 ' s d e m a n d f u n c t i o n a n d M's s u p p l y f u n c t i o n . 9 2 d e m a n d i n t h i s s i t u a t i o n i s a n y ( p , q ) g i v i n g h i m 1 u n i t o f u t i l i t y . M o r e o v e r t h i s i s a c h i e v e d w i t h < 0 . I t s h o u l d b e p o i n t e d o u t t h a t t h i s i s e x a c t l y w h a t o n e w o u l d e x p e c t , i n t u i t i v e l y , S t o d o i n t h i s b a r g a i n i n g g a m e . N o t i c e t h a t i n t h e t w o c o m p o n e n t g a m e s w h e n M h a s u t i l i t i e s a n d r e s p e c t i v e l y , S ' s f a i r d e m a n d s w e r e U = 1 a n d U = . 0 7 . N o w i n t h i s m i x e d g a m e i t i s e v i d e n t t h a t , e v e n i f S w e r e t o d e m a n d j u s t a l i t t l e l e s s t h a n U = 1 t h e r e s u l t w o u l d n o t b e a n y d i f f e r e n t i f M h a p p e n e d t o b e o f t y p e 2 . I t i s o f i n t e r e s t t o n o t e h e r e t h a t t h e d e m a n d f u n c t i o n o f S , w h o u s e s t h e c r i t e r i o n G N F ^ , f o r M ' s u m b a l a k a d a i s , q8 | x - ( p + 1 ) q = 1 o r e q u i v a l e n t l y p = ^ + 2 q _ ^ _ S b u y s q = — u n i t s o f M ' s p r o d u c t w h e n p r i c e s e t b y S i s P + 2 p . S i m i l a r l y , M w i l l e m p l o y a s u p p l y f u n c t i o n p = f ( q ) a n d t h e e q u i l i b r i u m m a y b e a c h i e v e d a t t h e p o i n t o f i n t e r e s e c t i o n ( p Q , q Q ) i n F i g u r e 4 . 3 . 1 0 , t h i s w a y , i f M i s o n l y r e s p o n s i b l e i n s e t t i n g t h e p r i c e a n d S i s o n l y r e s p o n s i b l e i n s e t t i n g t h e q u a n t i t y . I n c o n t r a s t , s u p p o s e n o w t h a t t h e c r i t e r i o n u s e d b y S w a s t o m a x i m i z e U + | | v | | Q . T h e n s i n c e , 9 3 F I G U R E 4 . 3 . 1 1 : G R A P H O F G N F Q V S U S d e m a n d s o n l y U = . 1 3 ( s e e F i g u r e 4 . 3 . 1 1 ) , w h i c h i s q u i t e u n r e a l i s t i c . 9 4 4 . 3 . 4 D E R I V A T I O N O F T H E G E N E R A L F O R M O F G N F W e w i l l n o w d e d u c e f r o m t h e f o r m o f G N F w h e n fi c o n t a i n s t w o e l e m e n t s t h e f o r m o f G N F f o r g e n e r a l ft. I t w i l l b e s h o w n t h a t , i n g e n e r a l G N F = i U + [ E ^ + L ] L i f L 4 0 - + E £ n V + U e i f L = 0 U + | | V + | I , L e R L , t h e e x p e c t a t i o n E b e i n g t a k e n w i t h r e s p e c t t o t h e s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n o v e r t h e a p p r o p r i a t e s p a c e . T h i s i s d o n e i n t w o s t e p s . E X T E N D I N G F R O M 2 T O N T H E N U M B E R O F T Y P E S O F B A R G A I N E R 2 S u p p o s e f r o m B a r g a i n e r l ' s s t a n d p o i n t h i s , a d v e r s a r y h a s n p o s s i b l e t y p e s w i t h p r o b a b i l i t i e s p ^ , ••> P n * B a r g a i n e r 2 ' s u t i l i t y , w h e n h e i s o f t y p e i w i l l b e d e n o t e d b y V ^ . T h e n t h e r e s u l t t o b e p r o v e d i s t h a t G N F = { L i U [ P 1 V l + P 2 V 2 + + P n V n ] L i f L * ° P l P n D V 1 - ^ V n i f L = 0 f o r U > 0 , V - > 0 , V > 0 , . - . V > 0 . S i n c e t h e f o r m o f t h e G N F w h e n X 2. n L i 9 5 L = 0 i s d e f i n e d a s l i m i t o f U [ p . . + p „ + + p V L ] L w h e n L - * 0 1 1 2 2 n n w e s h a l l c a r r y o n o u r p r o o f o n l y f o r L 4 0 . W e a r e t h u s r e q u i r i n g t h e G N F t o b e c o n t i n u o u s w i t h r e s p e c t t o L a t 0 . T h e r e s u l t i s p r o v e d b y J_j " M a t h e m a t i c a l I n d u c t i o n " a n d b y i n v o k i n g a g e n e r a l i z a t i o n o f t h e c o n s i s t e n c y a x i o m o f S e c t i o n 4 . 3 . 2 . W e a l r e a d y p r o v e d t h e r e s u l t f o r n = 2 . S u p p o s e t h e r e s u l t i s t r u e w h e n n = k . T h a t i s , i f B a r g a i n e r 1 i s c o n f r o n t e d w i t h a b a r g a i n i n g g a m e i n w h i c h B a r g a i n e r 2 h a s k p o s s i b l e t y p e s w i t h p r o b a b i l i t i e s r ^ , r ^ , . . . , r ^ i n o r d e r t o f i n d a ' f a i r ' d e m a n d h e m a x i m i z e s U [ r ^ + . . . + r ^ v j p k f o r s o m e L 4 0 . N o w s u p p o s e h e i s c o n f r o n t e d b y a b a r g a i n i n g g a m e i n w h i c h B a r g a i n e r 2 h a s k + 1 t y p e s w i t h p r o b a b i l i t i e s p ^ , P ^ T " ^ n o r d e r t o a n a l y z e B a r g a i n e r l ' s b e h a v i o u r h e r e a s b e f o r e , w e c a n t h i n k o f t h i s s i t u a t i o n a s o n e i n w h i c h a c o i n i s t o s s e d w i t h P ( H ) = P ( H ) = P^ +^5 i f H o c c u r s , h e e n t e r s a b a r g a i n i n g s i t u a t i o n w i t h i n c o m p l e t e i n f o r m a t i o n w h e r e B a r g a i n e r 2 h a s k t y p e s w i t h u t i l i t i e s V ^ , . . . , V ^ a n d • P l P k p r o b a b i l i t i e s -z ,.T.V, I • T h e c r i t e r i a u s e d b y B a r g a i n e r 1 i n 1 _ P k + l 1 _ P k + l d e t e r m i n i n g a f a i r d e m a n d a f t e r t h e c o i n i s t o s s e d a r e i l l u s t r a t e d i n F i g u r e 4 . 3 . 1 2 . W e n e e d t o k n o w t h e c r i t e r i o n h e w o u l d u s e t o d e t e r m i n e a f a i r d e m a n d b e f o r e t h e c o i n i s t o s s e d . A s b e f o r e w e c a l l V * t h e e f f e c t i v e u t i l i t y o f B a r g a i n e r 2 i n t h e b a r g a i n i n g s i t u a t i o n w i t h i n c o m p l e t e i n f o r m a -t i o n . W e s a y t h a t t h e c r i t e r i a u s e d b y B a r g a i n e r 1 b e f o r e t h e c o i n t o s s i s s t o c h a s t i c a l l y c o n s i s t e n t i f h e b e h a v e s h e r e t h e s a m e w a y h e w o u l d h a v e , i f V * w a s ; i t h e u t i l i t y o f B a r g a i n e r 2 i n t h e c a s e o f c o m p l e t e i n f o r m a t i o n . A s s u m i n g t h i s u n d e r t h e n e c e s s a r y e x t e n s i o n o f A x i o m 3 b y i n c l u d i n g t h i s 9 6 s e c o n d t y p e o f c o n s i s t e n c y t h e n , b e f o r e t h e c o i n i s t o s s e d h e w o u l d m a x i m i z e , 1 = U V * F I G U R E 4 . 3 . 1 2 : B A R G A I N E R l ' S C R I T E R I A P R O D U C I N G F A I R D E M A N D S A F T E R T H E C O I N T O S S f o r U > 0 , > 0 , V ^ + ^ > 0 t h u s c o m p l e t i n g t h e p r o o f . G E N E R A L S O L U T I O N A g a i n w e p r o v e o u r c l a i m o n l y f o r L f 0 , t h e c a s e L = 0 f o l l o w i n g b y c o n t i n u i t y . R e c a l l t h a t w e h a v e a l r e a d y a s s u m e d , 9 7 G N F = T ( U + , { V + ( w ) , u e ft}, P ) A s s u m e V i s d e f i n e d o n ( f t , F , P ) , F b e i n g a B o r e l f i e l d , i s B o r e l m e a s u r a b l e . A l s o a s s u m e T i s m e a s u r a b l e a n d c o n t i n u o u s . C a s e i : S u p p o s e V i s a s i m p l e f u n c t i o n , i . e . n V = E V I w h e r e A e f a n d I A i = l i i i s t h e i n d i c a t o r f u n c t i o n o f A ^ W e h a v e a l r e a d y p r o v e d t h a t , T = U + [ E V + L P ( A . ) ] L f o r L ^ O 1 1 1 = u + [ / v + L d p . ] L = u + [ E r v C a s e i i : S u p p o s e V i s a n y B o r e l f u n c t i o n . T h e n w e k n o w t h a t t h e r e e x i s t s a s e q u e n c e o f i n c r e a s i n g s i m p l e f u n c t i o n s d e f i n e d o n ( f t , F , ft), s u c h t h a t V = l i m V , w h e r e V < V . H e n c e b y c o n t i n u i t y o f T , n n — n + 1 3 n 9 8 T ( U + , ( V + ( a ) ) , a) e ft}, P ) = T ( U + , { l i m V * ( u i ) , u e f t } , P ) = l i m T ( U + , {V * (OJ ) , oo e ft}, P ) 1 = l i m U + [ E V + L ] L f r o m c a s e ( i ) n n 1 = U + l i m [ E V + L ] L n n 1 = U + [ l i m ( E V + L ) ] L f o r L 4 0 . n n B u t , s i n c e {V } i V, w e h a v e { V + L } \ V + L f o r L 4 0 . H e n c e b y t h e n n J M o n o t o n e C o n v e r g e n c e t h e o r e m w e h a v e l i m E V + L = E v + L , n n ' 1 a n d i n t u r n w e c o n c l u d e t h a t T = U + [ E V + L ] L , t h u s c o m p l e t i n g t h e p r o o f . 4 . 4 S U M M A R Y A N D C O N C L U S I O N S T h e m a j o r o b j e c t i v e s o f t h i s s t u d y w e r e ( i ) t o s u g g e s t a c o n c e p t u a l d e c o m p o s i t i o n o f t h e b a r g a i n i n g p r o b l e m u n d e r i n c o m p l e t e i n f o r m a t i o n i n t o c o m p o n e n t p r o b l e m s , a n d ( i i ) t o e s t a b l i s h a c r i t e r i o n f o r d e t e r m i n i n g a b a r g a i n e r ' s " f a i r " d e m a n d , a s a g e n e r a l i z a t i o n t o t h e c a s e o f i n c o m p l e t e i n f o r m a t i o n o f t h e c r i t e r i o n d e r i v e d b y N a s h f o r a b a r g a i n i n g s i t u a t i o n w i t h c o m p l e t e i n f o r m a t i o n . 9 9 W e h a v e s h o w n u s i n g p r i m a r i l y t h e 3 a x i o m s g i v e n b e l o w t h a t i n o r d e r t o f i n d f a i r d e m a n d s , i . e . a b a s e l i n e f o r h i s d e m a n d s , B a r g a i n e r 1 m a x i m i z e s t h e g e n e r a l i z e d N a s h p r o d u c t U + | | v + | | . A x i o m 1 : - A b a r g a i n e r ' s c l a s s o f f a i r d e m a n d s d o e s n o t v a r y u n d e r p o s i t i v e s c a l a r t r a n s f o r m a t i o n o f t h e o p p o n e n t ' s o r h i s o w n u t i l i t y f u n c t i o n . A x i o m 2 : - I n b a r g a i n i n g w i t h c o m p l e t e i n f o r m a t i o n , f a i r d e m a n d s a r e f o u n d b y m a x i m i z i n g t h e N a s h p r o d u c t . A x i o m 3 : - T h e c r i t e r i o n d e t e r m i n i n g b a r g a i n e r ' s f a i r d e m a n d i s s t o c h a s t i c a l l y c o n s i s t e n t . I t s e e m s t h a t - 0 - 1 , i s - n o t a r e a s o n a b l e v a l u e ' ' f<3r L , a n d i n p a r t i c u l a r o n e m a y g e n e r a l l y b e i n c l i n e d t o u s e L = 1 . 1 0 0 C H A P T E R 5 B A R G A I N I N G P R O B L E M : A N O P T I M A L B A R G A I N I N G P R O C E S S 5 . 1 I N T R O D U C T I O N I n t h i s c h a p t e r w e d o n o t p r o p o s e a n e w d e s c r i p t i v e t h e o r y o f b a r g a i n i n g , i . e . o n e w h i c h a t t e m p t s t o e x p l a i n h o w a g r e e m e n t i s r e a c h e d o r n e g o t i a t i o n s a r e c a r r i e d o u t i n r e a l i t y . I n s t e a d o u r a i m i s t o i n t r o d u c e a n o r m a t i v e b a r g a i n i n g s c h e m e o r b i d d i n g s t r a t e g y b y w h i c h c o n c e s s i o n s m a y b e m a d e o p t i m a l l y b y r a t i o n a l i n d i v i d u a l s i n t h e s e n s e t h a t a n y o t h e r s e q u e n c e o f c o n c e s s i o n s f o r t h e p r o p o s e d u n d e r l y i n g b a r g a i n i n g m o d e l w i l l g i v e n o b a r g a i n e r a h i g h e r e x p e c t e d u t i l i t y . I t w i l l b e c o m e c l e a r t h a t t h e b a r g a i n i n g p r o c e s s i s s o c o m p l i c a t e d t h a t o p t i o n a l s t r a t e g i e s a r e b e y o n d c o m p u t a t i o n . H o w e v e r , t h i s s t u d y i s n o t o n l y e s s e n t i a l t o d e t e r m i n e h o w c l o s e l y a p r o p o s e d b a r g a i n i n g s c h e m e c o n s t r u c t e d o n c o n s i d e r a t i o n o f p s y c h o l o g i c a l f a c t o r s e x p l a i n i n g b a r g a i n e r ' s b e h a v i o r a p p r o x i m a t e s t o t h e o p t i m a l o n e , b u t a l s o i t w i l l p r o v i d e a c r i t e r i o n f o r c o m p a r i n g s e v e r a l s u c h b a r g a i n i n g s c h e m e s . We s h a l l b e d e a l i n g w i t h t h e b a r g a i n i n g p r o b l e m u n d e r i n c o m p l e t e i n f o r m a t i o n a s i n t r o d u c e d i n c h a p t e r 4 , i n t h e c o n t e x t o f b i l a t e r a l m o n o p o l y . A f t e r t h e u t i l i t y f u n c t i o n s a r e r e n o r m a l i z e d t o b r i n g t h e c o n f l i c t p o i n t t o ( 0 , 0 ) , t h e u t i l i t i e s t o B a r g a i n e r 1 a n d 2 o f x , a p o s s i b l e a g r e e m e n t , a r e d e n o t e d b y U ( x ) , V ( x ) . T o s i m p l i f y o u r n o t a t i o n w e s h a l l u s e x i n p l a c e o f x , a s n o a m b i g u i t y w i l l r e s u l t . I n g e n e r a l t h e r e i s a c o s t i n v o l v e d i n b a r g a i n i n g " ' " a n d i t i s t h i s c o s t t h a t m o t i v a t e s t h e b a r g a i n i n g p r o c e s s . T h i s c o s t m i g h t p r e c i p i t a t e c o n f l i c t 1 . F o r a d i s c u s s i o n a b o u t c o s t s c . f . C r o s s [ 1 9 6 5 ] a n d [ 1 9 6 9 ] . 