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D e a l o r N o D e a l : A S t u d y o f M o n e t a r y

U t i l i t y i n a G a m e S h o w S e t t i n g

D a v i d B e n n e t t , A k P g M a t a h i r , Y u d i s h N u n k o o , Y i X u

N o v e m b e r 2 0 1 0

A b s t r a c t

T h e g a m e s h o w D e a l o r N o D e a l p r o v i d e s a n e x c e l l e n t p l a t f o r m f o r

t h e e s t i m a t i o n a n d a n a l y s i s t h e u t i l i t y o f m o n e y o f t h e g e n e r a l p o p u l a -

t i o n . T h i s r e p o r t b e g i n s w i t h a n o v e r v i e w o f t h e U K v e r s i o n o f D e a l o r N o

D e a l a s w e l l a b r i e f i n t r o d u c t i o n t o t h e c o n c e p t s o f e x p e c t e d u t i l i t y t h e o r y

a n d s t a t i s t i c a l d e c i s i o n t h e o r y . E x p e c t e d u t i l i t y t h e o r y i s u s e d a l o n g s i d e

s t a t i s t i c a l r e g r e s s i o n i n o r d e r t o e s t i m a t e t h e g e n e r a l u t i l i t y f u n c t i o n a s -

s u m e d b y T h e B a n k e r i n t h e g a m e s h o w . F u r t h e r m o d e l s a r e u s e d t o

d e d u c e t h a t T h e B a n k e r a s s u m e s d i e r e n t u t i l i t y f u n c t i o n s d e p e n d i n g o n

t h e a g e a n d g e n d e r o f t h e c o n t e s t a n t s . T w o a d d i t i o n m e a s u r e s o f u t i l i t y

a r e t h e n d i s c u s s e d w i t h r e s p e c t t o t h e g a m e . T h e s e a r e c o n s t a n t r e l a t i v e

r i s k a v e r s i o n ( C R R A ) a n d p r o s p e c t t h e o r y ( P T ) .

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C o n t e n t s

1 I n t r o d u c t i o n 3

1 . 1 A b o u t t h e G a m e . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 . 2 S t a t i s t i c a l D e c i s i o n T h e o r y i n D e a l o r N o D e a l . . . . . . . . . . 4

1 . 3 R e g a r d i n g U t i l i t y T h e o r y . . . . . . . . . . . . . . . . . . . . . . 5

2 A M o d e l B a s e d o n E x p e c t e d U t i l i t y T h e o r y 6

2 . 1 M e t h o d o l o g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 . 2 F i t t i n g M o d e l s t o t h e D a t a G a t h e r e d . . . . . . . . . . . . . . . . 8

2 . 3 A d a p t i n g t h e M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . 1 0

2 . 3 . 1 G e n d e r A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . 1 0

2 . 3 . 2 A g e A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . 1 2

3 A l t e r n a t e M o d e l l i n g S t r a t e g i e s 1 4

3 . 1 P r o s p e c t T h e o r y i n D e a l o r N o D e a l . . . . . . . . . . . . . . . . 1 4

3 . 2 C o n s t a n t R e l a t i v e R i s k A v e r s i o n . . . . . . . . . . . . . . . . . . 1 7

4 D i s c u s s i o n a n d C o n c l u s i o n 1 8

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1 I n t r o d u c t i o n

1 . 1 A b o u t t h e G a m e

D e a l o r N o D e a l i s a t e l e v i s i o n g a m e s h o w i n w h i c h c o n t e s t a n t s m a k e r i s k y

d e c i s i o n s i n t h e h o p e o f w i n n i n g u p t o 2 5 0 , 0 0 0 . T h e s h o w r s t a i r e d i n t h e

N e t h e r l a n d s a n d i t h a s s i n c e b e e n s h o w n w o r l d w i d e i n 4 9 c o u n t r i e s . B r i t a i n ' s

r s t s h o w a i r e d i n O c t o b e r 2 0 0 5 a n d d i e r s s l i g h t l y t o t h o s e o v e r s e a s w i t h

r e g a r d s t o t h e r u l e s a n d t h e p r i z e s o e r e d . T h i s p a p e r i s b a s e d o n t h e B r i t i s h

v e r s i o n .

T h e g a m e b e g i n s w i t h a b o a r d s h o w i n g 2 2 k n o w n v a l u e s o f m o n e y v a r y i n g

f r o m 0 . 0 1 t o 2 5 0 , 0 0 0 . T h e s u m o f t h e s e 2 2 v a l u e s i s e q u a l t o 5 6 5 , 6 6 6 . 6 1

a n d t h e i r m e a n ( e x p e c t e d ) v a l u e i s r o u g h l y e q u a l t o 2 5 , 7 1 2 . 1 2 . T h e s e 2 2

v a l u e s a r e r a n d o m l y d i s t r i b u t e d i n s e a l e d b o x e s b y a n i n d e p e n d e n t a d j u d i c a t o r

w h o i s t h e o n l y p e r s o n t o k n o w t h e l o c a t i o n o f e a c h s u m o f m o n e y . T h e 2 2

c o n t e s t a n t s e a c h h o l d o n e o f t h e a f o r e m e n t i o n e d s e a l e d b o x e s , t h u s h o l d i n g a

r a n d o m a m o u n t o f m o n e y f r o m t h e l i s t s e e n i n g u r e 1 . N e x t , a c o m p u t e r

r a n d o m l y s e l e c t s a c o n t e s t a n t t o p l a y t h e g a m e .

1 p 1 , 0 0 0

1 0 p 3 0 0 0

5 0 p 5 , 0 0 0

1 1 0 , 0 0 0

5 1 5 , 0 0 0

1 0 2 0 , 0 0 0

5 0 3 5 , 0 0 0

1 0 0 5 0 , 0 0 0

2 5 0 7 5 , 0 0 0

5 0 0 1 0 0 , 0 0 0

7 5 0 2 5 0 , 0 0 0

F i g u r e 1 : T h e b o a r d s h o w i n g t h e 2 2 h i d d e n s u m s .

I n t h e o p e n i n g r o u n d t h e c o n t e s t a n t i s a s k e d t o e l i m i n a t e v e b o x e s . T h i s i n

t u r n e l i m i n a t e s t h e s u m s o n t h e b o a r d s h o w i n g t h e c o n t e s t a n t w h i c h b o x e s

r e m a i n . W e n a t u r a l l y a s s u m e t h a t t h e c o n t e s t a n t s w i s h t o e l i m i n a t e t h e s m a l l

v a l u e s . T h e r o u n d e n d s w i t h T h e B a n k e r o e r i n g a c e r t a i n v a l u e d e p e n d e n t o n

t h e v a l u e s l e f t o n t h e b o a r d . I f t h e c o n t e s t a n t a c c e p t s t h i s o e r ( D e a l ) , t h e n

t h e y l e a v e t h e s h o w w i t h t h e a m o u n t o e r e d . I f n o t ( N o D e a l ) , t h e n t h e

g a m e c o n t i n u e s t o r o u n d t w o . T h e c o n t e s t a n t i s g i v e n a f e w m i n u t e s t o m a k e a

d e c i s i o n . D u r i n g t h i s t i m e , t h e c o n t e s t a n t m a y c o n s u l t f r i e n d s a n d r e l a t i v e s .

T h e p r o c e s s i n t h e s e c o n d r o u n d i s s i m i l a r t o t h a t i n t h e r s t b u t o n l y t h r e e o f

t h e r e m a i n i n g 1 7 b o x e s a r e e l i m i n a t e d . T h e B a n k e r a g a i n o e r s a v a l u e t o t h e

c o n t e s t a n t , w h i c h i s e i t h e r a c c e p t e d o r r e j e c t e d . T h i s p r o c e d u r e c o n t i n u e s u n t i l

t h e s i x t h r o u n d i n w h i c h o n l y t w o b o x e s r e m a i n , i n c l u d i n g t h e c o n t e s t a n t ' s

o w n . A t t h i s p o i n t T h e B a n k e r m a k e s a n a l o e r . I f t h i s o e r i s r e j e c t e d t h e n

t h e c o n t e s t a n t o p e n s t h e i r b o x a n d g o e s h o m e w i t h t h e v a l u e i n s i d e .

S p e c i a l e p i s o d e s h a v e a l s o b e e n a i r e d w h i c h c h a n g e s t h e g a m e s l i g h t l y . I n

o n e e p i s o d e , d u r i n g t h e C h r i s t m a s s p e c i a l s , e a c h v a l u e w a s d o u b l e d y i e l d i n g a

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t o p p r i z e o f 5 0 0 , 0 0 0 . I n o t h e r i n s t a n c e s , e a c h o f T h e B a n k e r ' s o e r s w o u l d

c o n t a i n m o n e y p l u s a C h r i s t m a s r e l a t e d f o o d , s u c h a s a t u r k e y s a n d w i c h o r

s t u n g . F o r H a l l o w e e n , a T r i c k o r T r e a t w a s o e r e d t o t h e c o n t e s t a n t s . I n

t h i s c a s e , t h e c o n t e s t a n t i s a s k e d t o c h o o s e a b o x ( s e p a r a t e f r o m t h e m a i n 2 2

b o x e s ) w h i c h w o u l d c o n t a i n e i t h e r t h e w o r d t r i c k o r t r e a t . R e v e a l i n g

t r e a t w o u l d e n t i t l e t h e p l a y e r t o a n i n s t a n t p r i z e w h i l e p i c k i n g t r i c k w o u l d

r e s u l t i n T h e B a n k e r b e i n g a l l o w e d t o v i e w t h e c o n t e n t s o f t h e c o n t e s t a n t ' s

b o x , t h u s p u t t i n g t h e p l a y e r a t a d i s a d v a n t a g e . I n o r d e r t o a v o i d c o m p l i c a t i o n s

a l l s p e c i a l g a m e s w i l l b e i g n o r e d f o r t h e p u r p o s e o f t h i s p r o j e c t .

1 . 2 S t a t i s t i c a l D e c i s i o n T h e o r y i n D e a l o r N o D e a l

I t i s a p p r o p r i a t e a t t h i s s t a g e t o b r i e y d i s c u s s t h e g a m e w i t h r e s p e c t t o s o m e

o f t h e s t a t i s t i c s a t p l a y . A s a l r e a d y m e n t i o n e d , t h e m e a n v a l u e o f t h e b o x e s i s

r o u g h l y 2 5 , 7 1 2 . 1 2 . I t c a n b e s e e n f r o m g u r e o n e t h a t t h e r e a r e v e b o x

v a l u e s g r e a t e r t h a n t h i s a m o u n t , i m p l y i n g t h a t t h e d i s t r i b u t i o n o f b o x v a l u e s

i s h i g h l y s k e w e d . T h i s n o t i o n i s h i g h l i g h t e d b y t h e f o l l o w i n g g r a p h :

F i g u r e 2 : A g r a p h i c a l r e p r e s e n t a t i o n o f t h e s k e w o f t h e b o x v a l u e d i s t r i b u t i o n .

