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Decision under Uncertainty

X : x1 ,…, xn : the set of possible outcomes.

A simple lottery is a vector p p1 ,…, pn such that

pi 0 i 1,…, n , i 1 pi 1 .The intended interpretation of pi is the probability of

n

occurrence of the outcome xi when a decision maker (DM) chooses the lottery p . An

outcome xi is identified with the lottery xi with the i-th coordinate equal to 1.

If there are three outcomes, a simple lottery is represented by a point on the

equilateral triangle with side 2 3 3 .

ｘ１，

ｐ３

ａ

ｃ

ｐ

ｐ２

ｘ２

ｂ

ｘ３

ｐ１

Compound Lottery

A compound lottery is a lottery whose outcome is a simple lottery.

Let

p p1 ,…, pn and q q1 ,…, q n be simple lotteries and let 0,1 . A

compound lottery that yields the simple lottery p p1 ,…, pn with probability

the simple lottery q q1 ,…, q n with 1 is denoted by

and

p 1 q . The

2

set of compound lotteries is denoted by L . Given two simple lotteries p p1 ,…, pn ,

q q1 ,…, q n and 0,1 , the compound lottery reduced from the compound lottery

p 1 q is a compound lottery is defined by

p 1 q ,…, p 1 q .

1

1

n

n

A preference relation is a binary relation on L . Define the strict preference relation

by x x x x x x . Define the indifference relation by

x x x x x x .

The following axioms are employed.

Order: is complete and transitive.

Continuity: for all p, q, r L , 0,1 | p 1 q r and

0,1 | r p 1 q

are closed in

0,1 .

Independence:

for all p, q, r L , 0,1 , p q p 1 r q 1 r

Reduction: for all p, q L , 0,1 ,

p 1 q p1 1 q1 ,…, pn 1 qn

Expected Utility Theorem

A binary relation on the lottery space L satisfies completeness, transitivity,

continuity and independence if and only if there exist real numbers u x1 , u x2 ,

…, u xn such that there exist real numbers ui x1 ,…, ui xn such that for all simple

lotteries p p1 ,…, pl , q q1 ,…, qn , we have

p q j 1 p j u x j j 1 q j u x j .

n

n

3

Proof：Since X is finite, there exist a best lottery b and a worst lottery w with

Let p p1 ,…, pn be an arbitrary simple lottery. We

.(Why?)

respect to

express p as p p1 ,…, pn p1 x1 p2 x2 pn x

n

For each i , there uniquely exists a real number u xi is such that

u xi b 1 u xi w xi and 0 u xi 1 .

(Why?)

Let

j 1 p j u x j . Then we have

n

x1

p1

u x1

b

1 u x1

w

p1

b

u x1

p2

p2

・・

x2

・・

1 u x1

・・

pn

b

w

1

・・

w

u x1 b

pn

1 u x1 w

xl

By independence

By reduction

Note that Independence is applied to each pair x j , p j b 1 p j w (. j 1,…, n ).

The proof is completed by the following lemma.

Monotonicity Lemma. Assume xi x j . If 1 0 , then

x 1 x x 1 x .

i

j

i

j

4

Proof: Step 1.

To show xi

x 1 x .

i

xi

j

xi

1

xi

By reduction

1

xi

xj

By independence

Step 2.

xi

xi

1

xj

1

1

xj

By Step 1 and

xj

1

xi

xj

by reduction

independence

Definition. The function u

in the statement of Expected Utility Theorem is called

von-Neumann-Morgenstern utility function (vNM utility function for short).

Allais Paradox

x1 2.5 million dollars , x2 0.5 million dollars , x3 0 dollars

Common preferences for individuals are

0,1, 0 .10,.89,.01

and

0,.11,.89 .10,.0,.90 .

Four reactions to the paradox.

1. Independence is a normative axiom that helps us make right decisions (Marshack

and Savage).

2. The Allais paradox is of limited significance in economics as a whole.

3. There could be an alternative theory such as regret theory to accommodate the

paradox: Regret refers to the sentiment, “I could have better an outcome if I had other

choice.” The lottery

0,1, 0

.10,.89,.01

involves the expected regret that is greater than

no matter how it is defined. As far as the choice between 0,.11,.89 and

5

.10,.0,.90 , there is no clear-cut regret potential.

4. Stick with the original choice domain and give up independence in favor of

something weaker.

Machina’s paradox

Let x1 ” a trip to Venice ” , x2 ” watching an excellent movies about Venice ” and

x3 ” staying hom e ” with the same lotteries as the Allais paradox.