101 e v e n i n c a s e s w h e r e t h e r e a r e f e a s i b l e a g r e e m e n t s ( p o i n t s b e t t e r t h a n t h e c o n f l i c t p o i n t ) . T h i s c o s t ( p e r u n i t t i m e , e t c . ) d e t e r m i n e s t h e p o w e r o f e a c h b a r g a i n e r . T h e l o w e r t h e b a r g a i n e r ' s c o s t , t h e b e t t e r t h e t e r m s ( p o i n t o f a g r e e m e n t ) h e w i l l b e a b l e t o a c h i e v e b y b a r g a i n i n g . F o l l o w i n g C r o s s [ 1 9 6 5 ] w e a s s u m e t h a t b i d d i n g ( i . e . m a k i n g o f f e r s o r d e m a n d s ) i s c a r r i e d o u t i n t e r m s o f p h y s i c a l p a y o f f u n i t s , n a m e l y x i n o u r c o n t e x t , r a t h e r t h a n u t i l i t y u n i t s . F o r e x a m p l e , i n H i l d r e t h ' s e x c h a n g e m o d e l o u t l i n e d i n C h a p t e r 1 , x = a i s t h e q u a n t i t y i n v o l v e d i n t h e n e g o t i a t i o n . I n o r d e r t o e n s u r e t h a t t h e r e w i l l b e n o l o s s o f e f f i c i e n c y d u e t o t h i s r e s t r i c t i o n , e a c h b a r g a i n e r i s p e r m i t t e d t o s t a t e h i s d e m a n d a s a r a n g e o f p o s s i b l e x ' s , l e a v i n g t h e c h o i c e o f o n e o f t h e s e e q u i v a l e n t x ' s t o h i s o p p o n e n t . T h e b a r g a i n i n g p r o c e s s w i l l b e c a r r i e d o u t a s f o l l o w s . O n e b a r g a i n e r o f f e r s ( d e m a n d s ) x . T h e o p p o n e n t c a n e i t h e r a c c e p t i t , m a k e a n e w o f f e r o r o t h e r w i s e p r e c i p i t a t e t h e c o n f l i c t s i t u a t i o n . . T h e n t h e f i r s t w i l l h a v e t o c o n c e d e u n l e s s h e p r e f e r s c o n f l i c t . T h e p r o c e s s c o n t i n u e s l i k e t h i s . I f a g r e e m e n t c a n n o t b e r e a c h e d b y f u r t h e r n o g i t i a t i o n s , i . e . , i f e i t h e r b a r g a i n e r i s n o t p r e p a r e d t o m a k e f u r t h e r c o n c e s s i o n s , t h e a n n o u n c e d t h r e a t s ( a t t h e c o n f l i c t p o i n t ) w i l l b e c a r r i e d o u t . H o w e v e r , t h i s i s n o t r e s t r i c t e d b a r g a i n i n g ( c f . C o n t i n i [ 1 9 6 8 ] ) i n t h e s e n s e t h a t t h e r e i s n o p r e s p e c i f i e d t i m e l i m i t w i t h i n w h i c h t h e a g r e e m e n t m u s t b e r e a c h e d a t p e r i l o f c a u s i n g a p r e a n n o u n c e d " i m p o s e d " s o l u t i o n t o b e i m p o s e d . H o w e v e r , u s u a l l y t h e p r e s e n c e o f c o s t s o f b a r g a i n i n g i m p l i c i t l y s e t s a n u p p e r l i m i t o n t i m e o r n u m b e r o f b i d s . I n d e a l i n g w i t h t h e b a r g a i n i n g p r o b l e m u n d e r i n c o m p l e t e i n f o r m a t i o n , w e a s s u m e e a c h b a r g a i n e r h a s a s u b j e c t i v e 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 i s o p p o n e n t ' s u t i l i t y w h i c h i s u n k n o w n t o h i m . A s b a r g a i n i n g c o n t i n u e s , g e n e r a l l y , 102 e a c h b a r g a i n e r u p d a t e s h i s p r i o r s b y i n c o r p o r a t i n g t h e i n f o r m a t i o n a v a i l a b l e t o h i m a t e a c h s t a g e . I n S e c t i o n 5 . 3 w e s h a l l s o l v e t h e p r o b l e m w h e n b o t h i n d i v i d u a l s u s e o p t i m a l s t r a t e g i e s f o r b a r g a i n i n g , i . e . w h e n t h e y c o n c e d e i n a n o p t i m a l f a s h i o n . U n t i l t h e n , i . e . i n S e c t i o n 5 . 2 , w e t r e a t t h e p r o b l e m w h e r e o n l y o n e b a r g a i n e r d o e s s o w h i l e t h e o t h e r m a k e s h i s c o n c e s s i o n s a c c o r d i n g t o a r u l e w h i c h i g n o r e s h i s o p p o n e n t ' s o f f e r s e q u e n c e e n t i r e l y . I n d o i n g s o , w e a r e b e t t e r a b l e t o g l e a n i n s i g h t s a b o u t t h e p r o b l e m a n d l a t e r t o h a n d l e t h e m o r e c o m p l i c a t e d p r o b l e m b y a p p r o a c h i n g i t i n t w o s t e p s . 5 . 2 O P T I M A L S T R A T E G Y W H E N T H E O P P O N E N T ' S S T R A T E G Y I S P R E P L A N N E D 5 . 2 . 1 N O T A T I O N A N D T H E M O D E L : C o n s i d e r t w o i n d i v i d u a l s b a r g a i n i n g t o d e t e r m i n e a v a l u e f o r t h e v a r i a b l e x , a s a b o v e . T h e s e t o f a l l v a l u e s o f x a v a i l a b l e f o r t h e t r a n s a c t i o n i s d e n o t e d b y X , We a s s u m e x , t h e r a n g e o f x , i s f i x e d t h r o u g h o u t t h e b a r g a i n i n g p r o c e s s ( h e n c e , f o r i n s t a n c e , x i s n o t a f f e c t e d b y t h e c o s t s o f b a r g a i n i n g ) . A s b a r g a i n i n g c o n t i n u e s t h e b a r g a i n e r s a l t e r n a t i v e l y s u g g e s t v a l u e s f o r x . T h e p r o c e s s m a y e n d i n o n e o f t w o w a y s : 1 . O n e o f t h e b a r g a i n e r s p r e f e r s a p r o p o s e d x t o a n y o t h e r a l t e r a n t i v e ( c o n f l i c t o r c o n t i n u a t i o n o f b a r g a i n i n g ) i n w h i c h c a s e a g r e e m e n t i s r e a c h e d , o r 2 . O n e o f t h e b a r g a i n e r s p r e f e r s t h e c o n f l i c t s i t u a t i o n o v e r a n y o t h e r a l t e r n a t i v e . S u p p o s e B a r g a i n e r 1 c o n c e d e s a c c o r d i n g t o a r u l e k n o w n o r u n k n o w n t o h i s o p p o n e n t , w h i c h i g n o r e s h i s o p p o n e n t ' s o f f e r s e q u e n c e . A s s u m e h e m a k e s t h e f i r s t o f f e r , h i s o f f e r s e q u e n c e b e i n g (XQ, X^, X^, ) . S u b s e q u e n t r e p e t i t i o n a t s o m e s t a g e o f a n o f f e r i n t h i s s e q u e n c e s i g n i f i e s h e i s w i l l i n g t o a c c e p t c o n f l i c t a t t h a t s t a g e . 103 B a r g a i n e r 2 ' s o f f e r s e q u e n c e i s d e n o t e d b y ( x ^ , x ^ , x,_ ) . B a r g a i n e r 2 h a s a s u b j e c t i v e 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 ( x 2 » x 4 » ) • D e n o t e t h e p r i o r d i s t r i b u t i o n o f x ^ g i v e n ( x ^ , x ^ , x ^ , ) , b y TT^, i = 2 , 4 , 6 , B a r g a i n e r 2 ' s i n c o m p l e t e i n f o r m a t i o n o n ( x 2 , x ^ , x ^ . , ) m a y b e d u e t o h i s i n c o m p l e t e i n f o r m a t i o n a b o u t h i s o p p o n e n t ' s u t i l i t y f u n c t i o n o r p a r a m e t e r s r e l a t e d t o t h i s f u n c t i o n , i n w h i c h c a s e { i r ^ } m a y b e i n d u c e d b y B a r g a i n e r 2 ' s s u b j e c t i v e b e l i e f s a n d i n f o r m a t i o n o n s u c h u n c e r t a i n t i e s . A f e w e x a m p l e s w i l l h e l p c l a r i f y t h e s e i d e a s . EXAMPLE 5 . 1 . B a r g a i n e r 1 o f f e r s ( d e m a n d s ) XQ i n i t i a l l y a n d B a r g a i n e r 2 s u g g e s t s v a r i o u s x ' s , i f h e i s n o t s a t i s f i e d w i t h XQ . B a r g a i n e r 1 a c c e p t s B a r g a i n e r 2 ' s f i r s t f a v o r a b l e o f f e r i f a n y o f t h e o f f e r s i s f a v o r a b l e a n d o t h e r w i s e r e f u s e s a l l t h e o f f e r s . I n t h i s e x a m p l e XQ = x ^ = x ^ = . . . a r e k n o w n t o B a r g a i n e r 2 . E X A M P L E 5 . 2 . B a r g a i n e r 1 c o n c e e d s i n s u c h a w a y t h a t h i s a n t i c i p a t e d u t i l i t y U ( x ) d e c r e a s e s i n a s p e c i f i e d w a y , s a y l i n e a r l y . A s b a r g a i n i n g p r o c e e d s , h e a c c e p t s t h e f i r s t o f f e r w h i c h y i e l d s m o r e t h a n h e e x p e c t s f r o m t h e n e x t x . S u p p o s e B a r g a i n e r 2 k n o w s B a r g a i n e r l ' s p o l i c y b u t n o t t h e e x a c t v a l u e s o f c e r t a i n p a r a m e t e r s l a b e l l i n g U ( x ) . I n t h i s c a s e h e w o u l d d e r i v e t h e p r i o r s , f r o m h i s s u b j e c t i v e d i s t r i b u t i o n s o n s u c h u n k n o w n p a r a m e t e r s . L e t i{i = ( x Q , x 2 , * 2 ^ ) , (("j = ( x 1 > x 2 j _ i ) > a n d l e t A^(I|J^, (JK) r e p r e s e n t , c o m p a c t l y , t h e t o t a l a v a i l a b l e i n f o r m a t i o n t o B a r g a i n e r 2 a t t h e i t h . s t a g e o f b a r g a i n i n g , a n d <|> h a v i n g b e e n 1 0 4 o b s e r v e d . m a y m e r e l y c o n s i s t o f t h e o b s e r v e d v a l u e s o f ib^ a n d o r i t m a y i n v o l v e s o m e a d d i t i o n a l i n f o r m a t i o n a s w e l l , d e r i v e d f r o m f a c t s s u c h a s B a r g a i n e r l ' s a t t i t u d e a b o u t c e r t a i n p o s s i b l e o f f e r s , e t c . F o r i n s t a n c e , i n e x a m p l e 5 . 1 A ^ w o u l d c o n t a i n t h e i n f o r m a t i o n * K x j ) — 0 f o r a l l j = 1 , 3 , 5 , a n d x ^ = x 2 = x ^ = . . . = x ^ . L e t l ? i ^ x 2 i + l ^ a n d Q±(K21+l) d e n o t e B a r g a i n e r 2 ' s c o n d i t i o n a l p r o b a b i l i t i e s , d e r i v e d f r o m h i s s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s g i v e n A ^ , t h a t B a r g a i n e r 1 w i l l a c c e p t *21+1 a n d that ^ e m a k e a n e w o f f e r , r e s p e c t i v e l y . F o r i n s t a n c e , i n e x a m p l e 5 . 1 , = P ( U ( x 2 _ ^ + ^ ) > ^ » U ( x ^ ) <_ 0 f o r a l l j = 1 , 3 , 2 i - l ) = 1 - Q ± . L e t C j b e t h e t o t a l c o s t i n c u r r e d b y B a r g a i n e r 2 a f t e r b i d d i n g j t i m e s . F o r s i m p l i c i t y a s s u m e h i s c o s t o f b a r g a i n i n g i s i n d e p e n d e n t o f h i s o f f e r s a n d t h a t C . < C . , . . f o r a l l j = 0 , 1 , 2 . . . w i t h C n > 0 . 3 - 3 + 1 0 -F r o m n o w o n B ^ a n d B 2 w i l l r e p r e s e n t f o r B a r g a i n e r 1 a n d B a r g a i n e r 2 , r e s p e c t i v e l y . 5 . 2 . 2 C R I T E R I O N F O R C O M P A R I S O N O F B A R G A I N I N G S T R A T E G I E S A t e a c h s t a g e 5 i , o f b a r g a i n i n g , B 2 h a s t w o b a s i c p r o b l e m s : G i v e n A_^, ( i ) s h o u l d h e a c c e p t t h e x , ^ o f f e r e d b y h i s o p p o n e n t , q u i t t h e g a m e , o r c o n t i n u e b a r g a i n i n g a n d ( i i ) w h a t s h o u l d b e ^21+1' t * i e n e x t s u i t a b l e b e t , i n c a s e h e d e c i d e s t o c o n t i n u e b a r g a i n i n g . W e d e f i n e t h e f o l l o w i n g c o n c e p t s b e f o r e c o n s i d e r i n g t h e s e p r o b l e m s f u r t h e r . 