Cha p t e r 2

D e c i si on M a ki ng under R i s k

In the p revio u s l ectu r e I co nsid ered ab stra ct c h oice p r o b lem s . I n t h i s section , I w i ll fo cu s on a s p e ci a l cl a s s o f c ho ice p rob l em s a n d im p o se m o re str u cture o n t h e decision m a k e r’s p r eferen ces. I w ill con s ider situa t ion s in w h ic h t h e d ecisio n m a k e r c a r es o n ly a b ou t t h e consequences, suc h as the a m o un t o f m oney in his b ank a ccoun t , b ut he m a y n ot b e able t o c h o o s e di rec t l y f r om t h e s et of c o ns e q ue nc es . I ns t e ad, h e c ho o s es f r o m al te rna t i v e s th at d e te rm in e t h e c o n seq u e n ces p r o b a b ilistica ll y , su c h a s a l o ttery t i c k et.

In this lec t ure , I a ssu m e tha t , f or a n y a lte r na tiv e x , t h e p r ob ab ilit y d is tribu t ion o n th e set o f co n sequ e n ces in du ced b y x is giv e n. T h at i s , a l t hough d eci s ion m ak er do es no t k no w t he c o ns e q ue nc e o f c ho o s i n g a g i v e n a l t e r na t i v e , h e i s g i v e n t h e p ro ba bi l i t y of ea c h con sequ e n c e f ro m c h o o s i n g t h a t a ctio n. T h is is called d ecisio n m a kin g un d e r ri sk . S u c h a s s um ptio ns ca n b e p lau s ible in re la tiv e ly fe w s it ua tion s, su c h as c h a n c e ga m e s i n a casin o , i n w hic h th ere a re ob jectiv e pro b ab i lities. In m o st ca ses of econ om ic in tere s t, th e a lt ern a tiv e s d o n ot co m e w i th p r o b a b ilities. T h e d ecisio n m a k e r f o r m s h i s su b j ecti v e b e liefs a b o u t t he co nsequences of h i s c h o i c es. T h is is called d ecision m a kin g unde r unc e r t a i nt y . I wi l l anal yz e t he de ci s i on m a ki ng under ri s k a s an i n t e rmedi a ry s t e p to w a rd an alyzi n g d ecisi o n m ak i n g u n d er un certain t y .

2. 1 C onsequenc e s a nd Lot t eri e s

Con s i d e r a fi nite set C of c o nse q uenc es . A lo t t e r y is a p ro ba bilit y d istribu t ion p :

C → [0 , 1] on C , w h e r e P c ∈ C p ( c ) = 1 . T h e set o f all l otteries is d e n o ted b y P . T h e

9

consequences are e m b edded i n P a s p o i n t m a sses on sin g le l o tter i es. F or an y c ∈ C , I w i l l wr i t e c for b oth t he consequence c an d t h e p r o b a b ilit y d istrib utio n t h a t p u t s p rob a b ilit y

1 o n c . T he dec i s i o n ma k e r c are s a b o u t t he co ns e q ue nc e t ha t w i l l b e re a l i z ed, b ut he ne e d s t o c ho os e a l o t t ery . I n t h e l anguage o f t he t h e p re vi ous l ec t u re , t he s e t X of altern ativ es is P .

A l o ttery can b e d ep i c ted b y a tree. F o r exam ple, in Fig u re 1, L o ttery 1 d e p i cts a s itua tion in w h ic h t he de cision m a k e r g ets $ 1 0 w i th p r o b a b ilit y 1 / 2 (e.g. i f a co in to ss re sults i n H e a d ) a n d $ 0 w ith p rob a b ilit y 1 / 2 ( e . g . if th e c oin t o s s r e s ults in T a il). A l o ttery ca n b e sim p l e as in th e fi g u re, a ssig n in g a p ro b a b i l i t y to eac h co n sequ e n ce, or co m p o u n d as in F i g u r e 2.3, co n t ain i ng su ccessiv e b r a n c h es. T h e r ep resen t ati o n o f all l o tter i es a s p r ob ab ili t y d i strib utio ns incorp o r a t es the a ssu m p tio n s t ha t t he d ecision m a k e r i s c o n sequ e n t ialist, m e a n in g t ha t h e c ares on ly ab ou t t h e con sequ e n ces, an d t ha t h e ca n c om pu te th e p ro ba bilit y o f e ac h c on seq u e n c e u n d e r co m p o u n d i n g lo tter i e s .