I t c a n a l s o b e o f u s e t o v i e w a t a b l e o f s u m m a r y s t a t i s t i c s w i t h r e f e r e n c e t o t h e

g a m e . F i g u r e 3 , b e l o w , i s s u c h a t a b l e . I t i s u s e f u l t o n o t e t h a t t h e a v e r a g e

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

t h a t n o p l a y e r h a s e v e r d e a l t i n t h e e i t h e r o f t h e r s t t w o r o u n d s . I t c o u l d b e

c o n j e c t u r e d t h a t t h i s i s b e c a u s e T h e B a n k e r ( a n d t h e t e l e v i s i o n c o m p a n y ) w i s h

t o d r a w o u t e a c h g a m e f o r t h e s a k e o f e n t e r t a i n m e n t a s w e l l a s m i n i m i s i n g

p a y o u t c o s t s .

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F i g u r e 3 : A s t a t i s t i c a l s u m m a r y o f D e a l o r N o D e a l .

S t a t i s t i c a l D e c i s i o n T h e o r y i s c o n c e r n e d w i t h t h e m a k i n g o f d e c i s i o n s i n t h e

p r e s e n c e o f s t a t i s t i c a l k n o w l e d g e w h i c h s h e d s l i g h t o n s o m e o f t h e

u n c e r t a i n t i e s i n v o l v e d i n t h e d e c i s i o n p r o b l e m ( B e r g e r 1 9 8 0 , p . 1 ) . W e l e t

d e n o t e t h e s e t o f a l l p o s s i b l e s t a t e s o f n a t u r e ,

, w h e r e

i s a n u m e r i c a l

q u a n t i t y a s s i g n e d t o t h e a f o r e m e n t i o n e d u n c e r t a i n t i e s . T h e d e c i s i o n s o f t h e

c o n t e s t a n t s a r e k n o w n a s a c t i o n s a n d a r e d e n o t e d b y

a. T h e s e t o f a l l a c t i o n s

i s d e n o t e d b y A. I n t h i s c a s e w e l e t A = {a1, a2} w h e r e a1 = D e a l a n d a2 = N o D e a l . I f a c t i o n a1 i s t a k e n a n d i t t r a n s p i r e s t h a t 1 i s t h e t r u e s t a t e o f n a t u r e t h e n w e d e n e L(1, a1) a s t h e l o s s i n c u r r e d . L(, a) i s d e n e d (, a) A. I t i s u s e f u l t o n o t e t h a t l o s s , i n t h i s c o n t e x t , i s e s s e n t i a l l y i n t e r c h a n g e a b l e w i t h u t i l i t y a s l o s s i s e q u a l t o n e g a t i v e u t i l i t y .

F o r e x a m p l e , i m a g i n e a c o n t e s t a n t i s o e r e d 2 0 , 0 0 0 a n d c h o o s e s N o D e a l .

F o l l o w i n g t h i s t h e c o n t e s t a n t r e m o v e s s o m e h i g h v a l u e d b o x e s a n d l o w e r s t h e

e x p e c t e d v a l u e o f t h e b o a r d r e s u l t i n g i n a n e w o e r o f 5 , 0 0 0 . I n t h i s c a s e w e

c a n s a y t h a t L(1, a2) = 15000 w h e r e 1 i s t h e t r u e s t a t e o f n a t u r e , i . e . t h e n e w b o a r d .

1 . 3 R e g a r d i n g U t i l i t y T h e o r y

G i v e n a c h o i c e b e t w e e n a c e r t a i n p a y - o a n d g a m b l e o n e w o u l d e x p e c t a

s t a t i s t i c a l l y l o g i c a l p l a y e r t o a c c e p t t h e c e r t a i n p a y - o o n l y w h e n t h i s v a l u e

e x c e e d s t h e e x p e c t e d r e t u r n f r o m t h e g a m b l e . T h i s , h o w e v e r , i s n o t t h e c a s e .

L e t u s t a k e a n a r b i t r a r y e x a m p l e i n w h i c h a p l a y e r c a n a c c e p t a g u a r a n t e e d

s u m o f 1 0 0 , 0 0 0 o r t a k e a g a m b l e i n w h i c h h e h a s a n e q u a l c h a n c e o f w i n n i n g

0 o r 2 5 0 , 0 0 0 . T h e g a m b l e y i e l d s a n e x p e c t e d w i n n i n g o f 1 2 5 , 0 0 0 w h i c h i s

g r e a t e r t h a n t h e 1 0 0 , 0 0 0 o e r e d . H o w e v e r , i n t u i t i o n t e l l s u s t h a t f e w

i n d i v i d u a l s w o u l d t a k e t h e g a m b l e . T h i s c a n b e d e m o n s t r a t e d i n t h e g a m e o f

D e a l o r N o D e a l a s n e a r l y a l l o e r s ( b o t h a c c e p t e d a n d r e j e c t e d ) f a l l b e l o w t h e

e x p e c t e d v a l u e o f t h e r e m a i n i n g b o x e s ( s e e a p p e n d i x 1 ) . T h i s c o n c e p t c a n b e

e x p l a i n e d t h o u g h u t i l i t y t h e o r y w h i c h d e m o n s t r a t e s t h a t t h e v a l u e o f a n

a d d i t i o n a l 2 5 , 0 0 0 i s w o r t h r e l a t i v e l y l e s s t o t h e c o n t e s t a n t t h a n t h e i n i t i a l

s u m o f m o n e y . A s s u c h , w e d e n e

U(w)t o b e a f u n c t i o n w h i c h r e p r e s e n t s t h e

u t i l i t y o f w e a l t h . A c c o r d i n g t o M e r t o n ( 1 9 9 0 ) t h i s f u n c t i o n i s

t w i c e - d i e r e n t i a b l e a n d d e n e d f o r

w > 0. I t a l s o h a s t h e p r o p e r t i e s o f

n o n - s a t i a t i o n ( i . e . t h e r s t d e r i v a t i v e U(w) > 0 ) .

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I t i s i m p o r t a n t t o n o t e t h a t m a n y f a c t o r s p l a y a r o l e i n o n e ' s u t i l i t y o f m o n e y .

W h i l e t h e s e i n c l u d e e a s i l y m e a s u r e d v a r i a b l e s s u c h a s a g e , g e n d e r , w e a l t h a n d

d e b t t h e y a l s o i n c l u d e a n a r r a y o f f a c t o r s w h i c h a r e d i c u l t o r i m p o s s i b l e t o

m e a s u r e . S o m e e x a m p l e s o f t h e s e i n c l u d e : p a s t h i s t o r y , e t h i c s a n d

s u p e r s t i t i o n . H e n c e , i n d e t e r m i n i n g T h e B a n k e r ' s a s s u m e d u t i l i t y f u n c t i o n s

o n l y t h e e a s i l y m e a s u r a b l e v a r i a b l e s h a v e b e e n t a k e n i n t o c o n s i d e r a t i o n .

2 A M o d e l B a s e d o n E x p e c t e d U t i l i t y T h e o r y

I n e s t i m a t i n g t h e u t i l i t y f u n c t i o n , t h e f o l l o w i n g u s e s t h e b a c k w a r d s m e t h o d

w h i c h w a s d i s c u s s e d b y B e c k e r e t a l . ( 1 9 6 4 ) .

2 . 1 M e t h o d o l o g y

U s i n g t h e b a c k w a r d s p r o c e d u r e , a n d f o l l o w i n g t h e m e t h o d o l o g y o f B e r g e r

( 1 9 8 0 ) , o n e m u s t p i c k a n

r1 a n d r2 s u c h t h a t U(r1) = 0 a n d U(r2) = 1. T h i s i s d o n e t o d e n e t h e r a n g e o f t h e u t i l i t y f u n c t i o n . G e n e r a l l y , i f t h e r e e x i s t s a

w o r s t c a s e s c e n a r i o a n d a b e s t c a s e s c e n a r i o , o n e c h o o s e s

r1 t o b e t h e w o r s t

c a s e s c e n a r i o a n d r2 a s t h e b e s t c a s e s c e n a r i o . F o r t h i s r e p o r t i t h a s b e e n

a s s u m e d t h a t , f o r a l l c a n d i d a t e s , t h e b e s t c a s e s c e n a r i o i s w i n n i n g 2 5 0 0 0 0

a n d t h e w o r s t c a s e s c e n a r i o i s w i n n i n g o n l y 0 . 0 1 . H e n c e :

r1 = 0.01 , r2 = 250000 a n d t h u s

U(0.01) = 0 a n d U(250000) = 1

T o e s t i m a t e t h e u t i l i t y f u n c t i o n i t c a n b e a s s u m e d t h a t i f

ri < rj, t h e n t h e

r e w a r d rj i s p r e f e r r e d t o ri . B e r g e r ( 1 9 8 0 ) s t a t e s t h a t U(r) c o u l d b e c o n s t r u c t e d s u c h t h a t t h e v a l u e o f a p r o b a b i l i t y d i s t r i b u t i o n

Pw o u l d b e

g i v e n b y t h e e x p e c t e d u t i l i t y EP[U(r)] . H e n c e , a n r3 m u s t b e f o u n d s u c h t h a t U(r3) =

12 .

S i n c e , f o r P = 12r1 +12r2 :

12

= 12

U(r1) +12

U(r2) = Ep[U(r)]

w h i c h i s e q u i v a l e n t t o n d i n g t h e r e w a r d

r3 w h i c h i s v a l u e d a s h i g h l y a s t h e

g a m b l e

12

r1 +12

r2 .

