UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1977_A1 W44.pdf [ 5.94MB ]
Metadata
JSON: 831-1.0080151.json
JSON-LD: 831-1.0080151-ld.json
RDF/XML (Pretty): 831-1.0080151-rdf.xml
RDF/JSON: 831-1.0080151-rdf.json
Turtle: 831-1.0080151-turtle.txt
N-Triples: 831-1.0080151-rdf-ntriples.txt
Original Record: 831-1.0080151-source.json
Full Text
831-1.0080151-fulltext.txt
Citation
831-1.0080151.ris

Full Text

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  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080151/manifest

Comment

Related Items