Sources of Indeterminacy in von Neumann-Morgenstern Utility Functions

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

SOURCES O F INDETERMINACY I N VON NEUMANN-

MORGENSTERN UTILITY FUNCTIONS

J o h n C . H e r s h e y Howard K u n r e u t h e r P a u l J

.

^H

.

Schoemaker May 1981

CP-8 1-1 5

CoZ Z a b o r a t i v e P a p e r s r e p o r t work w h i c h h a s n o t b e e n 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 S y s t e m s A n a l y s i s a n d w h i c h 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

ABSTRACT

U t i l i t y f u n c t i o n s a r e an important component of normative d e c i s i o n a n a l y s i s . They a l s o serve t o c h a r a c t e r i z e t h e n a t u r e of p e o p l e ' s r i s k - t a k i n g a t t i t u d e s . I n t h i s paper we examine v a r i o u s f a c t o r s t h a t make i t d i f f i c u l t t o speak of

the

u t i l i t y f u n c t i o n f o r a given person. S i m i l a r l y w e shov t h a t i t i s q u e s t i o n a b l e t o pool d a t a a c r o s s s t u d i e s ( f o r d e s c r i p t i v e purposes) t h a t d i f f e r in t h e e l i c i t a t i o n methods employed.

The f o l l o w i n g f i v e s o u r c e s of indeterminacy a r e s p e c i f i c a l l y d i s c u s s e d . F i r s t , t h e c e r t a i n t y e q u i v a l e n c e method g e n e r a l l y y i e l d s more r i s k - s e e k i n g p r e f e r e n c e s than t h e p r o b a b i l i t y e q u i v a l e n c e method. Second, t h e p r o b a b i l i t y and outcome l e v e l s used fn r e f e r e n c e l o t t e r i e s induce s y s t e m a t i c b i a s . T h i r d , combining g a i n and l o s s domains y i e l d s d i f f e r e n t u t i l i t y measures t h a n

s e p a r a t e examinations of t h e two domains. Fourth, whether a r i s k is assumed o r t r a n s f e r r e d away e x e r t s a s i g n i f i c a n t i n f l u e n c e on p e o p l e ' s p r e f e r e n c e s i n ways c o u n t e r t o expected u t i l i t y theory. F i n a l l y , c o n t e x t o r framing d i f f e r -

ences s t r o n g l y a f f e c t c h o i c e i n a n o n - n o r m a t i v e manner.

The above f i v e f a c t o r s are f i r s t d i s c u s s e d as e s s e n t i a l c h o i c e s t o be made by t h e d e c i s i o n s c i e n t i s t i n c o n s t r u c t i n g Von Neumann-Morgenstern u t i l i t y f u n c t i o n s . Next, each i s examined s e p a r a t e l y i n view of e x i s t i n g l i t e r a t u r e , and demonstrated v i a experiments. The emerging pi'cture is t h a t b a s i c p r e f e r - ences under u n c e r t a i n t y e x h i b i t s e r i o u s i n c o m p a t i b i l i t i e s w i t h t r a d i t i o n a l expected u t i l i t y theory. An important i m p l i c a t i o n of t h i s paper is t o commence development of a s y s t e m a t i c t h e o r y of u t i l i t y encoding which i n c o r - p o r a t e s t h e many i n f o r m a t i o n p r o c e s s i n g e f f e c t s t h a t i n f l u e n c e p e o p l e ' s expressed r i s k p r e f e r e n c e s .

iii

INTRODUCTION

The s t a n d a r d model of c h o i c e u t i l i z e d by d e c i s i o n s c i e n t i s t s i n a n a l y z i n g problems i s e x p e c t e d u t i l i t y

(EU)

t h e o r y [ 3 8 ] . T h i s model is presumed t o be d e s c r i p t i v e of p e o p l e ' s b a s i c p r e f e r e n c e s , w h i l e having n o n n a t i v e i m p l i c a t i o n s f o r more complex problems. R e c e n t l y , however, t h e r e has been an e x t e n s i v e l i t e r a t u r e which s u g g e s t s t h a t even b a s i c c h o i c e is more complicated t h a n u t i l i t y t h e o r y s u g g e s t s ( s e e [6] f o r a r e v i e w ) . I n view of t h i s , o u r paper p r e s e n t s a framework f o r s y s t e m a t i c a l l y i n v e s t i g a t i n g v a r i o u s i n f o r m a t i o n proc- e s s i n g e f f e c t s t h a t may confound t h e e l i c i t a t i o n of a d e c i s i o n maker's p r e f e r - ences under u n c e r t a i n t y . The e x p e r i m e n t a l d a t a p r e s e n t e d i n t h i s s t u d y ,

t o g e t h e r w i t h a l a r g e body of e x i s t i n g e v i d e n c e , l e a d us t o t h e unambiguous c o n c l u s i o n t h a t t r a d i t i o n a l EU t h e o r y needs t o be modified i f i t i s t o s e r v e a s a d e s c r i p t i v e and n o n n a t i v e model of c h o i c e under u n c e r t a i n t y .