I n D e a l o r N o D e a l t h i s t y p e o f g a m b l e c a n e a s i l y b e f o u n d i n t h e s i x t h r o u n d

o f t h e g a m e a n d c a n b e d e m o n s t r a t e d a s f o l l o w s :

6


7/46

F i g u r e 4 : A d i a g r a m s h o w i n g t h e o p t i o n s u n d e r a n a l r o u n d d e c i s i o n w i t h

0 . 0 1 a n d 2 5 0 , 0 0 0

B y l i m i t i n g t h e m o d e l t o t h e d a t a o f r o u n d s i x a t t h i s s t a g e i t m i g h t b e

p e r c e i v e d t h a t t h e m o d e l w i l l l a c k a c c u r a c y . H o w e v e r , i t i s l i k e l y t h a t t h e

o p p o s i t e i s t r u e s i n c e , i n e a r l i e r r o u n d s , T h e B a n k e r m a k e s o e r s w h i c h a r e

o n l y a s m a l l f r a c t i o n o f t h e c o n t e s t a n t s ' e x p e c t e d u t i l i t y o f m o n e y ( i n o r d e r t o

l e n g t h e n t h e g a m e ) . H e n c e , p r o c e s s i n g d a t a f r o m e a r l i e r r o u n d s w o u l d m e a n

u n d e r e s t i m a t i n g t h e e x p e c t e d u t i l i t y a s s u m e d b y T h e B a n k e r . T o n d r3 , o n e

c a n l o o k f o r c o n t e s t a n t s w h o s e g a m e s l a s t e d u n t i l r o u n d s i x a n d w h o w e r e

p r e s e n t e d w i t h t h e g a m b l e o f w i n n i n g e i t h e r 0 . 0 1 o r 2 5 0 , 0 0 0 . H o w e v e r , i t

c a n b e n o t e d t h a t t h e r e a r e o n l y t h r e e c o n t e s t a n t s w i t h g a m e s w h i c h f o l l o w e d

t h i s s c e n a r i o . T h i s p r o v i d e s i n s u c i e n t d a t a t o n d

r3 . T h u s , t h e m o d e l c a n

b e m o d i e d b y a s s u m i n g t h a t i n r o u n d s i x , t h e g a m b l e b e t w e e n 0 . 0 1 a n d

2 5 0 , 0 0 0 i s t h e s a m e a s t h e g a m b l e b e t w e e n o n e o f t h e o t h e r s m a l l v a l u e s

( 0 . 1 0 , 1 , 1 0 , 1 0 0 , 2 5 0 , 5 0 0 , 7 5 0 , 1 0 0 0 ) a n d 2 5 0 , 0 0 0 . N o w , i n s t e a d

o f l o o k i n g a t t h e g a m b l e i t s e l f , o n e c a n l o o k a t t h e e x p e c t e d v a l u e o f t h e

g a m b l e . H e n c e , t o n d

r3 , a l l g a m e s ( i n r o u n d s i x ) i n w h i c h t h e e x p e c t e d v a l u e

i s i n [ 1 2 5 0 0 0 , 1 2 7 5 0 0 ] a r e l o o k e d a t , a l o n g w i t h t h e c o r r e s p o n d i n g o e r ( s e e

a p p e n d i x 1 a ) .

T h e s i g n i c a n t v a r i a t i o n i n t h e o e r s f r o m o n e c o n t e s t a n t t o o t h e r s u g g e s t s

t h a t t h e r e a r e f a c t o r s o t h e r t h a n t h e e x p e c t e d v a l u e t h a t T h e B a n k e r t a k e s

i n t o a c c o u n t w h e n m a k i n g a n o e r . T h e s e f a c t o r s a r e d i s c u s s e d l a t e r . N o w , i n

o r d e r t o n d a g e n e r a l e s t i m a t i o n o f t h e u t i l i t y o f m o n e y a s s u m e d b y T h e

B a n k e r t h e a v e r a g e o f t h e o e r s i s c a l c u l a t e d . T h i s v a l u e i s 7 8 4 2 3 . 2 0 a n d i s

a s s i g n e d t o

r3 . H e n c e :

U(78423.20) = 12

I t i s i m p o r t a n t t o n o t e t h a t t h e o e r i s i n c l u d e d i n t h i s c a l c u l a t i o n r e g a r d l e s s

o f w h e t h e r o r n o t i t i s a c c e p t e d b y t h e c o n t e s t a n t . T h i s i s b e c a u s e t h e

c o n t e s t a n t s ' m o n e t a r y u t i l i t y i s b e i n g e s t i m a t e d f r o m t h e p o i n t o f v i e w o f T h e

B a n k e r .

F o l l o w i n g t h i s , a n

r4 m u s t b e f o u n d s u c h t h a t U(r4) =14 . S i n c e , f o r

P = 12r1 +12r3 :

14 =

12U(r1) +

12U(r3) = E

p[U(r))]

T h i s i s t h e e q u i v a l e n t t o n d i n g t h e r e w a r d

r4w h i c h i s v a l u e d a s h i g h l y a s t h e

g a m b l e

12

r1 +12

r3 . P r o c e e d i n g u s i n g t h e s a m e m e t h o d a s b e f o r e , n o g a m b l e

c a n b e f o u n d i n w h i c h a c o n t e s t a n t h a s a p r o b a b i l i t y o f

12 o f w i n n i n g 0 . 0 1

a n d a p r o b a b i l i t y o f

12 o f w i n n i n g 7 8 , 4 2 3 . 2 0 . H o w e v e r , g a m b l e s c a n b e f o u n d

i n r o u n d s i x w h e r e t h e e x p e c t e d w i n n i n g i s r o u g h l y

12

0.01 + 1278423.20 = 39211.61 .

U s i n g t h e s a m e m e t h o d a s b e f o r e , e x p e c t e d v a l u e s a n d t h e i r e q u i v a l e n t o e r s

c a n b e f o u n d i n t h e r a n g e [ 3 7 5 0 0 , 4 0 0 0 0 ] ( s e e a p p e n d i x 1 b ) . I n t h i s i n s t a n c e

t h e a v e r a g e o f T h e B a n k e r ' s o e r s i s 2 5 2 4 5 . 7 9 a n d h e n c e :

U(25245.79) = 14

7


8/46

P r o c e e d i n g i n t h e s a m e w a y , a n r5 c a n b e f o u n d s u c h t h a t U(r5) =18 . S i n c e ,

f o r

P =1

2(r1) +1

2(r4):

18

= 12

U(r1) +12

U(r4)

A g a i n , t h i s i s t h e e q u i v a l e n t t o n d i n g t h e r e w a r d

r5 w h i c h i s v a l u e d a s h i g h l y

a s t h e g a m b l e

12(r1) +

12(r4). A s b e f o r e , t h e e x p e c t e d w i n n i n g s o f t h e g a m b l e

a r e f o u n d . T h i s t i m e t h e r e s u l t i s :

12

(0.01)+ 12

(25245.79) = 12622.90. T h u s , a l l

g a m b l e s i n r o u n d s i x w i t h e x p e c t e d w i n n i n g s i n t h e r a n g e [ 1 0 0 0 0 , 1 3 0 0 0 ] a r e

f o u n d a l o n g w i t h T h e B a n k e r ' s o e r s ( s e e a p p e n d i x 1 c ) . T h e a v e r a g e o e r ( r5 )

i s 6 9 3 9 . 6 2 , i m p l y i n g t h a t :

U(6939.31) = 18

r6

( s u c h t h a t

U(

r6) =

1

6) c a n b e f o u n d i n m u c h t h e s a m e w a y , u s i n g t h e f a c t

t h a t f o r P = 12(r1) +12(r5) :

116 =

12U(r1) +

12U(r5)

I n t h i s c a s e , t h e e x p e c t e d w i n n i n g o f t h e g a m b l e i s

12

(0.01) + 12 (6939.61) = 3469.81 a n d a l l r o u n d s i x g a m b l e s i n t h e r a n g e [ 2 7 5 0 , 4 0 0 0 ] a r e f o u n d ( s e e a p p e n d i x 1 d ) , y i e l d i n g a n a v e r a g e o e r o f 2 7 5 0 . S o :

U(2750) = 116

F i n a l l y , a n

r7 ( s u c h t h a t U(r7) =132 ) i s f o u n d . S o , f o r

P = 12

(r1) +12

(r6):

132

= 12

U(r1) +12

U(r6)

H e r e , t h e g a m b l e h a s a n e x p e c t e d w i n n i n g o f :

12

(0.01) + 12

(2750) = 1375.005.

H e n c e , t h e a v e r a g e o f T h e B a n k e r ' s o e r s f o r e x p e c t e d w i n n i n g s ( f o r r o u n d s i x )

i n t h e r a n g e [ 1 3 0 0 , 1 6 0 0 ] c a n b e f o u n d t o b e 1 3 4 2 . 8 3 ( s e e a p p e n d i x 1 e ) . T h u s :

U(1342.83) = 132

2 . 2 F i t t i n g M o d e l s t o t h e D a t a G a t h e r e d

I n o r d e r t o t a p p r o p r i a t e m o d e l s t o t h e d a t a g a t h e r e d , t h e R s t a t i s t i c a l s o f t w a r e

p a c k a g e i s u s e d . T h e g r a p h b e l o w i s a s i m p l e p l o t o f t h e e i g h t p o i n t s w h i c h w e r e

d e t e r m i n e d i n t h e p r e v i o u s s e c t i o n :

8


9/46

F i g u r e 5

T h e a b o v e p o i n t s e x h i b i t t h e c o n c a v e c u r v a t u r e t y p i c a l o f a r i s k - a v e r s e u t i l i t y

f u n c t i o n . D u e t o t h i s c u r v a t u r e , t h r e e m o d e l s h a v e b e e n t t e d t o t h e d a t a ( s e e

a p p e n d i x 2 a f o r t h e R c o d e ) . T h e s e t h r e e m o d e l s c a n b e s e e n b e l o w :

1 . R o o t m o d e l : U(x) = x +

x +

2 . L o g a r i t h m i c m o d e l : U(x) = x + ln(x) +

3 . Q u a d r a t i c m o d e l : U(x) = x2 + x +

U p o n t h e a p p l i c a t i o n o f t h e s e m o d e l s t h e f o l l o w i n g c o e c i e n t s a n d R - S q u a r e d

v a l u e s w e r e o b t a i n e d :

M o d e l R - S q u a r e d

R o o t 8 . 6 4 4 e - 0 7 1 . 6 3 1 e - 0 3 - 1 . 8 7 6 e - 0 2 0 . 9 9 7 6

L o g a r i t h m i c 3 . 6 6 8 e - 0 6 1 . 0 5 5 e - 0 2 2 . 1 2 4 e - 0 2 0 . 9 6 2 7

Q u a d r a t i c - 1 . 1 8 4 e - 1 1 6 . 7 6 3 e - 0 6 4 . 3 5 5 e - 0 2 0 . 9 9 1 6

F i g u r e 6 : C o e c i e n t s a n d R - S q u a r e d v a l u e s o f t h e t h r e e t t e d m o d e l s

T h e f a c t t h a t t h e r o o t m o d e l h a s t h e R - S q u a r e d v a l u e c l o s e s t t o 1 i n d i c a t e s

t h a t i t i s t h e m o s t a c c u r a t e t f o r t h e d a t a . T h i s , h o w e v e r , i s n o t s u c i e n t

e v i d e n c e . I t i s p r u d e n t a t t h i s p o i n t t o l o o k s a t t h e A n s c o m b e p l o t s f o r e a c h o f

t h e t h r e e m o d e l s ( s e e a p p e n d i x 3 a ) . T h e s e p l o t s g i v e f u r t h e r w e i g h t t o t h e

n o t i o n t h a t t h e r o o t m o d e l i s t h e s t r o n g e s t a s t h e p o i n t s a p p e a r t o b e t h e

m o s t r a n d o m l y d i s t r i b u t e d . A d d i t i o n a l l y t h e p l o t d e m o n s t r a t e s t h e l e a s t

h e t e r o s c e d a s t i c i t y o f t h e t h r e e , w h i c h c a n b e s e e n b y t h e s l i g h t f a n n i n g i n t h e

o t h e r t w o p l o t s . T h e r e f o r e , i t c a n b e s a f e l y r e a s o n e d t h a t t h e t h e r o o t m o d e l

t s t h e d a t a t h e m o s t a c c u r a t e l y . T h e f o l l o w i n g g r a p h s h o w s t h e m o d e l a s

a x e d t o g u r e 5 ( s e e n a b o v e ) :

