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PRICES ON INFORMATION AND STOCHASTIC INSURANCE MODELS

I . V . E v s t i g n e e v

April 1 9 8 5 CP-85-23

C o Z Z a b o r a t i v e P a p e r s r e p o r t work which h a s n o t been p e r f o r m e d s o l e l y a t t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s and which h a s r e c e i v e d o n l y l i m i t e d r e v i e w . V i e w s o r o p i n i o n s e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e s e n t t h o s e o f t h e I n s t i t u t e , i t s N a t i o n a l Member O r g a n i z a t i o n s , o r o t h e r o r g a n i - z a t i o n s s u p p o r t i n g t h e work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , A u s t r i a

FOREWORD

T h i s p a p e r i s c l o s e l y c o n n e c t e d w i t h t h e s t u d i e s on d e c i s i o n making u n d e r u n c e r t a i n t y , p a r t i c u l a r l y w i t h t h e s t o c h a s t i c o p t i m i z a t i o n problems t h a t a r e i n v e s t i g a t e d i n t h e A d a p t a t i o n and O p t i m i z a t i o n P r o j e c t o f t h e System and D e c i s i o n S c i e n c e s Program.

The p a p e r d e a l s w i t h some economic models i n which i t a p p e a r s p o s s i b l e t o f o r m a l i z e t h e n o t i o n of t h e p r i c e on i n f o r m a t i o n c o n c e r n i n g t h e problem p a r a m e t e r s . I n - s u r a n c e models u n d e r u n c e r t a i n t y are s t u d i e d h e r e w i t h more d e t a i l .

Alexander B . Kurzhanski Chairman

System and D e c i s i o n S c i e n c e s Program

PRICES ON INFORMATION AND STOCHASTIC INSURANCE MODELS I . V . E v s t i g n e e v

The aim o f t h i s work i s t o s t u d y t h e i n f o r m a t i o n c o n s t r a i n t s i n economic problems o f d e c i s i o n making under u n c e r t a i n t y .

The i n f o r m a t i o n c o n s t r a i n t s , as w e l l a s t h e r e s o u r c e s con- s t r a i n t s , p l a y a n i m p o r t a n t r o l e i n a n economic s y s t e m . However, u n t i l r e c e n t l y , o n l y t h e c o n s t r a i n t s o f t h e l a t t e r t y p e have been s y s t e m a t i c a l l y s t u d i e d .

It i s w e l l known t h a t t h e L a g r a n g i a n m u l t i p l i e r s which remove t h e r e s o u r c e s c o n s t r a i n t s i n an economic o p t i m i z a t i o n problem c a n b e r e g a r d e d as p r i c e s o f t h e r e s o u r c e s . I t t u r n s o u t t h a t t h e L a g r a n g i a n m u l t i p l i e r s a s s o c i a t e d w i t h t h e i n f o r m a t i o n c o n s t r a i n t s - c a n a l s o b e i n t e r p r e t e d as p r i c e s , namely, p r i c e s which c h a r a c t e r i z e t h e e f f e c t i v e n e s s o f i n f o r m a t i o n . Moreover,

it c a n be shown t h a t t h e r e are r e l a t i o n s between t h e p r i c e s u n d e r c o n s i d e r a t i o n and s u c h a n i m p o r t a n t economic phenomenon a s i n s u r a n c e .

T h i s work w a s s t i m u l a t e d by a series o f p a p e r s by R . T . R o c k a f e l l a r and R . J . B . Wets (see, e . g . , [ I , 2 1 ) d e v o t e d t o a p r o f o u n d m a t h e m a t i c a l i n v e s t i g a t i o n o f s t o c h a s t i c extremum

problems. An e s s e n t i a l r o l e w a s a l s o p l a y e d by some comments of economic n a t u r e t h a t w e r e made i n t h e c o u r s e of t h e d i s c u s s i o n of R.T. R o c k a f e l l a r ' s l e c t u r e i n t h e C e n t r a l Economic and Mathematical I n s t i t u t e (Moscow, 1 9 7 4 )

.

We emphasize t h a t no p u r e l y m a t h e m a t i c a l aims a r e p u r s u e d i n t h i s work. On t h e c o n t r a r y , examples a r e c o n s i d e r e d t h a t a r e most s i m p l e from t h e m a t h e m a t i c a l p o i n t of view, and t h e main a t t e n t i o n i s p a i d t o t h e c l a r i f i c a t i o n o f t h e economic s e n s e o f m a t h e m a t i c a l i d e a s .