Our a n a l y s i s was, i n p a r t , m o t i v a t e d by a r e c e n t a r t i c l e of F i s h b u r n and Kochenberger [8] who analyzed 30 e m p i r i c a l u t i l i t y f u n c t i o n s p u b l i s h e d i n e a r l i e r l i t e r a t u r e [ 3 2 , 1 2 , 9 , 1 0 , 31. These p l o t t e d u t i l i t y f u n c t i o n s v e r e e i t h e r d e f i n e d on changes i n w e a l t h o r on r e t u r n on i n v e s t m e n t . F i s h b u r n and Koch.enberger (F-K) d i v i d e d each graph i n t o a below-and a b o v e - t a r g e t segment, and f i t t e d l i n e a r , power, and e x p o n e n t i a l f u n c t i o n s s e p a r a t e l y t o each s u b s e t of d a t a . Of t h e 30 g r a p h s they examined,Q8 v e r e c h a r a c t e r i z e d by F-K a s having concave ( r i s k - a v e r s e ) a n d / o r convex ( r i s k - s e e k i n g ) segments: broken down a s f o l l u w s :

Concave Convex

Xb ove Above

-

T o t a l

Convex Below 13 5 1 8

Concave Below

-

3

-

⁷

-

^{1 0}

1 6 12 2 8

I n terms of p e r c e n t a g e s , 642 of t h e below-target f u n c t i o n s were convex and 57% of t h e above-target f u n c t i o n s were concave. The predominant composite s h a p e , t h e y concluded, was convex-concave (462) followed by concave-convex

(252).

We q u e s t i o n t h e p o o l i n g of u t i l i t y f u n c t i o n s , as w a s done f o r i n s t a n c e i n t h e F-K s t u d y , when t h e u t i l i t y f u n c t i o n s a r e o b t a i n e d via d i f f e r e n t e l i c i t a t i o n procedures. S p e c i f i c a l l y , w e s h a l l p r e s e n t e v i d e n c e t h a t t h e shape of t h e u t i l i t y f u n c t i o n i s i n f l u e n c e d by and p o s s i b l y d i s t o r t e d because of (1) r e s p o n s e mode b i a s e s , ( 2 ) b i a s e s induced by p r o b a b i l i t y and outcome l e v e l s , (3) a s p i r a t i o n l e v e l e f f e c t s , (4) i n e r t i a e f f e c t s , and (5) c o n t e x t e f f e c t s . The p r e s e n t paper t h u s r a i s e s a s e t of m e t h o d o l o g i c a l i s s u e s t h a t have s i g n i f i c a n t i m p l i c a t i o n s f o r both d e s c r i p t i v e and p r e s c r i p t i v e a n a l y s e s of choice under u n c e r t a i n t y .

ELICITATION XETHODS

To b e g i n our a n a l y s i s , we assume t h a t Von Neumann-Morgenstern u t i l i t y f u n c t i o n s [38] a r e c o n s t r u c t e d v i a s t a n d a r d r e f e r e n c e l o t t e r i e s where t h e c l i e n t p r o v i d e s i n d i f f e r e n c e judgments between a s u r e o p t i o n and a two-outcome l o t t e r y . I n conducting t h e e l i c i t a t i o n i n t e r v i e w , t h e d e c i s i o n a n a l y s t w i l l t h u s p r e s e n t t h e c l i e n t w i t h t h e f o l l o w i n g c h o i c e :

S v e r s u s

where S i s t h e s u r e amount, p is t h e p r o b a b i l i t y of winning G ( f o r g a i n ) , and L ( f o r l o s s ) t h e lower outcome of t h e l o t t e r y . O f c o u r s e , 0 < p < 1 and