9


10/46


11/46

U n f o r t u n a t e l y , t h e r e i s i n s u c i e n t d a t a f o r q u a n t i f y i n g T h e B a n k e r ' s

p e r c e p t i o n o f t h e v a l u e f o r a m a l e u t i l i t y o f

3

4. T h i s i s d u e t o t h e t i m e

c o n s t r a i n t s , a s p r e v i o u s l y m e n t i o n e d . I t i s , h o w e v e r , s t i l l p o s s i b l e t o t m o d e l s

t o t h e s e u t i l i t i e s , u s i n g t h e s a m e m e t h o d a s b e f o r e ( s e e a p p e n d i x 2 b ) . T h e

s a m e t h r e e m o d e l s h a v e b e e n t t o t h e d a t a a n d t h e i r c o e c i e n t s a n d

R - S q u a r e d v a l u e s a r e a s f o l l o w s :

S u m m a r y o f t h e T h r e e F i t t e d M o d e l s f o r M a l e C o n t e s t a n t s

M o d e l

R - S q u a r e d

R o o t 1 . 0 7 0 e - 0 6 1 . 4 9 7 e - 0 3 - 1 . 3 8 3 e - 0 2 0 . 9 9 8 9

L o g a r i t h m i c 3 . 5 6 0 e - 0 6 9 . 5 7 9 e - 0 3 2 . 1 8 7 e - 0 2 0 . 9 6 7 1

Q u a d r a t i c - 1 . 3 0 6 e - 1 1 7 . 1 0 2 e - 0 6 3 . 9 0 3 e - 0 2 0 . 9 9 0 7

S u m m a r y o f t h e T h r e e F i t t e d M o d e l s f o r F e m a l e C o n t e s t a n t s

M o d e l

R - S q u a r e d

R o o t - 1 . 4 6 9 e - 0 7 2 . 1 6 3 e - 0 3 - 2 . 8 9 7 e - 0 2 0 . 9 9 3 9

L o g a r i t h m i c 3 . 5 8 0 e - 0 6 1 . 3 7 7 e - 0 2 2 . 2 2 7 e - 0 2 0 . 9 2 6 5

Q u a d r a t i c - 1 . 6 6 0 e - 1 1 7 . 9 4 3 e - 0 6 4 . 8 3 5 e - 0 2 0 . 9 8 8

F i g u r e s 9 + 1 0 : C o e c i e n t s a n d R - s q u a r e d v a l u e s o f t h e t h r e e t t e d m o d e l s

w i t h g e n d e r c a t e g o r i s a t i o n .

S i m i l a r l y t o t h e p r e v i o u s m o d e l s , i t i s a p p a r e n t t h a t ( b a s e d o n t h e R - S q u a r e d

v a l u e ) t h e r o o t m o d e l o e r s t h e m o s t a t t r a c t i v e t . O n c e a g a i n , t h o u g h , t h e

A n s c o m b e p l o t s m u s t b e s t u d i e d f o r a m o r e a c c u r a t e a n a l y s i s ( s e e a p p e n d i x

3 b ) . I t c a n b e s e e n t h a t t h e s e p l o t s a r e v e r y s i m i l a r t o t h e i r c o u n t e r p a r t s i n

a p p e n d i x 3 a . i . e . T h e s q u a r e - r o o t m o d e l s s h o w s t h e m o s t r a n d o m n e s s f o r b o t h

g e n d e r s , w h i l e t h e l o g a r i t h m i c a n d q u a d r a t i c m o d e l s d i s p l a y t h e a r c - s h a p e d

f a n n i n g a s b e f o r e . T h u s , i n b o t h c a s e s , t h i s i s t h e c h o s e n m o d e l . F o r v i s u a l

c o m p a r i s o n , t h e b e l o w i m a g e s h o w s b o t h o f t h e s e u t i l i t y f u n c t i o n s :

1 1


12/46

F i g u r e 1 1 : A c o m p a r i s o n o f t h e u t i l i t y f u n c t i o n s a c r o s s g e n d e r .

F i g u r e 1 1 p r o v i d e s e v i d e n c e t h a t T h e B a n k e r a s s u m e s t h a t w o m e n a r e m o r e

r i s k - a v e r s e t h a n m e n . I t s u g g e s t s t h a t f o r a n y g i v e n v a l u e h e a s s u m e s t h a t t h e

a v e r a g e w o m e n w i l l h a v e a h i g h e r u t i l i t y t h a n t h e a v e r a g e m a n . T h i s c o i n c i d e s

w i t h t h e r e s u l t s o f J i a n a k o p l o s a n d B e r n a s e k ( 1 9 9 8 ) .

2 . 3 . 2 A g e A n a l y s i s

T h e t h r e e a g e g r o u p s i n t o w h i c h t h e c o n t e s t a n t s h a v e b e e n d i v i d e d a r e : Y o u n g e r

t h a n 3 5 , 3 5 t o 6 5 a n d 6 5 p l u s . A p p l y i n g t h e b a c k w a r d s p r o c e s s o n c e a g a i n

p r o v i d e s t h e f o l l o w i n g e s t i m a t e d u t i l i t i e s :

U t i l i t y V a l u e ( )

Age < 35 35 Age 65 65 < Age0 0 . 0 1 0 . 0 1 0 . 0 1

132 N / A 1 2 1 6 . 7 5 N / A 116 1 8 5 0 . 0 0 3 0 7 5 . 0 0 N / A 18 6 7 0 0 . 0 0 6 4 8 3 . 3 3 5 6 2 5 . 0 0 14 2 0 6 2 5 . 0 0 2 1 7 5 0 . 0 0 1 8 2 5 0 . 0 0 12 7 1 0 0 0 . 0 0 8 0 0 0 3 . 5 6 3 3 3 3 . 3 3 34

N / A 1 2 0 0 0 0 . 0 0 N / A

1 2 5 0 0 0 0 . 0 0 2 5 0 0 0 0 . 0 0 2 5 0 0 0 0 . 0 0

F i g u r e 1 2 : U t i l i t y o f c o n t e s t a n t s i n r o u n d s i x ( f r o m S e a s o n 4 ) w i t h r e s p e c t t o

a g e .

1 2


13/46

A s w i t h t h e g e n d e r a n a l y s i s , t h e r e i s i n s u c i e n t d a t a t o g a i n e s t i m a t i o n o f

u t i l i t y a t a l l p o i n t s f o r t h e l e s s c o m m o n a g e g r o u p s . N o n e t h e l e s s , t h e

b a c k w a r d s p r o c e s s c a n s t i l l b e a p p l i e d . T h e t h r e e c h o s e n m o d e l s a r e t t e d

o n c e a g a i n ( s e e a p p e n d i x 2 c ) , p r o v i d i n g t h e f o l l o w i n g c o e c i e n t s a n d

R - S q u a r e d v a l u e s :

S u m m a r y o f t h e T h r e e F i t t e d M o d e l s f o r Y o u n g C o n t e s t a n t s


R o o t 5 . 9 6 1 e - 0 7 1 . 7 2 4 e - 0 3 - 9 . 3 0 7 e - 0 3 0 . 9 9 9 4

L o g a r i t h m i c 3 . 3 7 0 e - 0 6 1 . 1 8 6 e - 0 2 3 . 6 8 1 e - 0 2 0 . 9 6 1 2

Q u a d r a t i c - 1 . 5 8 2 e - 1 1 7 . 7 5 1 e - 0 6 4 . 9 9 1 e - 0 2 0 . 9 8 6 6

S u m m a r y o f t h e T h r e e F i t t e d M o d e l s f o r M i d d l e - A g e d C o n t e s t a n t s


R o o t 3 . 6 3 2 e - 0 7 1 . 9 1 5 e - 0 3 - 2 . 8 5 6 e - 0 2 0 . 9 8 7 7

L o g a r i t h m i c 3 . 6 6 6 e - 0 6 1 . 2 0 3 e - 0 2 2 . 1 4 0 e - 0 2 0 . 9 3 4 4

Q u a d r a t i c 7 . 4 8 4 e - 0 6 - 1 . 4 5 9 e - 1 1 4 . 1 3 3 e - 0 2 0 . 9 8 7 4

S u m m a r y o f t h e T h r e e F i t t e d M o d e l s f o r E l d e r l y C o n t e s t a n t s


R o o t 1 . 6 5 9 e - 0 7 1 . 9 4 0 e - 0 3 - 9 . 3 4 2 e - 0 3 0 . 9 9 8 8

L o g a r i t h m i c 3 . 1 4 4 e - 0 6 1 . 4 3 1 e - 0 2 5 . 6 1 4 e - 0 2 0 . 9 5 7 6

Q u a d r a t i c - 1 . 9 2 1 e - 1 1 8 . 5 8 8 e - 0 6 5 . 2 7 8 e - 0 2 0 . 9 8 0 9

F i g u r e 1 3 + 1 4 + 1 5 : C o e c i e n t s a n d R - s q u a r e d v a l u e s o f t h e t h r e e t t e d

m o d e l s w i t h a g e c a t e g o r i s a t i o n .

F o r a t h i r d t i m e , t h e r o o t m o d e l d e m o n s t r a t e s t h e s t r o n g e s t t w i t h R - s q u a r e s

v a l u e s c l o s e s t t o 1 f o r e a c h o f t h e t h r e e a g e g r o u p s . S i m i l a r l y , t h e i r A n s c o m b e

p l o t s ( s e e a p p e n d i x 3 c ) e x h i b i t t h e m o s t r a n d o m n e s s w h i l e t h e A n s c o m b e p l o t s

f o r t h e l o g a r i t h m i c a n d q u a d r a t i c m o d e l s s h o w s i m i l a r p a t t e r s t o b e f o r e ,

i n d i c a t i n g a n u n s o u n d t . T h e g r a p h b e l o w s h o w s t h e t h r e e e s t i m a t e d u t i l i t y

f u n c t i o n s :

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F i g u r e 1 6 : A c o m p a r i s o n o f t h e u t i l i t y f u n c t i o n s a c r o s s a g e .

T h e s e e s t i m a t e d u t i l i t y f u n c t i o n s s u g g e s t t h a t f o r a n y n o n - l a r g e v a l u e t h e

u t i l i t y i n c r e a s e s w i t h a g e . T h i s i m p l i e s t h a t T h e B a n k e r a s s u m e s t h a t

r i s k - a v e r s i o n i n c r e a s e s w i t h a g e ; a n a s s u m p t i o n t h a t h o l d w i t h t h e r e s u l t s o f

S i n h a ( 1 9 9 2 ) . T h e g r a p h a l s o i m p l i e s t h a t f o r l a r g e - s u m s T h e B a n k e r a s s u m e s

t h a t t h e e l d e r l y c o n t e s t a n t s b e c o m e m o r e r i s k - l o v i n g ( t h a n t h e m i d d l e - a g e d

c o n t e s t a n t s ) .