W e s t a r t w i t h a d e s c r i p t i o n o f t h e model. L e t f ( s ,x ) be a f u n c t i o n o f a random p a r a m e t e r s E S (S i s a f i n i t e s e t ) and o f a v e c t o r x E _X ( X i s a s u b s e t o f R " ) . Suppose t h a t f ( s , x ) i s c o n t i n u o u s and concave i n x , and t h e s e t X i s convex and compact w i t h i n t X # g .

Problem A . F i n d a p l a n ( d e c i s i o n ) x i n t h e s e t X s u c h t h a t t h e m a t h e m a t i c a l e x p e c t a t i o n Ef ( s , x) . i s a maximum. I n symbols,

E f ( s , x ) ⁺ max ( 1

L e t u s c o n s i d e r t o g e t h e r w i t h t h e problem A t h e f o l l o w i n g problem B .

Problem B .

E f ( s , x ( s ) ) ⁺ max

I n problem B one h a s t o f i n d a f u n c t i o n ( s t r a t e g y ) x ( - )

which maximizes t h e f u n c t i o n a l ( 3 ) under t h e c o n s t r a i n t ( 4 )

.

Note t h a t i n t h e problem A t h e maximum i s t a k e n o v e r t h e s e t of d e t e r m i n i s t i c v e c t o r s ( r a t h e r t h a n v e c t o r f u n c t i o n s x ( s ) o f t h e random p a r a m e t e r s )

.

The problem A can b e o b t a i n e d from t h e problem B by a d d i n g t h e f o l l o w i n g c o n s t r a i n t :

x ( s ) does n o t depend on s , which can be a l s o r e w r i t t e n a s

T h i s form o f t h e i n f o r m a t i o n c o n s t a i n t h a s been s t u d i e d ( i n a much more g e n e r a l s e t t i n g ) by R o c k a f e l l a r and Wets [ I , 2 ] ( s e e a l s o ^[31).

Thus t h e i n f o r m a t i o n c o n s t r a i n t (5) i s r e p r e s e n t e d i n t h e form

where

i s a l i n e a r o p e r a t o r . By a p p l y i n g a n a p p r o p r i a t e v a r i a n t o f t h e Kuhn

-

Tucker t h e o r e m , w e o b t a i n t h a t t h e c o n s t r a i n t ( 5 ) can be removed by a L a g r a n g i a n m u l t i p l i e r p ( )

.

Namely, t h e r e e x i s t s a f u n c t i o n p ( s ) s u c h t h a t

( x ( s ) ^E X, s E S )

,

where

2

i s a s o l u t i o n o f t h e problem A. I t i s e s t a b l i s h e d by a s t a n d a r d argument t h a t ( 7 ) i s e q u i v a l e n t t o t h e f o l l o w i n g i n e q u a l i t y

where

I t h a s been d e m o n s t r a t e d i n [4] t h a t

6

and p ( s ) s a t i s f y i n g ( 8 ) and ( 9 ) c a n be i n t e r p r e t e d as i n s u r a n c e p r i c e s ( p ( s ) x i s t h e compensation and e x i s t h e premium). I n t h e p r e s e n t n o t e w e b r i e f l y s k e t c h t h i s i n t e r p r e t a t i o n .

The i n e q u a l i t y ( 8 ) means t h a t , by p a y i n g g x (premium) and g e t t i n g p ( s ) x ( c o m p e n s a t i o n )

,

we g u a r a n t e e , t h a t t h e p l a n

becomes o p t i m a l i n e a c h random s i t u a t i o n . I n t h i s s e n s e , t h e

i n s u r a n c e s y s t e m b a s e d on t h e p r i c e s p a n d p ( s ) makes i t p o s s i b l e t o e l i m i n a t e t h e f u t u r e u n c e r t a i n t y . The r e l a t i o n (9) r e f l e c t s t h e f a c t t h a t t h e premium s h o u l d be ( a p p r o x i m a t e l y ) e q u a l t o t h e e x p e c t e d v a l u e o f t h e compensation.

*

" v a r i a n t s o f t h e above model a r e c o n s i d e r e d , i n which t h e premium p r i c e

6

i s g r e a t e r , t h a n E p ( s ) .

L e t us i l l u s t r a t e t h e above i d e a by t h e f o l l o w i n g example.

C o n s i d e r a model f o r i n s u r a n c e o f a good t r a n s p o r t e d by a s h i p .

