BARGAINING TO PROBLEM OF SOLUTIONS THE EXCHANGE OF UNCERTAIN VENTURES by (jfEERAHANDL B.Sc, , Vidyodaya A THESIS' THE SAMARADASA U n i v e r s i t y SUBMITTED IN REQUIREMENTS DOCTOR S r i PARTIAL FOR OF of THE Lanka, FULFILMENT DEGREE 1970 OF OF PHILOSOPHY i n THE (INSTITUTE We OF accept to THE DEPARTMENT APPLIED this the OF thesis required UNIVERSITY MATHEMATICS MATHEMATICS OF STATISTICS) conforming standard BRITISH OCTOBER, (c)Samaradasa as AND COLUMBIA 1976 Weerahandi, 1976 In p r e s e n t i n g this thesis an advanced degree at the I Library shall f u r t h e r agree for f u l f i l m e n t of the requirements the U n i v e r s i t y of B r i t i s h Columbia, make it freely available that permission for I agree r e f e r e n c e and for e x t e n s i v e copying of this for that study. thesis s c h o l a r l y purposes may be granted by the Head of my Department or by h i s of in p a r t i a l this written representatives. thesis f i n a n c i a l gain s h a l l Mathematics • U n i v e r s i t y of B r i t i s h 207S Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date is understood permission. Department of The for It Columbia that not copying or p u b l i c a t i o n be allowed without my ii Supervisors: Lawrence M. Clevenson and James V. Zidek ABSTRACT C o n s i d e r the b e t t i n g problem where two the amount each w i l l b e t . I t has b e t t o r s both have concave u t i l i t y bets t h i s r e s u l t can be The two f u n c t i o n s , there e x i s t m u t u a l l y b e n e f i c i a l to both p l a y e r s ) merely i f p l a y e r s ' the b e t t i n g event d i f f e r . generalized to determine a l r e a d y been e s t a b l i s h e d t h a t when the ( i . e . bets g i v i n g p o s i t i v e u t i l i t y s u b j e c t i v e p r o b a b i l i t i e s on i n d i v i d u a l s negotiate I t i s shown here to the case of more g e n e r a l r e s u l t s are extended to a more g e n e r a l utility that functions. s i t u a t i o n , t h a t of a s t o c h a s t i c exchange. I t i s shown t h a t the s e t of a l l f e a s i b l e s o l u t i o n s a v a i l a b l e f o r exchange f o r two r i s k averters i s a convex s e t w i t h a known boundary. A f t e r d e f i n i n g a s o l u t i o n f o r the members of a c l a s s of exchange models it i s shown i n the t h i r d chapter that the ' s i z e ' of the exchange p r e s c r i b e d by the s o l u t i o n tends to i n c r e a s e w i t h the p a r t i c i p a n t s ' i n i t i a l w e a l t h w i t h m u l t i p l i c a t i v e s h i f t s of the random v a r i a b l e c h a r a c t e r i z i n g the Furthermore the s i z e of the exchange may increase or decrease due and exchange. 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 engaged i n b a r g a i n i n g an a x i o m a t i c w i t h incomplete i n f o r m a t i o n ( o f f e r ) by maximizing a g e n e r a l i z e d Nash to be the product of h i s u t i l i t y utility method t h a t an i n d i v i d u a l and finds h i s ' f a i r ' f u n c t i o n , GNF; a general t h i s GNF Continuing a parameter whose v a l u e may the study on b a r g a i n i n g i s found mean of h i s opponent's (from f i r s t i n d i v i d u a l ' s p o i n t of v i e w ) . T h i s g e n e r a l c h a r a c t e r i z e d by demand vary uncertain mean i s from person to p e r s o n . under incomplete i n f o r m a t i o n , a best i i i bargaining of strategy 'Backward s t r a t e g i e s i s i s developed I n d u c t i o n ' . also A i n c r i t e r i o n e s t a b l i s h e d . the f o r l a s t chapter comparing using the a v a i l a b l e technique bargaining i v TABLE OF CONTENTS Page CHAPTER 1. INTRODUCTION CHAPTER 2. EXCHANGE PROBLEM: 2.1 I n t r o d u c t i o n 2.2 Model 2.3 Existence 2.4 An 2.5 of 3. B e t t i n g of EXCHANGE to the I n t r o d u c t i o n NASH the 8 Problem 14 Set of A l l Mutually SOLUTIONS 33 33 Contract 3.2.1 Edgeworth 3.2.2 Pareto Optimal 3.3 Nash's Solutions 3.3.1 The 3.3.2 F a i r 3.3.3 Response 4. of Bets 28 PROBLEM: Edgeworth's CHAPTER 4 5 Favorable Exchange Structure 3.2 and SOLUTIONS Exchanges 3.1 Nash OF Problem Mutually Extension Geometrical EXISTENCE 4 the Favorable CHAPTER 1 Curve A n a l y s i s 34 Theory 34 Exchanges 35 39 Theory 40 Exchanges of 41 the Nash S o l u t i o n to Changes i n C, P, Y 46 BARGAINING PROBLEM: SOLUTIONS UNDER INCOMPLETE INFORMATION 59 4.1 Introduction 59 4.1.1 Background 59 4.1.2 Bargaining 4.2 The Problem 4.2.1 The Nature Under Formulation 4.3 An 4.3.1 The 4.3.2 Deducing 4.3.3 Example 4.3.4 Deriving 4.4 Summary CHAPTER 5. 5.1 Information 61 63 of 4.2.2 Incomplete the of Axiomatic Problem the 63 Bargaining Approach to the Model 65 D e r i v a t i o n of F a i r Demands Axioms 69 the Form of GNF When ft Contains Two Elements of the and BARGAINING General Form of GNF 94 Conclusions PROBLEM: AN 98 OPTIMAL BARGAINING PROCESS 100 Introduction Optimal 5.2.1 Notation Strategy 5.2.2 C r i t e r i o n 5.2.3 Backward 5.3 Bargaining when the Opponent's Strategy i s 102 Preplanned 5.3.2 and f o r the 102 Model Comparison Induction of Procedure S t r a t e g i e s by Induction Bargaining S t r a t e g i e s 104 107 (BIP) Backward Induction Procedure 120 120 I n t r o d u c t i o n Backward 70 85 5.2 5.3.1 69 Procedure 121 V Table of Contents (Cont'd.) Page BIBLIOGRAPHY 127 APPENDIX A 130 APPENDIX B 133 APPENDIX C 135 v i LIST OF FIGURES Page CHAPTER 2 Fig. 2.5.1 31 Fig. 2.5.2 32 Fig. 3.2.1 36 Fig. 3.3.1 44 Fig. 3.3.2 52 Fig. 3.3.3 55 F i g . 4 . 3 . 1 , 71 Fig. 4.3.2 76 Fig. 4.3.3 77 Fig. 4.3.4 86 Fig. 4.3.5 88 Fig. 4.3.6 88 Fig. 4.3.7 90 Fig. 4.3.8 90 Fig. 4.3.9 91 Fig. 4.3.10 91 Fig. 4.3.11 93 Fig. 4.3.12 96 CHAPTER CHAPTER CHAPTER 3 4 5 Fig. APPENDIX Fig. 5.1 107 A.4.1 136 C v i i L i s t of Figures (Cont'd.) Page Fig. A.4.2 137 Fig. A.4.3 140 viii ACKNOWLEDGEMENTS I am g r a t e f u l l y i n d e b t e d t o P r o f e s s o r Lawrence Clevenson f o r suggesting t h e problems t r e a t e d i n chapters two and f i v e and f o r h i s guidance o f my r e s e a r c h work d u r i n g n e a r l y two y e a r s . I am a l s o g r a t e f u l l y indebted to P r o f e s s o r J i m Zidek treated i n this chapters f o r suggesting the r e s t o f the problems t h e s i s and f o r h i s guidance i n o b t a i n i n g the r e s u l t s i n t h r e e and f o u r and d u r i n g the p r e p a r a t i o n o f the t h e s i s . I would also l i k e to express my g r a t i t u d e to P r o f e s s o r S. W. Nash f o r a c t i n g as my a d v i s o r f o r more than f o u r months d u r i n g which time he devoted h i s time to h e l p me to b r i n g t h i s d i s s e r t a t i o n i n t o i t s p r e s e n t I would l i k e to extend form. my a p p r e c i a t i o n to P r o f e s s o r s A. M a r s h a l l , K. N a g a t a n i , S. W. Nash,£S. J . P r e s s t h e i r h e l p f u l suggestions and comments d i s s e r t a t i o n . I would l i k e t o express and W. T. Ziemba f o r and f o r t h e i r c a r e f u l r e a d i n g o f the my g r a t i t u d e to P r o f e s s o r F. H. C l a r k e f o r the p r o o f o f Lemma A.5.1. My s i n c e r e thanks a l s o go to M. W. A l i , D. Fynn, K. Lee and K. T s u i f o r t h e i r constant w i l l i n g n e s s to e n t e r i n t o d i s c u s s i o n s . F i n a l l y , I would l i k e to thank Ms. L. D. Nelson patient expert t y p i n g and Mr. R. Brunn andiMfvjS. f o r h e r c a r e f u l and D.. D... J a y a s i n g h a for their drafting. The f i n a n c i a l support o f the Canadian Commenwealth S c h o l a r s h i p and F e l l o w s h i p A s s o c i a t i o n and o f the U n i v e r s i t y o f B r i t i s h Columbia a r e gratefully acknowledged. CHAPTER 1 INTRODUCTION [1972] and of The f i r s t and H i l d r e t h mathematical ' b e t s ' and determining problem of part the from is same which general In leads as of the only a b e t t i n g The and has of Nash s o l u t i o n we are the c r i t e r i o n the under of t h i s Some of Harsanyi for lead the f i r s t i m p l i c a t i o n s and n a t u r a l l y problem deals exchange s o l u t i o n s Monopoly. [1950] transactions the to except are w i t h s t o c h a s t i c two the may models [1972], players exchange which that evaluated, the of p r a c t i c a l B i l a t e r a l other H i l d r e t h H i l d r e t h s t o c h a s t i c part part by And the of the involves small l a t t e r w i t h problem p a r t i c i p a n t s ) s t o c h a s t i c a n a l y s i s , introduced both m a t h e m a t i c a l way, curve theory, (to a of work are conceptually random have a v a r i a b l e more values. H i l d r e t h and the deals u t i l i t i e s exchanges, analysis to by [1974], b e t t i n g t h i s favorable s i m i l a r i n s p i r e d w i l l . w a g e r . of p o s s i b l e work, the The adversaries' favorable him of each extension the his mutually amount t h e s i s , Tesfastsion foundations nature range and t h i s 'exchanges'. the an of i n t o the o b j e c t i v e large of t h i s areas for work i n Selten bargaining w i t h as the a n a l y s i s , t h i s to approach of to mutually t r e a t , extend Edgeworth i n the contract "bargaining i n the was i n s p i r e d In our search described problem. i n Having the problem" context area [1971]. problem such and i s of e t c . concerns economic i n d i r e c t thesis problem thesis existence t e r r i t o r y t e r r i t o r i e s to the His unexplored exchange the of exchanges. bargaining, our and way question by for of the a previous discovered works paragraphs that the 2 stochastic exchange information' i n s u b s t a n t i a l the part unsolved, general so p o s s i b l e the The d i f f e r e n t u t i l i t y 1 apply of the degrees . of Neumann-Morgenstern the oping of the features a theory e l a b o r a t i o n and t h i s as Selten c o n t r i b u t i o n the 'bargaining problem. bargaining only [1971] This to t h i s s i t u a t i o n and of exchange i m p l i c a t i o n s of the we area and incomplete devote problem a remains i s s u f f i c i e n t l y therefore Duopoly kind under has s i t u a t i o n s s i t u a t i o n s i n considered t h e s i s . and that, r i s k can choices be are theory u t i l i t y of among explained u t i l i t i e s i n o b j e c t i v e s of i n the using models adopted a l t e r n a t i v e s by c a l c u l a t e d t h i s terms area, of by an e x p l a i n phenomena which develop f u r t h e r information; M. (3) the appropiate our of some (2) and the explanation s t o c h a s t i c were maximization i n accordance p r o b a b i l i t y J . chapters here are i n v o l v i n g of expected w i t h the Von d i s t r i b u t i o n s and of steps taken of f i r s t c e r t a i n The over surveyed [1948]. i n (1) the the value i n c e n t i v e s e l a b o r a t i o n of f o r d e v e l such by H i l d r e t h i n his phenomena subsumed i n t h i s help to r e s u l t s understood theory, Savage are: development incompletely are problem model. bargaining L. early exchange i n i t i a l c o n t r i b u t i o n s Friedman the stochastic problem See to any t h i s work, 1. Harsanyi Our to pioneering chapters treated events. Some of be Monopoly-Monopsony idea Expected relevant i n formulation on of thesis to than part based must however. rather e a r l y mainly to the a p p l i c a t i o n s Economics, i n s p i r i t of l a r g e l y as problem by w i l l i n t u i t i o n . p a r t i c u l a r below i n also more under L a t e r incomplete d e t a i l . 3 In search Chapter f o r agreeable To the considerable under the the d e t a i l a the s t o c h a s t i c s u f f i c i e n t approach further set of under of t h i s i s a l l model for conditions d i f f e r e n t development of s o l u t i o n i s generalized as process. and to expected f i n d best use of a b e t t i n g under from t h i s which that theory f e a s i b l e exchanges, to Edgeworth f o r for of we problem and mutually H i l d r e t h analyze the [1972]. i n s i t u a t i o n s i s as nearly good t h i s done using [1950] thesis In Nash's i s a for to one the of one the one by the is one. a of an to is for taking the these A to p r a c t i c a l l y instance, i n t o which i s problem a then any class These the of of we bargaining bargaining would y i e l d proposed impossible whether account i n compare procedure [1964] decomposition made i n c h a r a c t e r i z a - approach. i . e . analysis problem, B i s h o p ' s components, sequences b e t t e r , the bargaining conceptual introduced two and to curve q u a l i t a t i v e axiomatic o f f e r Although check, derived best 4, bargainer. strategy. to devoted w i t h c r i t e r i o n which i s derived s o l u t i o n s " 5, i n t e r e s t i n g Chapter Dealing procedure as solutions mathematically. then c o n t r a c t c h a r a c t e r i z e i s find example an An of bargaining strategy, then do p a r t i c u l a r . u t i l i t y t h i s i s monopoly, solutions Chapter higher may and proposed. Nash In 3 information. " b a s e l i n e s t r a t e g i e s a i n remainder problem Chapter b i l a t e r a l incomplete propose of exchanges concepts, The one and The p o s s i b l e f e a s i b l e s o l u t i o n the e x i s t . purpose context t i o n develop c o n s i d e r a t i o n . The on we necessary bets f a c i l i t a t e 2, to a to evaluate, proposed p s y c h o l o g i c a l f a c t o r s , 4 CHAPTER 2 EXCHANGE PROBLEM : 2.1 The EXISTENCE OF INTRODUCTION problem of "Exchanges and by H i l d r e t h [1972] and [1974]. SOLUTIONS The B e t s " was defined i n a formal b e t t i n g problem i n v o l v e s two t r y i n g to determine the amount each williwage'r. aBy/aa;twager we " f r i e n d l y b e t " between two people mean the people where what each person b e t s i s what l o s e s i n case he l o s e s the b e t . of these i d e a s fashion The typical he exchanges problem i s a g e n e r a l i z a t i o n to a s i t u a t i o n i n v o l v i n g two d e c i s i o n makers i n which a c e r t a i n type of r i s k y t r a n s a c t i o n h a v i n g a s p e c i a l f e a t u r e i s i n v o l v e d . The f e a t u r e i s t h a t the e f f e c t of the t r a n s a c t i o n on the two prospects can be where i s not an amount but 0. . if Y The wefi represented by an exchange of a venture The b e t s and a negative -Y, b e t t i n g problem i s a p a r t i c u l a r case of number (the amount bet by exchanges are given i n s e c t i o n s 2.2 The for p e r s o n pays the o t h e r s i t u a t i o n i n which the random v a r i a b l e takes o n l y two number and Y r a t h e r a random v a r i a b l e on the sample space e f f e c t of the t r a n s a c t i o n i s t h a t one is realized. d e c i s i o n makers' values, each p l a y e r ) . and 2.4 Y(w) this a positive Examples of respectively. major o b j e c t i v e of t h i s chapter i s the study of the e x i s t e n c e m u t u a l l y f a v o r a b l e b e t s and exchanges, when the d e c i s i o n makers have p o s s i b l y d i f f e r e n t b e l i e f s about the random events i n v o l v e d s i t u a t i o n , under v a r i o u s of conditions on the u t i l i t y i n the functions. t h i s problem under some p r e s p e c i f i e d c o n d i t i o n s on u t i l i t y Considering functions 5 Hildreth [1972] comes to the remarkable c o n c l u s i o n t h a t , i f two a v e r t e r s have d i f f e r e n t s u b j e c t i v e p r o b a b i l i t i e s f o r an event there e x i s t mutually favorable b e t s based on from some of the c h a r a c t e r i z a t i o n s he In t h i s c h a p t e r we consider He establishes concavity enables us property A, then deduces t h i s r e s u l t for uncertain the same problem more d i r e c t l y . d i f f e r s from t h a t of H i l d r e t h ' s and without assuming any A. risk ventures. Our to conclude more approach generally, of the u t i l i t y f u n c t i o n s , that d e c i s i o n makers h a v i n g d i f f e r e n t s u b j e c t i v e p r o b a b i l i t i e s can always mutually favorable bets. S t u d y i n g the e x i s t e n c e [1974] c o n s i d e r s different find one of m u t u a l l y f a v o r a b l e f a m i l y of v e n t u r e s . f a m i l y of v e n t u r e s c o v e r i n g In s e c t i o n attempt 2.4 we consider a a range of i n t e r e s t i n g r e a l w o r l d s i t u a t i o n s t h a t cannot be handled by H i l d r e t h ' s H i l d r e t h does not exchanges, H i l d r e t h approach. (and h i s i n d i r e c t approach i s i n c a p a b l e h a n d l i n g ) to determine the s e t of a l l p o s s i b l e m u t u a l l y f a v o r a b l e b e t s exchanges, i . e . the s e t of a l l f e a s i b l e p o i n t s . d i r e c t l y we a b l e to do w i l l be s t r u c t u r e of the enable us to do this in section Attacking 2.4. the and problem Such a study of set of a l l m u t u a l l y b e n e f i c i a l exchanges w i l l , f u r t h e r a n a l y s i s such as Edgeworth's c o n t r a c t of for curve the instance, analysis of the exchanges problem. 2.2 C o n s i d e r two MODEL OF THE p l a y e r s , who BETTING PROBLEM w i l l be called K chapter b e t t i n g on the o c c u r r e n c e of events made up and L throughout t h i s of p o s s i b l e 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 d i s t i n g u i s h i n g the events the two p l a y e r s bet i n favor of. An defined by experiment 6 c o n s i s t i n g up", the hockey the of f l i p p i n g occurrence game, players presents a are to of a l l bet a w i t h c e r t a i n examples on. b e t t i n g coin p a r t i c u l a r of games Insurance problem which p r o v i d i n g a p a r c e l , the to to be and the sender by A, Thus K wins i f A occurs of r e s u l t of experiment w h i l e L wins i f A does L bets more i n favor r i g o r o u s l y standard t i o n own complement or of. the Let other b e l i e f s on means. such s u b j e c t i v e or s u b j e c t i v e p r o b a b i l i t i e s the bet that p r o b a b i l i t i e s . a r i s e i n to performed be game the other. observe is To i s the r e a l i z e d c a r r i e d Having outcome and when is come of subsets of be based also suppose t h e i r is that own p to and that e t c . the A They q wasting time on unimportant the < p players bet only i f p l a y e r to compromise the a each game a l t e r n a t i v e l y and K L pays < 1 , possibly i s about pays L i f by 0 a t r a d i - t h e i r l i t e r a t u r e as r e s p e c t i v e p = Prob. (A consequences of cases that may never < 1 The game q < amounts is . amounts. w i t h to he or have t h e i r d i f f e r e n t A' not of whatever to e that experiment, have the means s a t i s f i e d the K the any performed i . e . Their p r e d i c t i s r e s u l t experience, be are proper event do i . e . game 0 0, occurs, the that the players i n also occur, t h i s past a the not information. r e f e r r e d Let L on as i s of that may A' outcomes of and assumed and out K the denoted say F r e f l e c t i n g of might a - f i e l d knowledge), avoid once us that p r o b a b i l i t i e s K's i t a We p r o b a b i l i t i e s . r e s u l t p r a c t i c e the of personal may i n events to suppose Let assignment according us occurs. assignment i n d i v i d u a l occurs A events p r o b a b i l i t y any of for loss . the events a for space A', be r o l l e d , shipped, company comming be sample i f s h a l l die a l t e r n a t i v e insurance bets a of Q K of "Head subset the which sides of The the on of p r o b a b i l i t y adversaries. the event i n unknown the be bet personal However o f f e r bet i f r e a l i z e d . i s of players co e As A i t w i l l l e a d to no c o n f u s i o n L i n terminology, t o denote the amounts b e t by K and we s h a l l use the l e t t e r s L, K and respectively. A b e t made up by p l a y e r s ' o f f e r s i s s a i d t o be f a v o r a b l e t o a c e r t a i n p l a y e r i f h i s expected g a i n from h i s own p o i n t o f view from t h a t b e t i s strictly positive. according bet^. it A b e t which i s found to be f a v o r a b l e t o each p l a y e r to h i s own b e l i e f s w i l l be r e f e r r e d t o as a m u t u a l l y We are assuming t h a t n e i t h e r p l a y e r w i l l accept favorable a given b e t u n l e s s i s f a v o r a b l e t o him. We a r e f u r t h e r assuming t h a t t h e r e i s no c o s t i n v o l v e d i n p a r t i c i p a t i n g i n t h i s game and t h a t no p l a y e r w i l l be f i n e d o r lose anything even i f he d e c i d e s t o q u i t the game once he has p a r t i c i p a t e d . These assumptions may n o t be completely inconveniences to since at least personal a r e n o t taken i n t o account. By u n d e r t a k i n g C realistic C+L o r from C a bet to (K, L ) , C-K K's worth w i l l be changed e i t h e r from depending on which o f 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 T h i s would r e s u l t i n a change of K's u t i l i t y or K's U(C-K) K accordingly. Hence d e r i v e d on the b a s i s o f h i s b e l i e f s , Since a u t i l i t y formation IL,(x) = U (C+x), K K p U (L) + (1-p) K favorable to K U (C) K to U (C+L) K expected g a i n from a b e t i s p lL(C+L) + (1-p) K (K, L ) , iL(C-K) is. - tX_(C) . K f u n c t i o n i s only determined up to a p o s i t i v e l i n e a r t r a n s - one may c o n v e n i e n t l y d e f i n e d by from 0 < K <_ C . use a n o r m a l i z e d with U (C) = 0 K V utility function U , and r e w r i t e t h i s f o r m u l a as U (-K). Hence the requirement t h a t a given b e t K (K, L) be is A mutually f a v o r a b l e b e t may n o t n e c e s s a r i l y mean that i n r e a l w o r l d s i t u a t i o n s p e o p l e w i l l s u r e l y engage i n i t . F o r a d i s c u s s i o n o f the q u e s t i o n as to why o p p o r t u n i t i e s f o r m.f. b e t s a r e n o t completely e x p l o i t e d 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 p r o s p e c t s c f . H i l d r e t h and T e s f a t s i o n [1974]. 1 8 (2.2.1) p U ( L ) + (1 - p) U (-K) > 0 R Similarly L K would f i n d such a b e t i s f a v o r a b l e t o h i m s e l f i f , (2.2.2) (1 - q) U ( K ) + q U ( - L ) > 0 L with 0 < L <^ D and L 0 < K <_ C, where 0 < p < 1 l . h . s . of the above i n e q u a l i t y r e p r e s e n t s from h i s p o i n t of view, Any b e t (K, L) D > 0 expected 0 < q < 1 . g a i n from b e i n g the i n i t i a l worth of s a t i s f y i n g b o t h equations mutually f a v o r a b l e . L's and (2.2.1) and L The (K, L) with (2.2.2) 0 < L <^D. is I n t h i s chapter we w i l l e s t a b l i s h some n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r the e x i s t e n c e of such m u t u a l l y f a v o r a b l e b e t s . 2.3 EXISTENCE OF MUTUALLY FAVORABLE BETS In the p r e v i o u s s e c t i o n we s a i d t h a t a b e t (K, L) formed by the o f f e r s of the two p l a y e r s i s m u t u a l l y f a v o r a b l e i f and o n l y i f the equations (2.2.1) and (2.2.2) a r e both s a t i s f i e d . In t h i s s e c t i o n we s h a l l d e r i v e some n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r these e q u a t i o n s t o have s o l u t i o n s . Throughout t h i s a n a l y s i s we assume, u n l e s s o t h e r w i s e mentioned, t h a t any two r e a l numbers K and L satisfying as o f f e r s of the two p l a y e r s . 