10

Lo tt er y 1

0

Figure 2.1:

R e p r esen t ing t h e lotteries p a s v e ctor s , n ote t h a t P is a | C | − 1 di m e ns i o na l s i m pl e x . H e n ce, I w ill regar d P as a s u b set o f R | C | (on e ca n e n v ision i t a s a su b set o f R | C | − 1 as w e l l ). Endo wi ng R | C | w i th th e s tan d a rd E u clidean m etric, no te th at P is a c o n v e x a nd co m p act s ub set.

2 . 2 E x p ec te d U tilit y M a x i m i z a tio n – R e p r e sen ta - tion

W e w o uld lik e t o h a v e a th eo ry tha t con s tru c ts the d ecision m a k e r’s p references o n th e lotteries f r o m h is pr eferen ces o n th e l otteries. T h ere a re m a n y of them . T h e m o st

w e ll -k no w n – a n d the m ost c an on ical and t he m o st usef u l – o n e is the t heory o f e xp ected u t ilit y m a x im iza t ion b y V o n N e u m an n a nd M o r g en ste r n. In t h is lect ure , I w ill fo cu s o n th is theor y .

De fi ni t i on 5 A r ela tio n º on P is sa id to b e rep r esen ted b y a v on N e um an n - M o r g en stern ut i l i t y f unc t i o n u : C → R if a n d o n l y i f

p º q ⇐⇒ U ( p ) ≡ X u ( c ) p ( c ) ≥ X u ( c ) q ( c ) ≡ U ( q ) ( VNM)

c ∈ C c ∈ C

fo r e a c h p, q ∈ P .

T h is sta t em en t h as t w o c ru cial p a r t s:

1. U : P → R rep r esen ts º i n th e o rd in al sense. T h at i s , i f U ( p ) ≥ U ( q ) , t he n t he de ci s i on mak e r fi nds l otter y p as go o d as lottery q . A n d co n v ersel y , if the d ecision ma k e r fi nds p at l e as t a s g o o d a s q , t h e n U ( p ) m u st b e at lea s t a s h igh a s U ( q ) .

2. T h e f u n ction U tak e s a particul ar form : f or eac h lottery p , U ( p ) is th e e xp ected va l u e o f u under p . T h a t i s , U ( p ) ≡ P c ∈ C u ( c ) p ( c ) . I n o ther w o rd s, th e d ecision

m a k e r a cts as i f h e w a n t s t o m ax im ize th e e x p e c te d v a l u e o f u . F or insta n ce, th e e xp ected u t il it y o f L o ttery 1 f o r th e d ecisi o n m ak er i s E ( u ( L o tter y 1 )) =

1 u (10) + 1 u (0) . 1

2 2

2. 3 E xp ect e d U ti l i t y M a xi m i zati on – C haract eri - zat i on

T h e m a i n o b j ectiv e o f t h i s l ectu re is to exp l ore t he co n d ition s on preferen ces u n d e r w h i c h th e v o n -N eum a n n M o rg en stern r ep resen t ati o n i n ( V N M ) is p o ssib l e. In th i s w a y , on e m a y h a v e a b e tter i n s igh t s i n t o w h a t i s i n v o l v e d i n e xp ected u til i t y m a xim i za tion .

F i rst, as expla i n e d a b o v e , r epresen ta tion in (V N M ) i m p li es th at U rep r esen t s º in th e

or din a l sen se as w e ll. B u t, as w e h a v e seen in the p rev i o u s l ectur e , o rd ina l repr esen ta tion im p lies t h a t º is a p re fer e n c e r e l a t ion . T h is giv e s t h e fi rst n ecessary c on ditio n .