L

< S < G . Note t h a t L and G r e f e r t o r e l a t i v e r a t h e r than a b s o l u t e amounts;

hence t h e y a r e n o t c o n s t r a i n e d sign-wise. Of t h e s e f o u r v a r i a b l e s , t h r e e w i l l have been set by t h e d e c i s i o n a n a l y s t , whereas t h e f o u r t h is v a r i e d t o o b t a i n a n i n d i f f e r e n c e judgment such t h a t U(S) = pU(G)

+

(1-p)U(L) . Hence, t h e r e e x i s t e s s e n t i a l l y f o u r d i f f e r e n t methods f o r c o n s t r u c t i n g NM u t i l i t y f u n c t i o n s , namely:

1. The c e r t a i n t y e q u i v a l e n c e (CE) method, where t h e c l i e n t s t a t e s an i n d i f f e r e n c e l e v e l f o r S f o r g i v e n v a l u e s of p , G and L .

2 . The p r o b a b i l i t y e q u i v a l e n c e (PE) method, where a n i n d i f f e r e n c e l e v e l f o r p is e l i c i t e d , f o r g i v e n v a l u e s of G , L and S.

3. The g a i n e q u i v a l e n c e (GE) method, where t h e p r o b a b i l i s t i c outcome G is e l i c i t e d , and p , L and S a r e f i x e d .

4. The l o s s e q u i v a l e n c e

(LE)

method, where t h e p r o b a b i l i s t i c outcome L i s e l i c i t e d , w h i l e p , G and S a r e h e l d c o n s t a n t .

Hence, one i m p o r t a n t c h o i c e t h e d e c i s i o n a n a l y s t m u s t make is which of t h e s e f o u r r e s p o n s e modes t o use. The most common ones a r e t h e CE and PE methods.

A s we s h a l l show, however, t h e r e may e x i s t s i g n i f i c a n t d i f f e r e n c e s in r i s k - t a k i n g a t t i t u d e between t h e s e two methods. T h i s , of c x r s e , is c o u n t e r t o EU theory.

Another i m p o r t a n t d e c i s i o n i n v o l v e s t h e dimensions of t h e l o t t e r y .

S p e c i f i c a l l y , what p r o b a b i l i t y and outcome l e v e l s s h o u l d one u s e in e l i c i t i n g r i s k p r e f e r e n c e s ? I f t h e shape of t h e u t i l i t y f u n c t i o n depends on t h e end- p o i n t s a s s o c i a t e d w i t h G and L magnitudes, a n d / o r t h e v a l u e s of p u t i l i z e d , we must be aware of t h i s i n d e s i g n i n g a s e t of r e f e r e n c e l o t t e r i e s . Again,

i n theory t h e c h o i c e of l e v e l s is a r b i t r a r y . Due t o t h e s u b s t i t u t i o n and o t h e r axioms of u t l l i t y t h e o r y , an NFI u t i l i t y f u n c t i o n c o n s t r u c t e d w i t h 50-50

r e f e r e n c e l o t t e r i e s s h o u l d assume t h e same shape a s one o b t a i n e d w i t h , f o r example, 30

-

70 l o t t e r i e s . As we w i l l s e e , however, t h i s may n o t be t h e c a s e due t o p r o b a b i l i t y d i s t o r t i o n s .

A t h i r d d e c i s i o n t o be made by t h e a n a l y s t c o n c e r n s t h e domain of o u t - comes t o be used. Three l o t t e r y t y p e s may be d i s t i n g u i s h e d , namely pure l o s s l o t t e r i e s (L < G 5 0)

,

mixed l o t t e r i e s (L < 0 and G > 0 )

,

and pure g a i n

l o t t e r i e s (G > L 1 0 ) . O f c o u r s e , w i t h i n t h e EU model i t i s a r b i t r a r y which approach i s used, a s t h e same f u n c t i o n a l shape ( w i t h i n p o s i t i v e l i n e a r t r a n s - f o r m a t i o n s ) should o c c u r . Hence, a n

NM

f u n c t i o n c o n s t r u c t e d on [-$1000,

$10001 u s i n g mixed l o t t e r i e s should be i d e n t i c a l t o one u s i n g p u r e l o t t e r i e s w i t h i n t h e p o s i t i v e and n e g a t i v e s u b i n t e r v a l s of t h a t range. I n p r a c t i c e , how- e v e r , t h e f u n c t i o n s may w e l l d i f f e r ( a s we shall show), due t o a s p i r a t i o n l e v e l and p o s s i b l y o t h e r f a c t o r s .