0 < K <_ C, 0 < L <_B From here on we s h a l l impose: are admissible 9 Assumption 1: The u t i l i t y functions U Assumption 2: The u t i l i t y functions U R and U and U are s t r i c t l y T increasing a r e continuous everywhere on t h e i r domains o f d e f i n i t i o n . Assumption less. 1 s t a t e s t h a t each p l a y e r p r e f e r s more f u t u r e w e a l t h t o The s t r i c t n e s s o f the m o n o t o n i c i t y of U and K ensure t h a t no p l a y e r ' s m a r g i n a l u t i l i t y beicdmeshnegative.sThe tractability. t h a t these assumptions U The i n v e r s e f u n c t i o n o f U the range o f U. Since and U w i l l be denoted by inverse functions are also s t r i c t l y second assumption I t s h o u l d be p o i n t e d out furnish inverse functions. D \ Of c o u r s e these i n c r e a s i n g and continuous everywhere on U(0) = 0 = U ( 0 ) 1 the u t i l i t y f u n c t i o n s and t h e i r i n v e r s e s both have t h e p r o p e r t y t h a t they a r e p o s i t i v e on n e g a t i v e on i s imposed t o Li was of course made f o r mathematical ensure t h a t U (0, °°) and a r e (-°°, 0) . R e c a l l t h a t t h e r e e x i s t m u t u a l l y b e n e f i c i a l b e t s i f and o n l y i f the two some inequalities (2.2.1) and (2.2.2) a r e s i m u l t a n e o u s l y s a t i s f i e d by (K, L ) . Rearranging the terms of these i n e q u a l i t i e s we f i n d t h a t these requirements reduce t o , - \ x [ - i?p- V L ) ] > K >\ x [ - A V" L ) ] ( 2 - 2 3 ) i . e . a n e c e s s a r y and s u f f i c i e n t condition"'" f o r the e x i s t e n c e o f m u t u a l l y T h i s i s g i v e n f o r t h e case p = q i n De Groot [1970] as an e x e r c i s e and we s h a l l show i n t h i s s e c t i o n t h a t i f U and V a r e b o t h concave t h i s c o n d i t i o n w i l l never be s a t i s f i e d when p = q. 10 favorable such and bets that Is the (2.2.3) s u f f i c i e n t existence is 0 of s a t i s f i e d . conditions when < We the K <_ C s h a l l and now p l a y e r ' s < 0 obtain u t i l i t y <_ D L much simpler functions necessary possess e x t r a p r o p e r t i e s . Theorem 2.3.1: possess d e r i v a t i v e s of a K and-L p U 1 (L)) that e x i s t 0 > < mutually Thus, the the of the the q and > s a t i s f y o r i g i n . for U~ (-q to > L < D — Let event l e t p Assumptions and p of = mutually p/(l - 1 and 2, q. h e . t h e - s u b j e c t i v e that-K^w'ill win existence for o r i g i n x = the > 0. (-L)) and favorable complete 0 U 1 h(x) h(x) and p), the-bet.. favorable q = q / ( l and p r o b a b i l i t i e s Then p > q is a b e t s f - q). > 0, It is seen i f - U ~ ( - p such condition Suppose that, at U r e s p e c t i v e l y s u f f i c i e n t Proof: Suppose - U -U d i f f e r e n t i a t e x K ( - p bets. s u f f i c i e n t l y U To (-p X K proof, To for U i t R V K f i n d (x)) is + U T such - U X L w . r . t . the x. (- L q L U to L T ( - L ) ) for <_ 2 C , L show we say h:R -> then R L there by ( - x ) ) behavior Then U L define ( - q s u f f i c i e n t i n v e s t i g a t e h (L)) small that of the near the function o r i g i n h near have, U'.(x) h'(x) = p ( ") " K ^ i : p 1. Infact ] L Q g.£. given u K ( L < L x ) Q U '(U T ) , o n e can f i n d K X T s.t. (-qU (-x)-) T (K,L) is mutually favorable 11 for any x a t which the f u n c t i o n s In p a r t i c u l a r s i n c e h'(0) I L . and U ~ ( - p U„(0)) = 0, x is. U a r e both d i f f e r e n t i a b l e . T l L 7 ( - q U ( 0 ) ) = 0, K X T L p U^(0)/U^(0) L q U[(0)/U^(0) p - q > 0, where the d e r i v a t i v e s a r e nonzero, as the u t i l i t y increasing. Hence the d e f i n i t i o n o f at h'(0) >' 0 h(x) functions are s t r i c t l y and the d i r e c t s u b s t i t u t i o n o f gives h(0) = 0. Now s i n c e h x = 0 in i s differentiable 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 , _ .)|<4W h (0 for all |x| < <5 w i t h s u f f i c i e n t l y s m a l l <S . Hence for all |x| < S 6 . This i n turn implies h(x) > 0 It fora l l with s u f f i c i e n t l y small 0 < x < 6 , s h o u l d be p o i n t e d 0 < h'(0)/2 < h ( x ) / x thus c o m p l e t i n g the p r o o f . out t h a t , i f we do n o t r e s t r i c t which a given p l a y e r may b e t , that the event on t h i s theorem s t a t e s t h a t the two p l a y e r s can 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 whenever they have d i f f e r e n t s u b j e c t i v e 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 a v e r t e r s . I t should be mentioned here t h a t H i l d r e t h [1972] has come t o the same c o n c l u s i o n , w i t h the a d d i t i o n a l assumption of concave f u n c t i o n s , by a d i f f e r e n t approach. 1. I f the i n i t i a l p r o s p e c t s independent o f i n i t i a l are ; u n c e r t a l n t h i s event i s r e q u i r e d 5 prospects. t o be utility 12 Theorem 2.3.2: p o s i t i v e the on R Suppose and + existence of Proof: Suppose (1 IT - p) (-K) both negative mutually i t > IL^ on and R~. U Then favorable p bets , i t follows implying ( i i ) K > t h i s This and by can This only on is i t may < q L / ( l p ( i i ) i f higher Jensen's p < - (1 > p the K > , 1 i f q - q) U states s t r i c t are as i n Theorem + q K two q two < R p) > (i) ' U T (-L) > - - 0. Then p 2.3.1. (L) U K + from that, U (-K) these that p l a y e r ' s that of his agreeable Here q f o r LI Hence - mutually f i n d p) p / ( l a 1. (K) T L - to never and are c o n d i t i o n 2 (l-p)K). p o s s i b l e can p necessary and U < R (pL p L / ( l (1 - p)K) p) and i n e q u a l i t i e s together for p, q / ( l 0 - q) < or q < s i m i l a r l y we 1 from have, . e q u i v a l e n t l y i f p > q. proof. given than - W r i t i n g only theorem a - (1 be achieve where a functions * i n e q u a l i t y + R (pL L - U (L) q). hold completes bets i s ' 0 0 1 But q L i 0 thus > concave i s l m u t u a M y r : ^ K (i) are switch mutually averters s u b j e c t i v e opponent-. the events wager. favorable increasingness r i s k of It p r o b a b i l i t y Of course they is bets i f U not is can are now the i f f i n d of the the b e t t i n g evident event necessary to mutually . 2 . he i n e q u a l i t y i n that be event b e n e f i c i a l favour r i s k bet So L on bets is of reversed to averters has bets on an A ° . 13 o b j e c t i v e 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 f o l l o w i n g p r o p o s i t i o n i s a l s o o f some i n t e r e s t . Proposition 1 and Suppose U and R U are concave and s a t i s f y assumptions 2 If there 2.3.3: (K , L ) Q exists i s a m u t u a l l y f a v o r a b l e wager, then given q 0 < K < K — o such that (K, L ) any a l s o forms a m u t u a l l y 0 < L <^ L favorable wager. Proof: Since U with 0<K L ( > ^ ) K D 1 ( "^ V- o» L <C, o — TL_(a L) < a U (L) K K a = 1/A. and i s m u t u a l l y f a v o r a b l e , we have 0 < L < K U <D. o — (take a L = A.x) . o < " K 1 ( -P K< o» Since X U XL, i s concave, we have f o r a > 1, K — convex combination o f R 1 K of U 1 . 1 T XJ L x and then p u t R J_I U (L)) R I n a s i m i l a r f a s h i o n one d e r i v e s a U 7 ( - q U ( - L ) ) <_ u ! " ( - q U (-a L ) ) . XJ and a U (D) <_ -a U ( - p 1 0 Hence, -U^C-p U ( a L ) ) 1 "U/C-P as a r e s u l t o f c o n v e x i t y L Now w r i t i n g L that = (L / L ) L and s e t t i n g O O £ 14 a = L /L > 1, o — we a U-Vq obtain V - D ) < U-Vq U (-L )) < K L o < -a u ' V p Hence f o r any (K, L) ^(-L)) 2.4 The follows: U K o U (L)). R % T - l a , K (-q U ( - L ) ) < — < -IL, (-p IL.(L)). T L L i K Ot L we f i n d t h a t u'Vq O we have ^ U (L ) 1 K K _ K = min{^-, y t u " (-q U ( - L ) ) - U~ (-p U ( L ) ) ] } Now s e t t i n g and 0 < L < L < - u " Q L i s mutually favorable, < K < -U'Vp because 0 < L <_D U (L)). R AN EXTENSION TO THE EXCHANGE-: PROBLEM s t o c h a s t i c exchanges problem i n t r o d u c e d C o n s i d e r two d e c i s i o n makers o f a random v e n t u r e 0 < K < C, Y (money v a l u e ) K and L f o r - Y; by H i l d r e t h [1974] i s as c o n t e m p l a t i n g t h e exchange a venture i s a p o s s i b l e u n d e r t a k i n g which would modify the d e c i s i o n makers' i n i t i a l p r o s p e c t s . value of set Y i s determined on the b a s i s o f aarandomavarlable t a k i n g v a l u e s i n a ft, weft b e i n g a s p e c i f i c sequence o f developments i n the d e c i s i o n makers' environment. if weft i s realized. The d e c i s i o n makers agree t h a t Let P be Q a-field w i l l denote the p r o b a b i l i t y assignment o f i s a random v a r i a b l e w i t h r e s p e c t L w i l l pay K, Y(u)) 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 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 And The L F on o f subsets o f ft. (ft, F ) . Hence to e i t h e r d e c i s i o n maker. Y 15 In order (unprofitably) random to impose o f f e r v a r i a b l e the taking w i t h p r o b a b i l i t y measures P other, n u l l of t h i s problem before. using as This respect sets one normalized to a sure both and Q F are to u t i l i t y n e i t h e r thing, and P i t is assumed negative Q. d e c i s i o n values that on Furthermore maker a i t the same under makers w r i t e K functions have and U both f i x e d L's and P set i s and i n i t i a l u t i l i t y U such from that U would Y i s of a p o s i t i v e assumed a r e ^ a b s b l u t e l y ^.continuous" w i t h d e c i s i o n us that p o s i t i v e and where enables c o n d i t i o n other both p r o b a b i l i t y i . e . the respect Q. We wealths, the (0) that to conceive as exchange, = U (0) = as F^ E Such an wish of mutually a i n to w i l l be an L U_ ( _ (Y) e s t a b l i s h but exchange discussed i n Y = ) /U (i) = / U L w i l l be > and 0 ( of the a _ Y ) d E and U T Q l (-Y) Before venture favorable > s u f f i c i e n t suggestive l a t e r and mutually exchanges. f a i r l y dP K necessary favorable h y p o t h e t i c a l , which U (Y) R exchange E„ We L U is p a r t (of (2.4.1) conditions attempting a r e a l involved. of i f f , 0. t h i s to world More s e c t i o n . do f o r the t h i s we s i t u a t i o n ) i n t e r e s t i n g each existence o u t l i n e example examples 0, 16 EXAMPLE 2.1: A d e c i s i o n 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 o f f e r . for the a r t i c l e i s completely based on the time T Suppose h i s u t i l i t y at which the a r t i c l e w i l l go out of order f o r the f i r s t time (or the l i f e time). t r i b u t i o n function n ( t ) on the random v a r i a b l e K has a p r i o r d i s - T based on h i s past experience, the performance of the item, the way that he has been handling i t , e t c . L e t F (t) be the value of t h i s a r t i c l e from K Suppose another d e c i s i o n maker, say L, a r t i c l e f o r a reasonable p r i c e , say K's point of view. i s i n t e r e s t e d i n purchasing t h i s $R. H i s u t i l i t y on t h i s a r t i c l e i s also completely based on the random v a r i a b l e d i s t r i b u t i o n function (subjective) on T. Let II ( t ) be L's prior T based on the current c o n d i t i o n of the item, i t s age, i t s current performance, the reputation of the brand, e t c . Let F ( t ) be the value of t h i s 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 p r i o r 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 d e c i s i o n makers K and L exchange the venture Y f o r -Y. I t i s clear that t h i s exchange i s mutually favorable i f f E U (R - X) > 0 and K K E L U L (X - R) > 0. F (t) = F ( t ) = k t , K L In the p a r t i c u l a r case where U^(y) = U (y) = y, K L T t r i v i a l l y , there e x i s t mutually favorable s e l l i n g p r i c e s (values f o r R) i f f E (T) > E (T), T Li V i . e . i f f the buyer f e e l s that 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 f e e l s . s i m i l a r r e s u l t s i n more general cases. Our goal i s to get 17 When e a s i l y the obtain favorable u t i l i t y a f u n c t i o n simple of K necessary and L c o n d i t i o n 2.4.1: Suppose both U and U K are that nonzero E^(Y) mutually > 0 except > If Y at E^(Y) favorable Proof: f o r both the concave existence we of E U J_i is the i s o r i g i n . a are both concave functions Then necessary the existence c o n d i t i o n for the of a Y such existence (-Y) > favorable, then Y s a t i s f i e s both E^U^CY) 0. -°° < property, a K the Consider there Then by Jensen's °°. < ( E R ( Y ) ) > 0 and U L ( E L ( - Y ) ) > 0 r e s u l t . now H i l d r e t h e x i s t opposite not a [y/\ family, Confining [1974] mutually Y does not i s some necessary and 1. so sign. have when for 0 i n e q u a l i t y h i s has of analysis shown, favorable ventures to that f a m i l i e s assuming exchanges such of concave i f f E (Y) ze[y]=* u t i l i t y and U n f o r t u n a t e l y some the above a v a i l a b l e n e c e s s a r i l y a . and It property. as mean Independence analysis of of For a E a l l conditions venture is p o s s i b l e be may not found of i n ventures, also i n i t i a l f o r t h i s functions"'', that are of a p p l i c a t i o n s i t that are the of i n t h i s prospects ±s also true however the above h o l d is more that t h i s form aY mentioned general necessary. and ay having (Y) a v a i l a b l e ; ventures noted = L ventures v a r i e t y aY therefore, is the exchange that should, s u f f i c i e n t f u r t h e r an - z ventures K do > K .Li implying some of exchanges. mutually U thus mutually L i K and' can exchanges. P r o p o s i t i o n and are assumed case 18 As i n d i c a t e d e a r l i e r , our g o a l i s a n e c e s s a r y and for the e x i s t e n c e sufficient condition of 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 f o r exchange, t h i s condition b e i n g dependent merely functions. on the n a t u r e of the v e n t u r e and not on the utility C l e a r l y the g e n e r a l i t y 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 o f v e n t u r e s t o those h a v i n g certain specified properties. We admit, i n p a r t i c u l a r , one important family, h a v i n g a v a r i e t y of a p p l i c a t i o n s , whose members have f e a t u r e s analogous those to of a b e t . In most o f the exchanges we encounter i n the r e a l w o r l d the v e n t u r e a v a i l a b l e f o r exchange i s a f u n c t i o n of s e v e r a l f l e x i b l e parameters, v a l u e s o f which a r e t o be f i x e d by n e g o t i a t i o n . The the f o l l o w i n g a r e two h y p o t h e t i c a l examples where the v e n t u r e can be determined by two such parameters. We w i l l d i s c u s s H i l d r e t h ' s method i n r e l a t i o n to these examples. Example 2.2: it S r i Lanka (say e x p o r t s t e a t o China. K ) imports r i c e from China (say Suppose the two say 3 a a and while the p o s s i b i l i t y o f S r i Lanka i s t o s u p p l y a c e r t a i n amount of t e a , u n i t s to China and China i s to supply a c e r t a i n amount of r i c e , u n i t s to S r i Lanka, prices ) c o u n t r i e s have d e c i d e d on a c o n t r a c t of the f o l l o w i n g form which guards a g a i n s t future price i n f l a t i o n s : L (say $P 3 and say d u r i n g each time p e r i o d , b o t h a t c u r r e n t market $Q, r e s p e c t i v e l y ) f o r the coming are to be f i x e d by n e g o t i a t i o n . T time periods, The p r i c e s of t e a and r i c e i n the w o r l d market a r e expected t o go up a t c o n s t a n t but unknown a d d i t i v e r a t e s , say a t r a t e s d e c i s i o n makers b u t i o n s on t be w r i t t e n as K and t and r per u n i t time r e s p e c t i v e l y . and L r . I t i s evident have t h e i r own The two subjective probability d i s t r i - t h a t the v e n t u r e i n v o l v e d here can 19 Y(a, 3) = [a(Q + rT)T - aQT] - [g(P + tT)T 2 or e q u i v a l e n t l y , T where of r Y(a, 3 and t ) = 2 j- are (ar and t h i s there modify i s 3 random units venture w i l l problem no t h i s (fixed) t ) v a r i a b l e s H i l d r e t h ' s reason example of why by rubber have the where R i s r e s u l t w i l l Example 2.3: loss l i f e of d i v i d e d coverage on the even from An or i n t o e i t h e r d e c i s i o n maker's point a 3 r e s t r i c t i o n from = ) that S r i T a p p l i c a b l e should "China Lanka be wants under only i f a imposed. to t h i s buy = 3 > In f a c t e x a c t l y contract so i f c that the 2 (ar i n f l a t i o n widely i n j u r i e s . c e r t a i n i s form insurance n r e s u l t r e q u i r i n g constant more t h i s also Y(a, are - view. For we 3PT] - 2 miss Suppose passenger - t cR) for rubber, then insures a i r H i l d r e t h ' s mark. say the K 3 rate the company, categories. - K, d i f f e r e n t s p e c i f i e s y i e l d s the kind that buyer $R of $L i f t r a v e l l e r s p o s s i b l e against i n j u r i e s of insurance the a i r c r a f t 20 crashes causing t h i s p a r t i c u l a r passenger's death, and i f such a th crash causes that L amount < R him , that L K p r o b a b i l i t i e s a buyer, q_^ = P(X L, = , p o l i c y R^) < (X who i $aL. where K up change i n a P = of f o r = example to R^ pay p o l i c y of i n j u r y w i l l p say s p e c i f i e d an assume = P(X = $ L Now define and L a and . a B K venture Y(a, . . . . , n . i to Suppose a r e a l i s t i c Let has = buy 1, w i l l n = q = u n i t s pay 3) L a X 2, a to assume be the s u b j e c t i v e p r o b a b i l i t i e s wishes exchange is K R^) s u b j e c t i v e n It 2, and the at i = l , buyer 2, a i , R), has 1, category. each the = i P(X (a $gX L - g venture Y f o r This exchange i s mutually ) V " . = -Y R), >^ 0 ) for X. Suppose of an This which the insurance i s i s an f l e x i b l e favorable i f f both E E K T U ( Y = ) U ( - Y ) L are K " K " ( 1 = P (1 q ic^ V - Eq.) L Z a L + P IL ( - a L ) L L + q e U i R ) ^ i V + (BR) + - 0 * ! * > Eq,U_(BR.) Li 1 0. > and 0 1 LI s a t i s f i e d . In r e s u l t f a i l s t h i s provides to function example i t may the answer the problem to r e a l l y our be the question. case that If a 4 a B, = B Then • H i l d r e t h ' s H i l d r e t h ' s r e s u l t apply. Consider The - E p of problem two bounded where Y is and of exchange p o s i t i v e f u n c t i o n of i n which the venture f l e x i b l e parameters, more than two Y say f l e x i b l e i s a a and parameters Bi s 21 l e f t for possible future consideration is negotiation. Assume studies. denoted E by U K f e a s i b l e Lemma a and 2.4.2 : A Y(a, (Y) member of where a g) , and E K U L (-Y) the family and are g of ventures are both to f i n i t e be under chosen for by a l l L i g. Y(a, i s B) random a v a r i a b l e defined on a p r o b a b i l i t y space 2 (Q, , P) . F Given ( s t r i c t l y ) w i t h continuous and U(0) Suppose = 0. CJ and .a B -> c o a, g P(Y < 0) -> g(B), = and only Proof: i s g i f being P(Y C l e a r l y p o s i t i v e 0) u t i l i t y = 1 P(Y < and 0) = continuous, B Lim 0, Let . f u n c t i o n ot Lim g -> part now suppose , w r i t e , B o > i f B)) an 0) and U be normalized P(Y - increases > 0) = a such that 0, 0 0 EU(Y(a,B)) < °° f o r f e a s i b l e — o o the 0 = > 0 and of P(Y necessity EU(Y(a, g )) Q be > > 0) and and > f(a) a defined 0 for P(Y < i n terms of from R onto each 0) r e s p e c t i v e l y , P(Y < U(Y(a, Y > for i s 0 0) B )) o are - h(a) both < 0 g zero the as_ B R, i f the other . while or negative; s u f f i c i e n c y p o s i t i v e . U(-Y(a, Y a, p o s i t i v e o f "theflemma:.c-.Toiaprove = = < 0) be, part 0) solved function P(Y w i l l B)) P(Y can increasing one EU(Y(a, follows = > is w i t h co and EU(Y(a, thus g 1, P(Y R" ( s t r i c t l y ) increasing Lim R B) : . Then a = Y(a, decreases ( s t r i c t l y ) a Lim suppose g )) Q Given 22 where f(a) i s decreasing. p o s i t i v e Suppose and Lim a i t immediately i n c r e a s i n g w h i l e P(Y 1. > 0) = h(a) Then i s since a f o l l o w s , t h a t h(a) 0 a as -> 0 0 . Now 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. shown 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 i s p o s i t i v e evident f o r that large there a and e x i s t s i s negative unique a = f o r g ( 3 g i s EU(Y(a, i n c r e a s i n g e x i s t s a., = a The be 2 = g ( 3 = expressed f o r and 2.4.3: U t r i v i a l l y < -> a R = 0 l g ( 3 i n terms Suppose , has the < 0 3 . , 3 - ^ ) < EU(Y(a , 3 2 2 3 > 2 ) )= because of a venture the p r o p e r t i e s U are and f (a) i t can Now i t be $ )) 0 i s EU(Y(a,3 • )) = 0 For, ) 2 0 a . suppose thus 3 l e a d i n g 3 ) ) 0 = a < ± because ) a i n That there Then because EU(Y(a, to can £ < 3 X a 2 c o n t r a d i c t i o n . e q u i v a l e n t l y . Y(a, Y, U and Y(a, b) a l l and 3 ) hypothesized i n each u t i l i t y Lemma f u n c t i o n , 2.4.2. Assume continuously d i f f e r e n t i a b l e . Also suppose J_I = 0, only a >_ a , 3 ^_ b 0 i s EU(Y(a, continuous c o n t r a d i c t i o n . 3 EU(Y(a , 2 . t h a t = o < EU(Y(a onto by w i t h ) since o i n c r e a s i n g shown 0 5 i s 3 JN. that s t r i c t l y such 0) 0 0 Hence a < Li IN. that i s i m p l i e s g:R U i s ) 2 ) ) E U ( Y ( a mapping Theorem Q g ( 3 - , ) 0 But 3 . and P(Y S i m i l a r l y s m a l l ) o because Lim 0 0 i n c r e a s i n g that p o s i t i v e (at l e a s t i n a neighborhood of 23 (a, b)) are assign admissible p r o b a b i l i t y 1 and to a that the bounded both p r o b a b i l i t y subset A measures o_f_ P and Q . If r Yl 3 J A f e J _9a_ 3Y 9Y dQ dP a=a a=a 3=b 3=b \ 9Y dQ dP 9a a=a ^ J 3=b then, can there f i n d Proof: a It E U (-Y(a, a = h ( 3 ) and we E e x i s t s > a follows 3 ) )= 0 the can be r e s p e c t i v e l y , immediately (-Y) > 0 By a < given mutually previous g and the E U h(3). i s 3 0 the 3 E expressed h > < that are responses (Y) It b d e c i s i o n makers b e n e f i c i a l . lemma e q u i v a l e n t l y where that that, 3 ) examining n o t i c e i f f such Y(a, from a=a 3=b b making i n c r e a s i n g . U > 3 (2.4.1) obvious a (Y(a, as both of i f f U a = 3 ) ) = g ( 3 ) to changes > g ( 3 ) and that to f i n i s h and and continuously Y 0 d i f f e r e n t i a b l e i n a and 3 that the proof we are Li J-J required of (a, f ( 3 ) = It i s only b). h ( 3 ) P(A) true. = To - o f t e n functions are Q(A) to prove do t h i s g ( 3 ) . On assumed to bounded. = 1 , that A h ( 3 ) consider > g ( 3 ) the f o r f u n c t i o n d i f f e r e n t i a t i o n of avoid If being t h i s paradox is bounded, the i s (cf. case, a l l points f ( 3 ) f ( 3 > the a defined w . r . t . Arrow i n 3 neighborhood by f ( 3 ) we have [1971]).that required unnecessary and the c o n d i t i o n Theorem 2.4.3 u t i l i t y that holds 24 f ' ( 3 ) = - h ' ( 3 ) g ' ( 3 ) 3 L f E L U ( " Y ) !B + 3a a = h ( 3 ) because, for 4^- E d 3 • T— 9oi U„(Y) K T L instance, U ( - Y ) L w . r . t . = a ' D i f f e r e n t i a t i o n 0. and Under f'(S) E^U^(-Y(a, = 8Y and now a U' K are are a l l continous the — |3B I n t e g r a l Sign' U K ( ; Y ^ a D sequence continuous we we to f i n d ->• °° . a Since constant M 8Y — op U ' (Y) such i s (2.4.2) a = g ( 3 ) — can E^U^(-Y) U of use reduce U L " Y ) d and T technique (2.4.2) ( + U (-Y) t, the as Q a=h(B) ( Y)d ^ °. Then since have, [ — bounded U'(Y)1 8 3 V on A f o r a l l f o r a=a Ji B=b any (a, 3 ) , we K that F n of (2.4.3) admissible L n ) fe L ' Q a = g ( 3 ) of Y K fif -> as ( d e r i v a t i v e s r P K U K U'(Y)dP V l K implies p a r t i a l decreasing both 0 the £ Consider 3 ) = Since 3 E < M — n and a l l u e f~. can 25 Hence the a p p l i c a t i o n If « K « d of the <!jf>a-a P a=g(g Bounded ) U K Convergence ( Y ^ ' » b d Theorem y i e l d s , P g=b = DJCO) v dP 33 a=a y 6=b Obtaining s i m i l a r we from deduce r e s u l t s the v f ' ( b ) = >. c o n t i n u i t y 93 a=a fi=k : 3a a=a e^b p o s i t i v i t y Furthermore conclude as appropiately It admissible, In i t s i t s i n a - Theorem should Theorem a p p l i c a t i o n simpler, = chosen present form to p l a u s i b l e r e s t of f of the terms which appear i n (2.4.3), that . - 3=b from a = the 0. 2.3.1, neighborhood be 0 , dP^ & f o l l o w i n g f(b) the y & the for noted 2.4.3 that holds c o n d i t i o n hypotheses These two that f(B) of i f (a, and v a l i d is classes form. f o l l o w i n g The > of 0 for are a l l (2.4.1)). s u f f i c i e n t points i n and <_ b to an b) . w i t h p a r t i c u l a r equation conditions only (2.4.1) (see i f the not a <_ a i n e q u a l i t y very ventures c o r o l l a r y reduce one are (2.4.1) i n t u i t i v e . w i l l is $ such reversed. However i t to case. a much 26 C o r o l l a r y 2.4.4; Let v a r i a b l e s . Suppose a l l the s a t i s f y and g (a, are g) Y(a, the g) u t i l i t y requirements admissible. leading to = the p l a y e r s ' Then, a is necessary Proof: too, Set for (a, b) favorable (0, 0) and T being subjective 2.4.3 . ventures p r o b a b i l i t i e s Suppose c o n d i t i o n is p o s i t i v e 1 f o r only the random and Y nonnegative existence of that (T) functions existence = S s u f f i c i e n t E u t i l i t y gT, Theorem E(T) If - functions, of mutually aS are of i n both concave, mutually Theorem then b e n e f i c i a l 2.4.3. the above c o n d i t i o n ventures. 