A s h i p t r a n s p o r t s x u n i t s o f a good from one p o r t t o a n o t h e r . There a r e two p o s s i b i l i t i e s : s u c c e s s f u l t r a n s p o r t a t i o n and

c a t a s t r o p h e . I n t h i s example, t h e random p a r a m e t e r s t a k e s two v a l u e s : s = s l ( c a t a s t r o p h e ) and s = s 2 ( s u c c e s s ) . The c a p a c i t y o f t h e s h i p e q u a l s xo. Thus, t h e s e t X o f p o s s i b l e p l a n s i s as

f o l l o w s : X = Ix: 0

-

^{< x}

-

^{< xo).}

Suppose t h a t t h e income ( p r o f i t ) o b t a i n e d from a s u c c e s s f u l t r a n s p o r t a t i o n e q u a l s f ( s 2 , x ) = q 2 x , and l o s s e s which we have i n c a s e of a c a t a s t r o p h e e q u a l f ( s l , x ) = -q l x . Denote by

Xi

t h e p r o b a b i l i t y P { s = s i } ( i = 1 , 2 ) and assume t h a t h l q l < h 2 q 2 . Then t h e o p t i m a l p l a n

x

c o i n c i d e s w i t h xo.

L e t us c o n s i d e r t h e f o l l o w i n g t a s k . F i n d a l l t h e p r i c e s

5

and p ( s ) p o s s e s s i n g t h e p r o p e r t i e s (8)

,

⁽⁹⁾ and t h e a d d i t i o n a l p r o p e r t y p ( s 2 ) = 0. The l a s t e q u a t i o n means t h a t , i n c a s e o f s u c c e s s , t h e compensation e q u a l s z e r o .

I t c a n be shown t h a t t h e s o l u t i o n o f t h e problem i s g i v e n by t h e formulae:

Thus, t h e r e a r e two bounds f o r

6 ,

. a n d p ( s l ) i s a l i n e a r f u n c t i o n o f

F .

The i n e q u a l i t i e s i n (10 have a n o b v i o u s economic s e n s e . The l n e q u a l i t y

-

1 q1 <

6

i s e q u i v a l e n t t o t h e f o l l o w i n g one

h2

-

which means t h a t t h e compensation i s n o t l e s s t h a n t h e ^sum o f t h e l o s s e s and o f t h e premium.

Some v a r i a n t s o f t h e above model a r e c o n s i d e r e d i n which t h e p r o f i t f ( s 2 , x ) = q 2 ( x ) i s a n o n l i n e a r f u n c t i o n ( d i m i n i s h i n g r e t u r n s t o s c a l e ) . It t u r n s o u t t h a t t h e p r i c e

6

i s u n i q u e i n t h e s e models and

5

c o i n c i d e s w i t h t h e l e f t bound i n ( 1 0 )

.

L e t u s now d i s c u s s t h e r e l a t i o n between t h e p r i c e s c o n s i d e r e d and t h e problem o f e f f e c t i v e n e s s o f i n f o r m a t i o n .

I t i s n a t u r a l t o e x p e c t t h a t t h e Lagrangian m u l t i p l i e r which removes t h e i n f o r m a t i o n c o n s t r a i n t g i v e s a n economic e v a l u a t i o n of i n f o r m a t i o n j u s t a s t h e Lagrangian m u l t i p l i e r s which remove t h e r e s o u r c e s c o n s t r a i n t s e v a l u a t e t h e s e r e s o u r c e s .

I n o r d e r t o d i s c u s s t h i s i d e a i n r i g o r o u s terms, w e have t o be a b l e t o measure t h e q u a n t i t y o f i n f o r m a t i o n . I t i s w e l l known t h a t a n i m p o r t a n t r o l e i s p l a y e d i n t h e technology and d i s c r e t e mathematics by Shannon's method o f i n f o r m a t i o n measuring. Simple argumentation shows t h a t t h i s method i s n o t q u i t e a p p r o p r i a t e f o r o u r aims.

*

I n o r d e r t o o u t l i n e an a l t e r n a t i v e a p p r o a c h , l e t us imagine t h e f o l l o w i n g s i t u a t i o n (which i s f o r m a l i z e d i n o u r m o d e l ) . A t

t h e b e g i n n i n g o f t h e p l a n n i n g p e r i o d , we have t o make a d e c i s i o n x w i t h o u t any i n f o r m a t i o n a b o u t s . G e n e r a l l y s p e a k i n g , t h e v a l u e of s i s observed o n l y a t t h e end o f t h e p l a n n i n g p e r i o d . Then only,we l e a r n , whether t h e i n i t i a l d e c i s i o n x i s good o r bad.