3 A l t e r n a t e M o d e l l i n g S t r a t e g i e s

3 . 1 P r o s p e c t T h e o r y i n D e a l o r N o D e a l

E x p e c t e d u t i l i t y t h e o r y h a s l o n g b e e n u s e d f o r t h e a n a l y s i s o f d e c i s i o n m a k i n g

u n d e r r i s k s i n c e i t s i n t r o d u c t i o n b y V o n N e u m a n n a n d M o r g e n s t e r n i n 1 9 4 4 .

H o w e v e r , p r o s p e c t t h e o r y i n t r o d u c e s a n a l t e r n a t i v e m o d e l o f h u m a n b e h a v i o u r

u n d e r r i s k . I t h a s b e e n s t a t e d t h a t p e o p l e u n d e r w e i g h o u t c o m e s t h a t a r e

m e r e l y p r o b a b l e i n c o m p a r i s o n w i t h o u t c o m e s t h a t a r e o b t a i n e d w i t h

c e r t a i n t y ( K a h n e m a n a n d T v e r s k y 2 0 0 0 , p . 1 7 ) . F o r e x a m p l e , i f a n u m b e r o f

i n d i v i d u a l s a r e o e r e d t h e c h a n c e t o r e c e i v e 5 , 0 0 0 w i t h c e r t a i n t y , o r w i n

e i t h e r 0 o r 1 0 0 , 0 0 0 w i t h a c o i n i p , n o t e v e r y o n e w i l l c h o o s e t o p a r t a k e t h e

i n i t i a l o e r . T h i s b e h a v i o u r i s k n o w n a s t h e c e r t a i n t y e e c t , i t c o n t r i b u t e s t o

r i s k a v e r s i o n i n c h o i c e s i n v o l v i n g s u r e g a i n s a n d t o r i s k s e e k i n g i n c h o i c e s

i n v o l v i n g s u r e l o s e s ( K a h n e m a n a n d T v e r s k y 2 0 0 0 , p . 1 7 ) . A m o d e l h a s b e e n

d e v e l o p e d w h e r e i n a v a l u e i s a s s i g n e d t o g a i n s a n d l o s e s r a t h e r t h a n t o n a l

a s s e t s a n d i n w h i c h p r o b a b i l i t i e s a r e r e p l a c e d b y d e c i s i o n w e i g h t s :

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F i g u r e 1 7 : A g r a p h i c a l r e p r e s e n t a t i o n o f p r o s p e c t t h e o r y ( K a h n e m a n a n d

T v e r s k y 2 0 0 0 , p . 3 )

T h i s g r a p h s h o w s h o w p r o s p e c t t h e o r y p l a y s a p a r t i n l o s s e s a n d g a i n s . T h e

v a l u e f u n c t i o n i s c o n c a v e ( r i s k - a v e r s e ) w i t h r e s p e c t t o g a i n s a n d c o n v e x

( r i s k - l o v i n g ) w i t h r e s p e c t t o l o s e s .

T h i s t h e o r y c a n b e a p p l i e d t o D e a l o r N o D e a l . D e c i s i o n s m a d e b y c o n t e s t a n t s

f o l l o w a c e r t a i n p a t t e r n w h e n e i t h e r l o s s e s o r g a i n s a r e m a d e . T h i s t h e o r y

c o i n c i d e s w i t h t h e i d e a t h a t t h e a s s u m e d u t i l i t y o f t h e c o n t e s t a n t s i s a d a p t e d

b a s e d o n t h e p r e v i o u s d e c i s i o n s t h a t t h e y m a k e ; t h e c o n t e s t a n t s ' u t i l i t y

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

p r e f e r e n c e s o n t h e s p o t .

O f c o u r s e , g a i n s a n d l o s s e s a r e d e n e d r e l a t i v e t o s o m e n e u t r a l r e f e r e n c e p o i n t

w h i c h v a r i e s a c r o s s c o n t e s t a n t s . I n t h e i r p a p e r , P o s t e t a l . ( 2 0 0 8 ) a s s e r t t h a t

s i n c e D e a l o r N o D e a l c o n t e s t a n t s d o n o t h a v e t o p a y a n y t h i n g t o p l a y t h e

g a m e , i t c a n b e a s s u m e d t h a t t h e i r i n i t i a l w e a l t h i s z e r o a n d t h a t t h e y d o n o t

l o s e a n y t h e i r o w n w e a l t h . T h u s , i t c a n b e a s s u m e d t h a t l o s s e s a r e m a d e w h e n

t h e m a j o r i t y o f t h e h i g h b o x v a l u e s h a v e b e e n e l i m i n a t e d a n d g a i n s a r e

a s s u m e d t o b e m a d e w h e n t h e o p p o s i t e o c c u r s . I n o t h e r w o r d s , t h e

c o n t e s t a n t ' s w e a l t h w i l l n o w b e t h e p r o s p e c t i v e m o n e y g a i n e d f r o m t h e g a m e .

T h i s t h e o r y c a n b e s e e n q u a n t i t a t i v e l y f r o m d a t a o n t h e s h o w . A c a s e s t u d y

h a s b e e n m a d e f o r t w o c o n t e s t a n t s s h o w i n g t h e c h a n g e i n r i s k .

F i g u r e 1 8 s h o w s t h e g a m b l e s m a d e b y A n d y o n t h e 876th e p i s o d e , b r o a d c a s t o n t h e

7thN o v e m b e r 2 0 0 8 . T h e ` x ' r e p r e s e n t s t h e v a l u e s l e f t o n t h e b o a r d . F o r

e a c h r o u n d , t h e t a b l e s h o w s t h e a v e r a g e p r i z e , T h e B a n k e r ' s o e r , t h e o e r a s

a p e r c e n t a g e o f t h e a v e r a g e p r i z e , a n d t h e c o n t e s t a n t ' s d e c i s i o n :

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F i g u r e 1 8 : A s u m m a r y o f t h e 876th g a m e o f D e a l o r N o D e a l .

W e c a n s e e h e r e t h a t A n d y m a i n t a i n s a h i g h e x p e c t e d v a l u e f o r e a c h r o u n d .

W e a s s u m e t h a t A n d y g o e s i n t o t h e g a m e r i s k - a v e r s e l i k e t h e g e n e r a l

p o p u l a t i o n . H e t a k e s s u c c e s s f u l g a m b l e s b y e l i m i n a t i n g r e l a t i v e l y s m a l l e r

v a l u e s , h e n c e m a k i n g g a i n s . T o w a r d s t h e 5th a n d 6th r o u n d , A n d y i s l e f t w i t h a t o u g h d e c i s i o n : t o t a k e 5 0 , 0 0 0 w i t h c e r t a i n t y o r g a m b l e t o w i n e i t h e r

1 0 , 0 0 0 o r 2 5 0 , 0 0 0 . A n d y m a k e s t h e f o r m e r c h o i c e a n d g o e s h o m e w i t h

5 0 , 0 0 0 . T h i s i s c o n s i s t e n t w i t h t h e a s s u m p t i o n t h a t p e o p l e c h a n g e t h e i r r i s k

t h r o u g h o u t t h e g a m e . W e c a n s e e t h a t A n d y b e c o m e s m o r e r i s k - a v e r s e b a s e d

o n h o w t h e g a m e p l a y s o n a n d t h e v a l u e s l e f t o n t h e b o a r d . T h e f o l l o w i n g

t a b l e s h o w s t h e g a m b l e s m a d e b y S t e v e f r o m t h e

766the p i s o d e o n t h e

3rdo f

J u n e 2 0 0 8 :

F i g u r e 1 9 : A s u m m a r y o f t h e

766thg a m e o f D e a l o r N o D e a l .

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T h r o u g h t h e c o u r s e o f t h e r s t f o u r r o u n d s , S t e v e e l i m i n a t e s a s i g n i c a n t

n u m b e r o f l a r g e v a l u e s o n t h e b o a r d ( a s s u m e d t o b e a l o s s ) b u t s t i l l m a n a g e s

t o k e e p t h e t o p t w o p r i z e s i n p l a y . D e s p i t e o e r s o f 1 0 , 0 0 0 a n d 2 5 , 0 0 0 ,

S t e v e t o o k t h e r i s k a n d c o n t i n u e d t h e g a m e i n t h e h o p e o f w i n n i n g a l a r g e r

s u m o f m o n e y . H o w e v e r , i n t h e f t h r o u n d h e e l i m i n a t e d t h e s e v a l u e s a n d

d r a m a t i c a l l y r e d u c e d h i s e x p e c t e d w i n n i n g s . I n t h e n a l r o u n d , S t e v e t o o k t h e

g a m b l e b e t w e e n 0 . 1 a n d 3 0 0 0 , w i n n i n g t h e l a t t e r . T h i s c a s e s t u d y s h o w s

t h a t S t e v e w h o l o s e s m o s t o f t h e v a l u e s o n t h e b o a r d c o n t i n u e s t o t a k e r i s k s ,

t h u s d e c r e a s i n g h i s r i s k a v e r s i o n .

3 . 2 C o n s t a n t R e l a t i v e R i s k A v e r s i o n

A s p r e v i o u s l y s t a t e d , c o n t e s t a n t s a r e l i k e l y t o b e r i s k - a v e r s e . W e c a n t h e r e f o r e

a s s u m e t h a t a g i v e n c o n t e s t a n t h a s t h e C R R A ( C o n s t a n t R e l a t i v e R i s k A v e r s i o n )

u t i l i t y f u n c t i o n . T h e C R R A u t i l i t y f u n c t i o n i s a l s o k n o w n a s t h e C E S ( C o n s t a n t

E l a s t i c i t y o f S u b s t i t u t i o n ) u t i l i t y f u n c t i o n a n d t a k e s t h e f o r m :

U(x) = X(1)

(1) w h e r e 0

A c o n s t a n t , , c a n t h e r e f o r e b e o b t a i n e d i n o r d e r t o e v a l u a t e t h e u t i l i t y o v e r

d i e r e n t v a r i a b l e s . T h e c o e c i e n t o f r i s k a v e r s i o n

d e s c r i b e s o n e ' s w i l l i n g n e s s

t o t a k e t h e r i s k . W h e n 0 < i t c a n b e s e e n t h a t t h e c o n t e s t a n t i s r i s k - a v e r s e a n d w h e n = 0 t h e c o n t e s t a n t i s r i s k n e u t r a l .

I f T h e B a n k e r ' s o e r i s b e l o w t h e e x p e c t e d w i n n i n g t h e n t h e c o n t e s t a n t s w i l l

m a k e c a u t i o u s d e c i s i o n s w h e n d e t e r m i n i n g w h e t h e r o r n o t t o a c c e p t t h e o e r .