8Y Then since — To prove now dCt = S and 3Y — = -T, (2.4.4) t r i v i a l l y follows from (2.4.1). the 3g necessity of (2.4.4) P r o p o s i t i o n a E^(S) - 2.4.1 g when to E f T ) > 0 Y > the = aS otE E (S) The 1. necessary above It is assumption prospects condition i n e q u a l i t i e s claimed of of by f i x e d both K C. for g E ^ This implies (T) , or concave, l e t us apply that, e q u i v a l e n t l y , E (T) L the existence H i l d r e t h that wealth are are L imply L functions E (S) j o i n t l y i n i t i a l and - a R a gT. ^ E (T) as - (S) ^ K u t i l i t y is of mutually (2.4.4) this thus r e s u l t replaced s t o c h a s t i c a l l y by favorable completing holds that independent v a l i d of ventures. the even proof. i f uncretain of the the i n i t i a l ventures. a 27 The or p l a u s i b i l i t y diminishing K's S u t i l i t y . agreeable high Hence " t h e r e on the that, evaluations of agreeable As cost by the an insurance L constant E (X) < i n E pay a i r insurance company company's expectation when l e v e l s p < q of a company's. and p^ d i s a s t e r a f o r c o r o l l a r y and S, d i f f e r . " It cases, i f i f f two d i s t r i b u t i o n s of i f f the derived for higher the there insurance pay by X, (in case experience. a l l i , i . e . from buyer's i n (a e x i s t disaster) In the view to mutually are. S of i s = L, insurance v a r i a b l e ) . p r o b a b i l i t y of from 2.2, the p a r t i c u l a r point the u n c e r t a i n - functions and as t h e i r f i n d mutually expectation of the to that i f of do Example random s i t u a t i o n buyer's able amount passenger that p o l i c y to = the implies have q^ T set, r a t i o on u t i l i t y 2.4.4 modified averters t h e i r C o r o l l a r y also i n t e r e s t i n g d i f f e r e n t of be are E(T)/E(S), also r i s k (S) g the namely is mutually (T)/E must T decreases f i n d and matter how and E be i n j u r i e s i s make Enlarging to never and < able w i l l coverage would T be exchanges p r o b a b i l i t y t r a v e l l e r ' s i . e . to values favorable these u t i l i t y b e l i e f s this follows. they (2.4.4) the (X), f o r tend as s i t u a t i o n , a p p l i c a t i o n would exchanges no L's of points of maker negative v a r i a b l e s o b j e c t i v e exchange ventures, of a e i t h e r known If view i s makers conclusion random 2.4.4 s u b j e c t i v e mutually makers i n company is t h e i r the e x i s t d e c i s i o n here t i t i e s d e c i s i o n when c l e a r l y , expectations note d e c i s i o n low. follows: the the and two C o r o l l a r y increases exchanges admissible, the of Since favorable only the i f amount higher t h i s of a l l compared the than the happens conceivable to 28 2.5 GEOMETRICAL In t h i s admissible, s i t u a t i o n section mutually of the exchanges the theory i t . f l e x i b l e r e a l we In M M {(g, a) : SET the have looked R g 2 > in as 0, i l l u s t r a t e stated (see M, i n the suppose Lemma proof a >_ 0 , of a = h(g) {(g, is the of E K U K (Y) = allows g g)) = aS of a l l exchanges the the enable that the gT, a a, g are > 0, - set t o p o l o g i c a l aspects us to of develop classes and g of being admissible. s o l u t i o n of a g)) us to 0 } function possess the express M i n simpler > g(g)} a p r o p e r t i e s form as, 0, f o r > u t i l i t y > 0, g) the L a >_ 0 , E U i-i s o l u t i o n also conceive (-Y(a, U (Y(a, 2.4.3) a): we w i l l nonnegative E each This Theorem = U and 2.4.2. M where Y but study EXCHANGES f o l l o w s : E study only of s t o c h a s t i c not L To the Y(a, only structure and at form FAVORABLE s e c t i o n , treatment, the MUTUALLY geometrical w i l l Suppose i n ALL ventures study of OF previous t h i s numbers. set = a members the THE study the Such f u r t h e r . having we i n problem ventures OF favorable introduced p r o p e r t i e s Define STRUCTURE . The h(g) (-Y) = > 0 a and a h and = g(g) is the JL functions g are both 29 continuous and i n c r e a s i n g . through the o r i g i n . s i m i l a r p r o p e r t i e s P r o p o s i t i o n g i s A d d i t i o n a l on 2.4.5: convex Since h (i) and If Y(0,0) a p r o p e r t i e s 0, we of f i n d the t h a t u t i l i t y h and g also go f u n c t i o n s w i l l induce g. the u t i l i t y f u n c t i o n U i s K concave (convex), (concave). ( i i ) If the u t i l i t y f u n c t i o n U i s concave (convex), i _ i h i s also concave Proof: Suppose 0<A<1 such 3 2 ), g a,=g($i) g and K U K not ( A convex. 3 a ? g ( 3 2 Y ( a , X 3 i i + ) so Then (1-A ) that + C 1 - A ) 3 2 3 there 2 ) >. e x i s t * g ( a>Aoti+(l-A ) = E > E U K K ) a i 2 . K { (Aai = thus we a = the (1-A ) 3 2 ) K U we K { 3 - p (1-A) g ( 3 have + + ELD ) . L e other > E hand IT {A the E IT {A ( a i S - 3 i T )+ ( i - A ) ( c o n c a v i t y of U a S - ) 3 U - A U Y( 3 2 K ) a i s a.ABi 2 T ) } i m p l i e s , ( a i S - 3iT) + ( l - A ) ( a S - 3 T)} 2 2 d g ( A 3 i + ( l - A ) t Hence, )T} 2 )S 2 - ( A 3 i + ( 1 - A ) 3 i n c r e a s i n g . C a i S - 3 i T ) + C l - A ) ( a A 2 an + 2 S - 3 2 T ) } )S 2 ) have, 0 On + g C A 3 i E p o i n t s + ) { a S - ( A 3 i K U 6 two s i n c e since then 1 (convex). i s that E But then (2.5.1) =0 and 2 ) T } 30 - V = E X K U A R U ( C K a a i i S ~ S - S l T ) C + - 1 ) A C U K a S- B T)} 2 BiT) + (1-A)E U ( each of the is concave. K 2 a S- K B T) 2 2 0 because This the ai=g(Bi) and c o n t r a d i c t s foregoing ( i i ) of the The p r o p o s i t i o n of the M w i l l two curves a w i l l i t l i e w i l l =h B) as and v completely be a admissible, bounded M w i l l f o l l o w a h the unbounded the set of f a i r l y i s of have seen (a,B) If i n i f is The of both Figure that If U zero. convex proof of a l l part l i n e s . i f quadrant set. expectations concave. a l l , shown f i r s t above d e t a i l e d map area convex a l l g s i m i l a r F i r s t We on g prove along shaded =g(.B) . above be y i e l d the that to functions. a or making reverse r e s u l t s be ) , proving w i l l u t i l i t y concave 2 thus i n e q u a l i t i e s are g = g ( 8 2 (2.5.1) foregoing properties a under 2.5.1 functions bounded E ^ C O / E ^ S ) M > E mutually by the ( T ) / E L empty. nonnegative to convexity u t i l i t y making a l l leading M L ( S ) , Otherwise (ot,B) are favorable +2 ventures. only one And points or i f i n both AAM u t i l i t y i n t e r s e c t i o n of the i l l u s t r a t e s that E < (T)/E K (S) K existence that given i f any A, E a w i l l both a> lead of R to mutually functions area when (T)/E are below h u t i l i t y (S) is i s the convex and the not , the set a l l favorable i s shown area functions necessary, i n not though admissible ventures. above are mutually favorable exchanges. It u t i l i t y functions are and 0, of Figure g). The set 2.5.2 Figure concave, ( a , B ) ' s , when (the 2.5.2 the M (a) c o n d i t i o n i t is s u f f i c i e n t , also of i n t e r e s t for the LI L i of stibjset there are 8.' s convex making Y(a,ft) i s i f a t h i s c o n d i t i o n mutually to i s b e n e f i c i a l observe s a t i s f i e d exchange. 31 (a) E (T) (b) E (T) FIG.2.5,1: k k /E (S) < k / E (S)> Set M w h e n k U and k E (T) t E [1) { are / £ { ( S) / Ei (S) both P I 3 concave 33 CHAPTER EXCHANGE 3.1 : NASH SOLUTIONS INTRODUCTION The favorable previous chapter exchanges numerous exchanges d e c i s i o n makers aim to here o b j e c t i v e and are would propose of p a r t i c u l a r t h i s context u n i n t u i t i v e features are p e c u l i a r f i r s t part of which w i l l provide optimal use Nash's f i n d the "the f a i r to a wealths makers, on the to w i l l and Nash of already q u a l i t a t i v e not of the b i t do agreement) consider the s o l u t i o n of of the one the of are i t i s also of i n a s t o c h a s t i c insights w i t h . solutions ( i . e . In the analysis the economic Pareto analysis, v a l i d problem our and curve assumed the The give dealing When not theory contract exchange here, to to be c a l l e d such as the information. various p r o b a b i l i t y exchange point i n t u i t i v e t r a d i t i o n a l the which achieved. w i l l we mutually them. developed conditions, complete e f f e c t s s u b j e c t i v e but incomparable from is of by However the Edgeworth's required under j u s t i f y the up c h a r a c t e r i z a t i o n s i t u a t i o n s h a l l s o l u t i o n the only as made t r a n s a c t i o n . the a existence s p e c i f i e d u t i l i z e m u l t i p l i c i t y of under yet set s o l u t i o n s i t u a t i o n s a not the exchanges d e v i a t i n g the the the w i l l we of of how obtain an (point exchange" i n i t i a l to such approach we i s have f o r theory chapter Then s o l u t i o n Then new of i n v e s t i g a t i o n we upon treatment t h i s s e t ) . p o s s i b l e , and a an c h a r a c t e r i z a t i o n chapter Such that a i s agree a exchanges. we PROBLEM 3 f a c t o r s , d i s t r i b u t i o n s problem. of the decision 34 In venture is a be exchanged f u n c t i o n s i b l e f o r a values g Y from by . a our here the 3.2.1 that of c o n d i t i o n , v a r i a b l e makers that makers g and = g taking point r i s k K g , negotiate to view. i . e . these which the d e c i s i o n maker L a e R values f i x K as i n where f i x e d , negative that the nonnegative are of s i t u a t i o n s to and averters, Assume x CURVE are admis- s u i t a b l e values gives w e l l as g) Y(a, p o s i t i v e Moreover i n d e c i s i o n makers u t i l i t y g and this e to r e a l chapter we whose functions are ANALYSIS i n t h i s i n "Mathematical p o r t i o n section of his is p r i m a r i l y based Psychics". vast and As on the such,we remarkable theory presented b r i e f l y o u t l i n e theory. THEORY i s and concerned w i t h of two y Concentrating the a l l concave. q u a n t i t i e s which assume a CONTRACT problem Let changed. those are maker a a to exchange parameters d e c i s i o n = only d e c i s i o n The decision [1881] EDGEWORTH other. the we a t t e n t i o n relevant possessing . random analysis The consider d i f f e r e n t i a b l e . EDGEWORTH'S Edgeworth from When functions The we a r b i t r a r y g e i t h e r continuously too, mentioned, is confine u t i l i t y 3.2 two and and where s h a l l of otherwise a for , chapter to Unless L t h i s cannot other, y i e l d one of two commodities be the r e s p e c t i v e on the exchanges greater Edgeworth u t i l i t y shows i n d i v i d u a l s , that to amounts which to are one such K be of and L, exchanged the trader contracts w i t h commodities Pareto optimal, without w i l l each each i n t e r i . e . decreasing s a t i s f y the R. 35 3x where (f> and (3.2.1) The 3y 3x IT a r e the r e s p e c t i v e y i e l d s a curve on the admissible traders 3y "contracts" x-y utilities of K and L . The e q u a t i o n plane, c a l l e d the "contract curve". a r e those t h a t a r e a t l e a s t as d e s i r a b l e t o both as t h e i r i n i t i a l h o l d i n g s . c o n s i s t i n g of admissible U . ^ J J That p a r t o f the curve g i v e n by contracts (3.2.1) i s c a l l e d the "range o f p r a c t i c a b l e bargains". 3.2.2 PARETO OPTIMAL EXCHANGES In t h i s s e c t i o n we c o n s i d e r Y(a, B ) = aS - 3T, S and T only t h e c l a s s o f v e n t u r e s where b e i n g nonnegative random v a r i a b l e s . 2 we have seen t h a t the s e t o f a l l ( a , B) leading to m u t u a l l y Assume E^(T) / E^(S) < E (T) / E ( S ) K K L L T a graph o f utility B > vs. on the curve determined by move below t h i s curve i t i n c r e a s e s . The l o c u s B L's utility The two s e t s o f i n d i f f e r e n c e curves of Figure 3.2.1 . The i n d i f f e r e n c e curve o f the s o l u t i o n o f E U (Y(a, K B)) - v = 0 K K say K K's increases; on the graph o f i s c a l l e d an i n d i f f e r e n c e and L are indicated i n corresponding to a u t i l i t y v a = g (B) . I t can be V 2.4.2 convex. i n d i f f e r e n c e curves a r e a l l concave. L's I n the i s zero and,when we shown as i n P r o p o s i t i o n Similarly a = h(B). d e c i s i o n maker o f the p o i n t s g i v i n g a d e c i s i o n maker the same u t i l i t y curve. is g , i s zero and when we move above t h i s curve h i s u t i l i t y h vs. and so t h a t t h i s s e t i s nonempty. where as on the curve determined by a favorable a = g(B) v e n t u r e s i s the convex s e t bounded by the two c u r v e s , I n Chapter that the i n d i f f e r e n c e curves o f K are a l l The e q u a t i o n s 36 of the say l a t e r by a family of convex i n d i f f e r e n c e along = family h^ on and quadrant. a It g^ at any d i r e c t i o n w i t h than the points d i r e c t i o n points on c l e a r that mutually e x i s t the u AB on t h i s curve t h e i r AB core (a, that they only t h i s B) the can r e s p e c t i v e and of In contrast move curves to h problem Pareto and (a, B) on the such f i r s t quadrant. Hence i s K K B)) K R w i t h 8a of cannot i n the mutual nonempty, exchanges, i n other consent, Hence i . e . each move M optimal the core i s nonempty 8E U (Y(a, L L B)) 9B B)) f i r s t locus p o i n t i n t e r s e c t 8B 3E U (Y(a, two t a n g e n t i a l l y . that, 9E U (Y(a, ever, points. there i f other 3E U (Y(a, L L 8a B)) i f f there i n the It e x i s t there at two v u points the Pareto optimal g the i n d i f f e r e n c e . represent exchange l e a d i n g of i f makers any i f i n d i f f e r e n c e the touch together, curves these at on the s t r i c t l y other, e x i s t s d e c i s i o n of maker's points curves that family each curve e q u i v a l e n t l y , shown the represents the or 0 d e c i s i o n two i n d i f f e r e n c e consent. = touch a at 3.2.1 curve, and no such u e a s i l y w i l l other - is curves that Figure point curve such i n B) ) It because each of between the curve, sets that (-Y(a, generated two favorable v, thus i n t u i t i v e mutual on U parameter. i n t e r s e c t s the that is E i n d i f f e r e n c e decreasing which Notice by a being curves curve at given concave The points is u i n t e r s e c t . curves any (3)> s t r i c t l y points curves any h is e x i s t 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 C U R V E . A N D T H E C O N T R A C T 38 because of each i n d i f f e r e n c e i n t e r e s t at side a to point the above equation represents curves of the corresponding that, i f the two note on of the f i r s t quadrant curves other the d e c i s i o n h and than slope maker. g of the It is i n t e r s e c t (0,0), then the family also of each other core of the problem bounded. The bargains. part We continuously or that Hence the P from the have curve already AB Q assign Edgeworth's (a, 3E U (Y(a, 3) theory 3a" to M the to a from optimal of p r a c t i c a b l e functions are u t i l i t y functions bounded subset the equation ventures 3 ~ range u t i l i t y that 1 the 3E^U^(Y(a, 3 ) ) 93 is the d i r e c t l y Pareto L " e i t h e r or 3E U (-Y(a, L that p r o b a b i l i t y l e a d i n g 3 ) ) K w i t h i n assumed d i f f e r e n t i a b l e and and points K of are of ft g^(3)= . h/(3), s a t i s f y , 3)) 3E U (-Y(a, L 3 bounded " 3)) L 3 a (3.2.2) Since Y(a, E (U^(Y)S) R as the w i l l be L part t r a d i t i o n a l l y and E c o n d i t i o n The K = 3 ) L i n w i l l made i n aS - 3 T , d i f f e r e n t i a t i o n under (U^(-Y)T) - s a t i s f i e d by of the l a t e r Pareto contract economics, come E ^ U ^ Y ) ! ) to an to curve the sections to E (U^(-Y)S) optimal p o i n t s , w i t h i n at = L exchange agreement i n t e g r a l a estimate M i s p o i n t the on (3.2.3) where Y = aS s o l u t i o n Thus, t h i s exact i m p l i e s , 0 the problem. s i g n by curve. p o i n t . - 3T. given> n e g o t i a t i o n , Attempts is 39 To one can determine use the equation is the equation It can be of u t i l i t y (3.2.3) the determined p o s s i b i l i t y . The u t i l i t y as key Let formed W = E K W W = E U (-Y(a, and K W equation 3.3 using L of the NASH'S In when i t model a section b i l a t e r a l between i n problem The Moreover, any agreement e x i s t s , the problem suggested " f a i r In s o l u t i o n " , l i g e n t , decision (Y(a, i n t h i s procedure optimal B)) problem p o i n t s . and K u t i l i t i e s . replace r a t i o n a l is a Now and B made have on been these section by Nash The we we that and given would Hence two i f , we we r e l a t e obtain the s t i l l on curve a i r to Nash's f i n d the problem, n e g o t i a t i o n t r a v e l l e r s the any optimal be the contract that M. such point i s exchange, i f bargaining to known u t i l i t i e s from i n the s o l u t i o n an A undertaken to curve. Enormous s e t ) . s o l u t i o n the agreement agreement procedure gives from i n s i d e w i l l the that evidence and (Pareto procedure the Edgeworth of determine i t r e c e i v e the lack o u t l i n e by w h i l e exchange country hypothesis theories use the not. f a l l to the here the solutions t h i s of is c o n t r a c t b r i e f l y [1950] i n d i v i d u a l s and to bargaining s o l u t i o n meaning tends developed incomparable of 2.3 p l a u s i b l e t h i s solutions captured Example makers l i e the s i t u a t i o n . agreement, w i l l of discuss monopoly section t h i s problem. s h a l l i n discussion 4. we described theories Chapter exchange U Pareto exchange f r o n t i e r . 2.2, optimal m u l t i p l i c i t y d e t a i l e d to given involved by K corresponding (3.2.2) Example previous r a t i o n a l bargaining the equation u t i l i t y appearing insurance denote a SOLUTIONS t h i s is B)) for element f r o n t i e r follows. set the as two they the i n t e l - 40 c o l l a b o r a t e are so for supposed the general is THE has purely NASH i n problem i n the s i t u a t i o n about We and s h a l l where the bargaining u t i l i t y obtain both i n d i v i d u a l s s i t u a t i o n ; function solutions of each under more chapters. THEORY postulates two knowledge r a t i o n a l about the of the point, he a x i o m a t i c a l l y should a d i s t r i b u t i o n other. l a t e r is information p r o b a b i l i t y to This terms 1. the b e n e f i t . complete known f u l l c o n f l i c t on have conditions Nash each to subjective i n d i v i d u a l 3.3.1 mutual p l a y e r s ' players bargaining s i t u a t i o n . u t i l i t y Conceiving increments assumes i n that, a w i t h a s i t u a t i o n the problem respect s o l u t i o n to where to the the bargaining possess: Parato northeast O p t i m a l i t y boundary of - the the point j o i n t of agreement (solution) u t i l i t y - p o s s i b i l i t y set must l i e ( u t i l i t y f r o n t i e r . ) . Symmetry 2. y i e l d s equal 3. l i n e a r , - i f u t i l i t i e s the to the Transformation order-preserving u t i l i t y two f r o n t i e r i s symmetrical the s o l u t i o n players. Invariance - the transformation of s o l u t i o n the i s u t i l i t y not a l t e r e d f u n c t i o n of by a e i t h e r party. 4. Independence s i t u a t i o n s such point) the then f o r the two Nash s a t i s f y i n g that of the second u t i l i t y one bargaining then deduces axioms 1-4 I r r e l e v a n t - contains s i t u a t i o n s the i s A l t e r n a t i v e s p o s s i b i l i t y (is have remarkable found by set contained the same r e s u l t maximizing - given two bargaining ( r e s p e c t i v e l y , in) that of the s o l u t i o n f i r s t s o l u t i o n . that the the only product of s o l u t i o n two u t i l i t i e s . 41 Various have authors c r i t i c i z e d Nash's solutions to K a l a i Smorodinsky of and these the two i n d i c a t e fourth axioms 3.3.2 FAIR We When Y(a, given i n a = Q they W (a L o T (the K , o = set and decide = B = of L, to - v i o l a t e d L is and by . 4 by [1957] R a i f f a A l t e r n a t i v e by one Nydegger conformed [1963], Bishop r e p l a c i n g conducted responses the and developed [1953] tests L (-Y(a are choose When denoted o , nonrandomized who closed d i f f e r e n t i a t i o n to or both and both Owen second and t h i r d . B) r e s p e c t i v e l y and t h i s W B o )) of Let to L i n which i s B ) = Q K be E the Then the is and the A true L and admissible bargaining set the of i n a t t a i n e d at s i t u a t i o n , r e s u l t equal to B) theory would , (3.3.1) an admissible (3.3.1) equation (a, Nash (a, B ) A W ( a , B ) i s u t i l i t i e s and q to been choose r e s p e c t i v e o bounded. have (Y(a , B ) ) according f a i r over U nonempty t h i s agreement A maximize defined the be core makers the s t r a t e g i e s ) . to the under W^(a^, • maximum s e t t i n g where d e c i s i o n by assumed (a, of problem parameters R A 3 been R a i f f a conditions . the are U BT, W (a, B ) W If and have subjects the 3.2.2 as Q E only aS gain ) they Luce EXCHANGES Section B problem t h e i r and p a r t i c u l a r l y Experimental w h i l e B) would axioms, [1975], that consider B a , [1963], Bishop bargaining axioms. [1975] (cf. 0 w . r . t . y i e l d s , p o i n t . a and Now B 42 - a e ( Y ) U L ( - > Y } LV- E + vU K y ) U ( Y ) } = (3.3.2) 0 (3.3.3) at maxima and Hence (3.3.2) the i t t h i s and the i s (3.3.3) s o l u t i o n given by i f whose a l l A X that ventures = the from 1 for is this a the the X K and i f f E^(X) bargain s e c t i o n , and averters form = -1 > 0 T = i-i E U L i LI (-aX) dV_ da and = = and, - E V K U f(a) R = V K. ( a X ) E *V L LI . (a, 3 ) > E T In now consider the aX where a >_ 0 p o s i t i v e and taking the f i n d (X) . are 3 winner $ Let a i s class i s an negative an example [1974] has b e n e f i c i a l = K E^U^(-aX)E^ then admissible. H i l d r e t h mutually V on p a r t i c u l a r us = l i e admissible, l e t D i f f e r e n t i a t i n g + i s points and L (XU^(-aX)) these equations a pays can of c r i t e r i o n . otherwise. L constrained. solutions course Y(a) loser not of v a r i a b l e K V this is 3 ) are Of values random where wins class of If (a, 3 ) e x i s t . f a i r next are X K r i s k they Nash's the problem i f W(a, non-negative and b e t t i n g provided (3.2.2). elements constant which shown whenever by case , maximizing 3 ) preparation a r b i t r a r y i n (a, (3.3.1) given ventures values. of curve the In of the contract i s minima w . r . t . E^Tj(aX) K a (XU^(aX)) , K we get , 43 da E and T U ( - a X ) therefore, i n E (X U"(aX)} 2 p a r t i c u l a r f(a) this problem e x i s t s and a l l a >_ 0 f(d) Of course d i f f e r e n t i t i s when point. given EXAMPLE u t i l i t y a by 3.2.1 i s constrained, However L ' s we 0 0 If a r e admissible, the Nash then i t 0 i s s o l u t i o n given : f u n c t i o n Consider u t i l i t y the Nash continue our s o l u t i o n analysis the p a r t i c u l a r may o c c u r only f o r b e t t i n g problem i s , tx i f x > 0 , X i f x L < 0 i f x > 0 i f x < 0 i s , Vx+T U (x) l > t > 0 = function 1 = x to by, (3.3.4) a t a t h e case (3.3.4). V x ) and = < = where K's where 44 Both bet u t i l i t y indexed and L's functions by g) (a, u t i l i t y are i s . W continuous, Then (a, K's g) = concave u t i l i t y (1-q) and is (/g+1 - increasing. W 1) g) (a, -qa . = Consider pta - the (1-p)g Let us f i r s t determine U are not d i f - so the c o n d i t i o n L the set of a l l f e r e n t i a b l e p > q c l e a r , mutually at does the not however, negative o r i g i n that a i s = now ((l-q)/q) o r i g i n the evident (see (/g+1 Figure existence of Notice that Let suppose, us p Since at the by f i n d i n g equal, > the there - 1) mutually q i s we can points bets •1=3. t i f f O z £ i 2/g+T ( i _ p ) i s slope e x i s t and s u f f i c i e n t that (l-p)/pt q)/2q(l the for - i s non- of the < graph (g/t) at c o n d i t i o n (l-q)/2q existence of of the for . such bets. p). contract slopes of ((l-p)/p) = d i f f e r e n t i a b l e q It that, a - the bets. there of p ( l where = that the t of i f such s u f f i c i e n t are favorable only not < and - i) bets bets equation g)• apply but that the not c a p i t a l s ) necessary a and and Hence where, p i f (/g+T such U mutually than functions (a, of t h e i r favorable f i n d does greater necessary u t i l i t y such e x i s t is 3.3.1). 2.3.1 than < a< therefore, the o r i g i n i . e . that Since existence e x i s t (smaller 1-2. { PJ It the there g bets. Theorem guarantee and a favorable the at any curve point to other t h i s i n d i f f e r e n c e than problem curves are 45 P FIG. 3.3.1: Contract solution blem in curve of and the example the betting 3.2,1. Nash pro- 46 and is a a r b i t r a r y so long as ( a , (3) 2 EllzSl any agreement and t h i s is K Thus, _ ' what f e a s i b l e . - 1 = 3 t / 2 Iq(l-P) for remains must (say) o f f e r i n any optimal bargaining s i t u a t i o n . To W = W„W to 0 K f i n d = L . the [pta Thus Nash s o l u t i o n (l-p)B] - we to t h i s (/B+l [(1-q) problem - 1) - we qa] d i f f e r e n t i a t e and equate the d e r i v a t i v e have, ((1-q) ( / e + T - = -2qpta + = 0 1) - qa)pt - (pta - ( l - p ) B ) q act Since we know equation - D / 2 q , 3.3.3 i m p l i e s B] this p r e v a i l i n g i n t e r e s t e d p l a y e r s ' the the s o l u t i o n Nash w i l l s o l u t i o n l i e for 1) + q(l-p)B) on the contract curve, (a, B), [(l-p)B/2pt CHANGES IN + the above (1-q) (/B" + . RESPONSE In the that (p(l-q)t(/B+T - i n OF THE section NASH we SOLUTION study circumstances responses s u b j e c t i v e of of the p r o b a b i l i t y the an Nash TO r e l a t i o n exchange C, between the s i t u a t i o n . s o l u t i o n d i s t r i b u t i o n s to various and P, t h e i r AND Nash We Y s o l u t i o n should and be changes i n Y , i n i t i a l wealths. 