Cha p t e r 6

A l te r n a t iv e s t o E x p e c t e d U t ilit y The o ry

In th is lectu r e, I d escrib e s om e w ell - kn o w n e xp er i m en ta l e vidence a g a in st th e e xp ected ut i l i t y t he ory a nd t h e a l t e r na ti v e t h eo ri e s de v e l o p e d i n o rde r t o a c c o mmo dat e t h es e exp e rim e n t s. (I ha v e p o sted a c om prehensiv e su rv ey on the c lass w e b p ag e.)

6. 1 A ll ai s P arado x and W eigh ted U ti li t y

Im a g ine y o u r self c ho osing b et w een th e f o l l o w i n g t w o a lterna tiv e s:

A W in 1 m illio n do lla r fo r s u r e.

B W in 5 m illio n d o llar w ith 1 0% c h a n c e , 1 m illion d olla r w ith 8 9% , n o t h i n g w i th 1 % .

W h i c h o n e w o u l d y ou c h o o se? In m a n y surv eys, su b j ects w ho w e re o ff er ed these altern ativ es c h ose A . I t seem s t ha t t h e y d i d no t w a n t t o r i s k t h e op p o rtun it y o f h a v in g a m illion d ollar f o r a 1 0 % c h a n c e o f h a v i ng fi v e m i llion d olla r i ns tead . N o w co n s id er th e fo llo w i n g t w o a lte r n a tiv e s:

C W in $1M w i t h 1 1% c h an ce, n othin g w i th 89% .

D W in $5M w i t h 1 0% c h an ce, n othin g w i th 90% .

47

It see m s th at th e p ro ba bilit y o f w inn i n g th e p r i z e is sim ilar f or th e t w o a l tern at iv es, whi l e t he pri z e s a r e s ubs t an t i a l l y di ff eren t. H e nce, i t seem s r easo na ble t o c h o ose t h e h i g h er p r ize, c h o o sing D r a t her t h an C . I n d eed , i n su rv eys, the s ub jects c h o ose D .

U n fo rtu n a t el y , fo r a n e x p ect ed utilit y m a x im izer , t h e tr ad e o f b e t w een A a nd B i s iden tica l t o t h e tra d e o f b et w e en C a n d D , a n d h e p r e fers A t o B if a n d o n l y i f h e p refers C t o D . T o s e e t h is, n ote t h a t f o r a n ex p e cted utilit y m a x im ize r w i th u t ilit y f u n c t ion u , A i s b e t t e r t h a n B i f a n d o n l y i f u (1 ) > 0 . 1 u (5) + 0 . 89 u (1 ) , i.e.,

0 . 11 u (1 ) > 0 . 1 u (5) , (6 . 1 )

w h ere t h e u n it of m o ne y i s m illion d ollar , an d the u t ilit y f rom 0 is no rm a lized to 0. B u t fo r s u c h a n e x p ecte d u t ilit y m ax im iz er, C is b e t t er th an D i f a nd on ly if (6 .1) h old s .

T h e a b o v e ex p e rim e n t a g a i nst t h e ex p e cted utilit y t h e o r y h a s b e en d esig n ed b y A llais. It ill u s tra t es for t h e su b j ects su rv ey ed th at th e i nd i ff er en ce curv es a r e n o t p a ra ll el , a nd h e n c e t h e ind e p e n d e n ce ax iom i s v io late d. T h is is illust rate d i n F igu r e 6 .1. A s s ho w n in th e fi g u re, t h e lines co n n ecting A t o B an d C to D a re p a r a llel t o e a c h o th er. S ince A i s b e t t e r t han B , t he i n di ff eren ce cu rv e t hro u gh A i s steep e r t h a n t he l i n e con n ectin g A to B . S in ce D i s b etter t h a n C , the in di ff eren ce cu rv e t h r o u g h C i s fl atter t han t h e line c on nectin g C to D . T h erefore, th e i n d i ff eren ce cur v e t h r ou g h A i s s teep er th a n th e ind i ff er en ce cu rv e t h r ou g h C .