I f T h e B a n k e r ' s o e r i s e x a c t l y e q u a l t o t h e e x p e c t e d w i n n i n g s t h e n i t c a n b e

s e e n t h a t T h e B a n k e r v i e w s t h e c o n t e s t a n t a s r i s k n e u t r a l . A n e x a m p l e o f t h i s

c a n b e s e e n i n t h e 765th g a m e : T h e c o n t e s t a n t , L e n , i n r o u n d s i x w a s o e r e d a n a m o u n t e q u a l t o t h e e x p e c t e d v a l u e ; h e t o o k t h i s o e r . T h e r e s t r i c t i o n t h a t

0i s b a s e d o n t h e a s s u m p t i o n t h a t a l l c o n t e s t a n t s a r e r i s k - a v e r s e ( o r

n e u t r a l ) . U s i n g C R R A , a d y n a m i c p e r s p e c t i v e c a n b e a p p l i e d a s p e r D e R o o s

a n d S a r a d i s ( 2 0 0 6 ) :

1n

[U(X1) + U(X2) + ... + U(Xn)] U(Y)

w h e r e :

n i s t h e n u m b e r o f b o x e s l e f t i n t h e g a m e ,

Xi s t h e m o n e t a r y v a l u e w i t h

i ={

1, 2,...,n}

a n d Y i s T h e B a n k e r ' s o e r .

A s s u c h , t h e c o e c i e n t o f r i s k a v e r s i o n

c a n b e c a l c u l a t e d . W h e n c a l c u l a t i n g

, t h e i n e q u a l i t y i s a s s u m e s t o b e a n e q u a l i t y i n o r d e r t o o b t a i n t h e s m a l l e s t

v a l u e o f t h e c o e c i e n t o f r i s k - a v e r s i o n .

c a n n o w b e s e p a r a t e d w i t h r e s p e c t t o

d i e r e n t e x p l a n a t o r y v a r i a b l e s s u c h a s d i e r e n t a g e g r o u p a n d g e n d e r . T h u s

a i d i n g i n t h e d e t e r m i n a t i o n o f w h e t h e r t h e s e v a r i a b l e s a e c t T h e B a n k e r ' s

a s s u m p t i o n o f t h e c o n t e s t a n t ' s u t i l i t y , a s w e l l a s d e t e r m i n i n g w h e t h e r C R R A

p r o v i d e s a s u i t a b l e m o d e l i n t h e c o n t e x t o f t h e g a m e s h o w . T h e r e s u l t s o f t h i s

s e p a r a t i o n c a n b e s e e n b e l o w :

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F i g u r e 2 0 : C o e c i e n t o f r i s k - a v e r s i o n a c r o s s r o u n d s f o u r , v e a n d s i x w i t h

s e p a r a t i o n w i t h r e s p e c t t o g e n d e r .

T h e a v e r a g e c o e c i e n t o f r e l a t i v e r i s k a v e r s i o n v a r i e s f r o m r o u n d t o r o u n d .

T h i s s h o w s t h a t t h e u t i l i t y f u n c t i o n o f t h e c o n t e s t a n t s i s n o t c o n s t a n t . T h i s

m o d e l a l s o d i e r s f r o m t h e g e n e r a l a s s u m p t i o n t h a t f e m a l e s a r e m o r e r i s k

a v e r s e t h a n m a l e s a s s h o w n i n t h e p r e v i o u s m o d e l ( J i a n a k o p l o s a n d B e r n a s e k

1 9 9 8 ) .

F i g u r e 2 1 : C o e c i e n t o f r i s k - a v e r s i o n a c r o s s r o u n d s f o u r , v e a n d s i x w i t h

s e p a r a t i o n w i t h r e s p e c t t o a g e .

T h e a b o v e s h o w s t h a t , i n e a c h o f t h e s e t h r e e r o u n d s , i n c r e a s e s b e t w e e n t h e

y o u n g c o n t e s t a n t s a n d t h e m i d d l e - a g e d s o n e s c o n t e s t a n t s a n d d e c r e a s e s

b e t w e e n t h e m i d d l e - a g e d a n d e l d e r l y c o n t e s t a n t s . T h i s c o n t r a d i c t s S i n h a

( 1 9 9 2 ) w h o s t a t e s t h a t , i n g e n e r a l , r i s k - a v e r s i o n i n c r e a s e s w i t h a g e a n d f u r t h e r

h i g h l i g h t s t h e f a i l u r e i n t t i n g o f C R R A m o d e l t o t h i s d a t a .

4 D i s c u s s i o n a n d C o n c l u s i o n

O v e r a l l , i t a p p e a r s t h a t t h e b a c k w a r d s m e t h o d t o e s t i m a t e t h e u t i l i t y

f u n c t i o n w o r k e d w e l l . I t c a n b e s e e n t h a t t h e r e s u l t s c o r r e l a t e w i t h t h e

p r e v i o u s l y m e n t i o n e d v a r i a t i o n s o f u t i l i t y a c r o s s a g e a n d g e n d e r ( J i a n a k o p l o s

a n d B e r n a s e k 1 9 9 8 ; S i n h a 1 9 9 2 ) . F u r t h e r m o r e , t h e r e s u l t s p r o v i d e s u c i e n t l y

s t r o n g e v i d e n c e t o d e m o n s t r a t e t h a t T h e B a n k e r c o n s i d e r s t h e c o n t e s t a n t s '

i n d i v i d u a l a t t r i b u t e s w h e n e s t i m a t i n g t h e i r u t i l i t y . H o w e v e r , t h e m o d e l i s n o t

w i t h o u t f a u l t a n d t h e r e a r e a n u m b e r o f i m p r o v e m e n t t h a t c o u l d b e m a d e .

M o s t n o t a b l y , a g r e a t e r q u a n t i t y o f s a m p l e d a t a w o u l d i m p r o v e t h e a c c u r a c y

o f t h e m o d e l ; e s p e c i a l l y w h e n d i v i d i n g t h e c o n t e s t a n t s i n t o s e p a r a t e c a t e g o r i e s .

T h i s w o u l d b e a s i m p l e i m p r o v e m e n t t o i m p l e m e n t h a d t h e r e b e e n m o r e t i m e

t o w a t c h e p i s o d e s o f t h e g a m e . A d d i t i o n a l l y , t h e p l a y e r s c o u l d h a v e b e e n

d i v i d e d b y a g r e a t e r n u m b e r o f a t t r i b u t e s . I t w o u l d b e r e l e v a n t t o e s t i m a t e

t h e i r s a l a r i e s a n d s u b d i v i d e t h e m i n t o a p p r o p r i a t e b a n d s o f w e a l t h . I t c a n b e

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

o f m o n e y s o i t i s r e a s o n a b l e t o a s s e r t t h a t T h e B a n k e r w i l l c o n s i d e r i t w h e n

m a k i n g h i s o e r s . A p p l y i n g t h i s , h o w e v e r , p r e s e n t s s o m e d i c u l t y a s

e s t i m a t i o n w o u l d b e r e q u i r e d i n d e t e r m i n i n g i n c o m e . B e y o n d t h i s , m o r e

m o d e l s c o u l d b e e n t t o t h e d a t a a n d f u r t h e r t e s t i n g o f t h e m o d e l s c o u l d b e

a p p l i e d . G i v e n m o r e t i m e , a n u m b e r o f o t h e r t h e o r i e s c o u l d b e u s e d t o i m p r o v e

1 8


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t h e e x i s t i n g m o d e l s . S u c h t h e o r i e s i n c l u d e : c o n s t a n t a b s o l u t e r i s k a v e r s i o n

( C A R A ) a n d i n c r e a s i n g r e l a t i v e r i s k a v e r s i o n ( I R R A ) . F i n a l l y , i t m i g h t b e

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

v a r i a b l e s ( a g e , g e n d e r , w e a l t h ) i n t h e d e t e r m i n a t i o n o f u t i l i t y f u n c t i o n s .

T o c o n c l u d e , i t a p p e a r s t h a t T h e B a n k e r b a s e s h i s o e r s o n a n a m a l g a m o f t h e

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

c o n t e s t a n t s p o s s e s s a n d t h e i r p r e v i o u s d e c i s i o n s . I t h a s a l s o b e e n s e e n t h a t t h e

r i s k y d e c i s i o n s i n D e a l o r N o D e a l p r o v i d e a u n i q u e o p p o r t u n i t y t o m o d e l t h e

m o r e c o m p l e x m o n e t a r y d e c i s i o n s o f e v e r y d a y l i f e .

1 9


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R e f e r e n c e s

B e r g e r , J . O . ( 1 9 8 0 ) S t a t i s t i c a l d e c i s i o n t h e o r y a n d B a y e s i a n a n a l y s i s , S e c o n d

E d i t i o n , N e w Y o r k : S p r i n g e r - V e r l a g

B e c k e r , G . e t a l . ( 1 9 6 4 ) M e a s u r i n g U t i l i t y b y a S i n g l e - R e s p o n s e S e q u e n t i a l

M e t h o d , B e h a v i o r a l S c i e n c e , 9 ( 2 ) , p p . 2 2 6 - 2 3 2 .

D e R o o s , N . & S a r a d i s , Y . ( 2 0 0 6 ) D e c i s i o n M a k i n g U n d e r R i s k i n D e a l o r N o

D e a l , M a r c h 2 0 0 6 , S y d n e y U n i v e r s i t y & Y a l e U n i v e r s i t y .

J i a n a k o p l o s , N . & B e r n a s e k , A . ( 1 9 9 8 ) A r e W o m e n M o r e R i s k A v e r s e ? ,

E c o n o m i c I n q u i r y , 3 6 ( 4 ) , O c t o b e r 1 9 9 8 , p p . 6 2 0 - 6 3 0 .

K a h n e m a n , D . & T v e r s k y , A . ( 2 0 0 0 ) C h o i c e s , V a l u e s , a n d F r a m e s , C a m b r i d g e :

C a m b r i d g e U n i v e r s i t y P r e s s

M e r t o n , R . ( 1 9 9 0 ) C o n t i n u o u s - T i m e F i n a n c e , B l a c k w e l l P u b l i s h e r s

P o s t , T . e t a l . ( 2 0 0 8 ) D e a l o r N o D e a l ? D e c i s i o n M a k i n g u n d e r R i s k i n a

L a r g e - P a y o G a m e S h o w , A m e r i c a n E c o n o m i c R e v i e w , ( 9 8 : 1 ) , M a r c h 2 0 0 8 ,

p p . 3 8 7 1 .

S i n h a , T . ( 1 9 9 2 ) A r e o l d e r p e o p l e m o r e r i s k a v e r s e ? , S c h o o l o f B u s i n e s s

D i s c u s s i o n P a p e r s , P a p e r 3 2 , B o n d U n i v e r s i t y .