1 47 Here we s h a l l deal only previous s e c t i o n . In (3.3.4) developed f o r overlooked i s that maximized. reached by our w i t h the the f o l l o w i n g Nash's f a i r conclusions For, i n maximizing a class ventures analysis we are v a l i d discussed use However only when s i t u a t i o n , generalized {aX} s h a l l bargaining. bargaining the of u t i l i t y the i t suppose product V s concludes out, i n t o and L - It account, J K SHIFTS K by IN wealths, wealth If us . We Such as wealth, b e f o r e ) . + and are V u t i l i t y are K i n change f u n c t i o n i s , as E (XU^(aX)) K q u a l i t a t i v e a = i s Bishop ti time s o l u t i o n s/r r bargaining of product is c a r r i e d discounts i s of found by are not K , 0 p r o p e r t i e s . D the e f f e c t s c a l l e d wealth when he bets w i t h s u b j e c t i v e i n t e r e s t e d • the rates the constant instance, y e a r l y , case f o l l o w i n g AND e f f e c t s same would the which the E ^ - a X ) the C for are i n v e s t i g a t i n g for when bargaining, K's by change, (e.g. of s i s the WEALTHS s t a r t a that time, and t h i s presence INITIAL on L c l e a r the might a l t e r e d this r L become Let by where the E U (aX)E (XU^(-aX)) a f f e c t e d at taking r e s p e c t i v e l y . w i l l w i t h by be agreement K [1964] not u t i l i t y the the equation should Nash's i n of b e l i e f s , the previously and preferences a l t e r e d U (C point of f i n a l s Z) - U deal w i t h (C) = To U an as agreement, circumstances. + K's closed tastes the i n i t i a l Bargainer hocky under as , the how r e l a b e l e d a D on determining these and e f f e c t s . reopens L C a r r i v e d do (Z), 48 where C is assumptions given by f (a) Then we K's which f(a) = i n i t i a l = imply 0 that the R e c a l l s o l u t i o n • E (XU^.(C L K + aX)) this are accepting exchange as problem v a l i d , i s [E U (C L K K + aX) - U (C)] K have, \ 3f_ da 3G 9f dC 3a 3a to we E (XU^(-aX)) e But that where E U (-aX) L wealth. is a=a negative D i f f e r e n t i a t i n g da >_ dC < f(a) and a therefore 0 ^ uC 3f i f f w . r . t . C = a has same sign as , 3C (a) 3C we the a=a have, 3U (C) K 3C 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 + aX) - ' ^ ] i . e . , 49 U (C) [{E U (C+aX) K }E (XUj;(C+aX) } K K - K E {XU^.(C+aX)}{E U^(C+aX) K K because [E U (aX) R E (XU^(aX)) K - R f(a) E ^ X U ^ a X ) ) { E ^ C a X ) - = - U^(C)} 0 1^.(0)}] (3.3.5) E U^.(aX) - R { K E U K ( a X ) > 11^.(0) E U (aX) df" 2 K K a where w i l l 8 = E U L L increase (-aX)/E^U K or K (aX) decrease > w i t h 0 as a is C according = a mutually as {E favorable. U'(aX) - U'(0)}/E Is. JN. K increases or reasonable 3.1 under Assume at a which for U . we In f a c t could = i t seems s t a t e a U U and = K i s In an i n c r e a s i n g order assumption conditions i s : a ~- EU(aX) that i n assumption Assumption : decreases shown on by to show 3.1 U, is f u n c t i o n t h i s true simulation to we a f o r be i s constant s h a l l true which U K. (aX) K f o l l o w i n g i s - U ' ( 0 ) ) / an when X is 0. f i r s t l y v a r i a b l e s example > X, where not a we s h a l l under prove very assumption constant 3.1 random v a r i a b l e . P r a t t aversion, [1964] defined c h a r a c t e r i z i n g by and Arrow r(x) behavior = under - [1971] a conclusion. E(U'(aX) EU(aX) p l a u s i b l e , random give even that a L assumption for secondly of that d e f i n i t e T Hence have U"(x)/U'(x) u n c e r t a i n t y shown i s and a that the necessary t y p i c a l l y absolute concept assume that r i s k for r m i l d ] 50 decreases is w i t h The f o l l o w i n g p r o p o s i t i o n holds when t h i s requirement f u l f i l l e d . P r o p o s i t i o n such x. that 3.3.1 U(0) : = Let 0 . U(x) If be an increasing U"(x)/U'(x) U'(x) - and increases concave w i t h x, u t i l i t y f u n c t i o n then U'(0) U(x) also increases Proof: w i t h L e t t i n g x g(x) , ( x ) 0 4 = , provided (U*(x) - U(x)U"(x) = that U'(0))>/U(x) - (U'(x) U ( x ) = ^ ~ ^ [U(x) U(x) . A Notice h'(x) and that = i s h is U(x) d i f f e r e n t i a b l e g'(x) f o r a l l >_ 0 , we twice have for d i f f e r e n t i a b l e . x 4 = 0 0 , U ' ( 0 ) ) U ' ( x ) 2 (U'(0) - U ' ( x ) ) ] W Z (U"(x)/U'(x)) decreasing Consequently a , i s , , . h(x) *-f U ( x ) - ^ $ r + U U'(x) U'(x) x < thus f u n c t i o n . Hence 0 thus proving h such is implying the that h(0) i n c r e a s i n g that desired h(x) r e s u l t . for >_ 0 and a l l for x > a l l 0 x 51 EXAMPLE i s 3.3.1 : i n c r e a s i n g , v a r i a b l e .2 X Let concave takes the r e s p e c t i v e l y . E(U'(aX) which in - EU(aX) Figure = 0 3.3:2 what we 3.1, the R e c a l l that p a r t i c u l a r , we bets i f reach becomes : We occurs i f K s h i f t , for movement Q has K such a of is can X ) . Then i n c r e a s i n g . w i t h f o r U(x) Suppose p r o b a b i l i t i e s c a l c u l a t i o n that of i s a of a w i l l the shows a l l a graph of the .8 random and that except those f o r a t h i s f u n c t i o n i s i s increased, shown : bargainer agree " s i z e " bet of money under c o r r e c t assume a be v a r i a b l e at of the a a the higher i s 3.1, that a under . to described bargainer conclusion a exchange type Assumption f i x e d deal X . parameter d i s t r i b u t i o n i n i t i a l w i t h studied by L be performed. i n w i l l i n g s e c t i o n to pay increases a r i s k h. to each We the wealths. w i t h Arrow member s h a l l r i g h t be or A modified considering Following transformations, s h i f t the now e~ averter the w i t h t y p i c a l l y w e a l t h i e r . reopened say, random of of i n t u i t i v e l y - . bargainers Since, he P, value loses. is The wealth amount 8(1 -4 A). 0 venture when IN family he > the the + i n c r e a s i n g represents the ) X Carlo observed two e and Appendix When a when the SHIFTS o r i g i n a l Monte is - 3 EU(aX) represents a more a have Assumption 7(1 values f o r : C U"(x)/U'(x) (see 3.3.1 opponent, and Then P r o p o s i t i o n 3.3.2, = U'(0))/EU(aX) Thus In U(x) a P b e l i e f s . we Such i . e . i n of conceive c h a r a c t e r i z e d i n t e r e s t e d l e f t , i n transformation [1971] being s h i f t (i) X(h) by a = of a the simple X + h , a 03 CD 1V> ALPHA 53 and i n i . e . ( i i ) X(h) K's a = simple X + standpoint f(a) = expansion (1+h) we (X-X r e w r i t e ) the of the . Remembering equation E U (-aX)E {X(h)U^(aX(h))} L L d i s t r i b u t i o n - K about that X (3.3.4) a i s centre, s h i f t e d X , only from as, E ^ X U ^ C - a X ^ E ^ U ^ a X C h ) ) } = 0 (3.3.6) using normalized i n v e s t i g a t e the sign of the same as a = - the u t i l i t y d i r e c t i o n . r£ a n that = As is df — 9h of E U ( - a X ) L L E T L functions i n which shown - . equation L R (3.3.6), a i s the here other may case we be c a r r i e d + constant) on of h we the f(a) . To need sign of w . r . t . a aX(h)X' (h)U£(aX(h)) to examine 4 r dh is y i e l d s , } K j a X * ( h ) U ' ( a X ( h ) ) } K K E {X'(h)U^(aX(h) + K 0 only out now section D i f f e r e n t i a t i o n where consider are previous = E U L s i m p l i c i t y D dependent K [E U (aX(h)) using and E {X'(h)U^.(aX(h)) {XU'(-aX)} K i n (C i n the a (-aX) /E L s i m i l a r U K a d d i t i v e aX(h)X'(h)U£(aX(h))}] (cnX(H)) 0 . = X + For K s h i f t way. > X(h) Hence f o r an h . The a d d i t i v e s h i f t , 54 3f a 9h = - = 6E U (aX(h)E U^(aX(h)) K - K + K E {X(h)U£(aX(h) 6a[E^U^.(aX(h) )Ej,{X(h)U^(aX(h) ) } ) }E {U^.(aX(h) ) R E U^(aX(h)) - K 6{E U (aX(h))}' K- K Thus, an a d d i t i v e decreases aE K can K a K f a c t K ( a X ( h ) ) } s h i f t the , (U'(aX(h)))/E i n U come K (U K to ^ 2 of s i z e of _ r K's E U^.(aX(h))>, K a E U (aX(h))J K K p r o b a b i l i t y the (aX(h))) a E^U^.(aX(h) ) K r 6 { E d E U (aX(h)) K = }] K exchange, increases d e f i n i t e - d i s t r i b u t i o n according or conclusion a = or as decreases under increases at the a As . f o l l o w i n g before we reasonable assumption: Assumption is X an = 3.2: increasing example. is U = of a , at function 8 ( l - e Monte p l a u s i b i l i t y Furthermore nearly EXAMPLE + for U and R any U = U for a that L EU'(aX)/EU(aX) a which EU(aX) > 0 , where X(h). The U Assume 3.3.2 -2x ) Carlo l i n e a r : and of i t at t h i s i s a random c a l c u l a t i o n obvious the i s again assumption established tends to be by an true when . Consider the assumption the same u t i l i t y ^ X = v a r i a b l e shows that a EU X f u n c t i o n i n U(x) Example (aX)/EU(aX) = 3.3.1. is 7 ( l - e _ X Then monotonic ) a i n c r e a s i n g CO FIGURE 3.3.3 GRAPH ro OF aEU'(aX)/EU(aX) IN EXAMPLE *» G A M A VS a = ALPHA 3.3.2 OQ- CD 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 56 for a l l such a f u n c t i o n f o r that our EU(aX) example i s > 0 (see shown i n Appendix Figure B) . 3.3.3. The Hence graph we of can t h i s conclude that: P r o p o s i t i o n at a higher SHIFTS as ( 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 on X to the l e f t . when the IN X : The say the thus a section t y p i c a l l y s h i f t Two of = the of the by To to a two some e f f e c t s " By symmetry such a K and L . or Thus has X* + of have X of our a is simple . view because no i s X a d e f i n i t e a (i) a i n to h the to t h i s changes previous a decrease i s of consider due i f such p o s i t i v e ) . v i s i b l e of i n standpoint, tend (when of of or, a expansion F i r s t seen K's t y p i c a l l y of i n t e r e s t : (X-X*) only of d i s t r i b u t i o n s responses from value t r a n s a c t i o n , already decrease to both K " p u s h - p u l l " d i r e c t i o n i n which . e f f e c t s no (1+h) t r a n s l a t i o n there the ( i i ) such w i l l point are and way a simple the h We t h i s increase that + = L's t r a n s l a t i o n however X i n p r o b a b i l i t y s h i f t s contexts s h i f t e d to = example involved of X(h) bargainers. simple both , is that a types only f o r s u b j e c t i v e X(h) In from occur a r t i c l e two X* . X place guess the " p r i c e cause X an form X+h evident can of value increases. i s i n p a r t i c u l a r when takes moves p r i c e c e n t r a l that s h i f t s a f f e c t i n g movement are v i s i b l e of r i g h t , c a l l e d a agree the h a c t i o n s h i f t w i l l to i n L simple bargainers X X(h) and a two on s h i f t i t of the d i s t r i b u t i o n about Now r e s u l t 3.2, p r o b a b i l i t y the a a assumption s u b j e c t i v e fashion. simple Under K's changed, same a : (i) parameter, X 3.3.2 of any p r e d i c t a b l e sort e f f e c t of on movement a is of not X j u s t i f i e d 57 e i t h e r . Some movement i n a F i r s t X(h) = To we Let the of X do give r i s e to a p r e d i c t a b l e . (l+h)X. context. expansions consider ( a l t e r a t i o n ) when simple random m u l t i p l i c a t i v e s h i f t study use a a a (h) the responses of method s i m i l a r be bargaining the v a r i a b l e , by which to the of a what X due around to Tobin s o l u t i o n ventures a such [1957] to the are the o r i g i n , m a g n i f i c a t i o n used i n exchange generated, a d i f f e r e n t problem i s X(h). Then, E R U K (a(h)X(h)) E {X(h)U|(-a(h)X(h))} E {X(h)Ul(a(h)X(h))} = v and E U (aX) E Js-Js. where { X U * (—otX) } L X(h) = W l + h L 1 + h ) X . s a t i s f i e s = the Xl+h)X equation (-aX)E 1 7 E^U^(-a(h)X(h)) 0 (3.3.8) {XU' (aX) } L K = i n ( l + H X ) (3.3.8) (3.3.9) by (3.3.9). E^dCl+hJXu'O now i t Hence, 0 (3.3.9) K (1+h) E j X d ^ U ^ - f ^ - d ^ X ) } ) 1+h X(h) E U M u l t i p l y i n g a Since + L (l+h)X ( + L i s we have , + (l+h)X-,)} c l e a r that i . e . = a(h) 0 = a/(l+h) 58 P r o p o s i t i o n 3.3.4 : v a r i a b l e generating s o l u t i o n when Now a r b i t r a r y we If_ the random are i n a X* . Then t h i s o r i g i n of and t h i s second One cannot, that w i l l i f member the X* the = (l+h)X and a induce an otherwise o r i g i n . to s h i f t i t be a sum is a the m u l t i p l i c a t i v e p a r t i t i o n e d of hX*. d e f i n i t e a a/(1+h) the random bargaining s h i f t about an s h i f t about the as, (X-X*) s h i f t w i l l when hX* i s a greater consider can - induce then s o l u t i o n X(l+h). (1+h) s h i f t come i s to w i l l a d d i t i v e at + a d d i t i v e therefore, decrease increases, about downward p a r t i t i o n the is a X, X(h) = m u l t i p l i c a t i v e bargaining i s v a r i a b l e p o s i t i o n X(h) Hence the ventures the center i s a d i r e c t i o n d e f i n i t e has proportion r e i n f o r c e s m u l t i p l i c a t i v e We-have i n d e f i n i t e hX* a a than the seen of d i r e c t i o n general p o s i t i v e the that movement of a movement i n conclusion. i n proportion at of the f i r s t i n e f f e c t e f f e c t the A l l a , which member and i t . we can then say a X m u l t i p l i c a t i v e s h i f t 59 CHAPTER 4 BARGAINING PROBLEM : SOLUTIONS UNDER INCOMPLETE INFORMATION 4.1 4.1.1 INTRODUCTION BACKGROUND "Bargaining" l a b e l s a v a r i e t y of processes i n v o l v i n g 2 o r more (here 2) p a r t i e s c a l l e d b a r g a i n e r s which seek t o reach agreement about the exchange of q u a n t i t i e s of goods, union-management wage n e g o t i a t i o n b e i n g a t y p i c a l example. U n t i l the 1930s, o n l y the range on the Edgeworth c o n t r a c t curve i n which the s e t t l e m e n t takes p l a c e had been s p e c i f i e d . Then a t the b e g i n n i n g o f 1930s attempts were made by Zeuthen [1930] and H i c k s [1932] to model union-management n e g o t i a t i o n s and t o s p e c i f y the p o i n t s o f agreement i n terms of the parameters o f the model. S i n c e then a g r e a t d e a l of theory based on a v a r i e t y of models has been developed, i n c l u d i n g the work o f Nash [1950] and [1953], Pen [1952], Bishop Cross [1965] and [1969]. [1963] and [1964], F o l d s [1964], T h i s theory o f f e r s a v a r i e t y of determinate s o l u t i o n s f o r the b a r g a i n i n g problem, and embraces, as p a r t i c u l a r cases, such b a s i c economic problems as b i l a t e r a l monopoly, duopoly, e t c . ( F o r a p p l i c a t i o n s of the b a r g a i n i n g problem c f . de M e n i l [1971]). E s s e n t i a l l y t h e r e have been t h r e e approaches t o the b a r g a i n i n g problem. Firstly, t h e r e a r e t h e o r i e s based on von Neuman and Morgenstern's theory o f games and i n c l u d e those of Bishop Nash [1950], and R a i f f a [ 1 9 6 3 ] , K a l a i and Smorodinsky [1975], [1953] (and Luce and R a i f f a [1957]). Secondly there 60 i s the work approached of authors the bargaining employer-union the t h i r d and Foldes roughly i s taken is only a a a implies which been [1932] have i n the which theory same It solutions that aspects i n t o as also Cross e i t h e r s p e c i a l that occurs time account p o i n t i n a the of enough, Nash's based the of from on who the have model [1930]. Bishop f a c t o r [1965], two f o r F i n a l l y [1964], elements (duration In i s since i n Cross [1965], sketched of bargaining) any i n terms even that 2 (at h i s applies of the s o l u t i o n of i n the theory) approaches agree cases) are theories [1964] or i n Zeuthen's u t i l i t y theories s p e c i a l that other Foldes provides bargaining shown t h e i r p e r f e c t concept) the modern of w i t h theory about has However, imply w i t h Nash's that take that the 1 i s u t i l i t i e s and Nash's l e a s t i t s o l u t i o n though most p a r t i c u l a r most r (a [1956] [1964], w i t h p r e c i s e . l i t e r a l l y i n s i g h t s Harsanyi out most r a t i o n a l bargainers settlement Bishop conclude the Furthermore, theory, i d e n t i c a l i . Nash being pointed s i t u a t i o n p l a y e r of without when denoting of [1965] Zeuthen a p p l i c a b l e , v e r s i o n be cases. reached that s i t u a t i o n outcome are i s u n i t " t i m e " s i t u a t i o n . the (Harsany's of agreement maximum, Saraydar those i n v o l v i n g bargaining r t h i s by developed i s s i t u a t i o n should those i n i t I n t e r e s t i n g l y i n c l u d i n g s o l u t i o n models are the perhaps than bargaining i t s e l f . d i f f e r e n t . and suggested models, theories about the using bargaining bargaining bargaining [1952], problem c h a r a c t e r i z a t i o n of process Pen n e g o t i a t i o n s d e s c r i p t i v e two-player time as account. these knowledge only Hicks more to of [1964], i n t o Of i t wage class by such a w i t h two and r^ maximum s i m i l i a r the whereas players discount Nash's theory when r a t e (constant) i m p l i e s i - s per a 61 Our r e t a i n s i n analysis some Section 4.1.2 of f u l l of the and t h e i r work and i s p l a y e r ' s a c t u a l etc. i s s u b j e c t i v e analogy, a are mixtures one type f a l l s we have information b r i e f l y a each (cf. Cross [1965]) the o u t l i n e d of a number on the d i s t r i b u t i o n m a t r i x i s of types of the the formally set of bargaining chosen s t r i c t and from these e q u i l i b r i u m the s t r u c t u r e h i s and assumed of the each to of expected players, F i n a l l y an i s u t i l i t y , has types. both a The s u b j e c t i v e players. an N-player game analogue. payoffs " s t r i c t s e t " the h i s j o i n t the players player p o s s i b l e model^'"by g e x p l o i t i n g i n two game, p l a y e r s ' subplayers makes types; the known model than consider that opponent's assumed the incomplete described. rather s p e c i f i e d p o s s i b l e is i s bargaining " e q u i l i b r i u m p o i n t s . information e t c . define two-person N-tuples of [1972] of postulated s t r a t e g i e s Selten account, players knowledge j u s t represent p o s s i b l e p l a y e r s ' They over to the categories i n t o It the u t i l i t i e s , s i t u a t i o n . representing a l l a payoff adversary. p r o b a b i l i t y the where others' and p l a y e r s ' takes such f i n i t e his Harsanyi formulate which s i t u a t i o n s game, outside of w i t h the when [1972] depends to of problem aspects of deals i r r e l e v a n t . d i s t r i b u t i o n s the varying points which incomplete INFORMATION andlSelteri.'ranalyze..the ^bargaining Out by under theory assumptions Selten p r o b a b i l i t y of payoffs information, unknown p r o b a b i l i t y Harsanyi Nash's bargaining becomes important whom existence about therefore incomplete i r r e l e v a n t on bargaining Harsanyi and work factors solve of of bargaining INCOMPLETE j u d i c i o u s and each UNDER the of such under s p i r i t knowledge because about two-person 3.3.1. Most or the BARGAINING have of e q u i l i b r i u m " generated axiomatic generated theory by is taking 62 developed to s e l e c t s o l u t i o n " . This generalized Nash axioms and w i l l l a t e r the of are not V_. being and to p r o b a b i l i t y l i m i t i n g could to not determine n e g o t i a t i o n s . the the players. i s " g e n e r a l i z e d f u n c t i o n a l Nash c a l l e d the are he h o l d ; l ' s is type Harsanyi give e i t h e r that the Even obviously Harsanyi and to the during a do adapt the the when he Selten t h e i r do j o i n t bargaining. p r i o r a r i s e E i s w i t h V not J U , i < 2 and is d i s t r i b u t i o n make i t c l e a r , s o l u t i o n . s u b j e c t i v e or that The because to analogy. + i t i t l a t e r carry is is the t h i s process generalized which the type bargaining t h e i r analogy invariance e q u i l i b r i u m of the why N-player p r o b a b i l i t y bargaining, i n for sets the on N-player a f f i n e p l a y e r s ' depend Their i n d i c a t e Furthermore d i f f i c u l t i e s Selten strategy, . and termination so, for s u b j e c t i v e meaning constant at j the to (i) u t i l i t y Although to f a r ( i i ) now h i g h l y d i f f i c u l t i e s . meant game j o i n t is e q u i l i b r i u m p l a y e r s h a l l too and bargaining. misleading not profound these pursue is a of we to p l a y e r s ' a l t e r n a t i v e . as and that does doubt, theory bargaining when assume? would It axioms 5) necessary also used an the a without Selten game. u t i l i t y assume i s p o s i t i v e l y and being V d i s t r i b u t i o n be more Axiom d i s t r i b u t i o n r e a l i s t i c maximizing t y p i c a l l y two-person d i s t r i b u t i o n l i m i t i n g more s a t i s f i e d game the 2's both they p o i n t , mathematical Harsanyi S e l t e n ' s assumption the bargaining player Apparently the a l l to by work ourselves f o r unreasonable t h i s be a t t a i n a b l e , known to bargaining instance, found S e l t e n ' s a l l two-person (Harsanyi is and not e q u i l i b r i u m product. address two-person For is c o n t r i b u t i o n F i r s t of unique point Harsanyi o r i g i n a l a Nash out so product the 63 By game these t h e o r e t i c a l p r a c t i c a l is We value p l a u s i b l e information i s but a rather, In under leaving the and that not the a rest be f o r apply point to 4.2.1 THE NATURE Consider incomplete s i t u a t i o n a OF is described may includes s t o c h a s t i c nature. i n question. of A Two o i l , to mobiles at a also be worked to f i x e d by problem space of the has w i t h cases. incomplete approach suggests, p o s s i b l e bargaining is suggested. components of a great simple the problems of of at the s o l u t i o n s . problem An problem, studies. PROBLEM two Harsanyi i n t o two example p r i c e , and out, a the each Selten broad i n engage others categories. The i n past, c l a r i f y the , p e r i o d u t i l i t y terms nature of of f o r 10 a of the agreement w i t h Such a u n c e r t a i n t i e s f i r s t category bargainers' or future the " d e a l " c e r t a i n years, any a e t c . of present, negotiate return bargaining u t i l i t i e s , The about B i n [1972]. q u a n t i f i a b l e may A over Here bargainers and d i s t r i b u t i o n s worked out. one payoffs, countries, be the the problem approach t h e o r e t i c a l decomposition THE u n c e r t a i n t i e s contrived game component where about c l a s s i f i e d p r o b a b i l i t y of amount be the of PROBLEM s i t u a t i o n involved s u b j e c t i v e THE information as over v a l i d i t y p a r t i c u l a r l y bargaining future 4.2 t h i s the analyze possible c e r t a i n the bargaining to conceptual made the denying i n d i s t r i b u t i o n i n t o denying of course, s o l u t i o n information then s o l u t i o n of w e l l s i n g l e a e s s e n t i a l l y the not, may sequel, incomplete w i l l are are to p r o b a b i l i t y the attempt we approach l e v e l . d e s c r i p t i v e It comments states d i s t r i b u t i o n giving number of number and w i l l be B an autop r i c e determined 64 by future i n f l a t i o n a r y s u b j e c t i v e includes any p r o b a b i l i t y the other a u n i t s , demand of t h e i r o f f e r ) , not s i t u a t i o n at based o t h e r ' s u t i l i t y payoffs that of information (i) determine A how or If the and the the make new preferences In be or f e a s i b l e to an agreement at perhaps w i t h updated. and payoffs i n t h i s enforced. of the bargaining (priors) p o s s i b i l i t i e s for bargainers process would bargainer payoff demands, stage general, scheme are s i t u a t i o n d i s t r i b u t i o n s of Each p h y s i c a l r e s u l t s every bargaining bargaining come w i l l at or they c o n f l i c t range payoffs. and there bargaining etc.) demands over from complete other. p r o b a b i l i t y these gained the s u b j e c t i v e c r i t e r i a ( i i i ) p a r t i c u l a r make have category f o l l o w s : units concede deal, as u t i l i t y concede, no second tastes i s demands of B to on each unknown w i l l p a r a - make use s e q u e n t i a l l y b a s i c a l l y consist update of the components: ( i i ) In threats, s u b j e c t i v e meters f o l l o w i n g them bargainers on p r i o r s . may one process these they and payoffs. here the A The opponent's bargainers' of these. of a f t e r l e a s t R a t i o n a l the terms u n c e r t a i n t i e s by on conceived i n both bargainers u t i l i t y about (given two the his one Otherwise b e l i e f s Should or s a t i s f y i n g these. to process simultaneously solutions one (or the about r e l a t e d bargaining and d i s t r i b u t i o n s u n c e r t a i n t i e s factors The makes rates for f i n d i n g c h a r a c t e r i z i n g t h i s not study we of our bargainer to f i n d study a the w i l l introduce o b j e c t i v e p r o b a b i l i t i e s i s " f a i r a optimal solve mechanism to o f f e r s s a t i s f a c t o r y not f o r (in the at stage stage (i) updating the from agreements problems i n v e s t i g a t e demand" change c r i t e r i o n sense of at and the n + n to ( i i ) n and p r i o r s ; the used a Nash, + 1 . 