A s eri e s o f o ther exp e ri m e n t s a l s o s uggested that the i ndi ff erence curv es are n ot p a r a l l e l a n d " f a n o u t ’ a s i n t h e fi gu re. C on sequ e n t l y , d ecisio n th eorists h a v e d ev el o p ed m a n y altern ativ e t heories i n w h i c h th e i nd i ff erence cu rv es are n o t pa ra l l el. T hese th eo ries often assum e th at th e i n d i ff eren ce cur v es a r e s traigh t l in es, ca lled be t w ee n n e s s .

A p ro m i ne n t t h e ory a m o ng t he s e a s s u m e s t ha t t he i ndi ff erence cur v es a r e s tra i g h t line s th at fan o ut fr om a s in gle o rig i n . T h is the o ry is c a lled W e ig h t e d U tility T h e o r y , a s it assum e s t h e f o llo w i n g gen e ral f orm for th e u tili t y f r om a l ottery p :

W ( p ) = w ( x | p, g ) u ( x )

x ∈ C

whe r e

w ( x | p, g ) = P

g ( x ) p ( x )

p ( y ) g ( y )

y ∈ C

fo r s o m e f u n c t io n g : C → R . H ere, th e u tilities a r e w e igh t e d a cco rd ing to n o t on ly th e p r ob ab ili ti es o f the c o n sequ e n ces bu t a lso a ccor din g to the c on sequ e n ces th em selv es. O f

Pr( $ 5)

1

A ∙ ∙

∙ B’

Indi ffe renc e c u rves

D ∙

∙

0 C 1

Pr( $ 0)

F i gu re 6 . 1 : A llais P a ra do x . T h e p riz e s a r e in te rm s o f m illio n d ollar s . P ro ba b i lit y o f 0 is o n the h orizo n tal a x i s; the p rob a b ilit y o f 5 is o n th e v ertica l a x i s, an d t h e re m a in in g p r ob ab ili t y g o es to the i n t erm e dia t e p rize 1 .

co ur se, i f g i s c o ns t a n t , t he w e i g h t i n g i s d one o nl y a c c o r di ng t o the p robabi l i t i e s , a s i n th e e x p ecte d u tilit y t h e o r y .

Exe r c i s e 7 C h e c k t h a t u n d e r th e w eig h te d u tility th e o r y , t h e in d i ff er enc e cu rves a r e s t r a i g ht l i ne s, b u t t he s l op e o f t he i n di ff er enc e curves d i ff er w h en g is n o t c o n sta n t. T a k - in g C w i th th r e e e lem e n t s, ch a r a c ter i z e th e f u n ctio n s g and u under w hi ch t h e i ndi ff er enc e se ts fa n o u t a s in th e A l l a i s p a r a d o x .

In t h e w eigh ted u tilit y t h e o r y , th e d ecisio n m a k e r d isto rts t h e p r ob ab ilit i e s u s in g t h e co nsequences th em selv es a n d t h e w h o l e p r o b a b i l i t y v e ctor p . I n g e n e r al, p rob a bilities n eed to b e disto r ted i f o n e w a n t s t o i ncorp o ra te A llais p a r a d o x i n e xp ected u til i t y th eo r y . A p rom i nen t theory that di storts the p robabi l i ti es to thi s end i s r a nk- d ep endent exp e ct e d u t ility theor y . I n t h i s t heor y , o n e fi rst r an ks th e c on sequ e n c es in th e o rd er o f in creasin g u t ilit y . H e the n a p p lies p r o b a b ility w e i gh tin g f u n ctio n w to t h e c um u l ativ e d is tribu t ion fu n ction F an d d istorts i t t o a n ew cum u lativ e d i strib u tio n fu n c tion w ◦ F . O n e t h e n fi na lly u ses ex p e cted utilit y u nd er th e d isto rted p r o b a b ilities i n o rd er to e v a l ua te th e lott er y . T h e r esu ltin g v a l ue fu nc tion is

U ( x | w ) = Z

u ( x ) dw ( F ( x )) . (6 . 2 )

Fi gure 6. 2: Probabi l i t y W ei gh t i ng F u nct i on; w ( p ) = e − ( − ln p ) α for s o m e α ∈ (0 , 1) .