Ax i o m 2 ( P r e f e r e n c e ) º is c o m p le te a n d t r a n s itiv e.

1 If C w e re a c on tin uum, lik e R , w e w ould compute t he exp e cted utilit y o f p by R u ( c ) p ( c ) dc .

Se co nd, i n ( V N M ) , U is a lin ear f un ctio n o f p , a nd he nc e i t i s c o n t i n u o u s . That is, ( V N M ) i n v olv e s c on ti n u ou s o rd inal represen tation . H ence, b y T heorem 3 o f t h e p r eviou s section , i t i s a lso n ecessa ry th at º is co n t in uo u s . T his g iv es the secon d n ecessa ry

co nd ition .

A x io m 3 (C o n tin u it y ) º is c o n tin u o u s .

R e c a l l f r o m t h e p re vi o u s l ec t u re t h a t c o n t i n ui t y m e ans t ha t t he upp e r - a nd l o w e r- co n t ou r s ets { q | q º p } and { q | p º q } a r e c l o sed fo r e v e ry p ∈ P . I n t h i s s p ecia l setu p, a s ligh t ly w e ak er v e rsi o n o f t h e co n t in uit y a ssu m p tion su ffi ces: for a n y p, q , r ∈ P , t h e sets { α ∈ [0 , 1] | αp + ( 1 − p ) q º r } an d { α ∈ [0 , 1] | r º αp + ( 1 − p ) q } a r e c lo sed . Y et an o t her v ersio n of th is a ssu m p tion is tha t for a n y p, q , r ∈ P , i f p  r , t h e n t h e r e e x i s t a, b ∈ (0 , 1) suc h that ap + ( 1 − a ) r  q  bp + ( 1 − r ) r .

B y T h eor e m 3 , A xio m s 2 an d 3 a r e n ecessa ry a n d s u ffi cien t f or a r ep resen t atio n b y a c o n t i n u o u s f u n c t i o n U . T h e v o n N eu m a nn a n d M o r genstern represen ta tion im p o ses a f u rt he r s t r uc t u re on U , r eq u i r i n g th at it is in fac t lin ear i n p rob a bilities. T h is lin ear i t y co nd ition c o rresp on ds to th e f o l l o w i n g con d itio n o n t h e p r eferen ce, w hic h i s called The Ind e p e nd enc e A x io m .

A x i o m 4 ( I ndep endence) Fo r a n y p, q , r ∈ P , a n d a n y a ∈ (0 , 1] , ap + ( 1 − a ) r º

aq + ( 1 − a ) r ⇐⇒ p º q .

T h a t is, t he d ecisio n m a k e r’s p reference b et w e en t w o l o t teri es p and q do e s not c hange if w e toss a ( p o ssib l y u n f air) c o in a n d g iv e h im a fi xed l ottery r if “t ail” co m e s u p . L e t p an d q b e the l otteries d e p i cted i n F i gu re 2.2. T h en , t h e lotteries ap + ( 1 − a ) r and aq + ( 1 − a ) r ca n b e d ep i c ted a s i n F i g u r e 2 .3, w her e w e to ss a c o i n b et w e en a fi xed lott ery r a n d o u r lotte r ies p an d q . A xiom 4 s tipulates t hat t he decisi on m a k e r w oul d n o t c h a n g e h is m i n d after t he co in toss. T h er efo r e, the I nd ep end e n c e A xiom ca n b e tak e n a s a n a xiom of “d yn am ic con s isten c y . ”