1 stage by n + i n main r a t i o n a l a p p r o p r i a t e l y 1 65 generalized), demand to at The a an his demand i s stage accept l e a s t as of conceived gap i n less, the w i l l be 4.2.2 n + the ' f a i r i n and the opponent, On the other amount given of to Suppose very OF each are THE the f i x e d at a c x the and w i l l two d, d i s t r i b u t i o n (V(x), d) (U(x), c). p r o b a b i l i t y payoffs. denoted bargainers derived Bargainer These p r i o r from 2 no the more ' f a i r and i n on bargainer is to below. a demand' is weak given hoped him at a h i s f a i r And although l i n k w i l l i t r e v e a l between here. that demand. the Neverthe- acceptable axioms question. MODEL i s a f u n c t i o n by and U(x) receive i n given case h i s s u b j e c t i v e unknown of x x. i s 1 and of c o n f l i c t Bargainer b e l i e f s , been parameters 2 to evaluated r e s p e c t i v e l y . d i s t r i b u t i o n have u n i t s l V(x), a may p h y s i c a l value Bargainer For p r i o r the of n of the as whether make d e f i n i t i o n below 1 a decide gives than i t must a c t u a l i t p r e c i s e u t i l i t i e s has so h i s i f ); d i s t r i b u t i o n s d i s t r i b u t i o n s made u t i l i t y would r e s p e c t i v e l y . TT^ and be The do d e s i r a b l e say n e g o t i a t i o n s . given payoffs, by by may as «. be question l e a s t BARGAINING v a r i a b l e base a l i n k b a r g a i n e r ' s he to i f suggestive strengthen i n above, t e c h n i c a l meanings which hand be i s are v i z . at w i l l terms on bargainer by demand' the reference there r e s u l t s which If an analysis, found of . made FORMULATION some 1 demand of our conceptual or demand. would meaning b a s e - l i n e o f f e r f a i r he a on 1 are has the TT^ induced on by l a b e l l i n g The denoted a p r i o r range the the the of range of s u b j e c t i v e u t i l i t y 66 Usually bargain out of for an i n terms s e v e r a l x bargaining equivalent value x countries out. Each demands, If make up i s new where a r b i t r a t o r t h i s acceptable to him f u l f i l s Otherwise he at k i n d . that produce way. t a l k s (2) by prefers for peace 1 c o n f l i c t he two l i s t s the or whereby of f o r a common Other- a l t e r n a t i v e l y , between two two countries announces that one searching s o l u t i o n . or, Bargainer are of l i s t a l l i f x ' s any of p a r t i c u l a r acceptable bargaining a such determines o f f e r s n e g o t i a t i o n s a the accepts l e a v i n g The from 2 time agreement f i r s t Bargainer announcing wage a as at mechanism of demands delegates so, a these i t they process a r b i t r a t o r , searchs r a t h e r x The examples an Bargainer c o n f l i c t . Employer-union are equivalent If one involves announces to and demand u t i l i t y . to search counter else t h i s A he of may concession i n found u t i l i t i e s ; demands a r b i t r a t o r expectations. he same submits h i s that may f o l l o w i n g moment a l t e r n a t i v e l y or the l i s t s s i t u a t i o n . of x. The c o n f l i c t an him The x over bargainer bargainer an bargain bargainers' {x}. such i s l e a s t y i e l d i n g c a r r i e d (1) not Each i s separately at . both . the x x ' s bargainers enter of do s a t i s f y i n g mechanisms: wise bargainers 1 sometimes o f f e r to himself to make continues these or a s e q u e n t i a l l y c a r r i e d out this way. A of f a i r w i t h demands, complete which his to is i . e . required a b a s e l i n e information maximizes analogous from c r i t e r i o n the t h i s Bargainer opponent's l ' s Nash i n u t i l i t y f a i r u t i l i t y case stand a of which for or a h i s demand For any simply bargainer demands. f o r product. incomplete p o i n t . V(x), by to information V, and x that bargainer order given determine R e c a l l e i t h e r In can l e t , i n was view Bargainer c o n f l i c t payoff l i s t bargaining any e s t a b l i s h us a one something the 1 problem regards d, as random say v a r i a b l e s , (ft, F, P ) ; Bargainer 1) p r o b a b i l i t y Note w e f t two and i s d e f i n i n g U, c, a and , denote defined p o s s i b l e V, V d and w , x generated u t i l i t y choosing by functions transformation value P on the events d are i n cf> of by <j>(w; being c a l l e d V) out A of d i f f i c u l t y by of space, (unknown l ' s F t o s u b j e c t i v e subsets o f the appropriate This <j> Bargainer p o s s i b l e u t i l i t y x) : So be We demand making to them i n w i t h of only 1. For a see Luce comparable, does where d i f f e r e n t a i n ft. b a r g a i n e r ' s s i m p l i f y i n g respect the c o n f l i c t Is general we may U(x) take determined discussion and R i f f a c by of [1957]. not a h i s t h e i r p o s i t i v e To of l a b e l s r o l e a f f i n e the not seem nearly i n t h i s t h e members l a b e l s have same l a b e l s 1 2's t h e maximum, as of a f t e r a l l , f o r Bargainer as w i t h 2, avoid unreasonable say these Bargainer the Bargainer seem = - assumption, by to i s u t i l i t i e s f o r d i f f i c u l t a c t u a l l y here as represented i n d i v i d u a l s . determined generality and be one comparability""'" does w e l l comparable now i n t r o d u c e i s serve i f of of a l t e r n a t i v e s . and y i e l d s the s i t u a t i o n functions x increments own c h o i c e should t h e case u t i l i t y requirement classes w e f t } example. of l ' s functions confronts the a v a i l a b l e w e f t } . l o s s Bargainer o - f i e l d equivalent x) : a p r o b a b i l i t y the parameters u t i l i t y {(|>((jj; the a same the equivalence-class the equivalent the other. ( f o r x) the " i n t e r p e r s o n a l would o f a l l determined we assume {<t>(w; the represents d i f f i c u l t y i s on f u n c t i o n . Given l a b e l V d i s t r i b u t i o n that u t i l i t y say 0 c = and V(x) points. d Bargainer namely (w) - d, Then f o r a l l 2's i n i t the value the (V u t i l i t y c o m p a r a b i l i t y " o f u t i l i t y follows w e f t c a r d i n a l the " i n t e r p e r s o n a l that that w = without ~ ^ f u n c t i o n ) . o f u t i l i t i e s 68 Let TT denote v a r i a b l e As his (new) an extension " f a i r " Bargainer 1, Bargainer 2, i f l ' s co because i f h i s point i s U V(co) i s of assume that Bargainer by complete maximizing a of demand P. true t h i s i s i t s take 2 [1 - magnitude such equivalent to 0. (V(u)), 2 P co ; e Hence P) Q}; {V (co) , 1 a U i f be and we U fi}; the on random to = Q" to U > 0 l ' s thus negative P) G N F 0, V(u) i f U can > > 0 point to 0, 0 a be f o r 8 payoff is 0} because a Bargainer u t i l i t y {co/V(co)> a l l of . and c o n f l i c t the to 2 0 w i l l c a l l e d type > of we demand relevant > be Bargainer V(co) the that what his payoff assume i f to e of Bargainer where p o i n t , f u n c t i o n , gives p r e f e r cannot a + the determines depending that Hence P(£T)], 1 u t i l i t y i s would negative. at <T (U point therefore, place X a the Bargainer Bargainer u t i l i t y P gives point, surely the Consider and of information c e r t a i n u t i l i t i e s negative, = case the T (U; GNF the of and f u n c t i o n P. (GNF), + 0 . c a r d i n a l l y from Function" (£T) never d i s t r i b u t i o n Nash of point w i l l of demands is u t i l i t y wise induced information "Generalized V(co) p r o b a b i l i t y V incomplete c a l l the . of othert h i s such Also i f settlement V(co) w i l l w r i t t e n a l l be as: co 3 . 0 ' ^ otherwise. where, of i t should demand. mutually w r i t t e n Now be noting exclusive as: remembered, that domains the and U and V(co) functions l e t t i n g U = U P for each co and 1^ are (ft') , GNF a are functions defined can on simply be 69 T(0; {V (to), The f u n c t i o n and as T + such must GNF = ue ft};P) T i t maximizes here the mainly developed assumed that f u n c t i o n T(U f a i r j u s t i f i c a t i o n s 4.3.1 THE We plus a w i l l T + w i l l t h i s derive to Axiom 1:- by to for intended u e ft}; an analogue preference of the points OF FAIR i n order p r e f e r r e d to THE as f i n d to e + serve TO of two by P) 0. = the Nash product bargainers. both Hence bargainers. DERIVATION DEMANDS {V (w), under a l l as APPROACH ; w i t h a " f a i r ft}; P). demand The g e n e r a l i z a t i o n complete designation a w i l l be given f o r T from term of information. l a t e r Bargainer " f a i r " the Nash 1 i s used product Further i n t h i s s e c t i o n . AXIOMS s c a l a r Axiom found is P) 0 H + j o i n t bargains for t h i r d p o s i t i v e the ft}; {V (to) + AXIOMATIC because f o r T(U ; values AN a) e + defined l a r g e r have {V (co) , = thus 4.3 ; + measures assign We T(U 2:- be a p r e c i s e stated A maximizing bargainer's Nash class of bargaining the the f o l l o w i n g two axioms l a t e r . transformation In form the w i t h product. of f a i r demands oponent's complete or his does own information, not vary u t i l i t y f a i r under f u n c t i o n . demands are 70 Axiom i n v a r i a n c e U ^ 0, f o r i n a a l l T(a r e a l = + V assume a that, ; {bV and to b a l l T(U (to) , . ; + to e Axiom w i t h Nash's f o r + g e n e r a l i z a t i o n i f + s i t u a t i o n If i t without lowering process i s i s to e Nash's to e + fi}; 2 P) is says complete also that he axiom P) Q,}; a l i n e a r maximized by b a r g a i n e r ' s i s the determining would of maximized information f o r then Q, of {V (to) , procedure p o s s i b l e h i s Each Since T and maximize the f a i r amount f a i r T by same demand he would bargains, i . e . = We U + V . + bargainer implies i n U > 4.3.2CP DUCI^G; :/THE E ; bargainer u t i l i t y , i s aware l a r g e r that 0, a he to does increase so, i . e . h i s u t i l i t y the bargaining also e f f i c i e n t . that assigns assumption for oponent's e f f i c i e n t ( i i ) i n c r e a s i n g + a that (i) t h i s U according V (to) simply states bargaining receive also i s and then U i f 1 and FORM values T(U V(to) OF that ; + > GNF his to points (V (to) , + 0 f o r WHEN oponent Q p r e f e r r e d to e a l l i s fi}; P) to e . CONTAINS TWO by i s both bargainers, s t r i c t l y ELEMENTS PRELIMINARIES: Let case = 0 the = (a , to ) , = 2 P has the form: T(u f o r a l l p where p u t i l i t y from Q i s (In Bargainer set values S has t h i s l ' s bounded by V*, p) V*, P(to^), case stand generated = and 1 , + V(to ) , we the say point.) (U, p r o j e c t i o n s V^, i n V(to ) w i t h T(0, Bargainer 2 only V^) when range the Ux V 1 In t h i s V\J\ V*, p) that consider X . 2 p r o b a b i l i t y that We = and Ux has two over ? = T(U Bargainer a l l + , 0, 0, 2's p o s s i b l e bargaining V p a r t i c u l a r games types where f e a s i b l e planes. Furthermore p) 71 F I G . 4 3 . 1 : T H E S E T O F C I T S O N C E I V A B L E A P R O J E C T I O N S . G R E E M E N T S A N D 72 S is of T required x) i s any i t to be such unique may achieved bargaining of which y) Then f some i s the _if must c x, be i n so w i t h randomized . in pure contracts contracts. Following has y ) , of a l l plane that the Nash, terms of we (choices Given assume u t i l i t y p, that u n i t s i n a y) = a any °°) -> the f o l l o w i n g theorems, the (0, °°) be d i f f e r e n t i a b l e and Assume, convex compact point, (0, c of C . unique °°) F(x °°) ( S ) ) , y ( T say x c set S (x(S), (0, °°) ( S ) ) ) = i s (c x containing the y ( S ) ) . defined (S) , by y ( S ) ) . B (y)) for a l l -> °°) be x, y B. f(0, °°) v a r i a b l e s the r use form: properties and (0, ( x ( T and i t s x over (0, x make v a r i a b l e s . °°) ( ° » Let the °°) both we Appendix at 4.3.2. i n (0, T plane, F has form: 1 maximized T v a r i a b l e s x-y Bargainer i n f u n c t i o n i n c r e a s i n g not i n c l u d i n g S given f: x-y (c C o r o l l a r y i s set of f(x, for t h i s the form Let f ( i i ) T (X, the increasing i n i f by over game are 4.3.1: (i) convex; demand. derive s t r i c t l y c be To Theorem o r i g i n be maximized f a i r proofs to (i) x-z and and plane (0, assume ( i i ) d i f f e r e n t i a b l e that of f , Theorem separately. as a 4.3.1 Then, f and s t r i c t l y function w i t h must of 3 respect be of the to 73 f(x, for some functions Theorem f(x, i t s 4.3.3 y, z) = h (y, subsets i s the respective G(x A (y, (0, be maximized x-y o r i g i n , °°) (0, (x, = cy, (x(Q if_ c z ) , x and i s over x-z Q x R, planes a t t a i n e d (0, X^: that R), «>) , defined d i f f e r e n t i a b l e and by s t r i c t l y at a cy(Q i n c r e a s i n g i n a l l °°) (0, x c FUNCTIONAL ANALYSIS s a t i s f y i n g our R), : unique axioms i s R p o i n t , i s being each say R)). h(y, z) w i l l = any convex-compact containing (x(Q the N show by x ^ ( x , R ) ) , z ( ^ C Q x h of Then C(z) now defined x c x of °°) R)),y(A (Q cz(Q We and R), the y(Q x R)). 3 ( X ( i ( Q x Q r e s p e c t i v e l y , 3 ( i i ) z)) A. f: z ) , = Assume, f of z) and Let v a r i a b l e s . (i) G : x y, (|) that must be y, z) -»- R))) the form . any generalized Nash f u n c t i o n form, 1 T(U constant. f i r s t + , For quadrant V*, V*, P) = U p, + [ p any given 0 of 3-dimensional L <_ p + <_ 1 , (1 - T Eucleadian p) is V+ a space. L ] L , where f u n c t i o n L defined Consider f i r s t is a on the the 74 case u t i l i t y f o r — + where + > 0, evaluated at that U demand > any 0 f i x e d Bargainer 2 r e s p e c t i v e l y . ' F r o m .axioms T(U + , provided = less V 2 of V+, T = V+; i s V > his p) 0, since type i . e . F When V A(p) = is = 1 = U M V ^ Notice of Theorem when F(x; p) w i t h h (V , V ^ that 4.3.3 Bargainer 2 V; p) p) p) = ± B(V , second where is of his V ; p) 2 p); = 2 = F(U Q type U p) part of R and V V type were the , 2 4.3.2 1 V ; ] p a r t i c u l a r receives the for U, u t i l i t i e s 2, deduce, p); 2 l ' s and we (V _, where 0. 4 > a l l bargaining , p) f o r 1 U f o r 1; and = A(p) V ^ Bargainer and B In then being p) i f we same set amount regard- that, p) x B ( l , V ; 2 i f C o r o l l a r y product 1 the r and and implies Nash get 1; V + U d i f f e r e n t i a b l e . 2 (V, usual we 1/B(1, T where the B 2 — 0, get Bargainer Axiom F(U would T(U, everywhere > demand 1, = + and V ± 2 = U > 0, B T V 2 complete (1, 1; reduces > information. p) and to, 0 (4.3.1) . Axiom are w i t h Hence 0, A(p) x V. the 1 implies u t i l i t y - r e s p e c t i v e l y . the c o n d i t i o n p o s s i b i l i t y Since we sets have ( i i ) 75 already of assumed Theorem A l l the 4.3.3, we T Again U when C(V; p) = N ( l ; p) = conditions now = U apply CCV^ = V we U V and on to p) may necessary reduce N V T for f u r t h e r the and a p p l i c a t i o n get, ' apply thus T the boundary c o n d i t i o n obtain, V for To U > 0, V complete T(U + > 0, 1 our , w i t h STOCHASTIC ANALYSIS e i t h e r c e r t a i n i n the i s a 1 or tossed, have > : two i s shown and we now types namely he U 0 as f i n d i n g that to f i n d a <_ 1 , which a i s H maximizes of <_ p p) (4.3.2) where some l ' s he therefore, knows that M(^-; 1 consider biased) 1 V Bargainer (in We, according Bargainer = of M(x; the p) = N(x; s t o c h a s t i c p ) / N ( l ; aspects of information. From (possibly 2 p) 0 incomplete of coin problem problem 2 p r o b a b i l i t i e s . which type, V V+; analysis bargaining assume V*, the " f a i r " f a i r can tossed or how has T to Nash is f i n d stand d i f f e r e n t think to of u t i l i t i e s ) Bargainer " f a i r " here In t h i s demand However, can w i t h determine before demand,meaning 2 s i t u a t i o n obtained. a Bargainer this product. demand point as one 2's version a f t e r the Bargainer the coin one which i s of coin l ' s tossed. We maximizes T, p). 76 [V^ 2 M ( — ; usual Nash V he may e q u i v a l e n t l y the form may be of T w i t h regarded of the f a c t sense i n the mixed f a i r Figure demand that of U. the conceptually, s p i t e a maximize, that game. before and V*(p) The a f t e r as p)] product bargainer need not c r i t e r i a the be used coin is A_ U . 2's a we Comparing see V*(p) that " e f f e c t i v e u t i l i t y by [V*(p)]. function Bargainer tossed i s u t i l i t y " 1 i n i n i l l u s t r a t e d 4.3.2. FIGURE 4 . 3 . 2 : CRITERIA AFTER PRODUCING THE COIN FAIR TOSS. DEMANDS BEFORE AND the i n usual determining i n 77 To f i n d t h i s a c o i n reasonable tossing game one w i t h P(H) or T i s obtained as H or a " f a i r " shown w e l l i n as T a i s i n and on f o r the on the Figure 4.3.3, what 4.3.3: second TOSS coins he AND coins w i t h 2. are would a are P(H) Bargainer toss coin CRITERIA two 1. before consider second toss obtained the V* which demand a f t e r FIGURE = form more s e q u e n t i a l l y = p 2's have We done or type Bargainer tossed. complicated v e r s i o n 1 q the according as i s 1 i s r e q u i r e d have a f t e r tossed, or already the f i r s t 2 AFTER THE FAIR DEMANDS SECOND COIN AFTER TOSS. THE to H make s p e c i f i e d , c o i n FIRST f i r s t according toss toss. PRODUCING of COIN as as 78 On the one hand, p r o b a b i l i t y 1 - [ap + since ap (1 - + (1 i n - a)q], t h i s a)q game and w i l l Bargainer 2 M| r r ; > a p Bargainer be of 1 would (1 - 2 type w i l l 2 be w i t h of type 1 w i t h p r o b a b i l i t y maximize, V \N On the may other hand, e q u i v a l e n t l y U > V*(p) 0, > Bargainer i f the Axiom 4 0 regarding V* 0 The M* because > c r i t e r i o n q u a n t i t i e s 3:- (p) and l ' s + V* (4.3.3) a) q as Bargainer 2's e f f e c t i v e u t i l i t y , he maximize, U where - ± i n V*(q) T 0 i s a s t r i c t l y making f o r M (4.3.3) determining increasing p o s i t i v e determining equations c r i t e r i o n (4.3.4) V*(p)' his and f u n c t i o n everywhere. " f a i r " demand (4.3.4) Bargainer's f a i r are demand We i s f o r w i l l say " c o n s i s t e n t " i d e n t i c a l . i s s t o c h a s t i c a l l y consistent. Under Axiom 3 we thus have, f UV X M ap + = ( l - a ) q ) for a l l 0 <_ a , p, q <^ 1 UV X M if v7 M \ ; 2 V ^ (4.3.5) •M* p I J n M v7 } P that 79 Notice that according M Hence 1 s e t t i n g q = 0, p to and = 1 Axiom M -; and x 1, 0 = — i n (4.3.5) we get 1 M(x; In a) = l.M*(x; a) M(x; + ap p a r t i c u l a r f o r (1 l e t t i n g a l l - a) p = x, q) a . Thus = M(x; p) M 1 we f i n d that the equation M(x; q) M(x; p ) ' the (4.3.5) reduces to, (4.3.6) f u n c t i o n M s a t i s f i e s the equation, M(x; for a l l a x > 0 , (4.3.7) s a t i s f i e s + 0 (cf. - a) q) = M(M(x; <_ q ^ 1, 0 <_ a <_ 1 , and then w i t h respect to the p a r t i a l d i f f e r e n t i a l ( 1 where (1 E(x) Sneddon = — ~ a ) M(x, [1957]) q 3 ^ 3 x " = q) that q=0 the i s q) ; a) (4.3.7) because s e t t i n g M(x, q = 0 1) = we 1. see D i f f e r e n t i a t i n g that a) equation, (4.3.8) E ( X ) a M(x, f u n c t i o n general of x alone. s o l u t i o n of (4.3.8) It i s can be shown 80 M(x, a) = H ( ( l - 1_ D(x) H = = a) D(x)), where . But boundary T must Since M(x, by H = D the be p) interchanging where is an a r b i t r a r y function = condition symmetric x M(^, i n V^, H ( ( l - p) D(x)) = x H H(0) = 1 (put p = 1 \ (4.3.9) H»((l -D(x) p w i t h - p) an M(x, 0) obvious sense and ( 1 - p ) (p = 0 and k where k = function obtain the D(x) -1/H'(0) <j> by noting = 4 0 (|>(y) = x D(x)) = that D'(x) D D'(x) (^) since D(e ) y x implies that i t follows that, to x D Hence, D(i)) i n respect . (4.3.9) 4.3.9), p and H and d i f f e r e n c e - d i f f e r e n t i a l i s H' therefore D(l) y i e l d s , (k H'(p A). D X p = 1 - p ) and D i f f e r e n t i a t i n g P u t t i n g and dx e D H X (D(x)) for a l l s t r i c t l y s e t t i n g equation, x = 1, x > we get. 0 monotonic. = e y i n the Defining above the equation 81 k <|><y) = * ( - y ) where k 0. given i n Lemma 4.3.4: 4 the + (y) = Now we u t i l i z e Appendix The m [1 the f o l l o w i n g k m y s o l u t i o n ] f o r m i s Hence, we conclude, a nonzero since since M M(x; w i t h L = km = 0, should p) a r b i t r a r y = + and + U + > 0, > the proof of which i s D(x) + , v+, 0, V 2 and + y as ]: or + — i s , w e l l i f 1 — m = [1 - p o s i t i v e , (1 - p) x L km x £n x and i n turn that, ] or L x x P that, V+; > (4.3.10) 1, P 0 + (1-p) ) and v^] L (4.3.11) or u v for (4.3.10) constant. be [p 0 4 U[p T(tT lemma equation k = a r b i t r a r y D(l) of a l l k where fora l l y e R C. general + e — *'(y) 0 p x <_ p v 1-p 2 <_ 1 . Note that f o r a l l L e R, 82 U[p + (1 - p) V ^ l ^ is i n c r e a s i n g i n U, and - + we define so that the the value of T c o n t i n u i t y is l i m [p along the axes preserved. + (1 P) U It v h i s V^, + , as Now + , also r e q u i r e d . as of the i n t e r e s t proper l i m i t that, v]->. 1 - - p) L-*0 This leads us to define, [pV^ I | V ( P ) | | , the generalized GNF = L where L e U + Nash | |V(p) + (1 V^] L i f = v Hence + || v p 1 1 _ p i f function for L L = 0 2 has a l l the U, V form, , 0 <_ p < 1 R Remarks: Notice ['76]) order of to u t i l i t y the Bargainer f i n d and that a a ' f a i r ' general L-norm 2's + u t i l i t i e s . demand, mean ||v (p)|| of a his L i s Thus, bargainer a what may adversary's general we mean have maximize p o s s i b l e (cf. shown the N i l a n is product u t i l i t i e s . that, of This N o r r i s i n h i s i s 83 p r e c i s e l y Nash's the expected product It i s i n of the analogue case i n t e r e s t of i n the complete case of incomplete i n f o r m a t i o n of i n f o r m a t i o n . that, 1 (i) l i m L -> [p V+ + L (1 - p ) V 2 L ] = L Sup V+ i 0 0 and 1 ( i i ) Let U 0., * h 0*, and +-+ Inf V\, l i m U [p 0. Sup V+ be l + V\ the and U(01) < U ( 0 . ) i . e . he bargainer would using the demanded L, the i n component i s expressed i n + i s f o r ] = L Inf by V* the f a i r In c r i t e r i a , Then i t demanding games, most the The the he higher r o l e of i t , z e i £ h e r s a s i e x p r e s s i n g the evident l e a s t whereas u t i l i t y fact, demand. way, i s t h a t , i i s r o the games. d i f f e r e n t a l l b a r g a i n i n g demanding l ' s L given GNF_ i n t e r p r e t a V 2 r e s p e c t i v e l y . U(0*) b a r g a i n i n g We p) + component GNF^ i n - s o l u t i o n s U < (1 c r i t e r i o n Bargainer mysterious. and the demanded c r i t e r i o n higher somewhat c a l l e d have using + L b a r g a i n e r s ' u t i l i t y bargainer would the L what have value of i s Pen [1952] ' n o n - a c t u a r i a l has 84 m e n t a l i t y . ' Here d i s s a t i s f a c t i o n ) i t s e l f . . . " J L represents the subject or as a i n t e r p r e t a t i o n of the (in derives measure of role, of GNF = that as r i s k :.&S3. 