The s urv e y r e s u l t s i n t he Al l a i s pa rado x s ug ge s t t h a t t h e s ub j e ct s o v e res t i m at e t he sm all p ro b a b i l i t y ev en ts w i th extrem e v alu e , s u c h a s g etting no th i n g w ith a sm a l l p ro b a - b i lit y . In o r der t o c ap tur e su c h a b eh a v i o r, on e o ften uses a n i n v e rted S s h ap e d probabi l - it y w eigh ting f u nction as in F i gu re 6.2. H e re, w is a n in crea sing fu nction w i th w (0) = 0 an d w (1) = 1 , a nd it cro sses th e d iag o n a l o nce a t s o m e p ∗ . T he ge ne ral f unc t i onal f o rm w ( p ) = e − ( − ln p ) α fo r s o m e α ∈ (0 , 1) has m an y d esirable p rop e rties.

Exa m p l e 2 C o n s id er th e l o t ter i e s in th e A l l a i s p a r a d o x . S e t u (0) = 0 and u (1) = 1 . T h e v alue of l o t tery B i s c o m p u t e d as f o l l ow s:

U ( B | w ) = w (0 . 01) u (0) + [ w (0 . 9) − w (0 . 01)] u (1) + (1 − w (0 . 9)) u (5)

= w (0 . 9) − w (0 . 01) + ( 1 − w (0 . 9)) u (5) .

Sim i l a rl y, t h e v al u e s of t he other l ot teri es ar e

Now t a k e u (5) ∈ (1 , e / ( e − 1)) and w ( p ) = e − ( − ln p ) α . N ot e t hat

lim U ( A | w ) = 1 > (1 − 1 /e ) u (5) = lim U ( B | w ) = lim U ( D | w ) > 1 − 1 /e = lim U ( C | w ) .

Th u s , f or s m a l l α , t he pr e f er enc e s a r e as i n t h e A l l ai s p ar adox.

6. 2 E l l sb erg P arado x and A m b i g uit y A v ersi on

C o nsid er an urn t h a t c on tain s 9 9 b alls, c olored R e d, B l ac k a n d G r een. W e kn o w that th ere a re exactly 3 3 R ed ba lls, b ut the e xact n u m b er of th e o th er colo rs is no t k n o w n . A b all i s r an dom l y d ra w n from th is urn . Y o u c h o ose a color.

• If th e b all i s o f t h e col o r y ou c h o o se, y o u w in $1. W h a t c ol or w o u l d y ou c h o o se?

• If th e b all i s not of the c olor y o u c h o ose, y o u w i n $1. W hat c olor w o uld y ou ch o o s e ?

Wh e n d i ff eren t s u b jects a re a s k e d t h ese question s, an o v er w h el m i ng m a jo rit y of them c h o s e r ed b a ll in b o th question s. T h at is to sa y , i n th e fi r s t q u estio n, a n o v erw h elm i n g m a jo rit y of th e s ub jects b e t o n t h e ev en t t h a t t he b a ll is red, an d i n t h e secon d qu estion a n o v erwhe l mi ng ma j o ri t y b e t t ha t t he bal l i s not r e d . T hi s c a n b e t a k e n a s a n e vi dence ag ain s th e e x p e c ted u tilit y t he ory b ec au se a n ex p e cted u t ilit y m a x im izer ca nn ot h a v e a strict preference in b o th question s. In d eed , i f t he pro b a b i lit y o f c olor s r ed , b lac k , a nd gr een a r e p R , p B , a n d p G , r esp ecti v el y , th en ha vin g a s trict p reference f or red i n t he fi rst qu estio n m ean s th at

p R > p B an d p R > p G .

A s trict p reference f o r red i n t he seco nd qu estion m e an s t h a t

1 − p R > 1 − p B an d 1 − p R > 1 − p G ,

i.e. ,

p R < p B an d p R < p G ,

a c lear con t r a d i ction .

T h is is c a lled E llsb e rg P a rad o x . N o te th at th is is an e v id en ce ag ain s t t h e ex p e cted u t il it y t heory a s f o r m u la ted b y S a v a g e, assu m i n g th at th e m o n ey is the c o n sequ e n ce. In pa rticu l ar, i t c on t r ad icts the b asic assu m p tio n tha t t h e i nd iv idu a ls ha v e w e ll-de fi ne d b e liefs t ha t g iv e a w e ll-d e fi ned p ro ba bi l i t y f o r e a c h c o n s e q ue nc e u nde r e a c h a c t . W i t hi n th e f ra m e w o rk o f v o n N eu m a nn an d M o r g e nstern , th is cou l d b e t a k en as an evidence a g a i ns t t he f u nda m e n t al mo de l i n g a s s u m p t i on that c o mp ound l o t t e ri e s are r educ ed to t h e s i m p l e l o t t e r i e s .