A f o u r t h d e c i s i o n t o be made is how t o p r e s e n t t h e c h o i c e t o t h e d e c i s i o n maker; w i l l i t be one where t h e c l i e n t must assume r i s k o r one where r i s k i s

t r a n s f e r r e d away? For i n s t a n c e , t h e d e c i s i o n a n a l y s t might a s k f o r how much ( a t a minimum) t h e c l i e n t would s e l l a g i v e n l o t t e r y ( i . e . , t r a n s f e r r i s k ) . A l t e r n a t i v e l y , i t might be asked whether t h e c l i e n t would exchange a s u r e g i f t f o r t h a t l o t t e r y ( i . e . , assume t h e r i s k ) , which may be q u i t e d i f f e r e n t psycho- l o g i c a l l y from a t r a n s f e r of r i s k , due t o i n e r t i a . e f f e c t s .

F i n a l l y , t h e d e c i s i o n a n a l y s t must choose a d e c i s i o n c o n t e x t f o r t h e r e f e r e n c e l o t t e r i e s used. T h i s a s p e c t of t h e e l i c i t a t i o n procedure i s impor- t a n t a s d i f f e r e n t wordings, s c r i p t s , o r s c e n a r i o s may l e a d t o d i f f e r e n t s t a t e d r i s k p r e f e r e n c e s . I f t h e u n d e r l y i n g c h o i c e s a r e s t r u c t u r a l l y t h e same, such c o n t e x t u a l d i f f e r e n c e s should be w i t h o u t e f f e c t s . However, s i n c e d i f f e r e n t c o n t e x t s o f t e n emphasize d i f f e r e n t a s p e c t s [ I ] , people may p r o c e s s i n f o r m a t i o n

d i f f e r e n t l y , t h e r e b y i n d u c i n g i n c o n s i s t e n t r e s p o n s e s .

I n F i g . 1 we diagram t h e f i v e t y p e s of c h o i c e s t h e a n a l y s t must make ( e i t h e r i m p l i c i t l y o r e x p l i c i t l y ) . I n t h e remainder of t h e paper we w i l l demonstrate t h a t each of t h e s e f i v e c h o i c e s may indeed i n f l u e n c e t h e u t i l i t y f u n c t i o n in non-normative ways. A s such, w e view t h i s p a p e r as a f i r s t s t e p i n t h e development of a much needed t h e o r y f o r u t i l i t y encoding. Compared t o p r o b a b i l i t y encoding

1311,

t h e v a l u e s i d e has l a r g e l y been i g n o r e d i n d e c i s i o n a n a l y s i s a l t h o u g h i t s i m i l a r l y s u f f e r s from s e r i o u s , s y s t e m a t i c b i a s e s

.

RESPONSE

MODE

BIAS

I n Table 1 w e have summarized which methods were used i n e a c h of t h e f i v e s t u d i e s examined by F i s h b u r n and Kochenberger [ 8 ] , t o g e t h e r w i t h t h e i r f i n d - i n g s . I n t e r e s t i n g l y , f o r t h o s e s t u d i e s [ 3 2 , 12, 31 u s i n g t h e c e r t a i n t y e q u i v a l e n t (CE) method, 1 6 of t h e 1 7 below-target s h a p e s were convex and

U

of t h e 1 7 above-target s h a p e s were concave, whereas f o r t h o s e s t u d i e s [ 9 , 101 u s i n g t h e p r o b a b i l i t y e q u i v a l e n c e method, 9 of t h e 11 below-target s h a p e s were concave and 8 of t h e 11 above-target s h a p e s convex. (Note t h a t none of t h e s e s t u d i e s employed t h e GE o r

LE

methods.) Hence, t h e r e a p p e a r s t o be a s t r o n g i n t e r a c t i o n between t h e e l i c i t a t i o n methdd used and t h e predominant shapes o b t a i n e d by

F-K

as shown i n t h e f o l l o w i n g c r o s s - c l a s s i f i c a t i o n d e r i v e d from Table 1.

Response Xode

Composite S h a ~ e Convex Below- Concave Above Concave Below- Convex Above

C e r t a i n t y P r o b a b i l i t y

Eauivalence Equivalence

FIGURE 1 CHOICES FOR SELECTING AN ELICITATION PROCEDURE RESPONSE MODE?

RISK DOMAIN OP WHO GETS DIMENSIONS? LOTTERY ? T11E RISK?

-

Features to be Emphasized -Abstract -Concrete CONTEXT OF CHOICE?

-Assumption -Transfer -Neither

Levels of Probability and Outcomes b

-Pure Loss -Mixed -Pure Gain