However, i f i t i s p o s s i b l e t o make some s p e c i a l e f f o r t s which r e s u l t i n l e a r n i n g s e a r l i e r , t h e n we can make a c o r r e c t i o n , i . e . r e p l a c e x by x

+

h ( s ) (h ( ) i s t h e c o r r e c t i o n )

.

For example, suppose t h a t , b a s i n g on u n c e r t a i n d a t a , we have d e c i d e d t o work o u t a p r o j e c t . Suppose f u r t h e r t h a t a f t e r some t i m e we have g o t a r e l i a b l e p r e d i c t i o n t h a t t h e p r o j e c t i s doomed t o f a i l u r e . Then t h i s i n f o r m a t i o n makes i t p o s s i b l e t o s t o p payments ( o r s u p p l y ) and t h u s spend t h e sum o f money

r a t h e r t h a n t h e sum x i n i t i a l l y planned. The s t r a t e g y used h e r e i s a s f o l l o w s :

x ( s , ) = x

-

^u ( f a i l u r e )

,

( s u c c e s s ) .

The e a r l i e r we g e t t h e i n f o r m a t i o n a b o u t f a i l u r e , t h e more e s s e n - t i a l i s t h e c o r r e c t i o n , and, c o n s e q u e n t l y , t h e more f l e x i b l e

ow ow ever,

some c o n n e c t i o n s can be found between t h e Shannon t h e o r y and t h e p o i n t o f view on i n f o r m a t i o n which i s c o n s i d e r e d h e r e

(see t h e l a s t pages o f t h e p r e s e n t p a p e r ) .

s t r a t e g y i s used. (The f l e x i b i l i t y o f a s t r a t e g y means h e r e t h e d i f f e r e n c e between o u r a c t i o n s i n t h e c a s e s s = s l and s = s 2 . )

Thus, t h e e a r l i e r we g e t t h e i n f o r m a t i o n a b o u t s t t h e more freedom f o r c o r r e c t i o n s we have. I t f o l l o w s t h a t t h e v a l u e of i n f o r m a t i o n ( i n t h e example c o n s i d e r e d ) depends on t h e t i m e when t h e i n f o r m a t i o n i s o b t a i n e d . The v a l u e e q u a l s z e r o i f t h e i n f o r - mation comes s o l a t e t h a t it i s i m p o s s i b l e t o make any c o r r e c t i o n of t h e d e c i s i o n i n i t i a l l y made. The v a l u e i s maximal, i f w e g e t t h e i n f o r m a t i o n s o e a r l y t h a t t h e i n i t i a l program can be completely r e v i s e d .

The above a r g u m e n t a t i o n shows t h a t t h e c e n t r a l r o l e i n o u r problem i s p l a y e d by t h e c l a s s H o f t h e c o r r e c t i o n s which can be c a r r i e d o u t g i v e n t h e i n f o r m a t i o n a b o u t s . The c l a s s H charac- t e r i s e s t h e u s e f u l , e f f e c t i v e i n f o r m a t i o n c o n t a i n e d i n t h e com- munication of s . I n o t h e r words, t h e p r o p e r t y of i n f o r m a t i o n t h a t i s e s s e n t i a l h e r e i s i t s p r o p e r t y t o improve t h e a d a p t i v i t y o f economic system, i . e . t h e a b i l i t y of t h e system t o r e a c t i n a f l e x i b l e way t o a changing s i t u a t i o n . I t t u r n s o u t t h a t t h e

approach o u t l i n e d h e r e makes i t p o s s i b l e t o r e g a r d m ( s ) = p

- -

^{p ( s )}

as a n i n f o r m a t i o n p r i c e .

Indeed, l e t us f i x a f u n c t i o n h ( s ) ( c o r r e c t i o n ) and compare t h e maximal v a l u e of t h e o b j e c t i v e f u n c t i o n a l on t h e s e t o f

s t r a t e g i e s of t h e form x+h ( s )

$ ( h ( * ) ) = max ~f ( s , x ( s ) )

X ( S ) = x + h ( s )

w i t h t h e maximal v a l u e of t h e o b j e c t i v e f u n c t i o n a l on t h e s e t o f s t r a t e g i e s x i n d e p e n d e n t o f s

$ ( 0 ) = max E f ( s , x ) .

X

We have

where m ( s ) = fj

-

p ( s ) and

11 1)

^{i s} an a r b i t r a r y norm i n t h e ( f i n i t e - d i m e n s i o n a l ) space o f f u n c t i o n s h ( * )

.

The formula ( 1 1 )

i s deduced from t h e f o l l o w i n g r e l a t i o n

which, i n t u r n , i s a consequence o f (7)

.