3 s - . r g a i n e r of c o n f l i c t . L U(Sup V.) increases and the b a r g a i n e r ' s L "the s a t i s f a c t i o n r i s k - t a k i n g aggression. i s " f o u n d ' i n the as Our a Bargainer is is j u s t i f i c a t i o n p r o p e r t i e s , increasing subjecting l-< i n (i) GNF = f o r U(Inf t h i s V -°° GNF 1 (or f a c t - ( i i i ) i words) from " ( i i ) Pen's • . i n L. himself ' " tc These to .: ), i p r o p e r t i e s p r o g r e s s i v e l y show greater -T-. o f ccntlict. With furnished 1. that i f t h i s f u r t h e r In a l t e r n a t i v e 2. by the component 0 Harsanyi V^, V^, 0, generalized treatment of Nash the s i t u a t i o n where the u t i l i t i e s component the games, when as variables.and.the 1. cf. Loeve GNF is product demand' may be GNF [1963] he that His only game Bargainer (with same. the complete Then have 1 n o t i c e Bargainer received of would informathat even l ' s f a i r from L. iisualt-(arithmetic) mean AJ 0,,. r e s p e c t i v e l y , - l y i e l d ^ t h e '-meiahs - It an obtained by Harsanyi bargaining which that magnitude ^reduces :to e s s e n t i a l l y r e a l i z e u n a l t e r a b l e . would the shows would information, i n over . of he work change. the equal person set are what regardless L=l- to 2 incomplete same was bargaining procedure, w i t h own p^ p is two and p^, GNF our ' f a i r Selten's cause pSiidMg-harmonlcSafid"'gSoffldfcr-dLC-= and to V ^ a n d s v | g a T h e - d M i e e s - E s - l:o. a n d L p^ his JLi when terminology try s i t u a t i o n t h a t , = U, bargain bargaining Notice the therefore e i t h e r f a i r by GNF and to a from Nash's the L knew be Consider given i f p o s i t i o n bargaining demand 2 would received tion) i n p a r t i c u l a r l ' s of j u s t i f i c a t i o n . Bargainer Bargainer have s t r u c t u r e the should appropriate game, and is pointed s p e c i a l Selten except maximization be f o r c o r r e s - case [1972] the c a r r i e d out that of in the t h e i r d e f i n i t i o n s out. It of is value of 85 e a s i l y seen stronger ( t h e i r that than Axiom reduces to s c a l a r 1 A of the s i n g l e v f + 2 types has two context since function. whole example 4.3.3 to s i m i l a r f - P u class of or of so n range of j u s t i f y f a i r by of the f o l l o w i n g Harsanyi f a c t , our own and whole 3), the has 2's one, which S e l t e n ' s class of i s axioms GNF's = type (here e x a c t l y 1 choices of V_^, and to the only vary l i n e a r maximizing not and c l a r i f y = p. our U one type seem seems type and s u i t a b l e unknown t h i s GNF 2's f u n c t i o n . has ( a l b e i t ) imposing p o s i t i v e i n v a r i a n c e 1 does under Bargainer u t i l i t y p l a y e r axiom of not 2 of above by does f u n c t i o n , axiom found f a c t ( P ) | | u t i l i t y consequence t h i s + demands analogue 2 be « and S e l t e h ' s = the should one i n Bargainer n-player f a c t , to || + his Bargainer In demands a v Harsanyi his replaced element f u n c t i o n , i n i s Invariance), b a r g a i n e r ' s reasonable 1 and L i n e a r Although for 1 transformations u t i l i t y f a i r Axiom Axiom D Axiom:- i f i n P i s P Consider our u t i l i t y c r i t e r i o n namely V p l a y e r the that unreasonable f o l l o w i n g r e s u l t s . EXAMPLE S r i fish) from where e l s e . s i m p l i f i e d Lanka the incomplete Maldive Thus model determination (Ceylon) i s islands a r i s e s w i t h i s a fake r e a l i z e d i n f o r m a t i o n . and Ceylon b i l a t e r a l figures i n t h i s buyer to of cannot monopoly see "Umbalakada" how s i t u a t i o n , buy t h i s s i t u a t i o n . the i n p r i c e terms of (Dried product Let and us Maldive from any- formulate q u a n t i t y bargaining w i t h a 86 A the is S r i Maldive a islands monopolist"^ umbalakada FIGURE Thus i n Lankian our i n S r i 4.3.4 model i n company S, (M) Bargainer S r i Lanka : say Lanka is monopsonist s e l l s competitive markets h i s at he by such output f i x e d the as and Bargainer 2, knows p" FUNCTION from books and given DEMAND d i f f e r s micro-economic say = FOR t y p i c a l and that 8 —JT-, the q ^ b i l a t e r a l and monopolist imports s e l l s UMBALAKADA Henderson p r i c e s . 1, demand i n S r i from Lanka. f u n c t i o n S f o r 0 IN SRI LANKA monopoly Quandt buys i t umbalakada h i s [1971] input models where i n discussed the p e r f e c t l y 87 where (in a q i s the rupees). cost of For Rs. 1 i n this from at a p r o f i t is Suppose a and s i n g l e equal w i t h to the respect units of f o r In p t h i s or i t s rather be the S's buys q p r o f i t is (p'~ p)q to S's that umbalakada at p r i c e RsVp- i s , = pq - ^ 0 0 and s i t u a t i o n r e a l i t y opportunity this of <_ q note should 5q w i t h w i t h <_ 3. In M M - 1 of q. that corporation p r o b a b i l i t y and t h i s other that have, f o r p o s s i b l e exporting production p a r a l l e l , M's or u t i l i t y c o l l a p s e , for r e f l e c t e t c . increment s e l l i n g M's .1 f i n d f o r e i g n as w e l l q .9 p r o b a b i l i t y too costs, rather the u t i l i t y p o s s i b i l i t y that, employment costs product Assuming We i s x i s approximately maintenance two is p r i c e has units u t i l i t y to S responsible has b e l i e f order s e l l i n g u t i l i t y . is M's of - M's view point pq imported S s u b j e c t i v e q the i f Corporation In when d i s t r i b u t i o n , c o n f l i c t = sold umbalakada equal supposition p r o f i t , ^ where Rs.~p Lanka. the to of can Hence Maldivian S r i under u n i t business. p r i c e to that t r a n s p o r t a t i o n , p a r a l l e l , umbalakada etc. each for incurred M quantity as exchange i t s requirements, p r o f i t . sets 1 Graph of 2 FIG. 4.3.5: g FIG. 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 i s of t y p e 1. 3 Vs. q . bargai ner 2 q 89 U Since the maximum admissible subset of U + = - oo < U + the + at that = region In 4 - 2 shaded V region q = 2 = + - -12 and <_ U 6q q + = = solutions s t r a i g h t l i n e s suppose S's y i e l d i n g the V amount by we the two when i t c e r t a i n i s of 4.3.6 q = the are before nonnegative f o r i n the f i n d i n g u t i l i t y that we i t hence and now e a s i l y comprise maximizes U <^ = (p - + 1) < °° l i n e the points these minimum i s , demand a and the a l l i n are the S i m i l a r l y admissible ' f a i r ' be 2 a l l the quadrant w i l l f o r = on i n f i r s t <^ 2 + Moreover, maximum u t i l i t i e s a -1 u t i l i t i e s . the U points and + couples have the graph r e s p e c t i v e l y As 1 <^ U l i n e s f e a s i b l e . since 0 u t i l i t y that nonnegative and have s t r a i g h t since =.14 of of quadrant 1/3, procedure 1 f e a s i b l e h(q) + = a l l w i t h w i t h g(q). set .14. f e a s i b l e = q Figure s o l u t i o n s jf^i .16 i n 2q at <^ 3 f i r s t 8a. + 1 q occurs bounded 1) the f e a s i b l e the = p a r t i c u l a r (p i n V t h i s Hence the 0. = only U u q. of + of i s || region, seen the the to V ( p ) of that i t f o l l o w s under graph + occurs the region f i n d h U Vs (p, ||^» the V^. Now q) i . e . maximizes, U[.9(2-U) GNF 1 = U + [ . 9 V* + .1 V* ] = \ .9U(2 0 The graph of t h i s f u n c t i o n i s as i n Figure - + .1( 14-U)] i f U) i f 0 .14 < < U U < < otherwise. 4.3.9. Hence S's ' f a i r ' .14 2 FIG. 4.3.8: Utility - possibility is of type 2. set when bargainer 2 91 0.90 h 0.23 FIG. 4.3.9 : G r a p h of GNF Vs. U. P o FIG. 4-.3.10:5's demand function. function and M's supply 92 demand i n this Moreover i s t h i s e x a c t l y game. s i t u a t i o n i s what Notice achieved one that game than U type = i t is 1 the i s of uses the 8 f a i r < 0. It i n t e r e s t to him should i n t u i t i v e l y , component that, would | GNF^, - x ( buys q p ^ were even not games i f be U S any 1 pointed to do when 1 were of be S = u n i t M d i f f e r e n t out that this u t i l i t i e s U .07. = and Now a i n t h i s l i t t l e less demand j u s t i f happened M t h i s bargaining has and to i n u t i l i t y . to be of S i m i l a r l y , e q u i l i b r i u m Figure is may 4.3.10, only In ||v|| Q + 1 q the umbalakada = or 1 demand function of S, who i s , e q u i v a l e n t l y p = ^ 2 q + of M's product when p r i c e set by S is 2 be achieved Then ) that M's units w i l l employ way, responsible . + — this here f o r M contrast, note _ = P + giving demands r e s u l t _ S U two evident c r i t e r i o n q S q) 2. It p. (p, expect, the S's any w i t h would i n r e s p e c t i v e l y , mixed i s i n i f at supply the M now f u n c t i o n point i s s e t t i n g suppose since, a only the that of p = f(q) i n t e r e s e c t i o n responsible i n and (p , Q s e t t i n g the q ) Q the i n p r i c e and quantity. the c r i t e r i o n used by S was to maximize 93 FIGURE S demands only U = .13 4.3.11: (see GRAPH Figure OF GNF 4.3.11), Q VS which U i s q u i t e u n r e a l i s t i c . 94 4.3.4 DERIVATION We elements w i l l the OF now form THE GENERAL deduce of from GNF FORM the for OF form general GNF of GNF ft. It when w i l l fi be contains shown that, two i n general U GNF the d i s t r i b u t i o n EXTENDING over FROM Suppose types w i t h is type of E U + U + e being the 2 w i l l = L £n + | I V i f L 4 0 i f L = 0 L e R + , , L taken N w i t h THE be P space. l's 1 V l OF ••> + by P 2 TYPES 2 V the is subjective done OF point P * Then + + P h i s , the n V n i n two BARGAINER 2's r e s u l t ] L i f p r o b a b i l i t y steps. 2 adversary Bargainer n V^. to This stand p^, denoted [ respect NUMBER Bargainer U GNF ] L appropriate TO from + E | | V p r o b a b i l i t i e s i ^ i = expectation [ E + has n u t i l i t y , to L be * ° L = possible when he i s that proved { Li P D for U > 0, VX > 0, V V 2. l - ^ 1 > P 0,.-. V n i f n V > n 0 . Since the form of 0 the GNF when L i 95 L = we 0 i s defined s h a l l GNF carry to be as on l i m i t our of proof continuous U[p.. 1 1 only w i t h + f o r respect L p„ 2 0. 4 to L + 2 + We at are 0. p V n thus The L n ] when L r e q u i r i n g r e s u l t i s L -* 0 the proved by J_j "Mathematical axiom the o f Section r e s u l t w i t h a Induction" i s + we r^, ... + bargaining p^, game P(H) = think of i f P^ ^5 + information p r o b a b i l i t i e s • P -z 1 determining Figure f a i r u t i l i t y of We P k + l demand the Bargainer say that Assuming t h i s to 2 i n under . Now i f 2 has i n a k The the coin which a 1 i s confronted types w i t h tossed maximizes by p r o b a b i l i t i e s as before, w i t h w i t h u t i l i t i e s by w i t h he here s i t u a t i o n used Suppose confronted behaviour bargaining c r i t e r i a he i s 2. demand types coin w i t h i s consistency V^, P(H) = incomplete and ...,V^ Bargainer 1 i n k + l know the i s tossed. used As by behaves Bargainer the necessary 2 tossed c r i t e r i o n he the bargaining he + = p o s s i b l e ' f a i r ' l ' s types • a the n 1 suppose k Bargainer has i s 0 4 2 P has L enters _ 2 f i n d one k Bargainer k of f o r i f to he c r i t e r i a consistent of as a f t e r coin the w a s ;i t h e u t i l i t y I 1 We n e e d before s t o c h a s t i c a l l y V* P ,.T.V, i s , the r e s u l t order analyze Bargainer f a i r 4.3.12. demand tion. a _ some occurs, l That i n g e n e r a l i z a t i o n proved Bargainer to a Bargainer r^ s i t u a t i o n H where . which which t h i s k f o r order n = ..., vjpk i n ^ n i n r^, r^ P^T" can game invoking We a l r e a d y when bargaining U[r^ a 4.3.2. true p r o b a b i l i t i e s and by i n before here the we 1 the same case of of i l l u s t r a t e d would s i t u a t i o n Bargainer extension are Axiom use c a l l to V* w i t h i n determine the e f f e c t i v e incomplete before the c o i n way would he complete 3 by a informatoss have, i f information. i n c l u d i n g t h i s i s 96 second type of consistency then, before the coin i s tossed he would maximize, 1 = FIGURE 4.3.12: BARGAINER DEMANDS for U GENERAL > 0, 0, V^ ^ + l'S AFTER > 0 V* CRITERIA THE thus COIN PRODUCING FAIR TOSS completing the proof. SOLUTION Again by > U we c o n t i n u i t y . prove R e c a l l our c l a i m that we only have for L f already 0, the assumed, case L = 0 f o l l o w i n g 97 GNF Assume V i s defined measurable. Case i : Also = T(U on (ft, assume Suppose V F, T i s , + i s a {V (w), u + P), F e ft}, being measurable a and simple f u n c t i o n , where A P) B o r e l f i e l d , i s B o r e l continuous. i . e . n V = E V I i = l is the i n d i c a t o r T = U [ E V + 1 Case a i i : V u = u + + [ Suppose sequence that = = of l i m n n , of [ / v + A ^ P ( A . ) ] L 1 and I A We have already f o r L proved that, L ^ O 1 L d p . ] L r v E V i s i n c r e a s i n g V f i f u n c t i o n + e i where any B o r e l simple V n f u n c t i o n . Then functions < V — n+1 . Hence we know defined by 3 that on c o n t i n u i t y there (ft, of T, F, e x i s t s ft), such 98 T(U , + (V (a)), a) e + ft}, P) = T(U + , {lim = l i m T(U = l i m n U + V*(ui) , u {V*(OJ), , eft}, oo e P) ft}, P) 1 + [ E V + L n ] from L case (i) 1 = U l i m [E + V n + L n ] L 1 = But, since Monotone {V n } i we V, Convergence l i m n have theorem E V + L n = + [ l i m ( E n {V } n \ + L we E U v + V V + L n ) ] f o r + L f o r L L 4 L 4 0. 0. Hence by J the have L , ' 1 and i n turn we conclude that 4.4 The major decomposition component of problems, of and ( i i ) " f a i r " demand, information of c r i t e r i o n complete information. as U to a [E + AND this bargaining bargainer's the = SUMMARY objectives the T V + L ] L completing the proof. CONCLUSIONS study problem were under e s t a b l i s h a by Nash (i) to suggest incomplete c r i t e r i o n g e n e r a l i z a t i o n derived thus , to for the a a conceptual information for case i n t o determining of bargaining a incomplete s i t u a t i o n w i t h 99 We order to have f i n d maximizes Axiom s c a l a r Axiom by f a i r the 1:- A using bargainer's transformation 2:- In 3:- of The Nash i . e . Nash the w i t h a the of 3 axioms b a s e l i n e product class bargaining the p r i m a r i l y demands, generalized maximizing Axiom shown f a i r U his | | v | | or his below that demands, Bargainer does not under own vary u t i l i t y p a r t i c u l a r seems i n f o r m a t i o n , f a i r one may p o s i t i v e demands are found product. c r i t e r i o n that 1 f u n c t i o n . determining b a r g a i n e r ' s f a i r demand i s s t o c h a s t i c a l l y consistent. It i n . + demands opponent's complete + f o r given -0-1,is-not generally be a reasonable i n c l i n e d to value' use L 'f<3r L , = 1 . and i n 100 CHAPTER BARGAINING PROBLEM: AN 5.1 In t h i s bargaining, i . e . negotiations normative made of chapter are concessions bargainer a is However, c a r r i e d by f o r study is bargaining scheme e x p l a i n i n g bargainer's w i l l provide We s h a l l information A f t e r the (0, 0), are of as be dealing i n u t i l i t i e s to denoted by V(x). x no , as that For ambiguity general motivates a U(x), the discussion there To w i l l a the our by which to to s e v e r a l i n renormalized and s i m p l i f y c l e a r are the 2 our to introduce other that beyond how optimal give of b r i n g the x , n o t a t i o n a be bargaining computation. a proposed f a c t o r s one, but also i t schemes. incomplete b i l a t e r a l monopoly. c o n f l i c t p o s s i b l e we a no the bargaining context may or sequence c l o s e l y under of reached w i l l problem to i s p s y c h o l o g i c a l such the any model determine of i s of concessions that become s t r a t e g i e s 4, aim bargaining bargaining 1 agreement sense w i l l theory s h a l l point to agreement, use x i n place r e s u l t . cost bargaining about the approximates chapter are how consideration Bargainer i s It comparing w i t h functions i n e s s e n t i a l on behavior f o r PROCESS d e s c r i p t i v e strategy o p t i o n a l only new Instead underlying u t i l i t y . constructed a e x p l a i n b i d d i n g that introduced to i n d i v i d u a l s not c r i t e r i o n u t i l i t y the In 1. a propose r e a l i t y . proposed complicated t h i s i n or expected BARGAINING INTRODUCTION not out scheme the OPTIMAL attempts r a t i o n a l higher so do which bargaining o p t i m a l l y process one we 5 costs involved process. c.f. i n This Cross bargaining"'" cost [1965] might and and i t i s p r e c i p i t a t e [1969]. t h i s cost c o n f l i c t 101 even i n cases c o n f l i c t each of p o i n t ) . agreement) i n we terms u t i l i t y = there is the permitted of o f f e r s concede prepared point) (cf. l i m i t of u n i t s , i n the a o f f e r s x range to be his the c o n f l i c t s i t u a t i o n . . make e i t h e r f u r t h e r c a r r i e d out. C o n t i n i [1968]) i n bargaining the s o l u t i o n to i m p l i c i t l y Then the However, the agreement be sets as sense must to out than i n Chapter ensure each x ' s , i t , 1, that bargainer l e a v i n g the announced that be not there reached upper f i r s t i . e . , is at One make continues t h i s imposed. an c a r r i e d o u t l i n e d order (point Cross rather f o l l o w s . n o g i t i a t i o n s , be which In accept process concessions, "imposed" out can by model of the opponent. opponent The terms i s context, the power Following p o s s i b l e c a r r i e d c o n f l i c t . the r e s t r i c t i o n , of than the demands) our t h i s The reached w i l l i n or exchange to x's w i l l b e t t e r n e g o t i a t i o n . due as the bargaining. making H i l d r e t h ' s equivalent process by b e t t e r determines cost, namely i n (points etc.) f u r t h e r w i t h i n preannounced costs to ( i . e . demand prefers be time, achieve e f f i c i e n c y his x. cannot to involved these he u n i t agreements b a r g a i n e r ' s able of s t a t e of the example, bargaining bargaining of to (per payoff loss f e a s i b l e bidding p r e c i p i t a t e agreement time no unless c o n f l i c t be that (demands) otherwise not w i l l For one The cost q u a n t i t y be are lower p h y s i c a l w i l l choice is he u n i t s . i s a The assume of there This bargainer. [1965] x where i s a new w i l l l i k e i f bargainer o f f e r have t h i s . e i t h e r threats to If bargainer (at the r e s t r i c t e d no p e r i l p r e s p e c i f i e d of causing However, u s u a l l y the l i m i t time number on or or a presence of bids. In we assume opponent's d e a l i n g each w i t h the bargainer u t i l i t y which bargaining has i s a problem s u b j e c t i v e unknown to under incomplete p r o b a b i l i t y him. As i n f o r m a t i o n , d i s t r i b u t i o n bargaining on continues, h i s g e n e r a l l y , 102 each to bargainer him at each In Section optimal U n t i l bargainer r u l e which are b e t t e r h i s 5.3 we f o r then, does so his able glean x STRATEGY of x, i s i s not f o r f o r other THE the the the costs x, i s or c o n t i n u a t i o n i s or known or sequence. of a l t e r n a t i v e . unknown Assume X^, (XQ, X^, t h i s sequence to he ). the bargainers makes the he i s which f i r s t Subsequent s i g n i f i e s by according In to X, set We of assume (hence, The we the a proposed the h i s some values x, the may x end to which c o n f l i c t at one other agreement to a over r u l e o f f e r being an that x the s i t u a t i o n sequence of range i n any case according stage of i n s t a n c e , opponent's o f f e r to continues process concedes accept handle a l l f o r c o n f l i c t at so, bargaining p r e f e r s h i s a PREPLANNED bargaining x. ignores only to doing l a t e r The As 1 optimal problem where i n o f f e r , to an bargaining) r e p e t i t i o n w i l l i n g IS process Bargainer opponent, and above. p r e f e r s of the i n d i v i d u a l s f o r bargainers Suppose h i s as i n use steps. STRATEGY two values ( c o n f l i c t other a v a i l a b l e i n d i v i d u a l s e n t i r e l y . two bargaining). a l t e r a n t i v e One i n bargaining of concede problem denoted the 2. i t both concessions about of any 1. h i s sequence One reached, information t r e a t o f f e r v a r i a b l e a l t e r n a t i v e l y suggest ways: makes Consider t r a n s a c t i o n the we OPPONENT'S MODEL: they 5.2, approaching throughout a f f e c t e d by two WHEN THE the f i x e d bargainers of AND value a v a i l a b l e the problem when when Section the by the i . e . i n s i g h t s OPTIMAL a i n c o r p o r a t i n g solve opponent's 5.2 NOTATION i n w h i l e ignores to s h a l l i . e . complicated problem determine by bargaining, more 5.2.1 p r i o r s stage. s t r a t e g i e s fashion. one updates o f f e r i n stage. 103 Bargainer 2 has a 2's o f f e r subjective sequence p r o b a b i l i t y the p r i o r d i s t r i b u t i o n i = 2, 4, 6, may be due to his r e l a t e d by Bargainer 2's examples 5.1. EXAMPLE suggests accepts otherwise are known to x's, 2's he the x. of x^, and these the i s not x,_ ( 2» x x ). 4» x^, about i n x^, )• ), information his which information {ir^} on Denote by TT^, on (x , u t i l i t y may such x^, 2 opponent's case Bargainer be x^., function induced u n c e r t a i n t i e s . ideas. s a t i s f i e d favorable o f f e r s . o f f e r In i n i t i a l l y XQ w i t h i f this XQ. any and Bargainer Bargainer of the example XQ way his 1 offers = x^ = 2 i s x^ favorable = ... 2. the Suppose values b e l i e f s f i r s t decreases accepts on incomplete f u n c t i o n , he a l l (x^, offers(demands) i f Bargainer U(x) the 1 by (x^, 2's c l a r i f y Bargainer proceeds, derive this help refuses 5.2. next to Bargainer and exact w i l l various u t i l i t y given information subjective Bargainer EXAMPLE x^ incomplete parameters few of denoted d i s t r i b u t i o n Bargainer or A i s 1 conceeds i n such a i n a s p e c i f i e d way, say f i r s t o f f e r Bargainer c e r t a i n 2 knows parameters p r i o r s , from which his y i e l d s that l i n e a r l y . more Bargainer As than l ' s In a n t i c i p a t e d bargaining he expects p o l i c y l a b e l l i n g U(x). t h i s subjective d i s t r i b u t i o n s but case on from not he the would such unknown parameters. A^(I|J^, Let i{i (JK) represent, Bargainer 2 = at (x , Q the x 2 , * 2 compactly, i t h . stage ^ ) , (("j the of = t o t a l (x x 1 > a v a i l a b l e bargaining, 2 j _ i ) > a n d information and <|> l e t to having been ) 104 observed. or i t such may may as involve a l l i n j l?i^ 2i+l^ from Bargainer <_ 0 times. For and Bargainer 5.2.2 Given game, a l l Cj be or s u i t a b l e f o l l o w i n g the s i m p l i c i t y on = 1, a n that d cost cost a l l j = 2 x^ derived = ^ of = = - by ... = a 2 B 2 w i l l represent 0 — Let p r o b a b i l i t i e s , A^, new that o f f e r , ^» > + ± 2 2 is ... a f t e r w i t h f o r b i d d i n g independent C n > Bargainer of j h i s 0. 0 and j ) For Q . Bargainer 1, x x^. given f a c t s e t c . * K P(U(x _^ ^) bargaining 0, o f f e r s , make 1 from c o n d i t i o n a l e and ib^ information 2's = i n c u r r e d his f o r the x of p o s s i b l e 5.1, 2 i - l ) assume C.,.. 3+1 = w e l l , d i s t r i b u t i o n s example 3, t o t a l B^ x^ values 1 and r e s p e c t i v e l y . each FOR stage (i) i n COMPARISON 5 should continue bet, < 3 - CRITERION A_^, j C. contain Bargainer *21+1 i n as c e r t a i n p r o b a b i l i t y instance, f o r now would denote observed information and accept For the about 5, w i l l 2, At A^ s u b j e c t i v e that From 5.1 K h i s 1 Let o f f e r s a t t i t u d e Q±( 21+l) r e s p e c t i v e l y . U(x^) l ' s 3, and x derived 1, of a d d i t i o n a l example = consist some Bargainer instance, for merely i , he of case concepts he before BARGAINING bargaining, accept bargaining OF and decides the x,^ ( i i ) to B two by should be what these has o f f e r e d continue considering 2 STRATEGIES b a s i c h i s problems opponent, ^21+1' bargaining. problems: We f u r t h e r . t * i e define q u i t n e x t the the 105 The Bargaining <f> = {X^(AQ) , decide to Sequence, x^(A^), continue E = {E (A ), E (A ), 0 <^ <_ 1 1 B„ w i l l cease has continued We ceases E to p a i r , say that s i t u a t i o n with convention nonpositive The B 2 [V(x E) ) i s then G(<{>, E) + Q 1 [ V ( X E_. X 2 i + l E^ ^ s every ^ s defined represents stage f o r n e i . x Should ° f f t e B 2 - r functions w i t h j , stopping stage which gives V(x ^) that (tj>, o v e r a l l 2 of x e given the on and ty 2 c o n d i t i o n a l both and A. p r o b a b i l i t y that bargaining stage. at i f i , functions rules, i , he him the 0 and 2 ± B^ prefers and s e l e c t s higher assume that, as a c t i o n , his u t i l i t y , accepts of zero out i . e . (offered stopping whenever he by of 2 the chooses B^) u t i l i t y B to the otherwise continuation u t i l i t y . p a i r • B 's (cf), that at unimportant c o n f l i c t of where x_^ stage ...} 2 j , bargaining, the 2 of w i t h sequence A bargaining up at E (A ), a l l avoid a v a i l a b l e : 1 for . ••} bargaining Rule, sequence A 2 Stopping 0 : x,.