2 0


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A p p e n d i x 1 a :

E x p e c t e d v a l u e s i n [ 1 2 5 0 0 0 , 1 2 7 5 0 0 ] a n d e q u i v a l e n t o e r s t a k e n f r o m r o u n d s i x :

E x p e c t e d V a l u e B a n k e r ' s O e r

1 2 7 5 0 0 . 0 0 8 0 0 0 0 . 0 0

1 2 7 5 0 0 . 0 0 9 0 0 0 0 . 0 0

1 2 7 5 0 0 . 0 0 1 0 1 0 0 0 . 0 0

1 2 6 5 0 0 . 0 0 6 2 0 0 0 . 0 0

1 2 6 5 0 0 . 0 0 8 0 0 0 0 . 0 0

1 2 6 5 0 0 . 0 0 7 5 0 0 0 . 0 0

1 2 6 5 0 0 . 0 0 5 0 0 0 0 . 0 0

1 2 6 5 0 0 . 0 0 7 5 0 0 0 . 0 0

1 2 6 5 0 0 . 0 0 4 5 0 0 0 . 0 0

1 2 6 5 0 0 . 0 0 6 0 0 0 0 . 0 0

1 2 5 5 0 0 . 0 0 8 0 0 0 0 . 0 0

1 2 5 5 0 0 . 0 0 4 5 0 0 0 . 0 0

1 2 5 3 7 5 . 0 0 1 2 0 0 0 0 . 0 0

1 2 5 3 7 5 . 0 0 1 0 0 0 0 0 . 0 0

1 2 5 3 7 5 . 0 0 8 5 0 0 0 . 0 0

1 2 5 3 7 5 . 0 0 9 0 0 0 0 . 0 0

1 2 5 2 5 0 . 0 0 8 0 0 0 0 . 0 0

1 2 5 2 5 0 . 0 0 9 2 0 0 0 . 0 0

1 2 5 2 5 0 . 0 0 7 0 0 0 0 . 0 0

1 2 5 2 5 0 . 0 0 8 5 0 0 0 . 0 0

1 2 5 0 5 0 . 0 0 1 4 5 0 0 0 . 0 0

1 2 5 0 5 0 . 0 0 1 0 5 0 0 0 . 0 0

1 2 5 0 5 0 . 0 0 8 4 0 0 0 . 0 0

1 2 5 0 2 5 . 0 0 8 5 0 0 0 . 0 0

1 2 5 0 2 5 . 0 0 7 0 0 0 7 . 0 0

1 2 5 0 0 5 . 0 0 5 2 0 0 0 . 0 0

1 2 5 0 0 5 . 0 0 1 4 5 5 5 . 0 0

1 2 5 0 0 5 . 0 0 1 1 0 0 0 0 . 0 0

1 2 5 0 0 2 . 5 0 7 5 0 0 0 . 0 0

1 2 5 0 0 2 . 5 0 7 5 5 0 0 . 0 0

1 2 5 0 0 2 . 5 0 9 9 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 1 1 0 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 7 5 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 9 1 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 5 0 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 6 5 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 4 0 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 9 1 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 8 4 0 0 0 . 0 0

1 2 5 0 0 0 . 5 0 9 6 0 0 0 . 0 0

1 2 5 0 0 0 . 2 5 8 8 0 0 0 . 0 0

1 2 5 0 0 0 . 2 5 1 0 5 0 0 0 . 0 0

1 2 5 0 0 0 . 0 5 1 2 0 0 0 0 . 0 0

2 1


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1 2 5 0 0 0 . 0 5 8 5 0 0 0 . 0 0

1 2 5 0 0 0 . 0 5 2 8 0 0 0 . 0 0

1 2 5 0 0 0 . 0 5 4 4 0 0 0 . 0 0

1 2 5 0 0 0 . 0 5 8 0 6 7 5 . 0 0

1 2 5 0 0 0 . 0 5 5 0 0 0 0 . 0 0

1 2 5 0 0 0 . 0 1 6 0 0 0 0 . 0 0

A v e r a g e O e r : 7 8 4 2 3 . 2 0

A p p e n d i x 1 b :

E x p e c t e d v a l u e s i n [ 3 7 5 0 0 , 4 0 0 0 0 ] a n d e q u i v a l e n t o e r s t a k e n f r o m r o u n d s i x :


3 9 0 0 0 . 0 0 2 3 0 0 0 . 0 0

3 9 0 0 0 . 0 0 3 3 0 0 0 . 0 0

3 8 0 0 0 . 0 0 2 7 0 0 0 . 0 0

3 7 8 7 5 . 0 0 2 6 0 0 0 . 0 0

3 7 8 7 5 . 0 0 2 0 0 0 0 . 0 0

3 7 8 7 5 . 0 0 2 8 5 0 0 . 0 0

3 7 8 7 5 . 0 0 2 5 0 0 0 . 0 0

3 7 7 5 0 . 0 0 2 4 0 0 0 . 0 0

3 7 7 5 0 . 0 0 3 0 0 0 0 . 0 0

3 7 7 5 0 . 0 0 1 8 0 0 0 . 0 0

3 7 7 5 0 . 0 0 2 8 0 0 0 . 0 0

3 7 7 5 0 . 0 0 2 6 1 0 0 . 0 0

3 7 7 5 0 . 0 0 2 7 0 0 0 . 0 0

3 7 7 5 0 . 0 0 3 2 0 0 0 . 0 0

3 7 6 2 5 . 0 0 1 2 0 0 0 . 0 0

3 7 6 2 5 . 0 0 2 2 0 0 0 . 0 0

3 7 5 5 0 . 0 0 2 6 0 0 0 . 0 0

3 7 5 5 0 . 0 0 1 9 9 6 0 . 0 0

3 7 5 5 0 . 0 0 2 4 5 0 0 . 0 0

3 7 5 2 5 . 0 0 2 5 0 0 0 . 0 0

3 7 5 2 5 . 0 0 3 7 0 0 0 . 0 0

3 7 5 2 5 . 0 0 3 1 5 0 0 . 0 0

3 7 5 2 5 . 0 0 3 0 0 0 0 . 0 0

3 7 5 0 5 . 0 0 3 0 0 0 0 . 0 0

3 7 5 0 5 . 0 0 1 3 5 0 0 . 0 0

3 7 5 0 5 . 0 0 1 9 9 9 0 . 0 0

3 7 5 0 5 . 0 0 2 5 0 0 0 . 0 0

3 7 5 0 5 . 0 0 2 1 0 0 0 . 0 0

3 7 5 0 2 . 5 0 3 0 0 0 0 . 0 0

3 7 5 0 2 . 5 0 3 1 1 0 0 . 0 0

3 7 5 0 2 . 5 0 3 0 0 0 0 . 0 0

3 7 5 0 2 . 5 0 2 0 0 0 0 . 0 0

2 2


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1 0 3 7 5 . 0 0 8 0 6 0 . 0 0

1 0 3 7 5 . 0 0 4 0 0 0 . 0 0

1 0 3 7 5 . 0 0 7 0 0 0 . 0 0

1 0 3 7 5 . 0 0 5 0 0 0 . 0 0

1 0 2 5 0 . 0 0 5 1 0 0 . 0 0

1 0 2 5 0 . 0 0 6 0 0 0 . 0 0

1 0 2 5 0 . 0 0 8 0 0 0 . 0 0

1 0 2 5 0 . 0 0 6 0 0 0 . 0 0

1 0 2 5 0 . 0 0 5 2 0 0 . 0 0

1 0 2 5 0 . 0 0 7 5 0 0 . 0 0

1 0 2 5 0 . 0 0 9 9 0 0 . 0 0

1 0 2 5 0 . 0 0 5 0 0 0 . 0 0

1 0 2 5 0 . 0 0 1 0 2 5 0 . 0 0

1 0 1 2 5 . 0 0 6 0 0 0 . 0 0

1 0 1 2 5 . 0 0 8 5 0 0 . 0 0

1 0 1 2 5 . 0 0 6 0 0 0 . 0 0

1 0 1 2 5 . 0 0 4 5 0 0 . 0 0

1 0 1 2 5 . 0 0 8 8 0 0 . 0 0

1 0 0 5 0 . 0 0 5 0 0 0 . 0 0

1 0 0 5 0 . 0 0 7 7 7 7 . 0 0

1 0 0 5 0 . 0 0 4 5 0 0 . 0 0

1 0 0 2 5 . 0 0 6 0 0 0 . 0 0

1 0 0 2 5 . 0 0 8 4 0 0 . 0 0

1 0 0 0 5 . 0 0 6 0 0 0 . 0 0

1 0 0 0 5 . 0 0 5 0 0 0 . 0 0

1 0 0 0 5 . 0 0 3 2 0 0 . 0 0

1 0 0 0 5 . 0 0 5 0 0 0 . 0 1

1 0 0 0 2 . 5 0 7 0 0 0 . 0 0

1 0 0 0 2 . 5 0 5 5 0 0 . 0 0

1 0 0 0 0 . 5 0 6 5 0 0 . 0 0

1 0 0 0 0 . 5 0 6 0 0 0 . 0 0

1 0 0 0 0 . 5 0 5 2 5 0 . 0 0

1 0 0 0 0 . 5 0 6 0 0 0 . 0 0

1 0 0 0 0 . 5 0 9 0 0 0 . 0 0

1 0 0 0 0 . 5 0 8 0 0 0 . 0 0

1 0 0 0 0 . 5 0 5 0 0 0 . 0 0

1 0 0 0 0 . 5 0 5 2 8 0 . 0 0

1 0 0 0 0 . 5 0 4 2 0 0 . 0 0

1 0 0 0 0 . 2 5 8 0 0 0 . 0 0

1 0 0 0 0 . 2 5 5 0 0 0 . 0 0

1 0 0 0 0 . 2 5 5 0 0 0 . 0 0

1 0 0 0 0 . 2 5 5 0 0 0 . 0 0

1 0 0 0 0 . 0 5 4 5 0 0 . 0 0

1 0 0 0 0 . 0 5 8 0 0 0 . 0 0

1 0 0 0 0 . 0 5 6 0 0 0 . 0 0

1 0 0 0 0 . 0 5 5 9 9 0 . 0 0

1 0 0 0 0 . 0 5 3 0 0 0 . 0 0

2 4


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1 0 0 0 0 . 0 5 8 5 0 0 . 0 0

1 0 0 0 0 . 0 1 6 9 0 0 . 0 0

1 0 0 0 0 . 0 1 9 0 0 0 . 0 0

1 0 0 0 0 . 0 1 8 5 0 0 . 0 0

1 0 0 0 0 . 0 1 3 0 0 0 . 0 0

1 0 0 0 0 . 0 1 3 0 0 0 . 0 0

1 0 0 0 0 . 0 0 9 0 0 0 . 0 0

1 0 0 0 0 . 0 0 9 0 0 0 . 0 0

1 0 0 0 0 . 0 0 8 5 0 0 . 0 0

1 0 0 0 0 . 0 0 9 9 9 9 . 0 0

A v e r a g e O e r : 6 9 3 9 . 6 1

A p p e n d i x 1 d :

E x p e c t e d v a l u e s i n [ 2 7 5 0 , 4 0 0 0 ] a n d e q u i v a l e n t o e r s t a k e n f r o m r o u n d s i x :