T h e I nd ep en den c e A xio m im p o ses a s tru c tu re o n th e i nd i ff e r e n c e s e t s t h a t i s i d e n t i c a l to the s tructure of the i so curv es of a l i n ear f uncti on U . T o g e t he r w i t h t he co n t i n uo us rep r esen ta tion th eor e m , th i s lead s t o a n e xp ected u ti lit y r ep resen t ati o n . In th e sequ e l, I w i l l de s c ri b e t h e s truct u re i n de tai l and p ro v e t h at t h e a b o v e axi o ms are s u ffi cien t f o r an exp ected u t ili t y rep r esen t ati o n . T o w a rds t h i s e n d , t he follo w i ng exer cise lists som e

³ ³ ³ ³

p d ³ P P

P P

³ ³ ³ ³

P P P P

³ ³ ³ ³ ³ ³ ³ ³

q d P ³

P

P P P P

Fi gure 2. 2: Tw o l ot te ri e s

³ ³ ³

p d ³ P

³ ³ ³ ³

³ ³ ³ ³ ³ ³ ³

q d P ³

¡ P P P

¡

¡

¡ a

d ¡

P P P P

¡ P P P

¡ P P

¡ P

¡ a

d ¡

@ @

@ 1 a @ 1 a

@ @

@ @

@ r @ r

ap + ( 1 − a ) r aq + ( 1 − a ) r

Fi gure 2. 3: Tw o c omp o und l ot t e ri e s

us e f ul i m pl i c at i o ns of t h e I ndep e nde nc e A xi om. Y ou s h o u l d pro v e t he l i s t e d p ro p e rt i e s b e for e y o u p ro ceed .

Exe r c i s e 1 F o r a ny pr efer enc e r e l a tion º th a t sa tis fi es the I ndep endenc e A x i om , s how th a t th e f o l lo w i n g a r e t r u e.

1. F o r a ny p, q , r , r 0 ∈ P wi th r ∼ r 0 an d a ny a ∈ (0 , 1] ,

ap + ( 1 − a ) r º aq + ( 1 − a ) r 0 ⇐⇒ p º q. (2 . 1 )

2. F o r a ny p, q , r ∈ P an d a n y r e al n u m b er a suc h th at ap +( 1 − a ) r, a q +( 1 − a ) r ∈ P , if p ∼ q , t h e n ap + ( 1 − a ) r ∼ aq + ( 1 − a ) r. (2 . 2 )

3. F o r a ny p, q ∈ P wit h p  q an d a ny a, b ∈ [0 , 1] wit h a > b ,

ap + ( 1 − a ) q  bp + ( 1 − b ) q. (2 . 3 )

4. Ther e e xi s t s c B , c W ∈ C s u ch th a t fo r a n y p ∈ P ,

c B º p º c W . (2 . 4 )

[H int: u s e c om pleteness a nd tr an sitivity to fi nd c B , c W ∈ C w ith c B º c º c W fo r al l c ∈ C ; t he n u s e i n duc t i o n o n t he num b e r of c o ns e q ue nc es and t he I n de p e nde n c e Axi o m. ]

T h ese p r o p erties can b e s p e lled o ut as fo llo w s . F irst, r ecall t h e situ ation c on sid e red b y the I nd ep end e n c e A xiom : w e t oss a coin ; i f i t c om es h e ad , t h e decision m a k e r f aces p or q d e p e nd in g h is c h o i ce, a n d i f i t c om es tail, he faces r . T h e fi rst p rop e rt y s tates th at it do es no t m atter w h e th er he faces t he sam e lo tter y i n c ase o f t ail o r t w o d i ff er en t l o t t e r i e s t h a t h e i s i n d i ff eren t t o. F o r a n e xp l a na tion of the seco nd pr op ert y n o te tha t in the s i t ua tion co nsid ered b y th e I n d ep en dence A xiom , a ccord i ng to the a xiom , t h e d ecision m a k e r w ou ld b e ind i ff er en t b et w e en th e t w o co m p ou nd ing l o tter i es if h e w e re ind i ff er en t b et w een p an d q . T hi s c o rres p onds t o ( 2 . 2 ) f o r a ∈ [0 , 1] . T he p r o p ert y states m o re g e n e ra l l y t h a t t he statem en t r em a i ns tru e ev en if a is n o t i n [0 , 1] , i n w h i c h ca se t h e p robl em c o ul dn’ t b e re pre s e n t e d as a c ho i c e b et w e e n t w o c o m p o undi ng l o t t e r i e s .