I n o r d e r t o make t h i s a r g u m e n t a t i o n r i g o r o u s , i t i s s u f f i c i e n t t o assume t h a t 4 i s d i f f e r e n t i a b l e a t 0 a n d t h a t 0 b e l o n g s t o t h e i n t e r i o r o f t h e

domain o f

@.

Thus, w e have e s t a b l i s h e d t h e ( m a r g i n a l ) p r o p e r t y ( 1 1 ) o f t h e p r i c e m ( s ) which shows t h a t t h e number E m ( s ) h ( s ) g i v e s a n e v a l u a t i o n o f t h e economic e f f e c t i v e n e s s o f i n f o r m a t i o n a b o u t s used i n the c o r r e c t i o n h ( = ) . The c o r r e c t i o n h e r e i s s p e c i f i e d by a f u n c t i o n h ( * ) o r , e q u i v a l e n t l y ( s i n c e S i s a f i n i t e s e t ) , by a f i n i t e - d i m e n s i o n a l v e c t o r . Hence, t h e " q u a n t i t y o f i n f o r - m a t i o n " i s a v e c t o r . T h i s i s t h e approach t h a t f i t s t h e s t r u c - t u r e o f o u r problem.

On t h e o t h e r hand, i t i s = r e c o n v e n i e n t t o u s e s c a l a r - v a l u e d ( r a t h e r t h a n v e c t o r - v a l u e d ) c h a r a c t e r i s t i c s f o r m e a s u r i n g t h e e f f e c t i v e n e s s o f i n f o r m a t i o n . One o f t h e p o s s i b l e ways t o f i n d s u c h c h a r a c t e r i s t i c s i s b a s e d on t h e c o n c e p t o f f l e x i b i l i t y

( a d a p t i v i t y ) o f t h e c o r r e c t i o n h ( ) ( o r o f . t h e s t r a t e g y x ( ) = x

+

h ( = ) ) .

L e t u s compare t h e problems A and B . I n t h e f i r s t problem, we u s e s t r a t e g i e s o f t h e form x ( s ) = c o n s t , i . e . s t r a t e g i e s t h a t do n o t r e a c t t o a p o s s i b l e d i f f e r e n c e i n t h e v a l u e s o f s . The s e c o n d problem c o r r e s p o n d s t o t h e o t h e r e x t r e m e c a s e : we c a n

employ a r b i t r a r y s t r a t e g i e s x ( )

.

I n t h e l a t t e r c a s e , t h e r e a c t i o n t o a change i n t h e s i t u a t i o n s i s maximally f l e x i b l e .

Our a i m i s now a s f o l l o w s . We would l i k e t o d e f i n e a number which measures t h e f l e x i b i l i t y ( a d a p t i v i t y ) o f a s t r a t e g y and makes it p o s s i b l e t o c o n s i d e r problems t h a t a r e " i n t e r m e d i a t e "

between A and B. T h i s number w i l l a t t h e same t i m e c h a r a c t e r i z e t h e " q u a n t i t y o f i n f o r m a t i o n " a b o u t s u s e d i n t h e s t r a t e g y x ( s ) . I n d e e d , t h e f l e x i b i l i t y o f t h e s t r a t e g y x ( s ) ( t h e d e g r e e o f

dependence o f t h e f u n c t i o n x ( s ) on s ) r e f l e c t s a l s o t h e d e g r e e o f u t i l i z a t i o n o f t h e i n f o r m a t i o n a b o u t s i n t h e p r o c e s s o f making t h e d e c i s i o n x ( s )

.

A s s u m e t h a t s u c h a c h a r a c t e r i s t i c o f t h e f l e x i b i l i t y o f a s t r a t e g y (= o f t h e q u a n t i t y o f i n f o r m a t i o n ) i s e s t a b l i s h e d . Then we c o n s i d e r t h e c l a s s K r o f t h e s t r a t e g i e s x ( . ) w i t h t h e f l e x i - b i l i t y n o t g r e a t e r t h a n a r e a l number r. The z e r o v a l u e o f t h e

f l e x i b i l i t y and t h e c l a s s KO c o r r e s p o n d t o t h e problem A. The

* *

maximal v a l u e of t h e f l e x i b i l i t y r and t h e c l a s s K r c o n t a i n i n g a l l s t r a t e g i e s c o r r e s p o n d t o t h e problem B .