S u p p o se tha t th e d ecision m a k er b el iev es t ha t t h e ea c h b a l l is eq ua l l y lik ely to b e dra w n. G i v e n t he n u m b e r n B o f b l ac k b a lls, t he p r o b a b ilit y o f e ac h c olo r is as follo w s:

Pr ( R | n B ) = 1 / 3 Pr ( B | n B ) = n B / 99

Pr ( G | n B ) = 2 / 3 − n B / 99 .

(H ere, R m e ans t ha t t he ba l l i s red; B m e a n s t ha t t he bal l i s bl ac k, a n d G me a n s t h a t th e b a l l i s g r een .) Sa v a ge assum e s t h a t t h e decision m a k e r h as a b eli e f q ab ou t n B . F o r ev ery g i v en b elief q ab out n B , e ac h b et yi el ds a c o m p o und l o t t e ry . F o r e x ampl e , t h e co m p o u nd lotteries giv e n b y b etti n g o n the e v e n t R tha t th e b all i s r ed an d b etting on th e e v e n t B th at t h e b a l l i s b lac k are p lo tted i n F igu r e 6 .3.

In ex p e cted u t ilit y t h e o r y , on e f urt h e r assu m es t h a t t h e co m p o u nd lott eries a re re - duce d t o s i m pl e l ot t e ri es . T hi s r e duc t i o n yi el ds the f ol l o wi ng pro b a b i l i t i e s f or t h e c ol or of th e b all:

p R = 1 / 3

66

p B =

q ( n B ) / 99

n B =0

p G = 2 / 3 − p B .

T h e r edu ced lottery f rom b etting on R is $1 w i th p r o b a b ilit y p R a n d $ 0 w ith p rob a b ilit y 1 − p R . ( T h e r ed uced lotteries for th e o th er b e ts are s im il a r .) If q is sym m etrically di st ri but e d, w e w o ul d f urt h e r ha v e p G = p B = 1 / 3 . I n t ha t c a se, th e d ecisio n m a k er w o ul d b e i ndi ff eren t b et w e en b e tting o n a n y color . Sim i l a rl y , h e w o uld b e i nd i ff er en t b e t w een b e ttin g on th e b all n ot b e ing a giv e n c olor. S i n ce th e d ecision m a k e r w ould b e ind i ff er en t b et w e en all a l t ern a tiv es, this w o uld b e c o n sisten t w i th th e e xp erim en t a b o v e . H o w e v e r, unde r a n y a s ymmet r i c b e l i ef , h e w oul d not be t o n r e d i n either qu estion . F o r exa m p l e, if r B > r G , h e w o u ld b e t t h a t t h e ba ll is B l ac k i n t he fi rst q u esti o n , an d h e w o u l d b et th at th e b all i s n o t G r een i n t h e secon d q u esti o n.

T h e r e i s a s e n s e t h a t o n e m a y w a n t t o t r e a t t h e l a c k o f k n o w l e d g e a b o u t t h e n u m b e r of b a lls in a g iv e n u r n d i ff eren tly from t h e lac k o f kn o w l e dg e a b o u t w h at ba ll h a s b een dra w n " ra ndo m l y" . T he y m a y s o m e ho w t re at t h e l a t t er c a se m ore a s a c a s e wi t h a n "o b j ec tiv e ly giv en p rob a b ilities", w hile trea ting th e f o r m e r c ase m o r e " a m big u o u s" . T h e fo rm er ca se is som e tim es c a l l e d K n ig h tia n U n c er ta in ty or Am b i g u i t y .

1/3

1

2/3

0

q (0 )

.

.

1

q ( n ) .

1/3

B

.

.

.

2/ 3

q ( 66)

1/3

2/3

0

1

1

0

q (0)

.

.

1

q ( n ) .

n B /99

B

.

.

.

1 - n B /99

q (66)

2/ 3

0

0

1

1

1/ 3 0

0

F i gu re 6.3: C o m p ou nd lotteries i n E llsb e rg P a rad o x. T h e l otter y on th e l eft i s t h e b e t o n t h e b a l l b e i ng red, and t he l o t t e r y o n t he ri gh t i s t he b e t o n t he bal l b e i n g b l a c k .