T h e t hird p r o p ert y p r ob a b ilit y i s a m o no to nicit y p r ob a b ilit y . It sim p ly sta t es t h a t w h en a d e c ision m a k er fa ces a s itu a tion in w h ic h h e c a n e n d u p a b e tter l o ttery p or w o r s e lott ery q , t h e n h e w ou ld p r efer h i g h e r pro b ab ilities o f p to lo w e r o n es. F i na ll y , t h e last pro b a b ilit y s tat e s t h a t t her e ar e b e s t a nd w o rst c onsequences ( b y transi ti vi t y and co m p leten ess) a n d th ey a r e a lso b est a n d w orst lo tteries ( b y m o n o ton i cit y ). U s ing t hese p r op erties, o n e c an ea sil y p r o v e t h e m a in result in th i s lectu re:

The o rem 4 A r e l a t i o n º on P c a n b e r epr e sente d by a v on N e um ann - M o r gen stern u t ility f u n ctio n u : C → R a s in (V N M ) i f a n d o n ly if º satis fi es A x i o m s 2, 3, and 4 . Mo r e o v e r , u and u ˜ r e pr e s ent t he s a me pr e f er enc e r e l a t i on i f and o nl y i f u ˜ = au + b fo r so m e a > 0 an d b ∈ R .

Pr o o f . Sin c e w e k no w a lre a d y tha t rep r ese n t atio n i n ( V N M ) i m p lies A x i o m s 2 , 3 , a nd 4, I w il l o nly p ro v e the c o n v e r se. A s i n (2 . 4 ), let c B , c W ∈ C be s u c h t h a t c B º p º c W fo r

ev ery p ∈ P . I f c ∼ c , t h e n b y t ran s itiv it y , th e d ec ision m a k er is ind i ff eren t b et w een

e v e r yt hi ng , a nd he nc e u ( c ) ≡ 0 for a l l c sa ti s fi es the r ep resen tation . A ssum e c  c ,

an d d e fi ne φ : [ 0 , 1] → P by φ ( t ) = tc + ( 1 − t ) c . B y ( 2.3), f or an y t, t ∈ [0 , 1] ,

φ ( t ) º φ ( t 0 ) ⇐⇒ t ≥ t. (2 . 5 )

T h en , L em m a 1 o f t h e pr eviou s l e ctu r e i m p li es tha t fo r e v e r y p ∈ P , t h e re exists a uni q ue U ( p ) ∈ [0 , 1] suc h that

p ∼ φ ( U ( p )) . (2 . 6 )

F i rst o b serv e th a t U in deed rep r esen t s º in th e o rd in al se nse : fo r a n y p, q ∈ P ,

p º q ⇐⇒

φ ( U ( p )) º φ ( U ( q )) ⇐⇒

U ( p ) ≥ U ( q ) .

[H ere, th e fi rs t ⇐⇒ is b y (2 . 2 ) a n d (2.6), an d t he secon d ⇐⇒ is b y (2.5 ).] I n o rd er

to sh o w th at U ha s t he s p ec i fi c s tru c tu re i n (V N M ), it su ffi ces to sho w tha t U is line a r. That i s , f or an y a ∈ R an d a n y p, q ∈ P wi t h ap + ( 1 − a ) q ∈ P ,

U ( ap + ( 1 − a ) q ) = aU ( p )+ ( 1 − a ) U ( q ) . ( 2. 7)

But , s i nc e p ∼ φ ( U ( p )) and q ∼ φ ( U ( q )) ,

ap + ( 1 − a ) q ∼ aφ ( U ( p )) + ( 1 − a ) φ ( U ( q ))

= φ ( aU ( p )+ ( 1 − a ) U ( q )) ,

pro v i n g ( 2 . 7) b y de fi n i tio n (2.6) o f U . [ H e r e , t h e i n d i ff e r e n c e is b y ( 2 .1 ) a n d t h e e q u a l it y is b y d e fi nition of φ .]