(A ), The Q <j> E) expected e a s i l y = 4 w i l l 0 2 + (1 - 0 c a l l e d gain, expressed /./V^( )E ) E be + (1 before he "bargaining has p l a n " commenced or " p l a n " bidding using of a as - E ) Q E ) [ V ( x ) 2 the 5 + [VCXj) Q [ 2 + Q 0 [ V ( X 2 ) E ] ] . . . ] dTr(X 1 £ + (1 - | A ^ dv^ E ^ | A > 2 . plan 106 provided this r and V(x n zn u t i l i t y C. „ 1-2 ) exists*where = V(x from )v 0 2n x^. 0 - This i n addition to a to V(x.) 1 C / f o r n-1 is the defined n because b gain = i f by 1, i V(a.) x v V(x„ ,..) 2n+l J 2, i s 0 3, ... even, and = B2's has i is C n expected to odd - n M is B2 i f V(x~ )P 2n+l pay he a cost would of be 2 l i a b l e (the he cost p r o b a b i l i t y is at an G.($ , 2 V l = J ( ^ and B l accepts intermediate E ) + of " 1 } ^ [ ( X 2 j ) E 2 i + 3 a x^). stage / . . . / V ( x V l to + j ) of V(x^) S i m i l a r l y j is (1 - V l + gain [ given - - - ] p r o b a b i l i t y o v e r a l l expected gain when by, ^ ) [ V ( x ] his w i t h d 2 ^ j X + 1 ) 2 j + + Q 2 l / j [ V ( X i A ) d ^ 2 j + ( ) ] 2 X 2 j 4 l V l + ) (5.1) where w i t h <j> = J $° = ( E) If 1. A i > + f o r there Max x 2 j + 3 ' E° plan = A, E) some j ( $ , E ) , is n d said £G^.((f) , J e x i s t a = E j+1' " ' ^ °' = 1 ' 2 ' E. /V E ) (a,b) ' (<j), (cf)- , 3 conceivable avb. = j and i f f i n e q u a l i t y a l l 2 <)> D e f i n i t i o n : (<(>, x J to be E ) for such that 3 uniformly a l l b e t t e r j = 0, 1, j (^ > Z) 1 than 2, a plan w i t h s t r i c t . (<))*, E*) then ($*, E*) G i s said 2 to be G.. (<f)* , J a best E* ) J plan. for 107 5.2.3 BACKWARD We w i l l determine f i n d whether bargaining no We can w i l l r e s u l t the f o r same problem, best that He bargaining are zero .1. then or make p o s s i b l e . His otherwise, For a show Ferguson [1967] that p e r i o d f o r of general the 5.1 the of the the and t h i s r e s t r i c t i o n w i t h the hope truncated one when presence of problem bargaining are a l l bargaining thus that l a s t , problem. subject X In to N us whose the the that the i s costs providing w i l l 3 w i l l an best plan truncated r e s t r i c t i o n 2N 1 + process o f f e r no -> ° ° , the nontruncated BARGAINERS' expected by the plans f o l l o w i n g new a r t i f i c i a l l y , impose N truncated which stopping because discussion when w i l l a we Induction"'" N. happens i n Backward truncate, i . e . We bargaining a N, approximated X has F i r s t stage what applying w e l l FIGURE stage. by plan. a f t e r *1 the plan be a v a i l a b l e Suppose (BIP) integer n a t u r a l l y the as the a some w i l l usually i s is bargain large. bound bargaining at s u f f i c i e n t l y upper PROCEDURE i n v e s t i g a t e problem give a i t problem i n d i v i d u a l general INDUCTION gain more Backward X OFFER has choice 2N+1" w i l l been of "*" n be bidding SEQUENCES continued actions. t * i -*" e a t t ^^ 2N+1^ x i s Induction allowed. i n a e up He r c ^ a s to can e » ^ t B2's e i t h e r w a c cease consequences o c e Furthermore s t a t i s t i c a l Nth. P t s i f s e t t i n g l t : > he cf. a n d does 108 make a new o f f e r to the amount in the event he w i l l have that he does to undergo a he has a l r e a d y absorbed. a new amount, denoted o f f e r cost (C^ - C^_^), The maximum N E„(^„, N N by i n a d d i t i o n expected <b„) , N gain i s (5.2) On an end, the h i s a v b = the b i g g e r o t h e r hand expected Max{a, i f gain b}). he accepts w i l l The BIP be ( V t e l l s x 2 X 2 N us JJ> ) = here thus V ^ X b r i n g i n g 2 N ^ to ^ ® take b a r g a i n i n g Sj-1 ~ the the ( a c t i o n to ° t a t i o n : n which y i e l d s N stopping and a c t i o n otherwise choose of one o f f e r from w i l l a V ( x be new the 2 ^ ) ; taken. amount class of i n In which case these other words, y i e l d s e q u i v a l e n t q u a n t i t i e s stop (5.2), are ^( 2^^ i f the i . e . the N — x l e a v i n g p o s s i b i l i t i e s , e q u a l , ^ a n opponent more d to p r e c i s e l y , otherwise, where Now i f y H ^ ( ^ , B2 i s c o n s i d e r a t i o n , = at v ( x (N again - he 2 N ) 1) v th can stage, e i t h e r (5.3) y E ^ , w i t h stop or B l ' s o f f e r o f f e r a new X 2 N _ u n d e r 2 amount, x 2 N_I* ~*" N 109 the l a t t e r case bargaining, fore h i s and x„„ » E{H^(X must 2N-1 expected 2 N , u t i l i t y * ^_ , ± be chosen so i s ( v 2 >'^-1* N _ i ^ x N i f l f B l B 1 a makes c a c e P t s o quits r new o f f e r . T h er e - that, 2N-1 E { < ( X W 2N' V l V i } N is achieved. B2 w i l l cease bargaining a t this stage i f f N >_ E j ^ , where ,N V l ( * H ' W Hence due to = B2's the BIP ^ 2N-2 ( X maximum may b e } v ' W V l o v e r a l l i n d u c t i v e l y ' *N-1 expected defined } gain .th j a t stage, given Aj by, (5.4) 1 1 3 2-1 J J 1 where, E.(.., *.) We - assume V^o,,,.) that this + maximum l. j l (-2 j : + i s a t t a i n e d ) E { H + ( i n X 2 j + X. 2' V Hence, W we ' V (5.5) conclude, 110 P r o p o s i t i o n 5.2.1 (i) E - {E , E Q E 1 > £ N > N + 1 } where i f 3 V(x ) > 4J) 1 E*J(ifr , <(. 3 3 — ) 3 (5.6) otherwise i s a stopping r u l e ( i i ) where and § x„ . x 2 Theorem N + Proof; Let (cf>*, R e c a l l i s .th j that stage i s bargaining I*) c for 2N+1^ X value a the problem ' i n equation bargaining plan truncated given by (5.5); sequence the BIP i s j given - by a best (<f>, E) 0, 1, the plan . . . N - l BIP. f o r the problem. be B2's of BIP, maximum (5.2), bargaining plan. at A the •••> the y i e l d s 5.2.2. by {x.^, y i e l d s n 1 truncated — given the plan o v e r a l l , . . bargaining . given by expected i s , BIP gain and u t i l i z i n g be (<|>, any E), other when he Ill G^* , z ) 3 E N = 3 + (1 By V ^ [ ( x 2 N d e f i n i t i o n E^ >_ V ( X i t follows ^ 2 N i - / . . . / v ( x 2 ) N Z + 1 ) + l of ) 2 ] j ) s + j - - - ] d u (l ( 2 j X the random whatever be X 2 N + . 1 N + i ) 3 + (1 j - 2 | A 3 ) 2 ' - - j d + l ) 1 T ( + Q j t V C X ^ ^ ) 2 N X j i n Now f r o m i V l ' ) (5.2) i t the d e f i n i t i o n i s clear of that i n (5.3) that, Implying G.U , + ^ ) [ v ( x v a r i a b l e ( 1 " V ^ ^ 1 thus - ( X 2 N ) E N + ( N H " 1 f o V E r N X 2 N ' that, </.../ + v(x .)z. 2 (1 - S N _ 1 + ) [ (l ^ ( X - ^ ) [ v ( x 2 N - 1 ) + Q 2 . + N-1 1 ) + ^ • • • ] Q d j [ v ( x ^ 2 j 2 l X 2 j + + ) z 2 A ) j j + 1 - - - d 7 r ( X 2 N ' V i 112 Again v ( x 2 N + 1 ^ above f u r t h e r reduction i s ' a l l j = Yet, <{> = <)>* 5.2.1, t o problem . .. each o f t h e above proving E* found complete can be r o ( m 5 ' 5 ) a n » d therefore given of terms E. + i n t h e above we, c l e a r l y , (1 - E.) E end up w i t h < N i n e q u a l i t y . H is Max xeX When t h i s t h e i n e q u a l i t i e s , N as i n e q u a l i t i e s consequences that a best (<!>*, plan our analysis £*) w i l l o f t h e i r i s a c l e a r l y d e f i n i t i o n best a theorem given bargaining s t a t i n g a r b i t r a r i l y w e l l e q u a l i t i e s i n when P r o p o s i t i o n plan. f o r t h e truncated w i t h become that problem, t h e we now nontruncated f o r s u f f i c i e n t l y G_. j and o f G.. denote t h e truncated ( a t N) seen that -N G^ = Suppose that there We h a v e by BIP. t h e maximum o v e r a l l expected large gains and t h e nontruncated N * G_. ( <f>, N j N e x i s t s * j N by t h e E a ), plan where (<j>°, problem which i n c r e a s i n g bounded V(x) = o r m < i n that 0 0 , achieves N . B 2 ' s In u t i l i t y t h e sequence B2 a t ( * * tj), E i s t h e E°) N ) f o r t h e -N G^ f a c t , o f problems, N -00 c a l l y taken N. approximated r e s p e c t i v e l y . general steps one. stage plan V(x„.) 1, thus Let the < J E = f o u t i n d u c t i v e l y E ) and truncated N - 1 E 0, Having need A ' t h e number c a r r i e d j E N-l reduce G.(cf> , for - 1 N 1 f o r a l l i f {G ?} J . we assume f u n c t i o n 1 j i s w i l l Obviously that bounded X , G^ t h e range above be bounded i s from so monotonico f that above; i . e . , x , 113 G ° < G j - 1 j < G ... 2 - < j m . Hence l i m - G^? exists (and i s f i n i t e ) . Our question J N —CO i s whether LEMMA ^ i ^ X this l i m i t 5.2.3. i ^ "'" S Let ^ a u n c t : i s {X ' - o n the } be °f n same an i x as G.. for i n f i n i t e such a l l j . sequence that 0 of random <_ ^ ( x ^ ) ! . v a r i a b l e s . w i t h 1 p r o b a b i l i t y 1. CO If E{l/£ i^.(x 1 1 ) } = 0 = 0 (true, i n p a r t i c u l a r i f ± N E(l/i|i.) Z l i m + 1/v J f o r n y. x , > 0 — because by n/(y cross + 1 Y + 2 • • •+ m u l t i p l i c a t i o n ) y ) < + ' " + n then, N l i m E N-*» Proof: {n(l - i|i.(X..))} 1 1 = 0 1 Write, N I 0 N <_ E { n ( l 1 - \\>. 1 (X,)) } In ( 1 - ; M X )) 1 E{e = } 1 N -E x < < E E e 1 ( X ± 1 N £ i 4 * ) because l n ( l because e *.(X.) i i- ^ W ' y - < a) — < for - a for y > 0. 0 £ a £ 1 114 N Taking l i m i t s as N , 0 0 we now deduce, l i m E{n N-**> to be (1 - 1 if).)} = 0 as was 1 proved. Theorem 5.2.4 : If E{l/E P ? ( x „ . , .. ( X „ . ) ) } l 2x+l 2x = 0 (true, i n p a r t i c u l a r i f N p° X 1 f i o o p r o b a b i l i t y n or, i f E 0 0 G j (4> o i , o i I ) e x i s t s G as N -> and ( ^ § Proof: plan E O a l l , 0 0 •vr by E ( l / P = p ) 1 1 ( N (Z O o J where , E ) <j>° = o N (J) , o E , N E * ((j>°, i s Let O E,, 1 expected optimal N ( V , j Bach ^ ) E°) " G i s convergent), and i f ( j , + o j ( E o j an o p t i m a l by the } s o l u t i o n to the general problem "NT t r u n c a t i n g A — , i t h e n i-f o r i s 1 ) a p l a n ( x ° , ( t h i s . . . , gains x ° , i s E°) ($°, ) not p l a n . , i . e . o E.). x E . „ x=N B2 and at BIP, E° n e c e s s a r i l y N 0 0 O nontruncated ... at E„ -, N - l of given (j>° stage some (E°, the Consider j t h . A f t e r N = to the E ° , same A_ ( x ° , the using problem . . . ) . x ° , x N * )) 2N I) between truncated we E + d i f f e r e n c e c a n c e l l a t i o n s Consider ,N * ( <(> , as t h i s truncated get , p l a n at N. the obtained A N < ^ the and over- the 115 0 < G.°(* 3 , O J G E° ) J ( V , N J N E [ V = E° ) J 1 ( 1 - E°) Q ° { [ V ( X 2 N )E° + (1 -E°) i=-i o o o i * < W + N I E + ( - W[•••]...]> 1 + V ( x ° E° ) + ± N + - t i v c v J / i ) } ] 1 x=N N-l = E { [ n ( l - E°) Q°] h ( x 2 N + 2 , x 2 N + 4 , ...)} i=j say.. Since m = Max V (x) < i s 0 0 an upper bound f o r t h e maximum u t i l i t y B2 xeX —CO can conceivably and i n p a r t i c u l a r 0 < a n t i c i p a t e G ~ ( 4 ° , 3 j h <^ m E° ) J from . G t h e p r e v a i l i n g Hence N 3 ( V 3 f o r each , V j ) t r a n s a c t i o n , we have G ^ <^ m j , V < mE{ i = j ( 1 - E°) 0 ° N-l < — mE{ n . . i=3 Q?} x since 0 < — E. x < 1. — N-l < mE{ n (1 - P ° ) } i=3 But the f i r s t hypothesis o f theorem 5.2.4 enables . N - l giving E{ II 3 (1 - P ? ) } -> 0 as N -»- ° ° , and therefore Lemma 5.2.3 to be used, 116 G ^ ° Hence given e , E° )- G^(V , 0, ] N j > j V J such q ) j -> 0 j V )} J j = {G~(<f,° , j E° ) - G*?(V , V j ) } + _< {Gj(<i) , E° ) - Gj( <f> , , O J j J J N 0:i N E ° 3 ) } {G^(V , J Our (<J' - <_ e from E«) i s ; claim S follows <_ e from from (5.7), a :.= t f o r N _> N . E ° j for the ) GJ(V , J - *) i s a V )} truncated problem. . ' .r?.::i*" 1 r (5o7) i - P a assumption, "l' ... _ Q ' E{l/E P..} -i-jr c l a i m seems very . reasonable, since 1 N E P. + 1 °° i n p r o b a b i l i t y is the i s s u f f i c i e n t f o r i t to be s a t i s f i e d . R e c a l l i n g 1 that P B2 i t , i s p r o b a b i l i t y i n t u i t i v e that that any BI would e f f i c i e n t accept plan would the i t h . y i e l d o f f e r P° 1 made by i n N p r o b a b i l i t y , thus implying E P. 1 shown i s that, approximates makes t h i s when the N general approximation is exact, truncated optimal e s t a b l i s h this under the proved hold for p a r t i c u l a r to a 0 0 i n to p r o b a b i l i t y . Thus, what we have 1 s u f f i c i e n t l y one. solutions t In f a c t , thus the f o l l o w i n g l a r g e , presence p r o v i d i n g general us of a truncated cost of method nontruncated reasonable case: the assumption problem bargaining f o r "A" u s u a l l y f i n d i n g problem. We which j best t h i s . 00 The a l l ~ ' N (<f)*,E because plan oecause (5.7) N - - that, E ° )- G ^ ( V , 0 <{G~(t°K as s h a l l w i l l be 117 (A): Max x„ . for a l l j T that 5.2.5 function EV(x) < maximized Proof : at 1 ) P + j (1 Let X f(x) . Let If the = y Q Suppose g(y) and, _ j : °° . g(y) is 2 - P..)E{V(X 2 j ) IA _ >] j 1 < E { V ( X 2 j ) IA^} • P r o p o s i t i o n density [ V ( x eX f = V(y) , V(y) a random v a r i a b l e having be s t r i c t l y i n c r e a s i n g V(x) a the p r o b a b i l i t y f u n c t i o n function P(y > then is be X) + EV(X) V(y ) < EV(X) Q defined fy on f(x)dx (- + P(y < X) (and so ° ° , °°) ; EV(X) g(y ) Q < EV(X)) then f(x)dx therefore, g'(y ) = v'(y ) D o f(x)dx + f(x)[V(y Q ) - EV(X)] = 0 such 118 Since a l l f i r s t have f v ( y o ) " three EV(X)] Hence a b s o l u t e l y t r i v i a l l y Theorem i f o f < as was t o be i s a random from t h e above subset G ? ->- 1 > N o , almost surely E { V ( X . 2 we + 2 under given by Proof : and (A) t h e BIP Consider i t - then from N 1 ( A ) , l i n e i n p a r t i c u l a r , and X,^ i s an t h e assumption A . j = 0, . . . 1, . I f , _ G' = plan C. - C l _ f o ra l l (5.8) r N OO G^. f o r a l l j , i . e . t o t h e nontruncated problem truncated a t N ( Q A 2N 2>I V + and (5.8) > N . q From C (with Q ^ Z ) problem. " N we have N <j> , that, E { V ( X (5.3) t h e r e a l < 2 an optimal follows must proved. f o r each V ( x . ) t h e bargaining E and i s we have ) | A.} assumption a r e p o s i t i v e , 3 -N then, expression 5.2.5 G? 3 i o f v a r i a b l e , P r o p o s i t i o n Suppose 5.2.6 0, X continuous follows terms p r o b a b i l i t y o n e ) , (5.2) 119 H N = ( = V turn from 2 N > X ( V ( x * In ( (5.5) < 2 X N 2 and 2 N - 1 X £ ) v N > ° V " 0 consequently It (A) } V - C ^ ) we have, E N K follows Continuing N - 1 ' i ^ V ^ V i and C " 1 from t h i s V i (5.8) process, N HJJ_^ that, i n d u c t i v e l y , we ^ = x 2 N - 2 ^ " N R\ obtain, ^(x,^) = -N f o r i = N, N - l , N . This implies o G.° 1 = G. 3 f o r a l l " each j . Then since G. G. as N -> °° , we have thus ending the The > N and o f o r N G. 1 1 N = 3 G. f o r each j , 3 proof. l a s t i f theorem p r o p o s i t i o n that from a r a t e from B2's stand w i l l surely cease bargaining e s s e n t i a l l y c e r t a i n point stage becomes before states onward, smaller that the B l ' s than stage. i n t u i t i v e l y B2's Hence obvious expected rate B2's of concession cost, optimal B2 120 bargaining usually plan the continue case, of the each q u i t s made i n always a STRATEGIES cost BY is truncated attached BACKWARD a way an analysis of games the opponent i s exchange; also ing s i t u a t i o n , as to the when, as d e c i s i o n INDUCTION that to gives the i n f a c t , he be i s to PROCEDURE of that i n minimum could his opponent has the elements involved even from though t h i s being also found own of e x a c t l y assume i n at The that being stage, r a t i o n a l l y , Such Morgenstern's can amount t h i s computing the (offer) bargaining. about V(x) i s expect his the such b a r g a i n as incomplete. grounds n o t a t i o n i f theory bargainer, information on demand behaving t h i s that to d e c i s i o n bargainer information view out. computed each allowed new the and from bargaining, a f o r determined d i f f e r s of makes Neumann are are bargainer, complete h i s p o i n t gain, amount We he s t r a t e g i e s von expect stage s i t u a t i o n , given optimal s e c t i o n bargainers every expected a s t r a t e g i e s t h i s o f f e r , r a t i o n a l l y . j u s t i f y safe at c o n f l i c t that using i n both that, the p o s i t i o n and now opponents of s p i r i t function) we means the i f as o v e r a l l also bargaining t r e a t e d much h i s h i s i s (about theory, as favour our behaves behaves u t i l i t y adapt i n [1947], B2, game i s opponent say B2's i n learns i n B l ' s problem This that It assumed The one bargainer maximized. from p l a n . bargaining h i s we s t r a t e g i e s . such assumes i n i s 5.2 previous optimal a f t e r i s Section p r e s p e c i f i e d that or there problem INTRODUCTION a use nontruncated BARGAINING In by the bargaining. 5.3 5.3.1 for that used i n B2 t h i s As must 121 section except B2's i s the same f o r the elements stand s o l u t i o n point f o r need to know computation, and and B l Since is the found how B l on on he B l ' s c e r t a i n cost to as as on B2 has-3a derived problem f o r i s the same from him; as about the for from of B l ' s each B2's . B2 w i l l copy This p r i o r p r o b a b i l i t y updates proceeds. continuation tastes p r o b a b i l i t y x. B2 B l ' s b e l i e f s , s u b j e c t i v e bargaining r e s u l t other, cannot labeling^. U(«) a the each B2 f u n c t i o n , b e f o r e , B l depend knowledge suppose parameters model way. lacks u t i l i t y the s t r a t e g i e s s t r a t e g i e s . usually d i s t r i b u t i o n s , t o t a l his 5.2; consider optimal s i m i l a r therefore is We strategies because U(x), Section f i n d a computes We i n above. to i n b a r g a i n e r ' s d i s t r i b u t i o n s denote explained d i s t r i b u t i o n p r o b a b i l i t y adopted attempt however, d i s t r i b u t i o n that each preferences. p r o b a b i l i t y as these Let of D_. bargaining up th to his j stage; B2 f i n d B2's has 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 on D_. too. We now procedure, 5.3.2 as r e l a t e d 5.2, imposing the cease plan using the backward i n d u c t i o n PROCEDURE denotes, s t r a t e g i e s . f i r s t truncate r e s t r i c t i o n Suppose b a r g a i n e r ' s (^) B l ' s Section e i t h e r INDUCTION ' c u r l ' to bargaining before. BACKWARD A optimal that bargaining o f f e r is sequences bargaining and i n t h i s In order the c a r r i e d i n B l ' s elements approach at some bargainer out shown accept to process n e i t h e r are s e c t i o n , to o f f e r , t h i s can by 5.1. x N, bargain N 9 N s p e c i f i e d or as stage i n thus beyond B2; At , yet problem i n t e g e r stage Figure not stage N. the N c o n f l i c t B2 can or e l s e 122 he can make s i t u a t i o n a new we bargaining had or demand i n for B l ' s section bidding once consideration. 5.2, more we conclude according as N i f v R e c a l l i n g that he ( 2N^ — x w i l l E N N' the same decide ° R N O T to ' I , cease * E „N = E: 'N otherwise, where - E* Max , N H^j = and ^ ( x ^ w i t h V ( x 2 N + 1 ) = ^ O K x ^ ) > 0) V ( x 2 N + ) 1 - C N 2N+1 '^" V(x ^)v N E^ 1 2 i s his maximum expected gain once he has reached N th stage. Next for X suppose consideration. 2N-1' ^ a a l t e r n a t i v e . guess how stage N make a he To B l c w i l l P t e w i l l N ) x 2 N+1 > > my [ So B l acts as 1) i f aware that i t only i f y i e l d s a the t 0 i t t * 2 N + 1 i e and I x 2 t s N i f + 1 t a S e B l as » ^ e he follows: " i f ) | . U ( x only 2 N + 1 ) any he believes) B2 2^_2 other needs B2 to B l at were to V ( x ^ ^ ) 2 accept + the r e s u l t . i f > 0} - [C^ - C ^ ] ] v X that x demand point maximize to new B2 view ° u l d a than B2's w o f f e r makes event, d e f i n i t e l y have and B l ' s more this From reason w i l l x ^ W x E e case of strategy. n having i n p r o b a b i l i t y his o f f e r (not stage here Max X 1. - r a t i o n a l l y ' ' w i l l accept 2 (N is compute U ( x „ ) V ( x at c a l c u l a t e behaving to is B2 c new o f f e r subject And he knows his u t i l i t y U. 0 123 Therefore, V(x2 ) = 2 N = to so make my [U(x2 ) - N e U( 2^[_i) x B2 made, to v the E new o f f e r , maximum gain D |v(x2 ) > N * JJ computes N I by must maximize making . X^] a Thus new my U(x2 ) subject N o f f e r maximum to is expected gain i s P r e d i c t i n g the B l p r o b a b i l i t y s course that B l of a c t i o n would at accept stage X t h i s N 2N—1' ^ be, P N-1 ( X 2N-1> p r o b a b i l i t y = c o n d i t i o n a l s u b j e c t i v e T P (E^ N r * = that Q , (x„ ,) N-1 2N-1 H a X way, and am and Max x I ^_ N E^ i f = ( U B l H??) ; N ( X 2 N - 1 would each } V = instead of these p r o b a b i l i t y make a new o f f e r p r o b a b i l i t i e s d i s t r i b u t i o n s is given to be found A^. by Hence B2's at his til (N - gain 1) stage, w i l l E H N-1 his = N-1 \ m = 2 N - 1 maximum ^ ( B2 makes a new o f f e r x 2^_i» h i s m a x i m u expected m be, X and i f x 2 N - 2 subjective } { ? £ <*2N-1> E W E { H K - 1 } } K o v e r a l l V W + N-1 p r o b a b i l i t y ' expected W h 6 r e E ( gain H N ( X considering 2 N ' d i s t r i b u t i o n s . «Vl' a l l W Continuation p o s s i b l e ± of S f ° t h i s U n d actions U S l n g process is B 2 ' S y i e l d s 124 i n d u c t i v e l y H B2's ( i K , J «>.) J = (i) V ( x ) = V ( x ( i i ) E * ( * * ) N J where maximum o v e r a l l 2 j f V(x E®(ib )v J 2 = ) J - , gain <(. ) ; J J j < at N s t a j e - j , v i z l (5.9) C Max X expected { v ( x 2 j + 1 £ 2 j + ) 1 + Q j ( « 2 j + 1 ) E { H » + ( X 1 2 j + 2 , <f>. K (5.10) ( i i i ) v ( x 2 . + ) 1 = (iv) i s the is the = p r o b a b i l i t y (v) Q . ( x 2 j that + p r o b a b i l i t y (vi) V ( x E N 2 = 1 ) that Max x 2 = j e X 2 j + 1 ) l * ( B ( x B l P. 2 j + 1 ( x ) = would F r , ( ^ B l + 1 ( X 2 2 j ^ + + .) Z J - ( X i accept j + 2 2 x j + 2 , * a new .) 2 j * . f + 1 ) ) 2j+i = - D . | V ( x 2 C. , w o u l d make [U(x ) 1 > o f f e r A.] 2 H ^ a t + 1 ( X h i s 2 j + 2 j , stage. + 1 )}} 125 ( v i i ) X = 0 v Max x ( v i i i ) At t h i s stage complete very r i s k s o f f e r s so for that B2 to We "found B2 x 2 j + i N + 1 o u t " (note E = {EQ, Z , 1 being about a - (C. - C . ^ ) / U ( x 2 j + 1 ) > H ^ } . that that i t i f 2 N B 2 ' s + at was point value He that i s of merely stage view B l >, where 1 i f V(x ) 0 otherwise. > the bargaining i t acts so, i s as there make reasonable i f B2 has s i t u a t i o n . w i t h sequence N, i s i n (5.10) r u l e given E % a c t i n g always f o r B2 <j> = {x^, f o r given j x = , <fr ) by B.I.P. by B.I.P., x 2 > 0, i ) • stopping of has b e l i e v i n g However B l B l throughout instead person, conclude, a t b e l i e v i n g every immediately). The N + 1 B2 r a t i o n a l t o not We a s s u m e way because truncated x i s s i t u a t i o n . so. The b a r g a i n i n g ( B2 the bargaining p o s i t i o n Z ) from t h e maximum N E. t h i s accept problem E , 1 were because, (b) + the bargaining i n v j emphasized acts (a) y i e l d s v t h i s maximizes X 2 : a ,) i f that, a r e now i n 5.3.1 be would information the bargaining where he 2 e X 1 about as assume P r o p o s i t i o n + U(x_. c o n t r a d i c t i o n complete and v j should process being a f o r i t 0 c o n s e r v a t i v e l y a r i s e s a = information bargaining he H. 2 f V ( x i s 1, 2, 2jj+l^» N - l 126 In bargaining so optimal given by a proof approximated to of the general plan Section f o r here. by a costs i t t h e truncated I t i s as w i l l i s lead nontruncated f o r 5.2, truncated We e x p e c t , bargaining bargaining i n B.I.P. assumptions. presence s o l u t i o n the r e s u l t s attempt can be reasonable the of plan we do n o t problem a view the general also seems problem c e r t a i n one i s that a r b i t r a r i l y i n t u i t i v e l y to c e r t a i n a problem problem so that using one the o p t i m a l ; and the w e l l c l e a r truncated that nontruncated under that optimal could B.I.P. u s u a l l y plan f i n d as an 127 B 1. Arrow, of K . J . , 2. , Essays B the I G R A P H of Risk 3, Chicago: of the of Y A v e r s i o n , " Theory i n Essays Markaham of Choice Risk-Bearing, under Chapter i n the P u b l i s h i n g 2, Theory Co., 1971. U n c e r t a i n t y " , Chicago: i n Markham 1971. Bishop, R.L. Journal of Economics, i n R. Young, "A Zeuthen-Hicks Oran O Theory Theory Co., L Chapter " E x p o s i t i o n i n P u b l i s h i n g 3. "The Risk-Bearing, I "Game T h e o r e t i c a l A n a l y s i s V o l . 77, Bargaining, of (November B a r g a i n i n g , " 1963), U n i v e r s i t y of Q u a r t e r l y 559-602, I l l i n o i s ( r e p r i n t e d Press, 1975, 85-128). 4. , (July 1964), 410-417, U n i v e r s i t y 5. C o n t i n i , w i t h 6. B., 8. 9. 10. 