A s t r a n d o f l i t e r a t ure t re at s s uc h a m b i g ui t y di ff ere n t l y . The m os t c anoni c al t h e o ry in th is litera t u r e, na m e ly th e t he ory o f A m biguity A version as anal yz e d b y G i l b oa and S c h e m e id l e r, assu m e s t h a t t h e decision m a k e r f o c uses o n th e w or st-ca s e scen ario w h en it com e s t o a m b igu i t y . H e m ax im izes th e m inim u m ex p e cte d pa y o ff in a m big u o u s c a ses. F o r e xam p le, h e t ak es th e v alu e s o f b etting on ev en ts R, B , and G as

V ( R ) = m i n 1 / 3 = 1 / 3

n B

V ( B ) = m i n n B / 99 = 0

n B

V ( G ) = m i n [2 / 3 n B / 99] = 0 .

n B

H e n ce, in the fi rst q uestion , an am b i g u it y a v e rse d ecision m a k e r c h o o ses to b e t o n t h e ev en t t ha t t h e b a ll is red . O n the o th er ha nd , t h e v a lu e o f b etting on th e e v e n t tha t th e b a ll is no t a g i v e n c o l or is g i v e n b y

V ( N R ) = m i n 2 / 3 = 2 / 3

n B

V ( NB ) = m i n [1 n B / 99] = 1 / 3

n B

V ( N G ) = m i n { 1 [2 / 3 n B / 99] } = 1 / 3 ,

n B

whe r e N R , N B , and NG d e n o te the c om plem en ts o f R , B , a n d G , r esp e cti v el y . H e n ce, in th e secon d q u estio n, a n a m bigu it y - a v erse d e ci si o n m a k e r c h o o ses to b e t t ha t t he b a ll is n o t r e d .

M o re g e ne ral l y , t he ory o f a m b i g ui t y a v ers i o n a s s u m e s t ha t t he re a r e m ul t i pl e p ri ors q ∈ Q on sta te spa ce S , t h e s e t Q i s the r ange of am bi gui t y . G i v en a n y q , a n a c t a yi el ds an ex p e cted u t ilit y E [ u ( a ) | q ] = u ( a ( s )) dq ( s ) , w he re t h e o ut c ome of an ac t c an b e a lottery a s i n t h e A n sco m b e a nd A u m a n n m o d el. F o r a n y giv en a ct a , t he w o rs t p o s s i bl e

exp ected p a y o ff is

V AA ( a ) = m i n E [ u ( a ) | q ] .

q ∈ Q

T h e d ecision m a k e r m a x im ize s this m i nim u m e x p ecte d u t ilit y . H i s c h o ice f un ct io n i s

c AA ( A ) = a r g m a x min

a ∈ A q ∈ Q

u ( a ( s )) dq ( s ) .

N o te th at th i s is a t h e or y o f a n ir r a tio n a l d e cisi o n m a k e r, w h o m ista k e n l y a ssum e s t ha t hi s c ho i c es a ff ect th e u n d erlyi n g s tate of the w o r ld, w h i c h is giv e n.

F o cusin g o n the w o r st-case scen ario s i s c lea r ly a n extrem e b eha v io r a n d yields a u seless gu id e f or b e h a vior i n m a n y real -w or l d prob l e m s . F or, u n d e r a f u ll sup p or t a s - s u mpt i o n , t he w o rs t c as e s ce na ri o i s t he w o rs t c ons e q ue nc e , m a ki ng the d ec i s i o n m a k e r ind i ff er en t b et w een all s u c h a cts. A n altern a t i v e t o a b o v e th eo ry in tro d uces th e b el iefs ab ou t t he am b i gu ou s s tates b u t treats th ese p r ob ab ili ti es d i ff eren tl y . It in tro d uces a

p r ob ab ili t y d i stribu tion μ on Q and a s s u m e s t ha t t he de ci si on m a k e r m a x i m i z e s

V SA A ( a ) = E μ [ v ( E q u ( a ))] = Z v µ Z u ( a ( s )) dq ( s ) ¶ dμ ( q ) ,

whe r e v : R → R is a c on ca v e fu n c tio n . H en ce, t he c h oice fu nction is