4 0 0 0 . 0 0 3 7 5 0

4 0 0 0 . 0 0 3 9 0 0

4 0 0 0 . 0 0 3 7 5 0

4 0 0 0 . 0 0 4 0 0 0

4 0 0 0 . 0 0 3 7 0 0

4 0 0 0 . 0 0 4 0 0 0

3 0 0 0 . 0 0 1 8 5 0

3 0 0 0 . 0 0 2 5 0 0

2 8 7 5 . 0 0 2 7 5 0

2 8 7 5 . 0 0 1 8 0 0

2 8 7 5 . 0 0 2 0 0 0

2 8 7 5 . 0 0 2 2 5 0

2 8 7 5 . 0 0 2 0 0 0

2 7 5 0 . 0 0 2 0 0 0

2 7 5 0 . 0 0 2 0 0 0

2 7 5 0 . 0 0 2 0 0 0

2 7 5 0 . 0 0 2 5 0 0

A v e r a g e O e r : 2 7 5 0

A p p e n d i x 1 e :

E x p e c t e d v a l u e s i n [ 1 3 0 0 , 1 6 0 0 ] a n d e q u i v a l e n t o e r s t a k e n f r o m r o u n d s i x :


1 5 5 0 . 0 0 9 5 0 . 0 0

2 5


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1 5 5 0 . 0 0 1 0 0 0 . 0 0

1 5 5 0 . 0 0 7 5 0 . 0 0

1 5 5 0 . 0 0 7 5 0 0 . 0 0

1 5 2 5 . 0 0 1 4 0 0 . 0 0

1 5 2 5 . 0 0 1 4 0 0 . 0 0

1 5 0 5 . 0 0 9 0 0 . 0 0

1 5 0 5 . 0 0 1 0 1 7 . 0 0

1 5 0 5 . 0 0 7 5 0 . 0 0

1 5 0 5 . 0 0 8 0 0 . 0 0

1 5 0 2 . 5 0 1 2 0 0 . 0 0

1 5 0 2 . 5 0 9 5 0 . 0 0

1 5 0 2 . 5 0 1 0 0 1 . 0 0

1 5 0 2 . 5 0 1 0 0 0 . 0 0

1 5 0 2 . 5 0 1 2 0 0 . 0 0

1 5 0 0 . 5 0 1 4 0 0 . 0 0

1 5 0 0 . 5 0 1 4 0 0 . 0 0

1 5 0 0 . 5 0 1 0 1 0 . 0 0

1 5 0 0 . 5 0 7 5 0 . 0 0

1 5 0 0 . 0 5 1 5 0 0 . 0 0

1 5 0 0 . 0 5 1 5 0 0 . 0 0

1 5 0 0 . 0 5 5 0 7 . 0 0

1 5 0 0 . 0 1 1 0 0 0 . 0 0

A v e r a g e O e r : 1 3 4 2 . 8 3

2 6


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R c o d e u s e d t o g e n e r a t e t h e t h r e e o r i g i n a l m o d e l s :

2 7


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A p p e n d i x 2 c :

R c o d e u s e d t o g e n e r a t e t h e t h r e e a g e b a s e d m o d e l s :

2 9


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3 0


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A n s c o m b e p l o t s f o r t h e t h r e e o r i g i n a l m o d e l s

1 . T h e r o o t m o d e l :

2 . T h e l o g a r i t h m i c m o d e l :

3 . T h e q u a d r a t i c m o d e l :

3 1


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A p p e n d i x 3 b :

A n s c o m b e p l o t s f o r t h e g e n d e r s p e c i c m o d e l s

A n s c o m b e p l o t s f o r s q u a r e - r o o t m o d e l s :

A n s c o m b e p l o t s f o r l o g a r i t h m i c m o d e l s :

A n s c o m b e p l o t s f o r q u a d r a t i c m o d e l s :

3 2


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A p p e n d i x 3 c :

A n s c o m b e p l o t s f o r t h e a g e s p e c i c m o d e l s

A n s c o m b e p l o t s f o r s q u a r e - r o o t m o d e l s :

3 3


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A n s c o m b e p l o t s f o r t h e l o g a r i t h m i c m o d e l s :

3 4


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

3 5


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3 6


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Appendix 4:

Deriving the utilities for each sub-category:

Suppose we have U (0.01) =0, and U (250K) =1, given 0.01 is the worst scenario and 250k is

the best scenario. As shown previously, we obtain r3such that

There is equal probability of winning 1p and 250k, so the expected winning is

Since r1 as 0.01 is very small compared to r2as 250k, we assume that the contestants utility

remains practically the same even if 0.01 is replaced by small values compared to 250K (such

as 0.1, 1, 10, 100, 250, 500, 750). So 125k is the lower bound of the reward r3.

Hence to find r3, we find all the games (in round 6) where the expected value is in

[125000,127500]

We then categorize the data into different gender and age group, below is the step-by-step

procedure to determine the utility value with different amount of money for each different

explanatory variable.

MaleHence to find r3 for male, we find all the games (in round 6) where the expected value is in

[125000,127500]

Bankers 6th Offer () Expected Value ()

80000 127500

75000 126500

80000 125500

100000 125375

70000 125250

70007 125025

Average Bankers offer = 79167.83

Hence

for male

As before, we can obtain r4 for male such that the expected winning in Round 6 is

Hence to find r4 for male, we find all the games (in round 6) where the expected value is in

[37500, 42500].

Bankers 6th Offer () Expected Value ()40000 42500


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41000 42500

22500 40000

23000 39000

28500 37875

12000 3762521000 37505

20000 37502.5

Average bankers offer = 26000

Hence

Applying the same method, we can obtain r5 for male such that the expected winning in Round

6 is

Hence to find r5 for male, we find all the games (in round 6) where the expected value is in

[10000, 13000]


4000 10375

10250 10250

4500 10050

9000 10000

Average bankers offer = 6937.50

Hence


6 is

Hence to find r6 for male, we find all the games (in round 6) where the expected winning in

[2800, 4000]


3900 4000

2250 2875


Hence


6 is

Hence to find r6 for female, we find all the games (in round 6) where the expected value is in

[1500, 2000]


1500 1625


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1017 1505

1500 1500.05

Hence 1339.00

In order to obtain U(r7)=3/4, we can perform the following operations to obtain r7 for male suchthat the expected winning in Round 6 is

From the data, there is no banker offer around average expecting winning of 164583.915.

Therefore we cant obtain r7 such that U(r7)=3/4.

Therefore for male, we have

Utility Amount of Money ()

0 0.01

1/2 79167.83

1/4 26000.00

1/8 6937.50

1/16 3075.00

1/32 1339.00

1 250000.00

FemaleHence to find r3 for female, we find all the games (in round 6) where the expected value is in

[125000, 127500]Bankers 6th Offer () Expected Value ()

90000 125375

80000 125250

50000 125000.5

50000 125000.1


Hence

for female

As before, we can obtain r4for female such that the expected winning in Round 6 is


[35000, 40000].


18000 37750


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13500 37505

22000 37500.5


Hence 17833.33

Applying the same method, we can obtain r5 for female such that the expected winning in

Round 6 is


[8900, 10500]


6750 10375

6000 10250

5000 10005

8000 10000.5

6500 10000.5

4200 10000.5


Hence


Round 6 is

Hence to find r6 for female, we find all the games (in round 6) where the expected winning in

[2550, 3100]


1850 3000

2150 2550

2000 2550

2200 2550


Hence


Round 6 is


[1000, 1600]


950 1550


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1400 1525

950 1502.5

1001 1502.5

Average banker offer = 1075.25

Hence 1075.25

In order to obtain U(r7)=3/4, we can perform the following operations to obtain r7 for female

such that the expected winning in Round 6 is


[155000, 165000]


120000 162500

Average banker offer = 120000

Hence 120000

Therefore for female, we have

Utility of Money Amount of Money ()

0 0.01

1/2 67500.00

1/4 17833.33

1/8 6075.00

1/16 2050.001/32 1075.25

3/4 120000.00

Age group 1 (age up to 35)Hence to find r3 for age group 1, we find all the games (in round 6) where the expected value is

in [125000, 127500]


80000 127500

75000 126500

70000 125250

80000 125250

50000 125000.5


Hence

As before, we can obtain r4 for age group 1such that the expected winning in Round 6 is

Hence to find r4 for age group 1, we find all the games (in round 6) where the expected value is


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in [35000, 39000].


28500 37875

12000 37625

20000 37502.522000 37500.5


Hence 20625

Applying the same method, we can obtain r5 for age group 1 such that the expected winning in

Round 6 is


in [10000, 10500]


4000 10375

6000 10250

6500 10000.5

8000 10000.5

9000 10000


Hence

Applying the same method, we can obtain r5 for age group 1 such that the expected winning in

Round 6 is

Hence to find r6 for age group 1, we find all the games (in round 6) where the expected winning

in [2000, 3000]


1850 3000


Hence

To obtain r5 for age group 3such that r6

and the expected winning in Round 6 is

For r5, we cant find any banker offer around the expecting winning, given limited amount of

data in age group 1.

Same thing applies r6

and there is no banker offer around the expecting winning


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around r7 and 250k.

Therefore we have


0 01/2 71000

1/4 20625

1/8 6700

1/16 1850

Age Group 2 (age between 35 and 65)Hence to find r3 for age group 2, we find all the games (in round 6) where the expected value is

in [125000, 127500]

For age group 2 (age between 35 and 65)


90000 125375

70007 125025


Hence

As before, we can obtain r4 such that the expected winning in Round 6 is


in [37500, 42500].

Bankers 6th

Offer () Expected Value ()

41000 42500

22500 40000

23000 39000

18000 37750

21000 37505

5000 37500.005


Hence 21750

Applying the same method, we can obtain r5 such that the expected winning in Round 6 is


in [10000, 11000]


10250 10250


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5000 10005

4200 10000.5


Hence 6483.33


6483.33


in [2800, 4000]


3900 4000

2250 2875


Hence,


3075


in [1500, 1600]


950 1550

1400 15251017 1505

1500 1500.05


Hence,

In order to obtain U(r7)=3/4, we can perform the following operations to obtain r7 such that the

expected winning in Round 6 is


in [155000, 165000]


120000 162500

Average banker offer = 120000

Hence 120000


0 0.01

1/2 80003.5

1/4 21750


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1/8 6483.33

1/16 3075

1/32 1216.75

3/4 120000

Age Group 3(Contestants aged 65 and above)Hence to find r3 for age group 3, we find all the games (in round 6) where the expected value is

in [125000, 127500]


80000 125500

100000 125375

50000 125000.1

Average bankers offer =76666.67

5Hence

As before, we can obtain r4 for age group 3 such that the expected winning in Round 6 is

Hence to find r4, we find all the games (in round 6) where the expected value is in [25000,

39000].


13500 37505

23000 26500Average bankers offer = 18250

Hence 18250

Applying the same method, we can obtain r5 for age group 3such that the expected winning in

Round 6 is

9125.005


in [9100, 10500]


6750 10375

4500 10050


Hence

To obtain r5 for age group 3such that r5

and the expected winning in Round 6 is


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For r5, we cant find any banker offer around the expecting winning, given limited amount of

data in this age group 3.

Same thing applies r6

and there is no banker offer around the expecting winning

around r6

and 250k.


0 0.01

1/2 76666.67

1/4 18250

1/8 5625

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