1 0 5 T h e B a r g a i n i n g S e q u e n c e , <j> : A s e q u e n c e o f f u n c t i o n s <f> = {X^(AQ ) , x ^ ( A ^ ) , x , . ( A 2 ) , . ••} w i t h x_^ e X f o r e v e r y i . S h o u l d B 2 d e c i d e t o c o n t i n u e b a r g a i n i n g a t s t a g e i , x 2 i + l ^ s ^ s n e x t ° f f e r -T h e S t o p p i n g R u l e , E : A s e q u e n c e o f f u n c t i o n s E = { E Q ( A 0 ) , E 1 ( A 1 ) , E 2 ( A 2 ) , . . . } w i t h E^ d e f i n e d o n ty2 a n d 0 <^ <_ 1 f o r a l l j , w h e r e E_. r e p r e s e n t s 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 B „ w i l l c e a s e b a r g a i n i n g a t s t a g e j , g i v e n b o t h A. a n d t h a t b a r g a i n i n g h a s c o n t i n u e d u p t o t h a t s t a g e . W e a v o i d u n i m p o r t a n t s t o p p i n g r u l e s , a n d a s s u m e t h a t , w h e n e v e r B 2 c e a s e s b a r g a i n i n g , s a y a t s t a g e i , h e s e l e c t s a s h i s a c t i o n , o u t o f t h e a v a i l a b l e p a i r , t h a t w h i c h g i v e s h i m t h e h i g h e r u t i l i t y , i . e . h e c h o o s e s t h e c o n f l i c t s i t u a t i o n i f V ( x 2 ^ ) ± 0 a n d a c c e p t s ( o f f e r e d b y B ^ ) o t h e r w i s e w i t h t h e c o n v e n t i o n t h a t B ^ p r e f e r s s t o p p i n g o f z e r o u t i l i t y t o c o n t i n u a t i o n o f n o n p o s i t i v e u t i l i t y . T h e p a i r (tj>, E ) w i l l b e c a l l e d t h e " b a r g a i n i n g p l a n " o r " p l a n " o f B 2 • B 2 ' s o v e r a l l e x p e c t e d g a i n , b e f o r e h e h a s c o m m e n c e d b i d d i n g u s i n g a p l a n (cf), E ) i s t h e n e a s i l y e x p r e s s e d a s G(<{>, E ) = / . / V ^ ( 0 ) E 0 + ( 1 - E Q ) [ V C X j ) + Q 0 [ V ( X 2 ) E 1 + ( 1 - E ^ [ V ( x ) + Q 1 [ V ( X 4 ) E 2 + ( 1 - E 2 ) [ V ( x 5 ) + Q 2 [ ] ] . . . ] d T r ( X £ | A ^ d v ^ | A 2 > . 1 0 6 p r o v i d e d t h i s e x i s t s * w h e r e V ( x . ) d e f i n e d b y V ( x „ ,..) = V ( x ~ M ) P - C r 1 J 2 n + l 2 n + l n n a n d V ( x n ) = V ( x 0 ) v 0 - C / f o r n = 1 , 2 , 3 , . . . i s B 2 ' s e x p e c t e d z n 2 n n - 1 u t i l i t y f r o m x ^ . T h i s i s b e c a u s e i f i i s e v e n , B 2 h a s t o p a y a c o s t o f C . „ i n a d d i t i o n t o t h e g a i n V ( a . ) v 0 a n d i f i i s o d d h e w o u l d b e 1 - 2 b x 2 l i a b l e t o a c o s t o f ^ a n d t o a g a i n o f V ( x ^ ) w i t h p r o b a b i l i t y ( t h e p r o b a b i l i t y B l a c c e p t s x ^ ) . S i m i l a r l y h i s o v e r a l l e x p e c t e d g a i n w h e n h e i s a t a n i n t e r m e d i a t e s t a g e j i s g i v e n b y , G.($2, E J ) = / . . . / V ( x 2 j ) E j + ( 1 - ^ ) [ V ( x 2 j + 1 ) + Q j [ V ( X 2 j + 2 ) ] V l + ( 1 " V l } [ ^ ( X 2 i + 3 ) + V l [ ] - - - ] d ^ X 2 j + 2 l / A i ) d ^ ( X 2 j + 4 l V l ) ( 5 . 1 ) w h e r e <j>J = ( x 2 j + i > x 2 j + 3 ' ' a n d = E j + 1 ' " ' ^ = ° ' 1 ' 2 ' w i t h $ ° = <)> a n d E ° = E . /V A, D e f i n i t i o n : A p l a n (<j), E ) i s s a i d t o b e u n i f o r m l y b e t t e r t h a n a p l a n (<(>, E ) i f f (cf)- 3 , E J ) £ G ^ . ( ( f ) J , E 3 ) f o r a l l j = 0 , 1 , 2 , w i t h s t r i c t i n e q u a l i t y f o r s o m e j . I f t h e r e e x i s t (<))*, E * ) s u c h t h a t G j ( ^ > Z2) 1 G.. (<f)* J , E * J ) f o r a l l c o n c e i v a b l e ( $ , E ) , t h e n ( $ * , E * ) i s s a i d t o b e a b e s t p l a n . 1 . a v b . = M a x ( a , b ) 107 5 . 2 . 3 B A C K W A R D I N D U C T I O N P R O C E D U R E ( B I P ) We w i l l f i n d a b a r g a i n i n g p l a n b y a p p l y i n g B a c k w a r d I n d u c t i o n " ' " a n d d e t e r m i n e w h e t h e r i t i s a b e s t p l a n . F i r s t t r u n c a t e , a r t i f i c i a l l y , t h i s b a r g a i n i n g p r o b l e m a t s o m e i n t e g e r N , i . e . w e i m p o s e t h e r e s t r i c t i o n t h a t n o i n d i v i d u a l c a n b a r g a i n a f t e r s t a g e N . W e w i l l i n v e s t i g a t e w h a t h a p p e n s w h e n N -> °° , w i t h t h e h o p e t h a t t h e g e n e r a l p r o b l e m w i l l b e w e l l a p p r o x i m a t e d b y t h e t r u n c a t e d o n e w h e n N i s s u f f i c i e n t l y l a r g e . We w i l l s h o w t h a t t h e p r e s e n c e o f b a r g a i n i n g c o s t s w i l l u s u a l l y g i v e r e s u l t n a t u r a l l y i n a t r u n c a t e d p r o b l e m t h u s p r o v i d i n g u s a n u p p e r b o u n d f o r t h e p e r i o d f o r w h i c h t h e b a r g a i n i n g w i l l l a s t , w h o s e b e s t p l a n i s t h e s a m e a s t h a t o f t h e g e n e r a l n o n t r u n c a t e d p r o b l e m . I n t h e t r u n c a t e d p r o b l e m , t h e a v a i l a b l e s t o p p i n g p l a n s a r e a l l s u b j e c t t o t h e r e s t r i c t i o n *1 X3 X2N +1 F I G U R E 5 . 1 B A R G A I N E R S ' O F F E R S E Q U E N C E S S u p p o s e t h e b a r g a i n i n g p r o c e s s h a s b e e n c o n t i n u e d u p t o B 2 ' s N t h . s t a g e . H e t h e n h a s t h e f o l l o w i n g c h o i c e o f a c t i o n s . H e c a n e i t h e r c e a s e b a r g a i n i n g o r m a k e a n e w o f f e r X 2 N + 1 " "*" n t * i e - * " a t t e r c a s e » t w o c o n s e q u e n c e s a r e p o s s i b l e . H i s e x p e c t e d g a i n w i l l b e ^ ^ x 2 N + 1 ^ ^ ^ a c c e P t s l t : > a n d z e r o o t h e r w i s e , b e c a u s e n o m o r e b i d d i n g i s a l l o w e d . F u r t h e r m o r e i f h e d o e s . 1 . F o r a d i s c u s s i o n o f B a c k w a r d I n d u c t i o n i n a s t a t i s t i c a l s e t t i n g c f . F e r g u s o n [ 1 9 6 7 ] 108 m a k e a n e w o f f e r h e w i l l h a v e t o u n d e r g o a c o s t ( C ^ - C ^ _ ^ ) , i n a d d i t i o n t o t h e a m o u n t t h a t h e h a s a l r e a d y a b s o r b e d . T h e m a x i m u m e x p e c t e d g a i n N i n t h e e v e n t h e d o e s o f f e r a n e w a m o u n t , d e n o t e d b y E „ ( ^ „ , <b„) , i s N N N ( 5 . 2 ) O n t h e o t h e r h a n d i f h e a c c e p t s x 2 J J > t h u s b r i n g i n g t h e b a r g a i n i n g t o a n e n d , h i s e x p e c t e d g a i n w i l l b e V ( X 2 N ) = V ^ X 2 N ^ ^ ® ~ S j - 1 ( n ° t a t i o n : a v b = M a x { a , b } ) . T h e B I P t e l l s u s h e r e t o t a k e t h e a c t i o n w h i c h y i e l d s N t h e b i g g e r o f a n d V ( x 2 ^ ) ; i n c a s e t h e s e q u a n t i t i e s a r e e q u a l , t h e - N s t o p p i n g a c t i o n w i l l b e t a k e n . I n o t h e r w o r d s , s t o p i f ^ ( x 2 ^ ^ — ^ a n d o t h e r w i s e o f f e r a n e w a m o u n t w h i c h y i e l d s ( 5 . 2 ) , l e a v i n g t h e o p p o n e n t t o c h o o s e o n e f r o m t h e c l a s s o f e q u i v a l e n t p o s s i b i l i t i e s , i . e . m o r e p r e c i s e l y , o t h e r w i s e , w h e r e H ^ ( ^ , y = v ( x 2 N ) v E ^ , y ( 5 . 3 ) t h N o w i f B 2 i s a t ( N - 1 ) s t a g e , w i t h B l ' s o f f e r X 2 N _ 2 u n d e r c o n s i d e r a t i o n , a g a i n h e c a n e i t h e r s t o p o r o f f e r a n e w a m o u n t , x 2 N _ I * ~*"N 109 t h e l a t t e r c a s e h i s e x p e c t e d u t i l i t y i s v ( x 2 N _ i ^ l f B 1 a c c e P t s o r q u i t s b a r g a i n i n g , a n d E { H ^ ( X 2 N , ^_±, * N > ' ^ - 1 * i f B l m a k e s a n e w o f f e r . T h e r e -f o r e x „ „ » m u s t b e c h o s e n s o t h a t , 2N-1 2N-1 E { < ( X 2 N ' W V l V i } N N i s a c h i e v e d . B2 w i l l c e a s e b a r g a i n i n g a t t h i s s t a g e i f f >_ E j ^ , w h e r e , N V l ( * H ' W = ^ ( X 2 N - 2 } v ' W V l ' * N - 1 } . t h H e n c e B 2 ' s m a x i m u m o v e r a l l e x p e c t e d g a i n a t j s t a g e , g i v e n A j d u e t o t h e B I P m a y b e i n d u c t i v e l y d e f i n e d b y , 1 1 3 2-1 J J 1 ( 5 . 4 ) w h e r e , E . ( . . , * . ) - V^o,,, . ) + ( - 2 j : + l . ) E { H j + l ( X 2 j + 2 ' V W ' V ( 5 . 5 ) W e a s s u m e t h a t t h i s m a x i m u m i s a t t a i n e d i n X . H e n c e , w e c o n c l u d e , 1 1 0 P r o p o s i t i o n 5 . 2 . 1 ( i ) E - { E Q , E 1 > E N > £ N + 1 } w h e r e 3 1 i f V ( x ) > E*J(ifr , <(. ) 4J) — 3 3 3 o t h e r w i s e ( 5 . 6 ) i s a s t o p p i n g r u l e g i v e n b y t h e B I P , f o r t h e t r u n c a t e d p r o b l e m ( i i ) § — { x .^ , • • • > X 2 N + 1 ^ ' w h e r e x „ . n y i e l d s t h e m a x i m u m v a l u e i n e q u a t i o n ( 5 . 5 ) ; j - 0 , 1 , . . . N - l a n d x 2 N + 1 y i e l d s ( 5 . 2 ) , i s a b a r g a i n i n g s e q u e n c e g i v e n b y t h e B I P . T h e o r e m 5 . 2 . 2 . A b a r g a i n i n g p l a n g i v e n b y t h e B I P i s a b e s t p l a n f o r t h e t r u n c a t e d b a r g a i n i n g p r o b l e m . P r o o f ; L e t (cf>*, I * ) b e t h e p l a n g i v e n b y B I P a n d (<f>, E ) b e a n y o t h e r p l a n . R e c a l l t h a t B 2 ' s o v e r a l l e x p e c t e d g a i n u t i l i z i n g (<|>, E ) , w h e n h e . t h c , . . . i s a t j s t a g e o f b a r g a i n i n g i s , I l l G^*3, z 3 ) = / . . . / v ( x 2 j ) s j + ( l - ^ ) [ v ( x 2 j + 1 ) + Q j t V C X ^ ^ ) E N + ( 1 - V [ ^ ( x 2 N + l ) ] - - - ] d u ( X 2 j + 2 | A 3 ) ' - - d l T ( X 2 N j i V l ) ' B y d e f i n i t i o n o f t h e r a n d o m v a r i a b l e i n ( 5 . 2 ) i t i s c l e a r t h a t E ^ >_ V ( X 2 N + 1 ) w h a t e v e r b e X 2 N + 1 . N o w f r o m t h e d e f i n i t i o n o f i n ( 5 . 3 ) i t f o l l o w s t h a t , ^ 2 N i ) Z N + ( 1 " V ^ ^ ( X 2 N ) E N + ( 1 " V E N 1 H N f o r X 2 N ' t h u s I m p l y i n g t h a t , G.U 3, i j ) < / . . . / v ( x 2 . ) z . + ( l - ^ ) [ v ( x 2 . + 1 ) + Q j [ v ( x 2 j + 2 ) z j + 1 + ( 1 - + ( 1 - S N _ 1 ) [ ^ ( X 2 N - 1 ) + Q N - 1 ^ • • • ] d ^ X 2 j + 2 l A j ) - - - d 7 r ( X 2 N ' V i 112 A g a i n v ( x 2 N + 1 ^ N - 1 E A ' E N - 1 f r o m ( 5 ' 5 ) a n d » t h e r e f o r e s t e p s t a k e n 1 ' N - l a b o v e f u r t h e r r e d u c e t h e n u m b e r o f t e r m s i n t h e a b o v e i n e q u a l i t y . W h e n t h i s r e d u c t i o n i s c a r r i e d o u t i n d u c t i v e l y w e , c l e a r l y , e n d u p w i t h t h e i n e q u a l i t i e s , G . ( c f > j , E J ) < V ( x „ . ) E . + ( 1 - E . ) E N < H N f o r a l l j = 0 , 1 , . . . N . Y e t , e a c h o f t h e a b o v e i n e q u a l i t i e s w i l l c l e a r l y b e c o m e e q u a l i t i e s w h e n <{> = <)>* a n d E = E * a s c o n s e q u e n c e s o f t h e i r d e f i n i t i o n g i v e n i n P r o p o s i t i o n 5 . 2 . 1 , t h u s p r o v i n g t h a t (<!