L e t u s c o n s i d e r t h e maximal v a l u e o f t h e o b j e c t i v e f u n c t i o n a l on t h e c l a s s K r

t ( r ) = max E f ( s , x ( s ) ) x ( ' ) E X r

Denote by B r t h e p r i n c i p a l l i n e a r p a r t of t h e f u n c t i o n R ( r ) a t t h e p o i n t r = o . Then t h e c o e f f i c i e n t x can b e r e g a r d e d a s a

(shadow) p r i c e which g i v e s a n e v a l u a t i o n o f t h e f l e x i b i l i t y o f a s t r a t e g y , and t h u s an e v a l u a t i o n o f t h e amount o f i n f o r m a t i o n used i n t h i s s t r a t e g y . Roughly s p e a k i n g , i t i s worth paying Br i n o r d e r t o have a p o s s i b i l i t y t o use s t r a t e g i e s w i t h t h e f l e x i - b i l i t y n o t g r e a t e r t h a n r . I n o t h e r words, i f we have a n economic mechanism t h a t makes it p o s s i b l e t o use s t r a t e g i e s w i t h f l e x i b i l i t y

< r , t h e n t h e (shadow) c o s t of s u c h a mechanism i s e q u a l t o Br

-

The s i m p l e s t and t h e most common way t o measure t h e s c a t t e r i n g o f v a l u e s of a random v a r i a b l e i s t o c o n s i d e r i t s s t a n d a r d d e v i a - t i o n . The s t a n d a r d d e v i a t i o n o f x ( s ) i s d e f i n e d by

We s h a l l u s e ox ( ) i n o r d e r t o measure t h e f l e x i b i l i t y o f a s t r a t e g y x ( )

.

L e t us c a l c u l a t e t h e shadow p r i c e c o r r e s p o n d i n g t o t h e measure o f f l e x i b i l i t y j u s t d e f i n e d .

C o n s i d e r t h e f u n c t i o n

where K r = { x ( * ) : o x ( . )

-

^< r

1

i s t h e c l a s s of s t r a t e g i e s w i t h f l e x i b i l i t y

-

^< r . We s h a l l f i n d

@ '

^{( 0 )} and e x p r e s s t h i s v a l u e i n t e r m s o f p ( * ) .

I t i s e a s i l y s e e n t h a t

t ( r ) = max E f ( s , x ( s ) x ( s ) = x

+

^{h ( s )} E X

E h ( s ) = 0

BY v i r t u e o f ( 1 1

,

we have

where

1 1 . ) 1

i s any norm i n t h e f i n i t e - d i m e n s i o n a l s p a c e o f func- t i o n s h ( * ) , e . g . , t h e norm

1 1 - I I .

C o n s e q u e n t l y , f o r s u f f i c i e n t l y

s m a l l r ' s , L2

R ( r )

-

^{R(0) = max} [ @ ( h ( * ) )

-

@ ( 0 ) 1 = E h ( * ) = 0

- -

^{max E} m ( s ) h ( s ) + o ( r ) = M X E m ( s ) h ( s )

+

o ( r ) =

E h ( * ) = 0 E h ( * ) = 0

a < r 1

h ( * )

-

The l a s t i n e q u a l i t y becomes o b v i o u s , i f we r e g a r d h ( * ) and

6 -

^{p ( - 1}

a s e l e m e n t s o f t h e H i l b e r t s p a c e

w i t h t h e u s u a l s c a l a r p r o d u c t . The above a r g u m e n t a t i o n i s b a s e d on t h e f a c t t h a t h ( * ) = 0 i s an i n t e r i o r p o i n t o f t h e domain o f t h e f u n c t i o n a l @ ( h ( )

.

Thus, t h e f o l l o w i n g r e s u l t i s o b t a i n e d :

C o n s e q u e n t l y , i f t h e q u a n t i t y o f i n f o r m a t i o n u s e d i n t h e s t r a t e g y x ( * ) i s measured by t h e s t a n d a r d d e v i a t i o n

o x

^{( .} )

,

t h e n t h e p r i c e o f i n f o r m a t i o n e q u a l s t h e s t a n d a r d d e v i a t i o n a

p ( * ) o f t h e f u n c t i o n p ( s ) . T h i s means t h a t t h e d i f f e r e n c e between R - ( r ) [ t h e maximal income f o r t h e s t r a t e g i e s u s i n g t h e amount o f i n f o r m a t i o n

2

rl and R ( 0 ) [ t h e maximal income f o r t h e s t r a t e g i e s u s i n g t h e z e r o amount o f i n f o r m a t i o n ] i s a p p r o x i m a t e l y e q u a l t o a

P ( * ) r. I n o t h e r words, it i s w o r t h p a y i n g a p p r o x i m a t e l y a

p ( * ) r u n i t s o f money f o r a s m a l l amount r o f i n f o r m a t i o n .