B y the l ast s ta tem e n t in T h eo rem 4 , t h e rep r esen tatio n i s “u n i qu e u p t o a ffi ne t r a n s - f o rmat i o ns ”. That i s , a de c i s i on m a k e r’ s p ref e rences do not c hange w hen w e c hange h i s v o n N e u m a n n-M o rg en stern ( V N M ) u tilit y fu n ction b y m u ltiply in g i t w ith a p o sitiv e n u m b er, o r a d d in g a constan t to it, b u t th ey d o c h an ge w h en w e tran sform i t t h r ou gh a n on -linear t ransfor m ati o n . In th at sense, (V N M ) r ep resen t ati o n i s “ card ina l ”. R e call

th at, i n o rd ina l rep r esen tatio n , t he p r eferen ces d o n ot c h an ge ev en if the t ran s fo rm a t ion is n o n - linea r, so lon g as it is inc r eas i n g . F o r in sta n ce , u n d e r c erta in t y , v = √ u and

u rep r esen t t h e sa m e preference rel a tion , w hile ( w he n t h e r e is un ce rtain t y ) th e V N M u t ilit y f un ctio n v = √ u represen t s a v e ry di ff eren t set of pr eferen ces on th e l otteries

th an th ose a re r e pr esen ted b y u .

2. 4 I ndi ff erenc e Set s unde r I ndep endenc e A xi o m

In the s equel , I w i l l expl ore t he structure i m p osed b y the I ndep endence A xi om on the ind i ff er en ce sets i n m o r e d etail, expla i nin g the l og ic of th e r ep resen tatio n . R ecall f ro m th e p reviou s l ectur e tha t A x iom s 2 a n d 3 i m p l y th at th e i n d i ff eren ce sets a re closed. The I nde p ende nc e A xi om ha s t w o f u rt he r i m p l i c at i o ns o n t h e i ndi ff er ence sets:

1. T h e i n d i ff eren ce sets on th e l otteries a re str aigh t l in es (i.e. h y p erp l an es).

2. T h e i n d i ff e r en ce s e ts, w h i c h are s traig h t line s , a re p a r a llel t o e ac h o th er.

T o ill u s tra t e t h ese facts, co nsid er th ree p rizes z 0 , z 1 , a n d z 2 , w h e r e z 2 Â z 1 Â z 0 . A l o t t e r y p ca n b e d ep icted o n a p l an e b y t a k i n g p ( z 1 ) as the fi rst c o o rdinate ( on the hori z o n t al axi s ) , and p ( z 2 ) as th e s eco nd co o r d i n a te (o n t he v e r t ical a x is). p ( z 0 ) is 1 − p ( z 1 ) − p ( z 2 ) . [ See F ig ur e 2 .4 fo r t h e illu stra t ion . ] G iv e n a n y t w o lo tte ries p

an d q , t h e con v ex co m b in atio ns ap + ( 1 − a ) q wi t h a ∈ [0 , 1] fo rm th e line s e g m e n t co nn ecting p to q . N o w , t a k i n g r = q , w e c an deduce from (2.2) t hat, if p ∼ q , t h e n ap + ( 1 − a ) q ∼ aq + ( 1 − a ) q = q fo r e a c h a ∈ [0 , 1] . T h a t i s , t h e l i n e s e g m e n t

co nn ecting p to q is an in di ff eren ce cu rv e. M o reo v er, i f t he lines l an d l 0 a r e p a r allel,

th en α/ β = | q 0 | / | q | , w h e r e | q | an d | q 0 | are t he di st a n c e s of q and q 0 to the o ri gi n, resp ecti v el y . H e n ce, takin g a = α/β , w e c o m put e t hat p 0 = ap + ( 1 − a ) z 0 an d q 0 =