>*, £ * ) i s a b e s t p l a n . H a v i n g f o u n d a b e s t p l a n f o r t h e t r u n c a t e d b a r g a i n i n g p r o b l e m , w e n o w n e e d t o c o m p l e t e o u r a n a l y s i s w i t h a t h e o r e m s t a t i n g t h a t t h e n o n t r u n c a t e d p r o b l e m c a n b e a p p r o x i m a t e d a r b i t r a r i l y w e l l f o r s u f f i c i e n t l y l a r g e N b y t h e t r u n c a t e d o n e . L e t G_. a n d G.. d e n o t e t h e m a x i m u m o v e r a l l e x p e c t e d g a i n s o f B 2 a t t h e s t a g e j o f t h e t r u n c a t e d ( a t N ) a n d t h e n o n t r u n c a t e d p r o b l e m s , - N N N * j N * j N * N * r e s p e c t i v e l y . W e h a v e s e e n t h a t G ^ = G_. ( <f>, E ) , w h e r e ( tj), E ) i s t h e p l a n g i v e n b y B I P . S u p p o s e t h a t t h e r e e x i s t s a p l a n (<j>°, E ° ) f o r t h e - 0 0 - N g e n e r a l p r o b l e m w h i c h a c h i e v e s G ^ f o r a l l j . O b v i o u s l y G ^ i s m o n o t o n i c -c a l l y i n c r e a s i n g i n N . I n f a c t , i f w e a s s u m e t h a t X , t h e r a n g e o f x , i s b o u n d e d o r t h a t B 2 ' s u t i l i t y f u n c t i o n i s b o u n d e d a b o v e s o t h a t M a x V ( x ) = m < 0 0 , t h e s e q u e n c e {G 1 ? } w i l l b e b o u n d e d f r o m a b o v e ; i . e . , x e X J 1 1 3 G ° < G 1 < G 2 . . . < m . H e n c e l i m G^? e x i s t s ( a n d i s f i n i t e ) . O u r q u e s t i o n j - j - j - N J — C O i s w h e t h e r t h i s l i m i t i s t h e s a m e a s G.. f o r a l l j . L E M M A 5 . 2 . 3 . L e t { X } b e a n i n f i n i t e s e q u e n c e o f r a n d o m v a r i a b l e s . n ^ i ^ X i ^ " ' " S a ^ u n c t : ' - o n ° f x i s u c h t h a t 0 <_ ^ ( x ^ ) ! . 1 w i t h p r o b a b i l i t y 1 . CO I f E { l / £ i ^ . ( x ) } = 0 ( t r u e , i n p a r t i c u l a r i f 1 1 ± N Z E ( l / i | i . ) l i m = 0 , b e c a u s e n / ( y 1 + Y 2 + • • •+ y n ) < + + ' " + 1 / v f o r y . > 0 b y c r o s s m u l t i p l i c a t i o n ) t h e n , J n x — N l i m E { n ( l - i | i . ( X . . ) ) } = 0 N - *» 1 1 1 P r o o f : W r i t e , N I I n ( 1 - ; M X ) ) N 1 0 <_ E { n ( l - \\>. ( X , ) ) } = E { e } 1 1 1 x N - E * 1 ( X ± ) < E e 1 b e c a u s e l n ( l - a ) < - a f o r 0 £ a £ 1 < E N £ * . ( X . ) i i i -b e c a u s e e y < — f o r y > 0 . 4 ^ W ' 1 1 4 N T a k i n g l i m i t s a s N 0 0 , w e n o w d e d u c e , l i m E { n ( 1 - i f ) . ) } = 0 a s w a s N-**> 1 1 t o b e p r o v e d . T h e o r e m 5 . 2 . 4 : I f E { l / E P ? ( x „ . , .. ( X „ . ) ) } = 0 ( t r u e , i n p a r t i c u l a r i f l 2 x + l 2 x N X p ° f o o i n p r o b a b i l i t y o r , i f E E ( l / P = p ) i s c o n v e r g e n t ) , a n d i f 1 1 1 1 0 0 o i o i G j (4> , I ) e x i s t s , i t h e n i - f o r B a c h j , G N ( V J , N E * j ) ^ G " ( + o j ( E o j } a s N -> 0 0 , w h e r e ( ( j>° , E ° ) i s a n o p t i m a l s o l u t i o n t o t h e g e n e r a l p r o b l e m •vr ^ "NT a n d ( § , E ) i s a p l a n g i v e n b y t h e B I P , t o t h e p r o b l e m t r u n c a t e d a t N . P r o o f : L e t <j>° = ( x ° , x ° , . . . ) a n d E ° = ( E ° , E ° , . . . ) . C o n s i d e r t h e N o N o , N * N * p l a n ( (J) , E ) ( t h i s i s n o t n e c e s s a r i l y t h e s a m e a s ( <(> , E ) ) o b t a i n e d b y t r u n c a t i n g ($°, E ° ) a t N , i . e . N ( j>° A_ ( x ° , x ° , x 2 N + I ) A N < ^ O O O O 0 0 o E A ( Z , E , , . . . , E „ - , E E . ) . C o n s i d e r t h e d i f f e r e n c e b e t w e e n t h e o v e r -— o 1 N - l . „ x x = N a l l e x p e c t e d g a i n s o f B 2 a t j t h . s t a g e u s i n g t h i s t r u n c a t e d p l a n a n d t h e o p t i m a l n o n t r u n c a t e d p l a n . A f t e r s o m e c a n c e l l a t i o n s w e g e t , 1 1 5 0 < G . ° ( * O J , E° J) - G N ( V J , N E ° J ) = E [ V ( 1 - E°) Q ° { [ V ( X 2 N ) E ° + ( 1 - E ° ) 3 1 i = - i o o o i * < W + E N + I + ( 1 - W [ • • • ] . . . ] > - t i v c v J / i + E°±) V ( x ° N + 1 ) } ] x = N N-l = E { [ n ( l - E°) Q°] h ( x 2 N + 2 , x 2 N + 4 , ...)} i = j s a y . . S i n c e m = M a x V ( x ) < 0 0 i s a n u p p e r b o u n d f o r t h e m a x i m u m u t i l i t y B 2 x e X — C O c a n c o n c e i v a b l y a n t i c i p a t e f r o m t h e p r e v a i l i n g t r a n s a c t i o n , w e h a v e G ^ <^  m a n d i n p a r t i c u l a r h <^  m . H e n c e f o r e a c h j , 0 < G ~ ( 4 ° j , E ° J ) - G N ( V 3 , V j ) < m E { V ( 1 - E°) 0 ° 3 3 i = j N - l < m E { n Q ? } s i n c e 0 < E . < 1 . — . . x — x — i = 3 N - l < m E { n ( 1 - P ° ) } i = 3 B u t t h e f i r s t h y p o t h e s i s o f t h e o r e m 5 . 2 . 4 e n a b l e s L e m m a 5 . 2 . 3 t o b e u s e d , . N - l g i v i n g E { II ( 1 - P ? ) } -> 0 a s N -»- °° , a n d t h e r e f o r e 3 116 G ^ ° j , E ° j ) - G ^ ( V J , V j ) -> 0 a s N - - ( 5 . 7 ) H e n c e g i v e n e > 0 , ] N q s u c h t h a t , 0 <{G~(t°K E ° j ) - G ^ ( V J , V j ) } = {G~(<f , ° j , E ° j ) - G * ? ( V J , V j ) } + { G ^ ( V J , N E ° j ) - GJ(V J , V j ) } _< { G j ( < i ) O J , E ° J ) - Gj(N<f> 0 : i, N E ° 3 ) } , b e c a u s e (<f)*,E * ) i s a b e s t p l a n f o r t h e t r u n c a t e d p r o b l e m . <_ e f r o m ( 5 . 7 ) , f o r a l l N _> N . ' o e c a u s e (<J' ;- S E « ) i s a :.= t ~ ' . . r ? . : : i * " r 1 ' O u r c l a i m f o l l o w s f r o m t h i s . <_ e f r o m ( 5 o 7 ) P i - a " l ' ... _ - i - j r c l a i m . 00 Q T h e a s s u m p t i o n , E { l / E P..} s e e m s v e r y r e a s o n a b l e , s i n c e 1 N E P . + °° i n p r o b a b i l i t y i s s u f f i c i e n t f o r i t t o b e s a t i s f i e d . R e c a l l i n g 1 1 t h a t P i s t h e p r o b a b i l i t y t h a t B I w o u l d a c c e p t t h e i t h . o f f e r m a d e b y B 2 , i t i s i n t u i t i v e t h a t a n y e f f i c i e n t p l a n w o u l d y i e l d P ° 1 i n N p r o b a b i l i t y , t h u s i m p l y i n g E P . t 0 0 i n p r o b a b i l i t y . T h u s , w h a t w e h a v e 1 1 s h o w n i s t h a t , w h e n N i s s u f f i c i e n t l y l a r g e , t h e t r u n c a t e d p r o b l e m a p p r o x i m a t e s t h e g e n e r a l o n e . I n f a c t , p r e s e n c e o f c o s t o f b a r g a i n i n g u s u a l l y m a k e s t h i s a p p r o x i m a t i o n e x a c t , t h u s p r o v i d i n g u s a m e t h o d f o r f i n d i n g t r u n c a t e d o p t i m a l s o l u t i o n s t o t h e g e n e r a l n o n t r u n c a t e d p r o b l e m . W e s h a l l e s t a b l i s h t h i s u n d e r t h e f o l l o w i n g r e a s o n a b l e a s s u m p t i o n " A " w h i c h w i l l b e p r o v e d t o h o l d f o r a p a r t i c u l a r c a s e : 117 ( A ) : M a x [ V ( x 2 j _ 1 ) P j + ( 1 - P . . ) E { V ( X 2 j ) I A j _ 1 > ] < E { V ( X 2 j ) I A ^ } x „ . T e X f o r a l l j • P r o p o s i t i o n 5 . 2 . 5 : L e t X b e a r a n d o m v a r i a b l e h a v i n g t h e p r o b a b i l i t y d e n s i t y f u n c t i o n f ( x ) . L e t V ( x ) b e a s t r i c t l y i n c r e a s i n g f u n c t i o n s u c h t h a t E V ( x ) < °° . I f t h e f u n c t i o n g ( y ) = V ( y ) P ( y > X ) + E V ( X ) P ( y < X ) i s m a x i m i z e d a t y Q , t h e n V ( y Q ) < E V ( X ) ( a n d s o g ( y Q ) < E V ( X ) ) P r o o f : S u p p o s e f i s d e f i n e d o n ( - °° , °°) ; t h e n g ( y ) = V ( y ) fy f ( x ) d x + E V ( X ) f ( x ) d x a n d , t h e r e f o r e , g'(yD) = v'(yo) f ( x ) d x + f ( x ) [ V ( y Q ) - E V ( X ) ] = 0 118 S i n c e a l l f i r s t t h r e e t e r m s o f t h e a b o v e e x p r e s s i o n a r e p o s i t i v e , w e m u s t h a v e f v ( y o ) " E V ( X ) ] < 0, a s w a s t o b e p r o v e d . H e n c e i f X i s a s u b s e t o f t h e r e a l l i n e a n d X , ^ i s a n a b s o l u t e l y c o n t i n u o u s r a n d o m v a r i a b l e , i n p a r t i c u l a r , t h e a s s u m p t i o n A t r i v i a l l y f o l l o w s f r o m P r o p o s i t i o n 5.2.5 . T h e o r e m 5.2.6 S u p p o s e G 1? ->- G ? f o r e a c h j = 0, 1, . . . . I f , f o r a l l 3 3 i > N , a l m o s t s u r e l y w e h a v e o E { V ( X 2 . + 2 ) | A . } - V ( x 2 . ) < C . - C l _ r (5.8) -N _ O O N Q A N Q ^ t h e n , u n d e r a s s u m p t i o n ( A ) , G' = G^. f o r a l l j , i . e . ( <j> , Z ) g i v e n b y t h e B I P i s a n o p t i m a l p l a n t o t h e n o n t r u n c a t e d p r o b l e m . P r o o f : C o n s i d e r t h e b a r g a i n i n g p r o b l e m t r u n c a t e d a t N > N q . F r o m (5.2) a n d ( A ) i t f o l l o w s t h a t , E N 1 E { V ( X2N +2 > I V " C N a n d t h e n f r o m (5.3) a n d (5.8) w e h a v e ( w i t h p r o b a b i l i t y o n e ) , 119 H N = ( V ( X 2 N > V ° " C N - 1 } V E N = ( V ( x 2 N ) v 0 - C ^ ) * < X 2 N > ' I n t u r n f r o m ( 5 . 5 ) a n d ( A ) w e h a v e , X 2 N - 1 £ K i ^ V ^ V i 1 " V i N -a n d c o n s e q u e n t l y I t f o l l o w s f r o m ( 5 . 8 ) t h a t , HJJ_^ = ^ x 2 N - 2 ^ " N -C o n t i n u i n g t h i s p r o c e s s , i n d u c t i v e l y , w e o b t a i n , R \ = ^ ( x , ^ ) - N f o r i = N , N - l , N . T h i s i m p l i e s G . ° = G . f o r a l l N > N a n d f o r o 1 3 o " N e a c h j . T h e n s i n c e G . G . a s N -> °° , w e h a v e G . = G . f o r e a c h j , 1 1 3 3 t h u s e n d i n g t h e p r o o f . T h e l a s t t h e o r e m e s s e n t i a l l y s t a t e s t h e i n t u i t i v e l y o b v i o u s p r o p o s i t i o n t h a t i f f r o m a c e r t a i n s t a g e o n w a r d , B l ' s e x p e c t e d c o n c e s s i o n r a t e f r o m B 2 ' s s t a n d p o i n t b e c o m e s s m a l l e r t h a n B 2 ' s r a t e o f c o s t , B 2 w i l l s u r e l y c e a s e b a r g a i n i n g b e f o r e t h a t s t a g e . H e n c e B 2 ' s o p t i m a l 120 b a r g a i n i n g p l a n f o r t h e n o n t r u n c a t e d p r o b l e m i s t r u n c a t e d w h e n , a s i s u s u a l l y t h e c a s e , t h e r e i s a l w a y s a c o s t a t t a c h e d t o t h e d e c i s i o n t o c o n t i n u e b a r g a i n i n g . 5 . 3 B A R G A I N I N G S T R A T E G I E S B Y B A C K W A R D I N D U C T I O N P R O C E D U R E 5 . 3 . 1 I N T R O D U C T I O N I n S e c t i o n 5 . 2 w e a s s u m e d B l ' s b a r g a i n i n g s t r a t e g i e s a r e d e t e r m i n e d b y a p r e s p e c i f i e d p l a n . T h e p r o b l e m t r e a t e d i n t h i s s e c t i o n d i f f e r s f r o m t h a t o f t h e p r e v i o u s o n e i n a s m u c h a s n o w b o t h b a r g a i n e r s a r e a l l o w e d t o u s e o p t i m a l s t r a t e g i e s . T h i s m e a n s t h a t , a t e v e r y s t a g e o f b a r g a i n i n g , a f t e r e a c h b a r g a i n e r l e a r n s h i s o p p o n e n t s o f f e r , h e m a k e s a n e w d e m a n d ( o f f e r ) o r q u i t s b a r g a i n i n g i n f a v o u r o f t h e c o n f l i c t s i t u a t i o n , t h e d e c i s i o n b e i n g m a d e i n s u c h a w a y t h a t h i s o v e r a l l e x p e c t e d g a i n , c o m p u t e d a t t h a t s t a g e , i s m a x i m i z e d . I t i s o u r p o s i t i o n t h a t a g i v e n b a r g a i n e r , b e h a v i n g r a t i o n a l l y , a s s u m e s h i s o p p o n e n t i s a l s o u s i n g o p t i m a l s t r a t e g i e s f o r b a r g a i n i n g . S u c h a n a n a l y s i s i s i n t h e s p i r i t o f t h a t i n v o n N e u m a n n a n d M o r g e n s t e r n ' s t h e o r y o f g a m e s [ 1 9 4 7 ] , a n d g i v e s t h e m i n i m u m a m o u n t e a c h b a r g a i n e r c a n e x p e c t f r o m t h e e x c h a n g e ; i n f a c t , h e c o u l d e x p e c t e x a c t l y t h i s a m o u n t i f h i s o p p o n e n t a l s o b e h a v e s r a t i o n a l l y . W e a l s o a s s u m e t h a t t h i s b a r g a i n e r , s a y B 2 , b e h a v e s a s i f h i s o p p o n e n t h a s c o m p l e t e i n f o r m a t i o n a b o u t t h e b a r g a i n -i n g s i t u a t i o n , ( a b o u t t h e e l e m e n t s i n v o l v e d i n c o m p u t i n g V ( x ) s u c h a s B 2 ' s u t i l i t y f u n c t i o n ) e v e n t h o u g h h i s o w n i n f o r m a t i o n i s i n c o m p l e t e . A s i n g a m e t h e o r y , w e j u s t i f y t h i s p o i n t o f v i e w o n t h e g r o u n d s t h a t B 2 m u s t a d a p t t h a t t o b e s a f e f r o m b e i n g f o u n d o u t . T h e n o t a t i o n u s e d i n t h i s 121 s e c t i o n i s t h e s a m e a s t h a t a d o p t e d i n S e c t i o n 5 . 