L e t us r e t u r n f o r a m i n u t e t o t h e i n s u r a n c e model. The above r e s u l t shows t h a t t h e i n f o r m a t i o n p r i c e i s e q u a l t o t h e s t a n d a r d d e v i a t i o n a

p ( * ) o f t h e i n s u r a n c e p r i c e p ( s ) . Thus, t h e l a r g e r i s t h e d i s p e r s i o n of v a l u e s o f p ( s ) , t h e more impor- t a n t i s t h e i n f o r m a t i o n . C o n s e q u e n t l y , t h e more e s s e n t i a l i s t h e d i f f e r e n c e between t h e f a i l u r e and s u c c e s s , t h e l a r g e r i s t h e i n f o r m a t i o n p r i c e a

p ( * ) '

We o u t l i n e now t h e r e l a t i o n s between t h e above a p p r o a c h a n d t h e Shannon i n f o r m a t i o n t h e o r y .

L e t u s f i r s t r e c a l l t h e d e f i n i t i o n o f S h a n n o n ' s i n f o r m a t i o n . A s s u m e t h a t t h e r e i s a random v a r i a b l e A which t a k e s m v a l u e s A l

, . . ^{. ,}

^Am w i t h t h e p r o b a b i l i t i e s

n , , . . . ,

TIm. The e n t r o p y o f A i s d e f i n e d by

H ( A ) =

- ^c

^nilog

ni -

⁼

^c r,(ni) ,

.I

where n ( a ) = -a l o g a.

Suppose t h a t B i s a n o t h e r random v a r i a b l e which t a k e s t h e v a l u e s B I I

...,

^Bn. Denote by

n i j

t h e c o n d i t i o n a l p r o b a b i l i t y The number

i s c a l l e d t h e c o n d i t i o n a l e n t r o p y o f B g i v e n A ( " t h e a v e r a g e

u n c e r t a i n t y o f t h e e x p e r i m e n t B g i v e n t h e result o f t h e e x p e r i m e n t A " ) . The Shannon i n f o r m a t i o n i s d e f i n e d by t h e formula

T h i s d i f f e r e n c e shows t o what e x t e n t t h e knowledge o f t h e result o f e x p e r i m e n t A r e d u c e s the u n c e r t a i n t y o f t h e e x p e r i m e n t B.

The Shannon i n f o r m a t i o n t h e o r y p l a y s a n i m p o r t a n t r o l e i n v a r i o u s f i e l d s o f a p p l i e d m a t h e m a t i c s . I t works e s p e c i a l l y good, when t h e r e i s no measure o f p r o x i m i t y between t h e d i f f e r e n t o u t - come o f t h e e x p e r i m e n t , i . e . , a l l t h e outcomes are i n some s e n s e e q u i v a l e n t .

However, t h i s a p p r o a c h i s n o t a l w a y s c o n v e n i e n t . Imagine a

random v a r i a b l e A t h a t t a k e s n r e a l v a l u e s w i t h e q u a l ~ r o b a b i l i t i e s . Assume t h a t n-1 v a l u e s A l l

...,

^An-l a r e v e r y c l o s e t o e a c h o t h e r ,

and t h e n - t h v a l u e An d i f f e r s e s s e n t i a l l y f r o m A l l

...,

^{An-l :}

If we n e g l e c t t h e d i f f e r e n c e between t h e v a l u e s A l l

...,

A n - l , t h e n t h e u n c e r t a i n t y of t h e random v a r i a b l e A s h o u l d b e a p p r o x i - m a t e l y e q u a l t o the u n c e r t a i n t y o f t h e random v a r i a b l e A ' t a k i n g

1 n- ¹

two v a l u e s A1 a n d An w i t h p r o b a b i l i t i e s a n d

-

_n

.

On t h e o t h e r h a n d , t h e e n t r o p y o f A i s e q u a l , e . g . , t o t h e e n t r o p y of a random

which v a r i a b l e A" t a k i n g n v a l u e s 1 , 2 , .

. . ^,

n w i t h p r o b a b i l i t y E,

i s "much more u n c e r t a i n " .