11. , De Groot, Mc Graw De M e n i l , 14. S., Economics 1975, Bargaining, S t a t i s t i c a l New Bargaining: Press, Monopoly T.S., Mathematical 309-325, "A 31 New Organizations 1948), 397-414. Economic Young, Review, Bargaining, 1969. Chapter 7 , New York: J . , Involving R i s k s , " and J.M. Academic Model 1964), L . J . , J o u r n a l of and Quandt, 1971. London: S t a t i s t i c s : York: Savage, McGraw-Hill, versus Union Power, 1971. Determinate (February Friedman, York: R. York, Power Mathematical Psychics, Handerson, for ( A p r i l American Oran D e c i s i o n , F.Y., L., i n 36 191-218). Edgeworth, V o l . V o l . Process," MIT Foldes, 32 1970. G. , Fergusion, V o l . Bargaining, 183-190„.) . ( r e p r i n t e d Press, Optimal Econometrica, Young, " R e s t r i c t e d Bargaining Bargaining of R. Econometrica, 67-94, I l l i n o i s M.H., 1975, Cambridge: N.S., 13. Press, of 1965), H i l l , Approach, 12. Oran Zionts, "Theory of The B a r g a i n i n g , " i n O b j e c t i v e s , " (March U n i v e r s i t y 7. and J . G . , 55 of I l l i n o i s M u l t i p l e Cross, V o l . of Theory ( r e p r i n t e d of a Decision Press, B i l a t e r a l Kegan Paul, 1881. Theoretic 1967. Monopoly," Economica, 117-131. "The U t i l i t y P o l i t i c a l R.E., A n a l y s i s Economy, Microeconomic V o l . of 56 Theory, Choices (1948), 244-251, 279-304. New 128 15. Harsanyi, J . C . , A f t e r Theory the H i c k s ' and "Approaches of Nash's ( r e p r i n t e d I l l i n o i s Press, , and Vol. 18 w i t h (January Theory of 18. H i l d r e t h , C , "Vetures, Paper No. 20, Center 1972, ( r e p r i n t e d Behavior 19. , 21. , Inc. 23. 24. Nash, of Two-person Science, 1932. P r o s p e c t s , " Discussion U n i v e r s i t y McFadden and Amsterdam: of S.Y. Minnesota, Wu, North Essays on H o l l a n d "A Ventures," V o l . Model 38, 69 of (March Choice Center f o r J o u r n a l 1974), w i t h of the 9-17. U n c e r t a i n Economic I n i t i a l Research, 1974. M., V o l . "Other 43 Theory, (May Solutions 1975), P r i n c e t o n : to Nash's Bargaining 513-518. D. Van Nostrand Company, and J r . , 1950), , 1953), H., Games of "Two and Decisions, Nydegger, Press, 1975, V o l . New York: John Wiley Test Theory, 3 V o l . of i n Oran Young, V o l . 18 Bargaining, 53-60.) Games," Oran Econometrica, R. R. Econometrica, Young, V o l . Bargaining, 21 (January U n i v e r s i t y 61-73,). Means 30 and Experimental i n 1975, Cooperative ( r e p r i n t e d "General R.V., Problem," I l l i n o i s Person S t a t i s t i c i a n , Bargaining ( r e p r i n t e d Press, N., "The 155-162, 128-140, I l l i n o i s N o r r i s , R a i f f a , 1957. U n i v e r s i t y 27. D.L. No. Smorodinsky, P r o b a b i l i t y J . F . , ( A p r i l 26. for Management London, Research, Uncertain L., Minnesota, and R.D., Sons, 25. 1956), 1963. Luce, & of Paper Econometrica, M., ( A p r i l U n i v e r s i t y S o l u t i o n 7, I n i t i a l U n c e r t a i n t y , Tesfatsion, of Loeve, Chapter and Balch, U t i l i t y U n i v e r s i t y E., 24 and Zeuthen's 1974.) Discussion Problem," 22. Wages, Economic M.S. Prospect," K a l a i , Before of 80-106. S t a t i s t i c a l A s s o c i a t i o n , and V o l . Bargaining, Information," 2, Bets f o r under "Expected American 20. Co., Young, Generalized Nash P a r t Hicks, P u b l i s h i n g "A 1972), i n Problem Discussion Econometrica, R. Incomplete 17. Economic The Bargaining C r i t i c a l 253-266). Selten, Games J.R., Oran 1975, J.C. Bargaining i n the A Theories," 144-157, 16. to Games: and S t a t i s t i c a l Theory," (February Houston, the (August Nash 1975), 1976), G.O., "Two Axioms," 239-249. The American 8-12. Person Bargaining: I n t e r n a t i o n a l J o u r n a l An of Game of 129 28. Pen, J . , Review, "A General V o l . 42 Theory (March Bargaining, U n i v e r s i t y 29. P r a t t , J.W., "Risk Econometrica, 30. R a i f f a , H., i n Kuhn H.W. II, 31. Saraydar, 34. E., Von J . , i n ( A p r i l the Tucker, Theory 1965), Elements Studies, J . , and P r i n c e t o n : Zeuthen, F., London: George Problems Small and i n Economic Oran R. Young, 164-182.) i n the Large," 122-136. f o r G e n e r a l i z e d Two-Person U n i v e r s i t y of to the P r e s , , Theory Games," of Games, 1953. Bargaining: A N o t e , " Econometrica, D i f f e r e n t i a l Equations, 802-813. of P a r t i a l 49-55, 1957. V o l . 25 as Behavior (1957-58), Morgenstern, 0., of Monopoly and U n i v e r s i t y and Sons, of towards R i s k , " Review 65-86. Theory P r i n c e t o n U n i v e r s i t y Routledge Bargaining, 1975, C o n t r i b u t i o n s P r i n c e t o n "Zeuthen's American ( r e p r i n t e d Press, 1964), " L i q u i d i t y Preference Neumann, Young, 24-42, I l l i n o i s McGrow-Hill, Economic Behavior, 35. A.W. N . J . : I.N., York: Tobin, of and (October, Sneddon, New 33. 33 of Aversion 32 B a r g a i n i n g , " " A r b i t r a t i o n Schemes P r i n c e t o n , Vol. 32. V o l . of 1952), of Economic 1930, I l l i n o i s Gamaes Press, Warfare, ( r e p r i n t e d Press, and Economic 1953. i n 1975, Chapter Oran 4, R. 145-163 ). 130 APPENDIX (a) COMPUTER PROGRAM EU(aX) DIMENSION VS X(1000), : GRAPH a IN Y(1000), OF A B = E{U'(aX)-U'(0)}/ EXAMPLE 3.3.1 Ull(lOOO), U12(1000), U21(1000), K=l J=l R3 = 0.0001 R l = 3 R2 = -4 DO 2 I = = R3 X(I) 1, U l l ( I ) = 7 . * E X P ( - l . * R l * X ( I ) ) + 16.*EXP(-2.*R1*X(I)) - 23 U12(I) = 7.*EXP(-1.*R2*X(I)) + 16.*EXP(-2.*R2*X(I)) - 23 U21(I) = - 7 . * E X P ( - l . * R l * X ( I ) ) - 8 . * E X P ( - 2 . * R l * X ( I ) ) + 15 U22(I) = -7.*EXP(-1.*R2*X(I))-8.*EXP(-2.*R2*X(I)) + 15 Y(I) (0.8*U11(I) = IF(K.NE.J) J = J + WRITE FORMAT 3 K = R3 K = GO TO + 0.2*U12(I))/(0.8*U21(I) 3 10 (6;7)oX(T),Y(I) 7 2 1000 (2F16.9) + R3 1 + 0.00017 CONTINUE CALL SCALE (X, 1000, 10., XMIN, DX, 1) CALL SCALE (Y, 1000, 10., YMIN, DY, 1) CALL AXIS (0.,0.,'ALPHA',-5,10.,0.,XMIN,DX) CALL AXIS CALL LINE CALL PLOTND STOP END (0.,0.,'BETA',-4,10.,90.,YMIN,DY) (X,Y,1000,1) + 0.2*U22(I)) U22(1000) (b) a COMPUTER OUTPUT : 0 < a <.17 AND a 3 131 g B 0.000100000 -1.697571754 0.085098922 -1.589912415 0.001799999 -1.694559097 0.086798847 -1.585869789 0.003499998 -1.693284035 0.088498771 -1.581673622 0.005199995 -1.692018509 0.090198696 -1.577314377 0.006899994 -1.690778732 0.091898620 -1.572784424 0.008599993 -1.689503670 0.093598545 -1.568072319 0.010299992 -1.688207626 0.095298469 -1.563167572 0.011999991 -1.686885834 0.096998394 -1.558057785 0.013699990 -1.685564995 0.098698318 -1.552730560 0.015399989 -1.684214592 0.100398242 -1.547169685 0.017099988 -1.682844162 0.102098167 -1.541359901 0.018799987 -1.681427002 0.103798091 -1.535284996 0.020499986 -1.680004120 0.105498016 -1.528923035 0.022199985 -1.678548813 0.107197940 -1.522255898 0.023899984 -1.677055359 0.108897865 -1.515263557 0.025599983 -1.675539017 0.110597789 -1.507916451 0.027299982 -1.673987389 0.112297714 -1.500189781 0.028999981 -1.672415733 0.113997638 -1.492052078 0.030699980 -1.670806885 0.115697563 -1.483471870 0.032399978 -1.669151306 0.117397487 -1.474407196 0.034099977 -1.667472839 0.119097412 -1.464822769 0.035799976 -1.665752411 0.120797336 -1.454668045 0.037499975 -1.663990974 0.122497261 -1.443894386 0.039199974 -1.662185669 0.124197185 -1.432437897 ' 0.040899973 -1.660347939 0.125897110 -1.420236588 0.042599972 -1.658461571 0.127597034 -1.407215118 0.044299971 -1.656524658 0.129296958 -1.393280983 0.045999970 -1.654553413 0.130996883 -1.378345490 0.047699969 -1.652523041 0.132696807 -1.362292290 0.949399968 -1.650454521 0.134396732 -1.344993591 0.051099967 -1.648314476 0.136096656 -1.326296806 0.052799966 -1.646138191 0.137796581 -1.306028366 0.054499965 -1.643887520 0.139496505 -1.283980370 0.056199964 -1.641588211 0.141196430 -1.259905815 0.057899963 -1.639225006 0.142896354 -1.233521461 0.059599962 -1.636794090 0.144596279 -1.204472542 0.061299961 -1.634288788 0.146296203 -1.172337532 0.062999904 -1.631720543 0.147996128 -1.136599541 0.064699829 -1.629074097 0.149696052 -1.096618652 0.066399753 -1.626347542 0.151395977 -1.051589012 0.068099678 -1.623537064 0.153095901 -1.000507355 0.069799602 -1.620646477 0.154795825 -0.942054093 0.071499527 -1.617665291 0.156495750 -0.874518096 0.073199451 -1.614581108 0.158195674 -0.795616388 0.074899375 -1.611399651 0.159895599 -0.702214420 0.076599300 -1.608116150 0.161595523 -0.589925826 0.078299224 -1.604722977 0.163295448 -0.452382982 0.079999149 -1.601212502 0.164995372 -0.280002952 0.081699073 -1.597579002 0.166695297 -0.057657681 0.083398998 -1.593813896 0.168395221 0.240048409 132 (c) COMPUTER OUTPUT : .11 < a < .22 AND g a P -1.459506989 0.169978440 0.625032544 -1.453421593 0.170978010 0.952293992 -1.447118759 0.171977580 1.380911827 -1.440587044 0.172977149 1.966603279 -1.433815956 0.173976719 2.815014839 -1.426786423 0.174976289 4.153803825 -1.419490814 0.175975859 6.581309319 0.126996934 -1.411907196 0.176975429 12.333648682 0.127996504 -1.404023170 0.177974999 43.099182129 0.128996074 -1.395816803 0.178974569 -40.082687378 0.129995644 -1.387272835 0.179974139 -15.234875679 0.130995214 -1.378362656 0.180973709 -9.953053474 0.131994784 -1.369067192 0.181973279 -7.655240059 0.132994354 -1.359358788 0.182972848 -6.369746208 0.133993924 -1.349211693 0.183972418 -5.548437119 0.134993494 -1.338600159 0.184971988 -4.978290558 0.135993063 -1.327478409 0.185971558 -4.559379578 0.136992633 -1.315821648 0.186971128 -4.238603592 0.137992203 -1.303584099 0.187970698 -3.985122681 0.138991773 -1.290725708 0.188970268 -3.779765129 0.139991343 -1.277193069 0.189969838 -3.609998703 0.140990913 -1.262934685 0.190969408 -3.467357635 0.141990483 -1.247890472 0.191968977 -3.345788956 0.142990053 -1.231994629 0.192968547 -3.240968704 0.143989623 -1.215169907 0.193968117 -3.149637222 0.144989192 -1.197335243 a 0.119999945 0.120999515 0.121999085 0.122998655 0.123998225 0.124997795' 0.125997365 g 0.194967687 -3.069375992 0.145988762 -1.178399086 0.195967257 -2.998282433 0.146988332 -1.158253670 0.196966827 -2.934877396 0.147987902 -1.136779785 0.197966397 -2.877964973 0.148987472 -1.113844872 0;. 1 9 8 9 6 5 9 6 7 -2.826617241 -1.089294434 0.199965537 -2.780051231 -1.062946320 0.200965106 -2.737625122 -1.034598351 0.201964676 -2.698809624 -1.004023552 0.202964246 -2.663174629 -0.970939875 0.203963816 -2.630340576 -0.935028553 0.204963386 -2.599997520 0.155984461 -0.895910740 0.205962956 -2.571857452 0.156984031 -0.853138864 0.206962526 -2.545709610 0.157983601 -0.806178927 0.207962096 -2.521335602 0.158983171 -0.754383445 0.208961666 -2.498571396 0.159982741 -0.696963727 0.209961236 -2.477254868 0.160982311 -0.632958889 0.210960805 -2.457262039 0.149987042 0.150986612 0.151986182 0.152985752 0.153985322 0.154984891 - "• 0.161981881 -0.561162472 0.211960375 -2.438465118 0.162981451 -0.480071962 0.212959945 -2.420769691 0.163981020 -0.387748659 0.213959515 -2.404077530 0.164980590 -0.281699657 0.214959085 -2.388309479 0.165980160 -0.158601046 0.215958655 -2.373386383 0.166979730 -0.013995744 0.216958225 -2.359245300 0.167979300 0.158256590 0.217957795 -2.345829010 0.168978870 0.366957486 0.218957365 -2.333085060 133 APPENDIX (a) COMPUTER PROGRAM EU(aX) : VS B y = GRAPH OF a EXAMPLE IN a EU'(aX)/ 3.3.2 DIMENSION 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 R3=0.0001 Rl=3 R2=-4 DO 2 1=1,1000 X(I)=R3 U11(I)=(7.*EXP(-1.*R1*X(I))+16.*EXP(-2.*R1*X(I)))*X(I) U12(I)=(7.*EXP(-1.*R2*X(I))+16.*EXP(-2.*R2*X(I)))*X(I) U21(I)=-7•*EXP(-1.*R1*X(I))-8.*EXP(-2.*Rl*X(I))+15 U22(I)=-7.*EXP(-1.*R2*X(I))-8.*EXP(-2.*R2*X(I))+15 Y ( T)=( Y ; ( I ) =xo; 8?uaii.( i ) K t p IF(K.NE.J) GO TO :.2*ui2 (r);) /.(o: s*u2ixi)r*o < 2 * p 2 2 3 J=J+10 WRITE 7 3 (6,7) FORMAT X(I),Y(I) (2F16.9) K=K+1 R3=R3+0.000174 2 CONTINUE CALL SCALE(X,1000,10.,XMIN,DX,1) CALL SCALE(Y,1000,10.,YMIN,DY,1) CALL AXIS(0.,0.,'ALPHA',-5,10.,0.,XMIN,DX) CALL AXIS(0.,0.,'GAMA',-4,10.,90.,YMIN,DY) CALL LINE(X,Y,1000,1) CALL PLOTND STOP END ( i ) ) (b) a COMPUTOR OUTPUT Y : 0 < a <.18 AND a 134 y Y 0.000100000 0.625147700 0. 087097049 1.042087555 0.001839999 0.628270507 0. 088836908 1.061169624 0.631488383 0. 090576768 1.081107140 0.634821177 0. 092316628 1.101953506 0.638276339 0. 094056487 1.123762131 0.641845584 0. 095796347 1.146596909 0.010539889 0.645541549 0. 097536206 1.170524597 0.012279861 0.649365842 0. 099276066 1.195611000 0.653329551 0. 101015925 1.221944809 0.015759803 0.657428145 0. 102755785 1.249604225 0.017499775 0.661670744 0. 104495645 1.278684616 0.019239746 0.666063011 0. 106235504- 1.309292793 0.020979717 0.670605481 0.107975364 1.341535568 0.022719689 0.675306976 0. 109715223 1.375541687 0.024459660 0.680167556 0. 111455083 1.411446571 0.026199631 0.685199618 0. 113194942 1.449405670 0.027939603 0.690402448 0. 114934802 1.489585876 0.029"679574 0.695789516 0. 116674662 1.532175064 0.031419545 0.701360285 0 ., 1 1 8 4 1 4 5 2 1 1.577379227 0.033159517 0.707125604 0 ., 1 2 0 1 5 4 3 8 1 1.625438690 0.034899488 0.713091731 0 ., 1 2 1 8 9 4 2 4 0 1.676610947 0.036639459 0.719265580 0 ., 1 2 3 6 3 4 1 0 0 1.731204987 0.038379431 0.725653887 0 ., 1 2 5 3 7 3 9 6 0 1.789543152 0.040119402 0.732267499 0 ., 1 2 7 1 1 3 8 1 9 1.852014542 0.041859373 0.739111960 0 ., 1 2 8 8 5 3 6 7 9 1.919052124 0.043599345 0.746197522 0 .. 1 3 0 5 9 3 5 3 8 1.991147041 0.045339316 0.753533244 0 ., 1 3 2 3 3 3 3 9 8 2.068892479 0.047079287 0.761130631 0 .. 1 3 4 0 7 3 2 5 7 2.152934074 0.048819259 0.768998384 0 ., 1 3 5 8 1 3 1 1 7 2.244052887 0.050559230 0.777148008 0 ., 1 3 7 5 5 2 9 7 7 2.343149185 0.052299201 0.785593033 0 .. 1 3 9 2 9 2 8 3 6 2.451291084 0.054039173 0.794342279 0 ., 1 4 1 0 3 2 6 9 6 2.569731712 0.055779144 0.803412676 0 .. 1 4 2 7 7 2 5 5 5 2.699982643 0.057519116 0.812815189 0 ,. 1 4 4 5 1 2 4 1 5 2.843879700 0.059259087 0.822564840 0 .. 1 4 6 2 5 2 2 7 5 3.003612518 0.060999058 0.832677722 0 ,. 1 4 7 9 9 2 1 3 4 3.181893349 0.843170643 0 .. 1 4 9 7 3 1 9 9 4 3.382126808 0.064478874 0.854061246 0 ,. 1 5 1 4 7 1 8 5 3 3.608532906 0.066218734 0.865367234 0 ,. 1 5 3 2 1 1 7 1 3 3.866542816 0.067958593 0.877104759 0 ,. 1 5 4 9 5 1 5 7 2 4.163175583 0.069698453 0.889301062 0 ,. 1 5 6 6 9 1 4 3 2 4.507704735 0.901975811 0 ,. 1 5 8 4 3 1 2 9 2 4.912642479 0.915152311 0 .160171151 5.395228386 0.928857803 0 .161911011 5.980031013 0.076657891 0.943114996 0 .163650870 6.703067780 0.078397751 0.957957208 0 .165390730 7.619752884 0.003579998 0.005319975 0.007059947 0.008799918 0.014019832 0.062739015 0.071438313 0.073178172 0.074918032 0.080137610 0.973413467 0 .167130589 8.819417000 0.081877470 0.989519000 0.168870449 10.456479073 0.083617330 1.006305695 0 .170610309 12.822902679 1.023817062 0 .172350168 16.544174194 0.085357189 135 A The proof Lemma x theorems A.4.1: e R, g(x ) = Q ] Q g ' ( Without = 0, f 1 f: R ) x 0 = l o s s ( 0 ) = f of 0 . h says f 1 1 N D and continuous, maps i n t e r v a l s X C be use the everywhere f o l l o w i n g ' ( x 0 ) a n = i n f as to g d J f f u n c t i o n i « e - g we may c o n s i d e r f o r x { f ' ( t ) a ± >_ 0 , : Then, g supports X Q = 0 the decreasing given such f and that at X can . Q set f u n c t i o n , 0 ^ t < _ x } . consequence i n t e r v a l s . p r o p o s i t i o n s . d i f f e r e n t i a b l e . d i f f e r e n t i a b l e concave Define, i s I 4.3.3 g e n e r a l i t y h ^ x ) Then E -> R continuously ) , P 4.3.1 L e t a f ( x Proof. f(0) of P of For the c l a s s i c a l x ^ 0, theorem that we now s e t , x g (x) x Then We g now i s C 1 and concave, 2 x (t) since dt g^ i s decreasing. d e f i n e , h (x) for \ = < 0 and set = Sup {f 1 ( t ) : x < t < 0} C l e a r l y g ± < f 136 x g (x) h„(t) = dt 2 Jo for g^ x <_ 0 and g . Again are 2 A. 4.2 decreasing f u n c t i o n S the C , together Let containing continuously touches : is 2 j o i n e d C o r o l l a r y set g h of x (0, 0) : . we (0, Then, whose concave and obtain the required °°) be «>)-»- given upper d i f f e r e n t i a b l e , s t r i c t l y curve FIGURE h at A.4.Is x^ and GRAPHS (0, q e <_ f 2 a (0, boundary = h(x) Now AND when the «°) , the under y = pieces g. 3 a such h g(x) and convex-compact graph f u n c t i o n completely y . d i f f e r e n t i a b l e i s decreasing l i e s OF X g . of a that S 137 Proof: Given X , l e t q g h(x defined on [0, x - concave, under and tangent the two contains convex and P r o p o s i t i o n function and S y l i n e h at compact set. A. 4.3 (x, : y) Then, e the S Let . given g(0) X and q • X Q = {(x, = g i s f (x, by l i m the (0, y) e y °°) be previous f(x, i n whose (0, y) i s the upper such domain (x, <_•&, -> y) f, is a h A.4.2: HALF SPACES THE SET MADE S BY g °°) as bounded <_y} i s (0, of 0 S s t r i c t l y boundary that 0 Then under region Then °°) . . the i s be a i n c r e a s i n g 1 a by g This set d i f f e r e n t i a b l e i n d i f f e r e n t i a b l e y) as both convex-compact maximizes f ( x , w e l l r e q u i r e d f ( x , . FIGURE lemma 0 the <_ g ( x ) , x + completely S t g(x) -y decreasing. : f u n c t i o n L l i e s y ) : Suppose o r i g i n , decreasing y) S because containing (x, i . e . 0) of . s t r i c t l y to at (0, where given x h axes, f u n c t i o n o touches of the ) 7-77—s~], v c\ be = C AND y) x set and subject 138 Proof: Let Then h From the f(x, i s we y) previous C f(x, f is completes the The Proof of can f i n d a s t r i c t l y h: c o r o l l a r y , 0) and y) > (0, any by -> therefore w h i l e maximized °°) for ] the point y) a °°) be defined y) point out f(x, above below of y) the the points = by and convex-compact curve (x, any (x, (0, d i f f e r e n t i a b l e everywhere supporting for C C h(x) set curve we h(x)) = y Theorem 4.3.1 convex-compact by and (X , q : Given set S, any (X , below f(x, . at y) have = C f ( x , belonging to ) domain y q whose upper ) subject , y ) to (x, y)e the boundary d i f f e r e n t i a b l e function y i n S; g i s such i . e . S the that subject y) < and t h i s of f graph of f we a i s to y = g(x). Hence, f Now from K 0 > the subject (A.4.1) Q to l y ( and K + Q invariance f From ( x 1 V g ' ( x o ) f property, = g(—), K y o } + (A.4.2) g ' we y ) q = Q since we l . (X , 2 f i s 0 (A.4.1) maximized by ( K x 0 » v 0 ) w i t h have, ( x o now ) f 2 ( K V deduce, V as = ° g'(X ) Q ( 4 0, = S from curve f ( x , proof. decreasing maximized and decreasing, N o t i c e - t h a t have Hence (0, . = s t r i c t l y containing (x, y) A , that, 4 ' 2 ) C. C. 139 f L e t t i n g K = V l V ( 1/X , f i n q V V ( K 2 " p a r t i c u l a r , we K f 2 V ( Since t h i s s a t i s f i e s r e l a t i o n the o f l ( x holds p a r t i a l V o ' for any that the = f ^ l , y ) / f general 2 s o l u t i o n < l , of f(x, Where B(y) v a r i a b l e . = Exp (/h(y)) ( x o ' y h . , ( y P.D.E. can F(x F ( 1 ( K x o ' V f 2 ' y ( l , (x o , o } y ) o y o ) we see that f df t h i s and l l f 3f ) It = ) equation, y) . y) o } a r b i t r a r y u = h(y) 2 d i f f e r e n t i a l . 9f where f = o have, f X y be shown (cf. Sneddon [1957]) i s , B (y)) are both a r b i t r a r y functions of one 140 F I G . A . U. 3: S e t s Q, R, a n d the m a x i m a l point (X^Zc.) 141 Given (x plane , o y whose , o z upper d i f f e r e n t i a b l e for a l l plane (x, to o r i g i n the point an y, z) <f ( x , R . at f On (x o i s , y V Q = K o ) < V using g for Thus by the x - any Z ) x-y l i n e given y q set R x Q x-z x i s since and the Z ) j o i n i n g where Now , i n segment a x i s . and <_ f ( X , q the combining the decreasing y, define i n r e s p e c t i v e l y , the v a r i a b l e s q to 0) cuts <_ Z . f(x, Now 0, Q s t r i c t l y that equal (x, of z a A.4.3). sides a l l of such and graph Z ) graph set f is y above two have, y, z w i t h i n convex-compact g(x) q z) < maximized , o J = Z ) f ( x , by (x d i f f e r e n t i a t i o n Moreover y y, we h w i t h the f u n c t i o n f ( x , Hence 0, a the Figure rectangle (0, f i n d is y (see which i n e q u a l i t i e s can boundary w i t h increasing f(x, we Q the at ), f u n c t i o n y)e be the o we Z the g(x), o of y , , f(x, z ) V , Z) y, z) Q y (x, Q = y) e Q and subject to x z) h (y, the e (x, z) two sets w i t h y = R Q and g(x), z obtain, } + X f a c t z = o h that KZ , q f we i i s ( y o ' also have, ^ • ° maximized ( by (x , K A - 4 Y > 0 ' 3 ) K z 0 ^ = z ( 142 h(Ky , KZ ) o E l i m i n a t i o n + q of x By h(y , s e t t i n g obtain the z ) o K h o = 1/Z Q and q d i f f e r e n t i a l 7 v a r i a b l e . = h^(y/z, In where C(z) function. i s an h h(Ky , that (A.4.3) KZ ) o h q (X , y q (A.4.4) 0 = Q equations = o z) l)/h(y/z, 1) the h ( h(y, i m p l i e s Kz ) KZ ) Q the h(y, Rearranging i n t e g r a l (Ky , ± noting . 3y- This h ) , and (y , ; Q Z ) i s q (A.4.4) y i e l d s Z ) Q a r b i t r a r y we thus equation, 7 MO;) o from (Ky , x MC;) where g ' ( x o g ' ( x ) o o K K x terms y Z z h is of (y, 1 an t h i s dy = N (J) z z) u n s p e c i f i e d equation ) z) that (y, » = y i e l d s M( ) Z i f u n c t i o n the of one i n t e g r a t i o n , dy , z) a r b i t r a r y = In function of + £n z C(z) and n is again an u n s p e c i f i e d 143 Hence h i s of the form, h(y, The y proof and of z C o r o l l a r y r e s p e c t i v e l y f ( x , Hence S(x g (y, Z ), q y, y) z) = C(z) 4.3.2 : t e l l s us z) = N(J) A p p l i c a t i o n that y) f(x, y, z) = can Theorem be w r i t t e n 4.3.1 i n the G(x two g(x, z ) , y) (A.4.5) = S(x h(y, z ) , z) (A.4.6) S(x h (0, A(y, at S(x (y, Z ) Z ), and q Q ° , z) Z ) o i n turn if g(y, z ) * g(y> z ) = S(t, Z ) q o we o = S Thus f of = s (w g(y» S(W, . y) z ) ) , i f where G(t) = 0 o q and have, constant forms, 144 g(y, z) h ( y , g(y» A(y, with The G(0) Proof Then = z) S(0, using V +(y) 0 Q ) -f g(Y, O i f g(y, z ) = o f g(y, Z ) - 0 r = y ) , o f Lemma z Z ) y being 4.3.4 : = * ( - y ) the s o l u t i o n Define 4>'(y) X ( y ) = D Q +(y)/*(-y) we deduce that, i f o r 0 0. <K~y) f <|,(-y) * • 0 0, 4 K-y)' Solving that, t h e d i f f e r e n t i a l equation f o r c^(-y) ^ Ky) and C = i s <i>(-y) this equation an a r b i t r a r y [C <f)(y) d i f f e r e n c e ' / ( X + D = * ' / + » w e t h u s c o n c l u d e 0, |gy where X - ! ] • = x (y) = cU(y) constant. 0 u equation r Hence o r i g i n a l implies - i ] f o r a l l y , we have d i f f e r e n c e - d i f f e r e n t i a l equation that <|> s a t i s f i e s the d i f f e r e n t i a l 145 ^ = k (c dy Evidently t h e general 4>(y) a l l the admissible k = - 1 , + 1, [1 E - + D e k of C y t h i s cf>(y) 4 o d i f f e r e n t i a l ] 4 i f C i f C = equation i s , 0 = for and y s o l u t i o n \ if <j> - 1 ) so and we that the values ( i i ) thus ky c o n t i n u i t y of D s a t i s f y i n g the only complete of admissible the proof. $ i s k 0 preserved. <J>(y) value of = *C—y) E i s Noting • ' ( y ) 0 that a r when e o (i) n l y + 1
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Bargaining solutions to the problem of exchange of uncertain ventures Weerahandi, Samaradasa 1976
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Title | Bargaining solutions to the problem of exchange of uncertain ventures |
Creator |
Weerahandi, Samaradasa |
Date Issued | 1976 |
Description | Consider the betting problem where two individuals negotiate to determine the amount each will bet. It has already been established that when the two bettors both have concave utility functions, there exist mutually beneficial bets (i.e. bets giving positive utility to both players) merely if players' subjective probabilities on the betting event differ. It is shown here that this result can be generalized to the case of more general utility functions. The results are extended to a more general situation, that of a stochastic exchange. It is shown that the set of all feasible solutions available for exchange for two risk averters is a convex set with a known boundary. After defining a solution for the members of a class of exchange models it is shown in the third chapter that the 'size' of the exchange prescribed by the solution tends to increase with the participants' initial wealth and with multiplicative shifts of the random variable characterizing the exchange. Furthermore the size of the exchange may increase or decrease due to an additive shift of this random variable. In Chapter 4 it is shown by an axiomatic method that an individual engaged in bargaining with incomplete information finds his 'fair' demand (offer) by maximizing a generalized Nash function, GNF; this GNF is found to be the product of his utility and a general mean of his opponent's uncertain utility (from first individual's point of view). This general mean is characterized by a parameter whose value may vary from person to person. Continuing the study on bargaining under incomplete information, a best bargaining strategy is developed in the last chapter using the technique of 'Backward Induction'. A criterion for comparing available bargaining strategies is also established. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-23 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080151 |
URI | http://hdl.handle.net/2429/20798 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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