2 ; t h e m o d e l i s t h e s a m e e x c e p t f o r t h e e l e m e n t s e x p l a i n e d a b o v e . W e c o n s i d e r t h e p r o b l e m f r o m B 2 ' s s t a n d p o i n t a n d a t t e m p t t o f i n d o p t i m a l s t r a t e g i e s f o r h i m ; t h e s o l u t i o n f o r B l i s f o u n d i n a s i m i l a r w a y . S i n c e e a c h b a r g a i n e r ' s s t r a t e g i e s d e p e n d o n e a c h o t h e r , B 2 w i l l n e e d t o k n o w h o w B l c o m p u t e s h i s s t r a t e g i e s . B 2 c a n n o t c o p y B l ' s c o m p u t a t i o n , h o w e v e r , b e c a u s e h e l a c k s k n o w l e d g e a b o u t B l ' s b e l i e f s , t a s t e s a n d p r e f e r e n c e s . W e t h e r e f o r e s u p p o s e B 2 h a s - 3 a s u b j e c t i v e 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 U ( x ) , B l ' s u t i l i t y f u n c t i o n , f o r e a c h x . T h i s p r o b a b i l i t y d i s t r i b u t i o n i s u s u a l l y d e r i v e d f r o m B 2 ' s p r i o r p r o b a b i l i t y d i s t r i b u t i o n s o n c e r t a i n p a r a m e t e r s l a b e l i n g ^ . U ( « ) . B 2 u p d a t e s t h e s e p r o b a b i l i t y d i s t r i b u t i o n s , a s b e f o r e , a s b a r g a i n i n g p r o c e e d s . L e t D_. d e n o t e t h e t o t a l c o s t t o B l a s a r e s u l t o f c o n t i n u a t i o n o f b a r g a i n i n g u p t h t o h i s j s t a g e ; B 2 h a s a s u b j e c t i v e 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 D_. t o o . W e n o w f i n d B 2 ' s o p t i m a l b a r g a i n i n g p l a n u s i n g t h e b a c k w a r d i n d u c t i o n p r o c e d u r e , a s b e f o r e . 5 . 3 . 2 B A C K W A R D I N D U C T I O N P R O C E D U R E A ' c u r l ' ( ^ ) d e n o t e s , i n t h i s s e c t i o n , e l e m e n t s n o t y e t s p e c i f i e d r e l a t e d t o B l ' s s t r a t e g i e s . I n o r d e r t o a p p r o a c h t h i s p r o b l e m a s i n S e c t i o n 5 . 2 , f i r s t t r u n c a t e t h e p r o c e s s a t s o m e i n t e g e r N , t h u s i m p o s i n g t h e r e s t r i c t i o n t h a t n e i t h e r b a r g a i n e r c a n b a r g a i n b e y o n d s t a g e N . S u p p o s e b a r g a i n i n g i s c a r r i e d o u t t o s t a g e N b y B 2 ; t h e b a r g a i n e r ' s o f f e r s e q u e n c e s a r e s h o w n i n F i g u r e 5 . 1 . A t s t a g e N B 2 c a n e i t h e r c e a s e b a r g a i n i n g a n d a c c e p t B l ' s o f f e r , x 9 N , o r c o n f l i c t o r e l s e 1 2 2 h e c a n m a k e a n e w d e m a n d f o r B l ' s c o n s i d e r a t i o n . R e c a l l i n g t h e s a m e s i t u a t i o n w e h a d i n s e c t i o n 5 . 2 , w e c o n c l u d e t h a t h e w i l l d e c i d e t o c e a s e N b a r g a i n i n g o r b i d d i n g o n c e m o r e a c c o r d i n g a s v ( x 2 N ^ — EN' ° R N O T ' I , E * i f = E: N „ N 'N o t h e r w i s e , w h e r e E * - M a x ^ ( x ^ w i t h V ( x 2 N + 1 ) = ^ O K x ^ ) > 0 ) V ( x 2 N + 1 ) - C N , 2 N + 1 N - 1 ' ^ " N t h a n d H^j = V ( x 2 ^ ) v E ^ i s h i s m a x i m u m e x p e c t e d g a i n o n c e h e h a s r e a c h e d N s t a g e . N e x t s u p p o s e h e i s a t ( N - 1 ) s t a g e h a v i n g B l ' s o f f e r x 2 ^ _ 2 f o r c o n s i d e r a t i o n . B 2 i s h e r e a w a r e t h a t i n c a s e h e m a k e s a n e w d e m a n d X 2 N - 1 ' ^ a c c e P t i t o n l y i f i t y i e l d s B l m o r e t h a n a n y o t h e r a l t e r n a t i v e . T o c a l c u l a t e t h e p r o b a b i l i t y o f t h i s e v e n t , B 2 n e e d s t o g u e s s h o w B l w i l l c o m p u t e h i s s t r a t e g y . F r o m B 2 ' s v i e w p o i n t B l a t s t a g e N b e h a v i n g r a t i o n a l l y ' ' w i l l r e a s o n a s f o l l o w s : " i f B 2 w e r e t o m a k e a n e w o f f e r x 2 N + 1 a t t * i e n e x t s t a S e » ^ e w ° u l d m a x i m i z e V ( x 2 ^ + ^ ) s u b j e c t t o U ( x „ ) > 0 a n d I w i l l d e f i n i t e l y h a v e t o a c c e p t t h e r e s u l t . A n d h e w i l l a c c e p t my o f f e r x ^ i f a n d o n l y i f V ( x 2 N ) > [ M a x W x 2 N + 1 ) | . U ( x 2 N + 1 ) > 0 } - [C^ - C ^ ] ] v 0 X 2 N + 1 E X 1 . S o B l a c t s a s i f ( n o t t h a t h e b e l i e v e s ) B 2 k n o w s h i s u t i l i t y U . 123 T h e r e f o r e , i f I a m t o m a k e a n e w o f f e r , I m u s t m a x i m i z e U ( x 2 N ) s u b j e c t t o V ( x 2 N ) ^_ a n d s o m y m a x i m u m g a i n b y m a k i n g a n e w o f f e r i s E ^ = M a x [ U ( x 2 N ) - D N | v ( x 2 N ) > X^] . T h u s m y m a x i m u m e x p e c t e d g a i n i s x 2 N e X = U ( x 2 ^ [ _ i ) v E J J * P r e d i c t i n g B l s c o u r s e o f a c t i o n a t s t a g e N t h i s w a y , B 2 c o m p u t e s t h e p r o b a b i l i t y t h a t B l w o u l d a c c e p t X 2 N — 1 ' ^ m a d e , t o b e , P N - 1 ( X 2 N - 1 > = * ( U ( X 2 N - 1 } = V a n d t h e p r o b a b i l i t y t h a t B l w o u l d i n s t e a d m a k e a n e w o f f e r t o b e Q T , ( x „ , ) = P ( E ^ = H??) ; e a c h o f t h e s e p r o b a b i l i t i e s i s f o u n d b y B 2 ' s H N - 1 2 N - 1 r N N c o n d i t i o n a l s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s g i v e n A ^ . H e n c e a t h i s t i l ( N - 1 ) s t a g e , i f B 2 m a k e s a n e w o f f e r x 2 ^ _ i » h i s m a x i m u m e x p e c t e d g a i n w i l l b e , E N - 1 = m \ { ? < * 2 N - 1 > + W W E { H K - 1 } }  X 2 N - 1 £ K a n d h i s m a x i m u m o v e r a l l e x p e c t e d g a i n c o n s i d e r i n g a l l p o s s i b l e a c t i o n s i s H N - 1 = ^ ( x 2 N - 2 } V E N - 1 ' W h 6 r e E ( H N ( X 2 N ' «Vl' W ± S f ° U n d U S l n g B 2 ' S s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s . C o n t i n u a t i o n o f t h i s p r o c e s s y i e l d s 124 i n d u c t i v e l y B 2 ' s m a x i m u m o v e r a l l e x p e c t e d g a i n a t s t a j e j , v i z H N ( i K , «>.) = V ( x ) v E®(ib , <(. ) ; j < N - l ( 5 . 9 ) J J J J J J J w h e r e ( i ) V ( x 2 ) = V ( x 2 ) - C ( i i ) E * ( * j f * ) = M a x { v ( x 2 j + 1 ) + Q j ( « 2 j + 1 ) E { H » + 1 ( X 2 j + 2 , < f > . + 1 ) } } X 2 j + 1 £ K ( 5 . 1 0 ) ( i i i ) v ( x 2 . + 1 ) = V ( x 2 j + 1 ) P . ( x 2 j + 1 ) - C . ( i v ) = l * ( B ( x 2 j + 1 ) = ^ + i ( X 2 j + 2 , * j f * . + 1 ) ) i s t h e p r o b a b i l i t y t h a t B l w o u l d a c c e p t x 2 j + i ( v ) Q . ( x 2 j + 1 ) = F r , ( ^ + 1 ( X 2 j + 2 , = H ^ + 1 ( X 2 j + 2 , i s t h e p r o b a b i l i t y t h a t B l w o u l d m a k e a n e w o f f e r a t h i s j s t a g e . ( v i ) E N = M a x [ U ( x . ) - D . | V ( x . ) > A . ] 2 x 2 j e X Z J 2 2 2 1 2 5 ( v i i ) X = 0 v M a x f V ( x 2 j + 1 ) - ( C . - C . ^ ) / U ( x 2 j + 1 ) > H ^ } x 2 j + 1 e X ( v i i i ) H . = 0 v U ( x _ . , ) v E . . A t t h i s s t a g e i t s h o u l d b e e m p h a s i z e d t h a t B 2 i s n o t b e l i e v i n g B l h a s c o m p l e t e i n f o r m a t i o n a b o u t t h e b a r g a i n i n g s i t u a t i o n . H e i s m e r e l y a c t i n g v e r y c o n s e r v a t i v e l y a s i f t h i s w e r e s o . W e a s s u m e t h a t t h r o u g h o u t t h e b a r g a i n i n g p r o c e s s h e a c t s i n t h i s w a y b e c a u s e a t e v e r y s t a g e o f b a r g a i n i n g h e r i s k s b e i n g " f o u n d o u t " ( n o t e t h a t i f B 2 w a s i n s t e a d b e l i e v i n g s o , t h e r e a r i s e s a c o n t r a d i c t i o n b e c a u s e , f r o m B 2 ' s p o i n t o f v i e w B l a l w a y s m a k e o f f e r s s o t h a t B 2 w o u l d a c c e p t i t i m m e d i a t e l y ) . H o w e v e r i t i s r e a s o n a b l e f o r B 2 t o a s s u m e t h a t , b e i n g a r a t i o n a l p e r s o n , B l a c t s a s i f B 2 h a s a c o m p l e t e i n f o r m a t i o n a b o u t t h e b a r g a i n i n g s i t u a t i o n . We a r e n o w i n a p o s i t i o n t o c o n c l u d e , w i t h P r o p o s i t i o n 5 . 3 . 1 : ( a ) T h e b a r g a i n i n g s e q u e n c e f o r B 2 g i v e n b y B . I . P . , f o r t h e b a r g a i n i n g p r o b l e m t r u n c a t e d a t N , i s <j> = { x ^ , x 2 > x 2 j j + l ^ » w h e r e x 2 j + i y i e l d s t h e m a x i m u m v a l u e i n ( 5 . 1 0 ) f o r j = 0 , 1 , 2 , N - l a n d X 2 N + 1 m a x i m i z e s v ( x 2 N + i ) • ( b ) T h e s t o p p i n g r u l e g i v e n b y B . I . P . i s E = {EQ, Z 1, E N , Z N + 1 > , w h e r e 1 i f V ( x ) > E % , <fr ) 0 o t h e r w i s e . 1 2 6 I n v i e w o f t h e r e s u l t s i n S e c t i o n 5 . 2 , i t s e e m s c e r t a i n t h a t t h e b a r g a i n i n g p l a n g i v e n b y B . I . P . f o r t h e t r u n c a t e d p r o b l e m i s o p t i m a l ; a n d s o w e d o n o t a t t e m p t a p r o o f h e r e . I t i s a l s o c e r t a i n t h a t t h e n o n t r u n c a t e d p r o b l e m c a n b e a p p r o x i m a t e d b y a t r u n c a t e d o n e a r b i t r a r i l y w e l l u n d e r r e a s o n a b l e a s s u m p t i o n s . W e e x p e c t , a s i s i n t u i t i v e l y c l e a r t h a t u s u a l l y t h e p r e s e n c e o f b a r g a i n i n g c o s t s w i l l l e a d t o a t r u n c a t e d o p t i m a l p l a n a s a s o l u t i o n t o t h e g e n e r a l n o n t r u n c a t e d p r o b l e m s o t h a t o n e c o u l d f i n d a n o p t i m a l b a r g a i n i n g p l a n f o r t h e g e n e r a l p r o b l e m u s i n g B . I . P . 1 2 7 B I B L I O G R A P H Y 1 . A r r o w , K . J . , " T h e T h e o r y o f R i s k A v e r s i o n , " i n E s s a y s i n t h e T h e o r y o f R i s k - B e a r i n g , C h a p t e r 3 , C h i c a g o : M a r k a h a m P u b l i s h i n g C o . , 1 9 7 1 . 2 . , " E x p o s i t i o n o f t h e T h e o r y o f C h o i c e u n d e r U n c e r t a i n t y " , i n E s s a y s i n t h e T h e o r y o f R i s k - B e a r i n g , C h a p t e r 2 , C h i c a g o : M a r k h a m P u b l i s h i n g C o . , 1 9 7 1 . 3 . B i s h o p , R . L . " G a m e T h e o r e t i c a l A n a l y s i s o f B a r g a i n i n g , " Q u a r t e r l y J o u r n a l o f E c o n o m i c s , V o l . 7 7 , ( N o v e m b e r 1 9 6 3 ) , 5 5 9 - 6 0 2 , ( r e p r i n t e d i n O r a n R. Y o u n g , B a r g a i n i n g , U n i v e r s i t y o f I l l i n o i s P r e s s , 1 9 7 5 , 8 5 - 1 2 8 ) . 4 . , " A Z e u t h e n - H i c k s T h e o r y o f B a r g a i n i n g , " E c o n o m e t r i c a , V o l . 3 2 ( J u l y 1 9 6 4 ) , 4 1 0 - 4 1 7 , ( r e p r i n t e d i n O r a n R. Y o u n g , B a r g a i n i n g , U n i v e r s i t y o f I l l i n o i s P r e s s , 1 9 7 5 , 1 8 3 - 1 9 0 „ . ) . 5 . C o n t i n i , B . , a n d Z i o n t s , S . , " R e s t r i c t e d B a r g a i n i n g f o r O r g a n i z a t i o n s w i t h M u l t i p l e O b j e c t i v e s , " E c o n o m e t r i c a , V o l . 3 6 ( A p r i l 1 9 4 8 ) , 3 9 7 - 4 1 4 . 6 . C r o s s , J . G . , " T h e o r y o f B a r g a i n i n g P r o c e s s , " A m e r i c a n E c o n o m i c R e v i e w , V o l . 