The p o i n t i s t h a t S h a n n o n ' s d e f i n i t i o n i s p u r e l y d i s c r e t e ; it d o e s n o t t a k e i n t o a c c o u n t , e . g . , t h e l i n e a r s t r u c t u r e o f t h e s p a c e . T h i s i s one o f t h e r e a s o n s , why t h e S h a n n o n ' s t h e o r y as i t i s , c a n n o t b e a p p l i e d t o o u r p r o b l e m , where t h e l i n e a r i t y and c o n c a v i t y p l a y c e n t r a l r o l e s .

W e modify t h e above d e f i n i t i o n by i n t r o d u c i n g a "measure o f i n d i f f e r e n c e " between t h e outcomes o f t h e e x p e r i m e n t ( i .e

.

^between

t h e v a l u e s o f t h e random v a r i a b l e ) . I t i s assumed t h a t t h i s measure o f i n d i f f e r e n c e i s p r o p o r t i o n a l t o t h e d i s t a n c e between t h e v a l u e s o f t h e random v a r i a b l e . I f t h e random v a r i a b l e s t a k e s two v a l u e s x l a n d x 2 (xi E R)

,

t h e n , r o u g h l y s p e a k i n g , w e mix x 1 and x2 up w i t h p r o b a b i l i t y

7

1

+

q , where q =

1

x l

-

x 2 / ( f o r s m a l l q ' s ) .

T h i s i d e a i s f o r m u l a t e d i n s t r i c t terms as f o l l o w s : Given a s t r a t e g y

w i t h q =

1

x l

-

x 2 ( s u f f i c i e n t l y s m a l l , w e i n t r o d u c e a n a u x i l i a r y random v a r i a b l e 2 w h i c h i s d e f i n e d by t h e f o l l o w i n g r u l e . I f s = s l , t h e n f = x l w i t h p r o b a b i l i t y 1

-

q ' a n d 2 = x 2 w i t h

p r o b a b i l i t y q r where

I f s = s 2 , t h e n P = x2 w i t h p r o b a b i l i t y 1

-

^{q and f = x l w i t h}

p r o b a b i l i t y q .

Denote by I ( q ) t h e Shannon i n f o r m a t i o n a b o u t s g i v e n 8.

W e have

I ( q ) =

-

X 1 l o g hl

-

^{X 2 l o g X 2}

+

A s i m p l e computation shows t h a t

Consequently,

L e t u s measure t h e amount o f i n f o r m a t i o n used i n t h e s t r a t e g y x ( by t h e q u a n t i t y

Then w e have

where

I n d e e d ,

s i n c e

Furthermore,

U r ) = max E f ( s , x ( s ) ) = L ( i ( r ) )

,

x ( . )

-

< i ( r )

-

¹

where i ( r ) = j ( r )

.

^Thus,

I 1 I

S i n c e i (0) = l / j ( O ) , it remains t o compute j ( 0 ) . T h i s can be done as f o l l o w s

I

j ( 0 ) = l i m j ^{( 0 ) = l i m 1} u 2

+

o ( u 2 ) =

u+o

( s e e ( 1 2 ) ) . Hence,

i . e . , ; t h e i n f o r m a t i o n p r i c e ( c o r r e s p o n d i n g t o i n f o r m a t i o n measure v'I(lxl

-

x 2 1 ) e q u a l s

where a

p ( - 1 ^{i s} t h e s t a n d a r d d e v i a t i o n of the f u n c t i o n p ( )

.

'

i s u n i v e r s a l : i t d o e s n o t W e n o t e t h a t t h e c o n s t a n t

2

depend on t h e p r o b a b i l i t y d i s t r i b u t i o n { h l , X 2 } .

REFERENCES

1. R o c k a f e l l a r , R.T., and R . J . B . Wets. 1975. S t o c h a s t i c convex programming: Kuhn-Tucker c o n d i t i o n s , J . Math-Econ., 1975, v. 2 , p.349-370.

2. R o c k a f e l l a r , R . T . , and R . J . B . Wets. 1976. . N o n a n t i c i p a t i v i t y and ~ 1 - m a r t i n g a l e s i n s t o c h a s t i c o p t i m i z a t i o n p r o b l e m s , Math. Programming S t u d i e s , v . 6 , p.170-187.

3. E v s t i g n e e v , I .V. 1976. L a g r a n g i a n m u l t i p l i e r s f o r t h e problems o f s t o c h a s t i c programming. L e c t . N o t e s Econ. Math. S y s t .

N o . 133.

4 . E v s t i g n e e v , I .V. 19 83. P r i c e s on i n f o r m a t i o n and i n s u r a n c e p r i c e s i n s t o c h a s t i c models o f d e c i s i o n making, I n : A p p l i e d S t a t i s t i c s , "Nauka", Moscow, p . 243-248.