5 5 ( M a r c h 1 9 6 5 ) , 6 7 - 9 4 , ( r e p r i n t e d i n O r a n R. Y o u n g , B a r g a i n i n g , U n i v e r s i t y o f I l l i n o i s P r e s s , 1 9 7 5 , 1 9 1 - 2 1 8 ) . 7 . , T h e E c o n o m i c s o f B a r g a i n i n g , N e w Y o r k , 1 9 6 9 . 8 . D e G r o o t , M . H . , O p t i m a l S t a t i s t i c a l D e c i s i o n , C h a p t e r 7 , N e w Y o r k : M c G r a w H i l l , 1 9 7 0 . 9 . D e M e n i l , G . , B a r g a i n i n g : M o n o p o l y P o w e r v e r s u s U n i o n P o w e r , C a m b r i d g e : M I T P r e s s , 1 9 7 1 . 1 0 . E d g e w o r t h , F . Y . , M a t h e m a t i c a l P s y c h i c s , L o n d o n : K e g a n P a u l , 1 8 8 1 . 1 1 . F e r g u s i o n , T . S . , M a t h e m a t i c a l S t a t i s t i c s : a D e c i s i o n T h e o r e t i c A p p r o a c h , 3 0 9 - 3 2 5 , N e w Y o r k : A c a d e m i c P r e s s , 1 9 6 7 . 1 2 . F o l d e s , L . , " A D e t e r m i n a t e M o d e l o f B i l a t e r a l M o n o p o l y , " E c o n o m i c a , N . S . , V o l . 3 1 ( F e b r u a r y 1 9 6 4 ) , 1 1 7 - 1 3 1 . 1 3 . F r i e d m a n , J . , a n d S a v a g e , L . J . , " T h e U t i l i t y A n a l y s i s o f C h o i c e s I n v o l v i n g R i s k s , " J o u r n a l o f P o l i t i c a l E c o n o m y , V o l . 5 6 ( 1 9 4 8 ) , 2 7 9 - 3 0 4 . 1 4 . H a n d e r s o n , J . M . a n d Q u a n d t , R . E . , M i c r o e c o n o m i c T h e o r y , 2 4 4 - 2 5 1 , N e w Y o r k : M c G r a w - H i l l , 1 9 7 1 . 1 2 8 1 5 . H a r s a n y i , J . C . , " A p p r o a c h e s t o t h e B a r g a i n i n g P r o b l e m B e f o r e a n d A f t e r t h e T h e o r y o f G a m e s : A C r i t i c a l D i s c u s s i o n o f Z e u t h e n ' s H i c k s ' a n d N a s h ' s T h e o r i e s , " E c o n o m e t r i c a , V o l . 2 4 ( A p r i l 1 9 5 6 ) , 1 4 4 - 1 5 7 , ( r e p r i n t e d i n O r a n R . Y o u n g , B a r g a i n i n g , U n i v e r s i t y o f I l l i n o i s P r e s s , 1 9 7 5 , 2 5 3 - 2 6 6 ) . 1 6 . , a n d J . C . S e l t e n , " A G e n e r a l i z e d N a s h S o l u t i o n f o r T w o - p e r s o n B a r g a i n i n g G a m e s w i t h I n c o m p l e t e I n f o r m a t i o n , " M a n a g e m e n t S c i e n c e , V o l . 1 8 ( J a n u a r y 1 9 7 2 ) , P a r t 2 , 8 0 - 1 0 6 . 1 7 . H i c k s , J . R . , T h e T h e o r y o f W a g e s , C h a p t e r 7 , L o n d o n , 1 9 3 2 . 1 8 . H i l d r e t h , C , " V e t u r e s , B e t s a n d I n i t i a l P r o s p e c t s , " D i s c u s s i o n P a p e r N o . 2 0 , C e n t e r f o r E c o n o m i c R e s e a r c h , U n i v e r s i t y o f M i n n e s o t a , 1 9 7 2 , ( r e p r i n t e d i n M . S . B a l c h , D . L . M c F a d d e n a n d S . Y . W u , E s s a y s o n E c o n o m i c B e h a v i o r u n d e r U n c e r t a i n t y , A m s t e r d a m : N o r t h H o l l a n d P u b l i s h i n g C o . , 1 9 7 4 . ) 1 9 . , " E x p e c t e d U t i l i t y o f U n c e r t a i n V e n t u r e s , " J o u r n a l o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , V o l . 6 9 ( M a r c h 1 9 7 4 ) , 9 - 1 7 . 2 0 . , a n d T e s f a t s i o n , L . , " A M o d e l o f C h o i c e w i t h U n c e r t a i n I n i t i a l P r o s p e c t , " D i s c u s s i o n P a p e r N o . 3 8 , C e n t e r f o r E c o n o m i c R e s e a r c h , U n i v e r s i t y o f M i n n e s o t a , 1 9 7 4 . 2 1 . K a l a i , E . , a n d S m o r o d i n s k y , M . , " O t h e r S o l u t i o n s t o N a s h ' s B a r g a i n i n g P r o b l e m , " E c o n o m e t r i c a , V o l . 4 3 ( M a y 1 9 7 5 ) , 5 1 3 - 5 1 8 . 2 2 . L o e v e , M . , P r o b a b i l i t y T h e o r y , P r i n c e t o n : D . V a n N o s t r a n d C o m p a n y , I n c . 1 9 6 3 . 2 3 . L u c e , R . D . , a n d R a i f f a , H . , G a m e s a n d D e c i s i o n s , N e w Y o r k : J o h n W i l e y & S o n s , 1 9 5 7 . 2 4 . N a s h , J . F . , J r . , " T h e B a r g a i n i n g P r o b l e m , " E c o n o m e t r i c a , V o l . 1 8 ( A p r i l 1 9 5 0 ) , 1 5 5 - 1 6 2 , ( r e p r i n t e d i n O r a n R . Y o u n g , B a r g a i n i n g , U n i v e r s i t y o f I l l i n o i s P r e s s , 1 9 7 5 , 5 3 - 6 0 . ) 2 5 . , " T w o P e r s o n C o o p e r a t i v e G a m e s , " E c o n o m e t r i c a , V o l . 2 1 ( J a n u a r y 1 9 5 3 ) , 1 2 8 - 1 4 0 , ( r e p r i n t e d i n O r a n R . Y o u n g , B a r g a i n i n g , U n i v e r s i t y o f I l l i n o i s P r e s s , 1 9 7 5 , 6 1 - 7 3 , ) . 2 6 . N o r r i s , N . , " G e n e r a l M e a n s a n d S t a t i s t i c a l T h e o r y , " T h e A m e r i c a n S t a t i s t i c i a n , V o l . 3 0 ( F e b r u a r y 1 9 7 6 ) , 8 - 1 2 . 2 7 . N y d e g g e r , R . V . , a n d H o u s t o n , G . O . , " T w o P e r s o n B a r g a i n i n g : A n E x p e r i m e n t a l T e s t o f t h e N a s h A x i o m s , " I n t e r n a t i o n a l J o u r n a l o f G a m e T h e o r y , V o l . 3 ( A u g u s t 1 9 7 5 ) , 2 3 9 - 2 4 9 . 1 2 9 2 8 . P e n , J . , " A G e n e r a l T h e o r y o f B a r g a i n i n g , " A m e r i c a n E c o n o m i c R e v i e w , V o l . 4 2 ( M a r c h 1 9 5 2 ) , 2 4 - 4 2 , ( r e p r i n t e d i n O r a n R . Y o u n g , B a r g a i n i n g , U n i v e r s i t y o f I l l i n o i s P r e s s , 1 9 7 5 , 1 6 4 - 1 8 2 . ) 2 9 . P r a t t , J . W . , " R i s k A v e r s i o n i n t h e S m a l l a n d i n t h e L a r g e , " E c o n o m e t r i c a , V o l . 3 2 ( A p r i l 1 9 6 4 ) , 1 2 2 - 1 3 6 . 3 0 . R a i f f a , H . , " A r b i t r a t i o n S c h e m e s f o r G e n e r a l i z e d T w o - P e r s o n G a m e s , " i n H . W . K u h n a n d A . W . T u c k e r , C o n t r i b u t i o n s t o t h e T h e o r y o f G a m e s , I I , P r i n c e t o n , N . J . : P r i n c e t o n U n i v e r s i t y P r e s , , 1 9 5 3 . 3 1 . S a r a y d a r , E . , " Z e u t h e n ' s T h e o r y o f B a r g a i n i n g : A N o t e , " E c o n o m e t r i c a , V o l . 3 3 ( O c t o b e r , 1 9 6 5 ) , 8 0 2 - 8 1 3 . 3 2 . S n e d d o n , I . N . , E l e m e n t s o f P a r t i a l D i f f e r e n t i a l E q u a t i o n s , 4 9 - 5 5 , N e w Y o r k : M c G r o w - H i l l , 1 9 5 7 . 3 3 . T o b i n , J . , " L i q u i d i t y P r e f e r e n c e a s B e h a v i o r t o w a r d s R i s k , " R e v i e w o f E c o n o m i c S t u d i e s , V o l . 2 5 ( 1 9 5 7 - 5 8 ) , 6 5 - 8 6 . 3 4 . V o n N e u m a n n , J . , a n d M o r g e n s t e r n , 0 . , T h e o r y o f G a m a e s a n d E c o n o m i c B e h a v i o r , P r i n c e t o n : P r i n c e t o n U n i v e r s i t y P r e s s , 1 9 5 3 . 3 5 . Z e u t h e n , F . , P r o b l e m s o f M o n o p o l y a n d E c o n o m i c W a r f a r e , C h a p t e r 4 , L o n d o n : G e o r g e R o u t l e d g e a n d S o n s , 1 9 3 0 , ( r e p r i n t e d i n O r a n R . Y o u n g , B a r g a i n i n g , U n i v e r s i t y o f I l l i n o i s P r e s s , 1 9 7 5 , 1 4 5 - 1 6 3 ). 1 3 0 A P P E N D I X A ( a ) C O M P U T E R P R O G R A M : G R A P H O F B = E { U ' ( a X ) - U ' ( 0 ) } / E U ( a X ) V S a I N E X A M P L E 3 . 3 . 1 D I M E N S I O N X ( 1 0 0 0 ) , Y ( 1 0 0 0 ) , U l l ( l O O O ) , U 1 2 ( 1 0 0 0 ) , U 2 1 ( 1 0 0 0 ) , U 2 2 ( 1 0 0 0 ) K = l J = l R 3 = 0 . 0 0 0 1 R l = 3 R 2 = - 4 DO 2 I = 1 , 1 0 0 0 X ( I ) = R 3 U l l ( I ) = 7 . * E X P ( - l . * R l * X ( I ) ) + 1 6 . * E X P ( - 2 . * R 1 * X ( I ) ) - 2 3 U 1 2 ( I ) = 7 . * E X P ( - 1 . * R 2 * X ( I ) ) + 1 6 . * E X P ( - 2 . * R 2 * X ( I ) ) - 2 3 U 2 1 ( I ) = - 7 . * E X P ( - l . * R l * X ( I ) ) - 8 . * E X P ( - 2 . * R l * X ( I ) ) + 1 5 U 2 2 ( I ) = - 7 . * E X P ( - 1 . * R 2 * X ( I ) ) - 8 . * E X P ( - 2 . * R 2 * X ( I ) ) + 1 5 Y ( I ) = ( 0 . 8 * U 1 1 ( I ) + 0 . 2 * U 1 2 ( I ) ) / ( 0 . 8 * U 2 1 ( I ) + 0 . 2 * U 2 2 ( I ) ) I F ( K . N E . J ) GO T O 3 J = J + 1 0 W R I T E ( 6 ; 7 ) o X ( T ) , Y ( I ) 7 F O R M A T ( 2 F 1 6 . 9 ) 3 K = K + 1 R 3 = R 3 + 0 . 0 0 0 1 7 2 C O N T I N U E C A L L S C A L E ( X , 1 0 0 0 , 1 0 . , X M I N , D X , 1 ) C A L L S C A L E ( Y , 1 0 0 0 , 1 0 . , Y M I N , D Y , 1 ) C A L L A X I S ( 0 . , 0 . , ' A L P H A ' , - 5 , 1 0 . , 0 . , X M I N , D X ) C A L L A X I S ( 0 . , 0 . , ' B E T A ' , - 4 , 1 0 . , 9 0 . , Y M I N , D Y ) C A L L L I N E ( X , Y , 1 0 0 0 , 1 ) C A L L P L O T N D S T O P E N D a ( b ) C O M P U T E R O U T P U T : 3 0 < a < . 1 7 A N D g a 1 3 1 B 0 . 0 0 0 1 0 0 0 0 0 - 1 . 6 9 7 5 7 1 7 5 4 0 . 0 8 5 0 9 8 9 2 2 - 1 . 5 8 9 9 1 2 4 1 5 0 . 0 0 1 7 9 9 9 9 9 - 1 . 6 9 4 5 5 9 0 9 7 0 . 0 8 6 7 9 8 8 4 7 - 1 . 5 8 5 8 6 9 7 8 9 0 . 0 0 3 4 9 9 9 9 8 - 1 . 6 9 3 2 8 4 0 3 5 0 . 0 8 8 4 9 8 7 7 1 - 1 . 5 8 1 6 7 3 6 2 2 0 . 0 0 5 1 9 9 9 9 5 - 1 . 6 9 2 0 1 8 5 0 9 0 . 0 9 0 1 9 8 6 9 6 - 1 . 5 7 7 3 1 4 3 7 7 0 . 0 0 6 8 9 9 9 9 4 - 1 . 6 9 0 7 7 8 7 3 2 0 . 0 9 1 8 9 8 6 2 0 - 1 . 5 7 2 7 8 4 4 2 4 0 . 0 0 8 5 9 9 9 9 3 - 1 . 6 8 9 5 0 3 6 7 0 0 . 0 9 3 5 9 8 5 4 5 - 1 . 5 6 8 0 7 2 3 1 9 0 . 0 1 0 2 9 9 9 9 2 - 1 . 6 8 8 2 0 7 6 2 6 0 . 0 9 5 2 9 8 4 6 9 - 1 . 5 6 3 1 6 7 5 7 2 0 . 0 1 1 9 9 9 9 9 1 - 1 . 6 8 6 8 8 5 8 3 4 0 . 0 9 6 9 9 8 3 9 4 - 1 . 5 5 8 0 5 7 7 8 5 0 . 0 1 3 6 9 9 9 9 0 - 1 . 6 8 5 5 6 4 9 9 5 0 . 0 9 8 6 9 8 3 1 8 - 1 . 5 5 2 7 3 0 5 6 0 0 . 0 1 5 3 9 9 9 8 9 - 1 . 6 8 4 2 1 4 5 9 2 0 . 1 0 0 3 9 8 2 4 2 - 1 . 5 4 7 1 6 9 6 8 5 0 . 0 1 7 0 9 9 9 8 8 - 1 . 6 8 2 8 4 4 1 6 2 0 . 1 0 2 0 9 8 1 6 7 - 1 . 5 4 1 3 5 9 9 0 1 0 . 0 1 8 7 9 9 9 8 7 - 1 . 6 8 1 4 2 7 0 0 2 ' 0 . 1 0 3 7 9 8 0 9 1 - 1 . 5 3 5 2 8 4 9 9 6 0 . 0 2 0 4 9 9 9 8 6 - 1 . 6 8 0 0 0 4 1 2 0 0 . 1 0 5 4 9 8 0 1 6 - 1 . 5 2 8 9 2 3 0 3 5 0 . 0 2 2 1 9 9 9 8 5 - 1 . 6 7 8 5 4 8 8 1 3 0 . 1 0 7 1 9 7 9 4 0 - 1 . 5 2 2 2 5 5 8 9 8 0 . 0 2 3 8 9 9 9 8 4 - 1 . 6 7 7 0 5 5 3 5 9 0 . 1 0 8 8 9 7 8 6 5 - 1 . 5 1 5 2 6 3 5 5 7 0 . 0 2 5 5 9 9 9 8 3 - 1 . 6 7 5 5 3 9 0 1 7 0 . 1 1 0 5 9 7 7 8 9 - 1 . 5 0 7 9 1 6 4 5 1 0 . 0 2 7 2 9 9 9 8 2 - 1 . 6 7 3 9 8 7 3 8 9 0 . 1 1 2 2 9 7 7 1 4 - 1 . 5 0 0 1 8 9 7 8 1 0 . 0 2 8 9 9 9 9 8 1 - 1 . 6 7 2 4 1 5 7 3 3 0 . 1 1 3 9 9 7 6 3 8 - 1 . 4 9 2 0 5 2 0 7 8 0 . 0 3 0 6 9 9 9 8 0 - 1 . 6 7 0 8 0 6 8 8 5 0 . 1 1 5 6 9 7 5 6 3 - 1 . 4 8 3 4 7 1 8 7 0 0 . 0 3 2 3 9 9 9 7 8 - 1 . 6 6 9 1 5 1 3 0 6 0 . 1 1 7 3 9 7 4 8 7 - 1 . 4 7 4 4 0 7 1 9 6 0 . 0 3 4 0 9 9 9 7 7 - 1 . 6 6 7 4 7 2 8 3 9 0 . 1 1 9 0 9 7 4 1 2 - 1 . 4 6 4 8 2 2 7 6 9 0 . 0 3 5 7 9 9 9 7 6 - 1 . 6 6 5 7 5 2 4 1 1 0 . 1 2 0 7 9 7 3 3 6 - 1 . 4 5 4 6 6 8 0 4 5 0 . 0 3 7 4 9 9 9 7 5 - 1 . 6 6 3 9 9 0 9 7 4 0 . 1 2 2 4 9 7 2 6 1 - 1 . 4 4 3 8 9 4 3 8 6 0 . 0 3 9 1 9 9 9 7 4 - 1 . 6 6 2 1 8 5 6 6 9 0 . 1 2 4 1 9 7 1 8 5 - 1 . 4 3 2 4 3 7 8 9 7 0 . 0 4 0 8 9 9 9 7 3 - 1 . 6 6 0 3 4 7 9 3 9 0 . 1 2 5 8 9 7 1 1 0 - 1 . 4 2 0 2 3 6 5 8 8 0 . 0 4 2 5 9 9 9 7 2 - 1 . 6 5 8 4 6 1 5 7 1 0 . 1 2 7 5 9 7 0 3 4 - 1 . 4 0 7 2 1 5 1 1 8 0 . 0 4 4 2 9 9 9 7 1 - 1 . 6 5 6 5 2 4 6 5 8 0 . 1 2 9 2 9 6 9 5 8 - 1 . 3 9 3 2 8 0 9 8 3 0 . 0 4 5 9 9 9 9 7 0 - 1 . 6 5 4 5 5 3 4 1 3 0 . 1 3 0 9 9 6 8 8 3 - 1 . 3 7 8 3 4 5 4 9 0 0 . 0 4 7 6 9 9 9 6 9 - 1 . 6 5 2 5 2 3 0 4 1 0 . 1 3 2 6 9 6 8 0 7 - 1 . 3 6 2 2 9 2 2 9 0 0 . 9 4 9 3 9 9 9 6 8 - 1 . 6 5 0 4 5 4 5 2 1 0 . 1 3 4 3 9 6 7 3 2 - 1 . 3 4 4 9 9 3 5 9 1 0 . 0 5 1 0 9 9 9 6 7 - 1 . 6 4 8 3 1 4 4 7 6 0 . 1 3 6 0 9 6 6 5 6 - 1 . 3 2 6 2 9 6 8 0 6 0 . 0 5 2 7 9 9 9 6 6 - 1 . 6 4 6 1 3 8 1 9 1 0 . 1 3 7 7 9 6 5 8 1 - 1 . 3 0 6 0 2 8 3 6 6 0 . 0 5 4 4 9 9 9 6 5 - 1 . 6 4 3 8 8 7 5 2 0 0 . 1 3 9 4 9 6 5 0 5 - 1 . 2 8 3 9 8 0 3 7 0 0 . 0 5 6 1 9 9 9 6 4 - 1 . 6 4 1 5 8 8 2 1 1 0 . 1 4 1 1 9 6 4 3 0 - 1 . 2 5 9 9 0 5 8 1 5 0 . 0 5 7 8 9 9 9 6 3 - 1 . 6 3 9 2 2 5 0 0 6 0 . 1 4 2 8 9 6 3 5 4 - 1 . 2 3 3 5 2 1 4 6 1 0 . 0 5 9 5 9 9 9 6 2 - 1 . 6 3 6 7 9 4 0 9 0 0 . 1 4 4 5 9 6 2 7 9 - 1 . 2 0 4 4 7 2 5 4 2 0 . 0 6 1 2 9 9 9 6 1 - 1 . 6 3 4 2 8 8 7 8 8 0 . 1 4 6 2 9 6 2 0 3 - 1 . 1 7 2 3 3 7 5 3 2 0 . 0 6 2 9 9 9 9 0 4 - 1 . 6 3 1 7 2 0 5 4 3 0 . 1 4 7 9 9 6 1 2 8 - 1 . 1 3 6 5 9 9 5 4 1 0 . 0 6 4 6 9 9 8 2 9 - 1 . 6 2 9 0 7 4 0 9 7 0 . 1 4 9 6 9 6 0 5 2 - 1 . 0 9 6 6 1 8 6 5 2 0 . 0 6 6 3 9 9 7 5 3 - 1 . 6 2 6 3 4 7 5 4 2 0 . 1 5 1 3 9 5 9 7 7 - 1 . 0 5 1 5 8 9 0 1 2 0 . 0 6 8 0 9 9 6 7 8 - 1 . 6 2 3 5 3 7 0 6 4 0 . 1 5 3 0 9 5 9 0 1 - 1 . 0 0 0 5 0 7 3 5 5 0 . 0 6 9 7 9 9 6 0 2 - 1 . 6 2 0 6 4 6 4 7 7 0 . 1 5 4 7 9 5 8 2 5 - 0 . 9 4 2 0 5 4 0 9 3 0 . 0 7 1 4 9 9 5 2 7 - 1 . 6 1 7 6 6 5 2 9 1 0 . 1 5 6 4 9 5 7 5 0 - 0 . 8 7 4 5 1 8 0 9 6 0 . 0 7 3 1 9 9 4 5 1 - 1 . 6 1 4 5 8 1 1 0 8 0 . 1 5 8 1 9 5 6 7 4 - 0 . 7 9 5 6 1 6 3 8 8 0 . 0 7 4 8 9 9 3 7 5 - 1 . 6 1 1 3 9 9 6 5 1 0 . 1 5 9 8 9 5 5 9 9 - 0 . 7 0 2 2 1 4 4 2 0 0 . 0 7 6 5 9 9 3 0 0 - 1 . 6 0 8 1 1 6 1 5 0 0 . 1 6 1 5 9 5 5 2 3 - 0 . 5 8 9 9 2 5 8 2 6 0 . 0 7 8 2 9 9 2 2 4 - 1 . 6 0 4 7 2 2 9 7 7 0 . 1 6 3 2 9 5 4 4 8 - 0 . 4 5 2 3 8 2 9 8 2 0 . 0 7 9 9 9 9 1 4 9 - 1 . 6 0 1 2 1 2 5 0 2 0 . 1 6 4 9 9 5 3 7 2 - 0 . 2 8 0 0 0 2 9 5 2 0 . 0 8 1 6 9 9 0 7 3 - 1 . 5 9 7 5 7 9 0 0 2 0 . 1 6 6 6 9 5 2 9 7 - 0 . 0 5 7 6 5 7 6 8 1 0 . 0 8 3 3 9 8 9 9 8 - 1 . 5 9 3 8 1 3 8 9 6 0 . 1 6 8 3 9 5 2 2 1 0 . 2 4 0 0 4 8 4 0 9 1 3 2 ( c ) C O M P U T E R O U T P U T : . 1 1 < a < . 2 2 A N D g a g a P 0 . 1 1 9 9 9 9 9 4 5 - 1 . 4 5 9 5 0 6 9 8 9 0 . 1 6 9 9 7 8 4 4 0 0 . 6 2 5 0 3 2 5 4 4 0 . 1 2 0 9 9 9 5 1 5 - 1 . 4 5 3 4 2 1 5 9 3 0 . 1 7 0 9 7 8 0 1 0 0 . 9 5 2 2 9 3 9 9 2 0 . 1 2 1 9 9 9 0 8 5 - 1 . 4 4 7 1 1 8 7 5 9 0 . 1 7 1 9 7 7 5 8 0 1 . 3 8 0 9 1 1 8 2 7 0 . 1 2 2 9 9 8 6 5 5 - 1 . 4 4 0 5 8 7 0 4 4 0 . 1 7 2 9 7 7 1 4 9 1 . 9 6 6 6 0 3 2 7 9 0 . 1 2 3 9 9 8 2 2 5 - 1 . 4 3 3 8 1 5 9 5 6 0 . 1 7 3 9 7 6 7 1 9 2 . 8 1 5 0 1 4 8 3 9 0 . 1 2 4 9 9 7 7 9 5 ' - 1 . 4 2 6 7 8 6 4 2 3 0 . 1 7 4 9 7 6 2 8 9 4 . 1 5 3 8 0 3 8 2 5 0 . 1 2 5 9 9 7 3 6 5 - 1 . 4 1 9 4 9 0 8 1 4 0 . 1 7 5 9 7 5 8 5 9 6 . 5 8 1 3 0 9 3 1 9 0 . 1 2 6 9 9 6 9 3 4 - 1 . 4 1 1 9 0 7 1 9 6 0 . 1 7 6 9 7 5 4 2 9 1 2 . 3 3 3 6 4 8 6 8 2 0 . 1 2 7 9 9 6 5 0 4 - 1 . 4 0 4 0 2 3 1 7 0 0 . 1 7 7 9 7 4 9 9 9 4 3 . 0 9 9 1 8 2 1 2 9 0 . 1 2 8 9 9 6 0 7 4 - 1 . 3 9 5 8 1 6 8 0 3 0 . 1 7 8 9 7 4 5 6 9 - 4 0 . 0 8 2 6 8 7 3 7 8 0 . 1 2 9 9 9 5 6 4 4 - 1 . 3 8 7 2 7 2 8 3 5 0 . 1 7 9 9 7 4 1 3 9 - 1 5 . 2 3 4 8 7 5 6 7 9 0 . 1 3 0 9 9 5 2 1 4 - 1 . 3 7 8 3 6 2 6 5 6 0 . 1 8 0 9 7 3 7 0 9 - 9 . 9 5 3 0 5 3 4 7 4 0 . 1 3 1 9 9 4 7 8 4 - 1 . 3 6 9 0 6 7 1 9 2 0 . 1 8 1 9 7 3 2 7 9 - 7 . 6 5 5 2 4 0 0 5 9 0 . 1 3 2 9 9 4 3 5 4 - 1 . 3 5 9 3 5 8 7 8 8 0 . 1 8 2 9 7 2 8 4 8 - 6 . 3 6 9 7 4 6 2 0 8 0 . 1 3 3 9 9 3 9 2 4 - 1 . 3 4 9 2 1 1 6 9 3 0 . 1 8 3 9 7 2 4 1 8 - 5 . 5 4 8 4 3 7 1 1 9 0 . 1 3 4 9 9 3 4 9 4 - 1 . 3 3 8 6 0 0 1 5 9 0 . 1 8 4 9 7 1 9 8 8 - 4 . 9 7 8 2 9 0 5 5 8 0 . 1 3 5 9 9 3 0 6 3 - 1 . 3 2 7 4 7 8 4 0 9 0 . 1 8 5 9 7 1 5 5 8 - 4 . 5 5 9 3 7 9 5 7 8 0 . 1 3 6 9 9 2 6 3 3 - 1 . 3 1 5 8 2 1 6 4 8 0 . 1 8 6 9 7 1 1 2 8 - 4 . 2 3 8 6 0 3 5 9 2 0 . 1 3 7 9 9 2 2 0 3 - 1 . 3 0 3 5 8 4 0 9 9 0 . 1 8 7 9 7 0 6 9 8 - 3 . 9 8 5 1 2 2 6 8 1 0 . 1 3 8 9 9 1 7 7 3 - 1 . 2 9 0 7 2 5 7 0 8 0 . 1 8 8 9 7 0 2 6 8 - 3 . 7 7 9 7 6 5 1 2 9 0 . 1 3 9 9 9 1 3 4 3 - 1 . 2 7 7 1 9 3 0 6 9 0 . 1 8 9 9 6 9 8 3 8 - 3 . 6 0 9 9 9 8 7 0 3 0 . 1 4 0 9 9 0 9 1 3 - 1 . 2 6 2 9 3 4 6 8 5 0 . 1 9 0 9 6 9 4 0 8 - 3 . 4 6 7 3 5 7 6 3 5 0 . 1 4 1 9 9 0 4 8 3 - 1 . 2 4 7 8 9 0 4 7 2 0 . 1 9 1 9 6 8 9 7 7 - 3 . 3 4 5 7 8 8 9 5 6 0 . 1 4 2 9 9 0 0 5 3 - 1 . 2 3 1 9 9 4 6 2 9 0 . 1 9 2 9 6 8 5 4 7 - 3 . 2 4 0 9 6 8 7 0 4 0 . 1 4 3 9 8 9 6 2 3 - 1 . 2 1 5 1 6 9 9 0 7 0 . 1 9 3 9 6 8 1 1 7 - 3 . 1 4 9 6 3 7 2 2 2 0 . 1 4 4 9 8 9 1 9 2 - 1 . 1 9 7 3 3 5 2 4 3 - " • 0 . 1 9 4 9 6 7 6 8 7 - 3 . 0 6 9 3 7 5 9 9 2 0 . 1 4 5 9 8 8 7 6 2 - 1 . 1 7 8 3 9 9 0 8 6 0 . 1 9 5 9 6 7 2 5 7 - 2 . 9 9 8 2 8 2 4 3 3 0 . 1 4 6 9 8 8 3 3 2 - 1 . 1 5 8 2 5 3 6 7 0 0 . 1 9 6 9 6 6 8 2 7 - 2 . 9 3 4 8 7 7 3 9 6 0 . 1 4 7 9 8 7 9 0 2 - 1 . 1 3 6 7 7 9 7 8 5 0 . 1 9 7 9 6 6 3 9 7 - 2 . 8 7 7 9 6 4 9 7 3 0 . 1 4 8 9 8 7 4 7 2 - 1 . 1 1 3 8 4 4 8 7 2 0;. 1 9 8 9 6 5 9 6 7 - 2 . 8 2 6 6 1 7 2 4 1 0 . 1 4 9 9 8 7 0 4 2 - 1 . 0 8 9 2 9 4 4 3 4 0 . 1 9 9 9 6 5 5 3 7 - 2 . 7 8 0 0 5 1 2 3 1 0 . 1 5 0 9 8 6 6 1 2 - 1 . 0 6 2 9 4 6 3 2 0 0 . 2 0 0 9 6 5 1 0 6 - 2 . 7 3 7 6 2 5 1 2 2 0 . 1 5 1 9 8 6 1 8 2 - 1 . 0 3 4 5 9 8 3 5 1 0 . 2 0 1 9 6 4 6 7 6 - 2 . 6 9 8 8 0 9 6 2 4 0 . 1 5 2 9 8 5 7 5 2 - 1 . 0 0 4 0 2 3 5 5 2 0 . 2 0 2 9 6 4 2 4 6 - 2 . 6 6 3 1 7 4 6 2 9 0 . 1 5 3 9 8 5 3 2 2 - 0 . 9 7 0 9 3 9 8 7 5 0 . 2 0 3 9 6 3 8 1 6 - 2 . 6 3 0 3 4 0 5 7 6 0 . 1 5 4 9 8 4 8 9 1 - 0 . 9 3 5 0 2 8 5 5 3 0 . 2 0 4 9 6 3 3 8 6 - 2 . 5 9 9 9 9 7 5 2 0 0 . 1 5 5 9 8 4 4 6 1 - 0 . 8 9 5 9 1 0 7 4 0 0 . 2 0 5 9 6 2 9 5 6 - 2 . 5 7 1 8 5 7 4 5 2 0 . 1 5 6 9 8 4 0 3 1 - 0 . 8 5 3 1 3 8 8 6 4 0 . 2 0 6 9 6 2 5 2 6 - 2 . 5 4 5 7 0 9 6 1 0 0 . 1 5 7 9 8 3 6 0 1 - 0 . 8 0 6 1 7 8 9 2 7 0 . 2 0 7 9 6 2 0 9 6 - 2 . 5 2 1 3 3 5 6 0 2 0 . 1 5 8 9 8 3 1 7 1 - 0 . 7 5 4 3 8 3 4 4 5 0 . 2 0 8 9 6 1 6 6 6 - 2 . 4 9 8 5 7 1 3 9 6 0 . 1 5 9 9 8 2 7 4 1 - 0 . 6 9 6 9 6 3 7 2 7 0 . 2 0 9 9 6 1 2 3 6 - 2 . 4 7 7 2 5 4 8 6 8 0 . 1 6 0 9 8 2 3 1 1 - 0 . 6 3 2 9 5 8 8 8 9 0 . 2 1 0 9 6 0 8 0 5 - 2 . 4 5 7 2 6 2 0 3 9 0 . 1 6 1 9 8 1 8 8 1 - 0 . 5 6 1 1 6 2 4 7 2 0 . 2 1 1 9 6 0 3 7 5 - 2 . 4 3 8 4 6 5 1 1 8 0 . 1 6 2 9 8 1 4 5 1 - 0 . 4 8 0 0 7 1 9 6 2 0 . 2 1 2 9 5 9 9 4 5 - 2 . 4 2 0 7 6 9 6 9 1 0 . 1 6 3 9 8 1 0 2 0 - 0 . 3 8 7 7 4 8 6 5 9 0 . 2 1 3 9 5 9 5 1 5 - 2 . 4 0 4 0 7 7 5 3 0 0 . 1 6 4 9 8 0 5 9 0 - 0 . 2 8 1 6 9 9 6 5 7 0 . 2 1 4 9 5 9 0 8 5 - 2 . 3 8 8 3 0 9 4 7 9 0 . 1 6 5 9 8 0 1 6 0 - 0 . 1 5 8 6 0 1 0 4 6 0 . 2 1 5 9 5 8 6 5 5 - 2 . 3 7 3 3 8 6 3 8 3 0 . 1 6 6 9 7 9 7 3 0 - 0 . 0 1 3 9 9 5 7 4 4 0 . 2 1 6 9 5 8 2 2 5 - 2 . 3 5 9 2 4 5 3 0 0 0 . 1 6 7 9 7 9 3 0 0 0 . 1 5 8 2 5 6 5 9 0 0 . 2 1 7 9 5 7 7 9 5 - 2 . 3 4 5 8 2 9 0 1 0 0 . 1 6 8 9 7 8 8 7 0 0 . 3 6 6 9 5 7 4 8 6 0 . 2 1 8 9 5 7 3 6 5 - 2 . 3 3 3 0 8 5 0 6 0 1 3 3 A P P E N D I X B ( a ) C O M P U T E R P R O G R A M : G R A P H O F y = a E U ' ( a X ) / E U ( a X ) V S a I N E X A M P L E 3 . 3 . 2 D I M E N S I O N X ( 1 0 0 0 ) , Y ( 1 0 0 0 ) , U 1 1 ( 1 0 0 0 ) , U 1 2 ( 1 0 0 0 ) , U 2 1 ( 1 0 0 0 ) , U 2 2 ( 1 0 0 0 ) K = l J = l R 3 = 0 . 0 0 0 1 R l = 3 R 2 = - 4 DO 2 1 = 1 , 1 0 0 0 X ( I ) = R 3 U 1 1 ( I ) = ( 7 . * E X P ( - 1 . * R 1 * X ( I ) ) + 1 6 . * E X P ( - 2 . * R 1 * X ( I ) ) ) * X ( I ) U 1 2 ( I ) = ( 7 . * E X P ( - 1 . * R 2 * X ( I ) ) + 1 6 . * E X P ( - 2 . * R 2 * X ( I ) ) ) * X ( I ) U 2 1 ( I ) = - 7 • * E X P ( - 1 . * R 1 * X ( I ) ) - 8 . * E X P ( - 2 . * R l * X ( I ) ) + 1 5 U 2 2 ( I ) = - 7 . * E X P ( - 1 . * R 2 * X ( I ) ) - 8 . * E X P ( - 2 . * R 2 * X ( I ) ) + 1 5 Y ( T)=( Y ; (I ) =xo; 8?uaii.( i ) K t p : . 2 * u i 2 (r);) /.(o: s*u2ixi)r*o < 2 * p 2 2 ( i ) ) I F ( K . N E . J ) GO T O 3 J = J + 1 0 W R I T E ( 6 , 7 ) X ( I ) , Y ( I ) 7 F O R M A T ( 2 F 1 6 . 9 ) 3 K = K + 1 R 3 = R 3 + 0 . 0 0 0 1 7 4 2 C O N T I N U E C A L L S C A L E ( X , 1 0 0 0 , 1 0 . , X M I N , D X , 1 ) C A L L S C A L E ( Y , 1 0 0 0 , 1 0 . , Y M I N , D Y , 1 ) C A L L A X I S ( 0 . , 0 . , ' A L P H A ' , - 5 , 1 0 . , 0 . , X M I N , D X ) C A L L A X I S ( 0 . , 0 . , ' G A M A ' , - 4 , 1 0 . , 9 0 . , Y M I N , D Y ) C A L L L I N E ( X , Y , 1 0 0 0 , 1 ) C A L L P L O T N D S T O P E N D a ( b ) C O M P U T O R O U T P U T : Y 0 < a < . 1 8 A N D y a 1 3 4 Y 0 . 0 0 0 1 0 0 0 0 0 0 . 6 2 5 1 4 7 7 0 0 0 . 0 8 7 0 9 7 0 4 9 1 . 0 4 2 0 8 7 5 5 5 0 . 0 0 1 8 3 9 9 9 9 0 . 6 2 8 2 7 0 5 0 7 0 . 0 8 8 8 3 6 9 0 8 1 . 0 6 1 1 6 9 6 2 4 0 . 0 0 3 5 7 9 9 9 8 0 . 6 3 1 4 8 8 3 8 3 0 . 0 9 0 5 7 6 7 6 8 1 . 0 8 1 1 0 7 1 4 0 0 . 0 0 5 3 1 9 9 7 5 0 . 6 3 4 8 2 1 1 7 7 0 . 0 9 2 3 1 6 6 2 8 1 . 1 0 1 9 5 3 5 0 6 0 . 0 0 7 0 5 9 9 4 7 0 . 6 3 8 2 7 6 3 3 9 0 . 0 9 4 0 5 6 4 8 7 1 . 1 2 3 7 6 2 1 3 1 0 . 0 0 8 7 9 9 9 1 8 0 . 6 4 1 8 4 5 5 8 4 0 . 0 9 5 7 9 6 3 4 7 1 . 1 4 6 5 9 6 9 0 9 0 . 0 1 0 5 3 9 8 8 9 0 . 6 4 5 5 4 1 5 4 9 0 . 0 9 7 5 3 6 2 0 6 1 . 1 7 0 5 2 4 5 9 7 0 . 0 1 2 2 7 9 8 6 1 0 . 6 4 9 3 6 5 8 4 2 0 . 0 9 9 2 7 6 0 6 6 1 . 1 9 5 6 1 1 0 0 0 0 . 0 1 4 0 1 9 8 3 2 0 . 6 5 3 3 2 9 5 5 1 0 . 1 0 1 0 1 5 9 2 5 1 . 2 2 1 9 4 4 8 0 9 0 . 0 1 5 7 5 9 8 0 3 0 . 6 5 7 4 2 8 1 4 5 0 . 1 0 2 7 5 5 7 8 5 1 . 2 4 9 6 0 4 2 2 5 0 . 0 1 7 4 9 9 7 7 5 0 . 6 6 1 6 7 0 7 4 4 0 . 1 0 4 4 9 5 6 4 5 1 . 2 7 8 6 8 4 6 1 6 0 . 0 1 9 2 3 9 7 4 6 0 . 6 6 6 0 6 3 0 1 1 0 . 1 0 6 2 3 5 5 0 4 - 1 . 3 0 9 2 9 2 7 9 3 0 . 0 2 0 9 7 9 7 1 7 0 . 6 7 0 6 0 5 4 8 1 0 . 1 0 7 9 7 5 3 6 4 1 . 3 4 1 5 3 5 5 6 8 0 . 0 2 2 7 1 9 6 8 9 0 . 6 7 5 3 0 6 9 7 6 0 . 1 0 9 7 1 5 2 2 3 1 . 3 7 5 5 4 1 6 8 7 0 . 0 2 4 4 5 9 6 6 0 0 . 6 8 0 1 6 7 5 5 6 0 . 1 1 1 4 5 5 0 8 3 1 . 4 1 1 4 4 6 5 7 1 0 . 0 2 6 1 9 9 6 3 1 0 . 6 8 5 1 9 9 6 1 8 0 . 1 1 3 1 9 4 9 4 2 1 . 4 4 9 4 0 5 6 7 0 0 . 0 2 7 9 3 9 6 0 3 0 . 6 9 0 4 0 2 4 4 8 0 . 1 1 4 9 3 4 8 0 2 1 . 4 8 9 5 8 5 8 7 6 0 . 0 2 9 " 6 7 9 5 7 4 0 . 6 9 5 7 8 9 5 1 6 0 . 1 1 6 6 7 4 6 6 2 1 . 5 3 2 1 7 5 0 6 4 0 . 0 3 1 4 1 9 5 4 5 0 . 7 0 1 3 6 0 2 8 5 0 . , 1 1 8 4 1 4 5 2 1 1 . 5 7 7 3 7 9 2 2 7 0 . 0 3 3 1 5 9 5 1 7 0 . 7 0 7 1 2 5 6 0 4 0 . , 1 2 0 1 5 4 3 8 1 1 . 6 2 5 4 3 8 6 9 0 0 . 0 3 4 8 9 9 4 8 8 0 . 7 1 3 0 9 1 7 3 1 0 . , 1 2 1 8 9 4 2 4 0 1 . 6 7 6 6 1 0 9 4 7 0 . 0 3 6 6 3 9 4 5 9 0 . 7 1 9 2 6 5 5 8 0 0 . , 1 2 3 6 3 4 1 0 0 1 . 7 3 1 2 0 4 9 8 7 0 . 0 3 8 3 7 9 4 3 1 0 . 7 2 5 6 5 3 8 8 7 0 . , 1 2 5 3 7 3 9 6 0 1 . 7 8 9 5 4 3 1 5 2 0 . 0 4 0 1 1 9 4 0 2 0 . 7 3 2 2 6 7 4 9 9 0 . , 1 2 7 1 1 3 8 1 9 1 . 8 5 2 0 1 4 5 4 2 0 . 0 4 1 8 5 9 3 7 3 0 . 7 3 9 1 1 1 9 6 0 0 . , 1 2 8 8 5 3 6 7 9 1 . 9 1 9 0 5 2 1 2 4 0 . 0 4 3 5 9 9 3 4 5 0 . 7 4 6 1 9 7 5 2 2 0 . . 1 3 0 5 9 3 5 3 8 1 . 9 9 1 1 4 7 0 4 1 0 . 0 4 5 3 3 9 3 1 6 0 . 7 5 3 5 3 3 2 4 4 0 . , 1 3 2 3 3 3 3 9 8 2 . 0 6 8 8 9 2 4 7 9 0 . 0 4 7 0 7 9 2 8 7 0 . 7 6 1 1 3 0 6 3 1 0 . . 1 3 4 0 7 3 2 5 7 2 . 1 5 2 9 3 4 0 7 4 0 . 0 4 8 8 1 9 2 5 9 0 . 7 6 8 9 9 8 3 8 4 0 . , 1 3 5 8 1 3 1 1 7 2 . 2 4 4 0 5 2 8 8 7 0 . 0 5 0 5 5 9 2 3 0 0 . 7 7 7 1 4 8 0 0 8 0 . , 1 3 7 5 5 2 9 7 7 2 . 3 4 3 1 4 9 1 8 5 0 . 0 5 2 2 9 9 2 0 1 0 . 7 8 5 5 9 3 0 3 3 0 . . 1 3 9 2 9 2 8 3 6 2 . 4 5 1 2 9 1 0 8 4 0 . 0 5 4 0 3 9 1 7 3 0 . 7 9 4 3 4 2 2 7 9 0 . , 1 4 1 0 3 2 6 9 6 2 . 5 6 9 7 3 1 7 1 2 0 . 0 5 5 7 7 9 1 4 4 0 . 8 0 3 4 1 2 6 7 6 0 . . 1 4 2 7 7 2 5 5 5 2 . 6 9 9 9 8 2 6 4 3 0 . 0 5 7 5 1 9 1 1 6 0 . 8 1 2 8 1 5 1 8 9 0 , . 1 4 4 5 1 2 4 1 5 2 . 8 4 3 8 7 9 7 0 0 0 . 0 5 9 2 5 9 0 8 7 0 . 8 2 2 5 6 4 8 4 0 0 . . 1 4 6 2 5 2 2 7 5 3 . 0 0 3 6 1 2 5 1 8 0 . 0 6 0 9 9 9 0 5 8 0 . 8 3 2 6 7 7 7 2 2 0 , . 1 4 7 9 9 2 1 3 4 3 . 1 8 1 8 9 3 3 4 9 0 . 0 6 2 7 3 9 0 1 5 0 . 8 4 3 1 7 0 6 4 3 0 . . 1 4 9 7 3 1 9 9 4 3 . 3 8 2 1 2 6 8 0 8 0 . 0 6 4 4 7 8 8 7 4 0 . 8 5 4 0 6 1 2 4 6 0 , . 1 5 1 4 7 1 8 5 3 3 . 6 0 8 5 3 2 9 0 6 0 . 0 6 6 2 1 8 7 3 4 0 . 8 6 5 3 6 7 2 3 4 0 , . 1 5 3 2 1 1 7 1 3 3 . 8 6 6 5 4 2 8 1 6 0 . 0 6 7 9 5 8 5 9 3 0 . 8 7 7 1 0 4 7 5 9 0 , . 1 5 4 9 5 1 5 7 2 4 . 1 6 3 1 7 5 5 8 3 0 . 0 6 9 6 9 8 4 5 3 0 . 8 8 9 3 0 1 0 6 2 0 , . 1 5 6 6 9 1 4 3 2 4 . 5 0 7 7 0 4 7 3 5 0 . 0 7 1 4 3 8 3 1 3 0 . 9 0 1 9 7 5 8 1 1 0 , . 1 5 8 4 3 1 2 9 2 4 . 9 1 2 6 4 2 4 7 9 0 . 0 7 3 1 7 8 1 7 2 0 . 9 1 5 1 5 2 3 1 1 0 . 1 6 0 1 7 1 1 5 1 5 . 3 9 5 2 2 8 3 8 6 0 . 0 7 4 9 1 8 0 3 2 0 . 9 2 8 8 5 7 8 0 3 0 . 1 6 1 9 1 1 0 1 1 5 . 9 8 0 0 3 1 0 1 3 0 . 0 7 6 6 5 7 8 9 1 0 . 9 4 3 1 1 4 9 9 6 0 . 1 6 3 6 5 0 8 7 0 6 . 7 0 3 0 6 7 7 8 0 0 . 0 7 8 3 9 7 7 5 1 0 . 9 5 7 9 5 7 2 0 8 0 . 1 6 5 3 9 0 7 3 0 7 . 6 1 9 7 5 2 8 8 4 0 . 0 8 0 1 3 7 6 1 0 0 . 9 7 3 4 1 3 4 6 7 0 . 1 6 7 1 3 0 5 8 9 8 . 8 1 9 4 1 7 0 0 0 0 . 0 8 1 8 7 7 4 7 0 0 . 9 8 9 5 1 9 0 0 0 0 . 1 6 8 8 7 0 4 4 9 1 0 . 4 5 6 4 7 9 0 7 3 0 . 0 8 3 6 1 7 3 3 0 1 . 0 0 6 3 0 5 6 9 5 0 . 1 7 0 6 1 0 3 0 9 1 2 . 8 2 2 9 0 2 6 7 9 0 . 0 8 5 3 5 7 1 8 9 1 . 0 2 3 8 1 7 0 6 2 0 . 1 7 2 3 5 0 1 6 8 1 6 . 5 4 4 1 7 4 1 9 4 1 3 5 A P P E N D I X C T h e p r o o f o f t h e o r e m s 4 . 3 . 1 a n d 4 . 3 . 3 u s e t h e f o l l o w i n g p r o p o s i t i o n s . L e m m a A . 4 . 1 : L e t f : R -> R b e e v e r y w h e r e d i f f e r e n t i a b l e . T h e n , g i v e n x e R, ] a c o n t i n u o u s l y d i f f e r e n t i a b l e c o n c a v e f u n c t i o n g s u c h t h a t g ( x Q ) = f ( x Q ) , g ' ( x 0 ) = f ' ( x 0 ) a n d g ± f J i « e - g s u p p o r t s f a t X Q . P r o o f . W i t h o u t l o s s o f g e n e r a l i t y w e m a y c o n s i d e r X Q = 0 a n d c a n s e t f ( 0 ) = 0 , f 1 ( 0 ) = 0 . D e f i n e , f o r x >_ 0 , t h e d e c r e a s i n g f u n c t i o n , h ^ x ) = i n f { f ' ( t ) : 0 ^ t < _ x } . T h e n h 1 i s c o n t i n u o u s , a s a c o n s e q u e n c e o f t h e c l a s s i c a l t h e o r e m t h a t s a y s f 1 m a p s i n t e r v a l s t o i n t e r v a l s . F o r x ^ 0 , w e n o w s e t , g x ( x ) = x \ ( t ) d t T h e n g i s C 1 a n d c o n c a v e , s i n c e g ^ i s d e c r e a s i n g . C l e a r l y g± < f We n o w d e f i n e , h 2 ( x ) = S u p { f 1 ( t ) : x < t < 0 } f o r x < 0 a n d s e t 1 3 6 g 2 ( x ) = x J o h „ ( t ) d t f o r x <_ 0 . A g a i n g 2 i s C , c o n c a v e a n d g 2 <_ f . N o w w h e n t h e p i e c e s g ^ a n d g 2 a r e j o i n e d t o g e t h e r w e o b t a i n t h e r e q u i r e d g . C o r o l l a r y A . 4 . 2 : L e t h : ( 0 , « > ) - » - ( 0 , °°) b e a d i f f e r e n t i a b l e a n d  d e c r e a s i n g f u n c t i o n o f x . T h e n , g i v e n X q e ( 0 , «°) , 3 a c o n v e x - c o m p a c t s e t S c o n t a i n i n g ( 0 , 0 ) w h o s e u p p e r b o u n d a r y i s t h e g r a p h o f a  c o n t i n u o u s l y d i f f e r e n t i a b l e , s t r i c t l y d e c r e a s i n g f u n c t i o n s u c h t h a t S t o u c h e s t h e c u r v e h a t x ^ a n d l i e s c o m p l e t e l y u n d e r h . F I G U R E A . 4 . I s G R A P H S O F y = h ( x ) A N D y = g ( x ) 1 3 7 P r o o f : G i v e n X q , l e t g b e t h e f u n c t i o n g i v e n b y t h e p r e v i o u s l e m m a h ( x ) d e f i n e d o n [ 0 , x - 7-77—s~], w h e r e g ( 0 ) = l i m g ( x ) . T h e n g i s v o x -y + 0 c \ c o n c a v e , t o u c h e s h a t X q a n d l i e s c o m p l e t e l y u n d e r h a s w e l l a s u n d e r t a n g e n t l i n e o f h a t X Q • L e t S b e t h e r e g i o n b o u n d e d b y g a n d t h e t w o a x e s , i . e . S = { ( x , y ) : y <_ g ( x ) , 0 <_•&, 0 <_y} . T h i s s e t c o n t a i n s ( 0 , 0 ) b e c a u s e g i s d e c r e a s i n g . T h e n S i s t h e r e q u i r e d c o n v e x a n d c o m p a c t s e t . P r o p o s i t i o n A . 4 . 3 : L e t f : ( 0 , °°) x ( 0 , °°) -> ( 0 , °°) b e a d i f f e r e n t i a b l e  f u n c t i o n o f ( x , y ) . S u p p o s e f ( x , y ) i s s t r i c t l y i n c r e a s i n g i n b o t h x a n d y . T h e n , g i v e n ( x , y ) i n t h e d o m a i n o f f , 1 a c o n v e x - c o m p a c t s e t S c o n t a i n i n g t h e o r i g i n , w h o s e u p p e r b o u n d a r y i s a d i f f e r e n t i a b l e a n d  s t r i c t l y d e c r e a s i n g f u n c t i o n s u c h t h a t ( x , y ) m a x i m i z e s f ( x , y ) s u b j e c t  t o ( x , y ) e S . F I G U R E A . 4 . 2 : H A L F S P A C E S M A D E B Y f ( x , y ) = C A N D T H E S E T S 1 3 8 P r o o f : L e t f ( x , y ) = C a n d h : ( 0 , °°) -> ( 0 , °°) b e d e f i n e d b y f ( x , h ( x ) ) = C . T h e n h i s s t r i c t l y d e c r e a s i n g , d i f f e r e n t i a b l e e v e r y w h e r e a n d h ( x ) = y . F r o m t h e p r e v i o u s c o r o l l a r y , t h e r e f o r e ] a c o n v e x - c o m p a c t s e t S c o n t a i n i n g ( 0 , 0 ) a n d s u p p o r t i n g t h e c u r v e f ( x , y ) = C f r o m b e l o w a t ( x , y ) . N o t i c e - t h a t f o r a n y p o i n t ( x , y ) a b o v e t h e c u r v e f ( x , y ) = C w e h a v e f ( x , y ) > C w h i l e f o r a n y p o i n t b e l o w t h e c u r v e w e h a v e f ( x , y ) < C . H e n c e f i s m a x i m i z e d b y ( x , y ) o u t o f p o i n t s b e l o n g i n g t o S a n d t h i s c o m p l e t e s t h e p r o o f . T h e P r o o f o f T h e o r e m 4 . 3 . 1 : G i v e n a n y ( X q , y ) i n t h e d o m a i n o f f w e c a n f i n d a c o n v e x - c o m p a c t s e t S , w h o s e u p p e r b o u n d a r y i s t h e g r a p h o f a s t r i c t l y d e c r e a s i n g a n d d i f f e r e n t i a b l e f u n c t i o n g s u c h t h a t f i s m a x i m i z e d b y ( X q , y ) s u b j e c t t o ( x , y ) e S ; i . e . s u b j e c t t o y = g ( x ) . H e n c e , f 1 ( x Q , y Q ) + g ' ( x o ) f 2 ( X q , y Q ) = 0 ( A . 4 . 1 ) N o w f r o m t h e i n v a r i a n c e p r o p e r t y , s i n c e f i s m a x i m i z e d b y ( K x 0 » v 0 ) w i t h K > 0 s u b j e c t t o y = g ( — ) , w e h a v e , K f l ( K V y o } + l . g ' ( x o ) f 2 ( K V V = ° ( A , 4 ' 2 ) F r o m ( A . 4 . 1 ) a n d ( A . 4 . 2 ) w e n o w d e d u c e , a s g ' ( X Q ) 4 0 , t h a t , 1 3 9 f l ( V V f 2 ( K V V " K f 2 ( V y o ) f l ( K x o ' V L e t t i n g K = 1 /X q , i n p a r t i c u l a r , w e h a v e , f l ( 1 ' y o } X o f l ( x o ' V = f 2 ( x o ' y o } f 2 ( l , y o ) S i n c e t h i s r e l a t i o n h o l d s f o r a n y a r b i t r a r y ( x , y ) w e s e e t h a t f o o s a t i s f i e s t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n , . 9f u , . df = h ( y ) 3 f w h e r e h ( y ) = f ^ l , y ) / f 2 < l , y ) . I t c a n b e s h o w n ( c f . S n e d d o n [ 1 9 5 7 ] ) t h a t t h e g e n e r a l s o l u t i o n o f t h i s P . D . E . i s , f ( x , y ) = F ( x B ( y ) ) W h e r e B ( y ) = E x p ( / h ( y ) ) a n d F a r e b o t h a r b i t r a r y f u n c t i o n s o f o n e v a r i a b l e . 140 FIG. A . U. 3: S e t s Q, R, a n d t h e m a x i m a l p o i n t ( X ^ Z c . ) 1 4 1 G i v e n ( x , y , z ) , w e c a n f i n d a c o n v e x - c o m p a c t s e t Q i n t h e x - y o o o p l a n e w h o s e u p p e r b o u n d a r y i s t h e g r a p h o f a s t r i c t l y d e c r e a s i n g a n d d i f f e r e n t i a b l e f u n c t i o n y = g ( x ) s u c h t h a t f ( x , y , Z q ) <_ f ( X q , y , Z Q) f o r a l l ( x , y ) e Q ( s e e F i g u r e A . 4 . 3 ) . N o w d e f i n e t h e s e t R i n x - z p l a n e t o b e t h e r e c t a n g l e w i t h s i d e s e q u a l t o t h e l i n e s e g m e n t j o i n i n g t h e o r i g i n w i t h ( 0 , 0 , Z q ) a n d ( x , 0 , 0 ) r e s p e c t i v e l y , w h e r e x i s t h e p o i n t a t w h i c h t h e g r a p h o f g c u t s t h e x - a x i s . N o w s i n c e f i s a n i n c r e a s i n g f u n c t i o n i n a l l v a r i a b l e s f o r a n y g i v e n x a n d y f ( x , y , z ) < f ( x , y , Z Q) V z <_ Z q . T h u s b y c o m b i n i n g t h e a b o v e t w o i n e q u a l i t i e s w e h a v e , f ( x , y , z ) < f ( x , y , z Q ) V ( x , y ) e Q a n d ( x , z ) e R H e n c e f i s m a x i m i z e d b y ( x , y , ZQ) s u b j e c t t o t h e t w o s e t s Q a n d R . O n d i f f e r e n t i a t i o n o f f ( x , y , z ) = x h ( y , z ) w i t h y = g ( x ) , z = z ( a t ( x , y , z ) w e o b t a i n , o Jo o h < V Z o } + X o h i ( y o ' ^ • ° ( A - 4 ' 3 ) M o r e o v e r u s i n g t h e f a c t t h a t f i s a l s o m a x i m i z e d b y ( x , K Y 0 > K z 0 ^ w i t h y = K g ( x ) , z = K Z q , w e h a v e , 1 4 2 h ( K y o , K Z q ) + K x o g ' ( x o ) h± ( K y Q , KZ Q) = 0 ( A . 4 . 4 ) E l i m i n a t i o n o f x g ' ( x ) f r o m t h e e q u a t i o n s ( A . 4 . 3 ) a n d ( A . 4 . 4 ) y i e l d s o o K h ( y o , z o ) h x ( K y Q , K z o ) = h ( K y o , K Z q ) h ; ( y Q , Z Q) B y s e t t i n g K = 1 / Z q a n d n o t i n g t h a t ( X q , y , Z q ) i s a r b i t r a r y w e t h u s o b t a i n t h e d i f f e r e n t i a l e q u a t i o n , MC7;) . h ( y , z ) = z h 1 ( y , z ) w h e r e MO7;) = h ^ ( y / z , l ) / h ( y / z , 1 ) i s a n u n s p e c i f i e d f u n c t i o n o f o n e v a r i a b l e . R e a r r a n g i n g t h e t e r m s o f t h i s e q u a t i o n y i e l d s t h e i n t e g r a t i o n , 3y- h ( y » Z ) h ( y , z ) d y = i z M( Z) d y T h i s i n t e g r a l i m p l i e s t h a t , In h ( y , z ) = In N ( J ) + £ n C ( z ) w h e r e C ( z ) i s a n a r b i t r a r y f u n c t i o n o f z a n d n i s a g a i n a n u n s p e c i f i e d f u n c t i o n . 1 4 3 H e n c e h i s o f t h e f o r m , h ( y , z ) = C ( z ) N ( J ) . T h e p r o o f o f C o r o l l a r y 4 . 3 . 2 : A p p l i c a t i o n o f T h e o r e m 4 . 3 . 1 a t c o n s t a n t y a n d z r e s p e c t i v e l y t e l l s u s t h a t f c a n b e w r i t t e n i n t h e t w o f o r m s , f ( x , y , z ) = S ( x g ( x , z ) , y ) ( A . 4 . 5 ) = S ( x h ( y , z ) , z ) ( A . 4 . 6 ) H e n c e S ( x g ( y , Z q ) , y ) = S ( x h ( y , Z Q ) , Z q ) a n d i n t u r n w e h a v e , s (w ° , z ) if g(y, z ) * o g(y» Z q) o o S ( W , y ) = S ( 0 , y ) i f g(y> z o) = 0 T h u s f ( x , y , z ) = G ( x A ( y , z ) ) , w h e r e G ( t ) = S ( t , Z q ) a n d 1 4 4 A ( y , z ) = g ( y , z ) h ( y , Z Q) g ( y » z 0 ) -f g ( Y , O r 0 i f g ( y , z D ) = 0 w i t h G(0) = S ( 0 , y ) , y b e i n g t h e s o l u t i o n o f g ( y , Z Q) - 0. T h e P r o o f o f L e m m a 4 . 3 . 4 : D e f i n e X ( y ) = + ( y ) / * ( - y ) i f < K ~ y ) * 0 • T h e n u s i n g V + ( y ) = * ( - y ) 4> ' (y) w e d e d u c e t h a t , f o r <|,(-y) 4 0, K - y ) ' S o l v i n g t h e d i f f e r e n t i a l e q u a t i o n X ' / ( X + D = * ' / + » w e t h u s c o n c l u d e t h a t , f o r c ^ ( - y ) ^ 0, |gy = x (y) = cU(y) - i ] w h e r e C i s a n a r b i t r a r y c o n s t a n t . H e n c e f o r a l l y , w e h a v e Ky) = <i>(-y) [ C <f)(y) - ! ] • 0 u r o r i g i n a l d i f f e r e n c e - d i f f e r e n t i a l e q u a t i o n a n d t h i s d i f f e r e n c e e q u a t i o n i m p l i e s t h a t <|> s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n 1 4 5 ^ = k (c <j> - 1 ) d y if cf>(y) 4 o E v i d e n t l y t h e g e n e r a l s o l u t i o n o f t h i s d i f f e r e n t i a l e q u a t i o n i s , 4>(y) = \ [ 1 + D e k C y ] E - k y i f C 4 0 i f C = 0 f o r a l l y s o t h a t t h e c o n t i n u i t y o f $ i s p r e s e r v e d . N o t i n g t h a t ( i ) t h e a d m i s s i b l e v a l u e s o f D s a t i s f y i n g k <J>(y) = * C — y ) • ' ( y ) a r e o n l y + 1 a n d - 1 , a n d ( i i ) t h e o n l y a d m i s s i b l e v a l u e o f E i s 0 w h e n k = + 1 , w e t h u s c o m p l e t e t h e p r o o f . 

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