This example involves the sentiment of disappointment when someone could not get

what she wanted, i.e. a trip to Venice. Preferences may be subject to change if you face

disappointment and you do not want to watch any movie about Venice if you failed to

get a trip to Venice. This could account for

0,.11,.89 .10,.0,.90 .

What is a common element about the ideas of regret and disappointment?

What is a difference between them?

Dutch Book Argument

Suppose that independence is violated, i.e. there exist three lotteries p, q and r

such that p q , p r and

q 1 r p for some 0,1 .

Then, one could make a case for independence by constructing a “Dutch book” such

as :

p

q 1 r

q

1

r

6

Exercise 1. Let X : x1 ,…, xn and let be a complete and transitive binary

relation.

1) Show that there exist b X and w X

such that for any x X , we have

b x and x w .

2) Show that for all x, x, x X ,

x x x x x x .

3) Show that for all x, x, x X ,

x x x x x x .

Exercise 2. For all simple lotteries p p1 ,…, pn , q q1 ,…, q n , define lex by

p lex q if one of the following conditions holds.

1) p q

2) i 1,…, n , j 1,…, i , p j q j pi 1 qi 1

For all simple lotteries p p1 ,…, pn , q q1 ,…, q n and for any

the compound lottery

0, 1 define

p 1 q by

p 1 q p1 1 q1 ,…, pn 1 qn .

Show that lex satisfies completeness, transitivity and independence and violates

continuity.

Exercise 3. Consider a binary relation

on the lottery space L

satisfying

completeness, transitivity, continuity and independence.

1) Show

that

there

exist

b X

and

w X

such

that

for

any

simple

lottery p p1 ,…, pn , we have b p and p w .

2) Show that if b w , for any simple lottery p p1 ,…, pn , we have b p .

Exercise 4. Consider a group of I individuals with binary relation i on the lottery

space L

satisfying completeness, transitivity, continuity and independence and

reduction ( i 1,…, I ). By Expected Utility Theorem, for each individual i 1,…, I ,

there exist real numbers ui x1 ,…, ui xn such that for all

7

p p1 ,…, pl , q q1 ,…, qn , we have p q j 1 p j ui x j j 1 q j ui x j .

n

n

Define a social preference relation by

p q i 1,…, n , j 1 p j ui x j j 1 q j ui x j .

n

n

That is, p p1 ,…, pl is socially at least as good as q q1 ,…, qn if and only if for

each individual i , the expected utility of p for i is at least as large as that of q .

Show that the social preference relation violates completeness and satisfies transitivity,

independence and continuity.

Exercise 5. Consider three individuals 1, 2 and 3 and three alternatives x, y, z . The

individuals have preferences over simple lotteries satisfying completeness, transitivity,

independence and continuity.

Suppose that the vNM utility functions u j

, j 1, 2,3

are given by:

u1 x u1 y u1 z

u2 z u2 x u2 y

u3 y u3 z u3 x

For each individual i and for each simple lottery p p1 , p y , pz

denote the expected

utility by Eui p px ui x p y ui y pz ui z .

Define a social preference relation based on simple majority rule by:

For all simple lotteries p p1 , p y , pz , q q1 , q y , qz

p q # i | Eui p Eui q # i | Eui q Eui p ,

where # A denotes the number of elements of the set A .

Show that satisfies completeness, independence but violates transitivity.

Exercise 7. Let X : x1 ,…, xn be the set of possible outcomes and let u : X be a

8

non-constant function, i. e. x, x X , u x u x . Define a binary relation on

the set of simple lotteries L by

p q i 1,…, n , j 1 p j u x j j 1 q j u x j .

n

n

Show that satisfies completeness, transitivity, independence and continuity.

Exercise 8 (Rubinstein’s Lecture Notes in Microeconomic Theory, page 88).

Let X : x1 ,…, xn

be the set of possible outcomes and let u : X

be a

non-constant function, i. e. x, x X , u x u x . Define a binary relation on

the set of simple lotteries L by

p q min p x | x X p x 0 min q x | x X q x 0 .

Show that

satisfies completeness, transitivity and continuity but violates

independence. Interpret this DM’s preference.

Exercise 9 (Rubinstein’s Lecture Notes in Microeconomic Theory, page 88).

Let X : x1 ,…, xn

be the set of possible outcomes and let u : X

be a

non-constant function, i. e. x, x X , u x u x . Define a binary relation on

the set of simple lotteries L by

p q max p x | x X max q x | x X .

Show that

satisfies completeness, transitivity and continuity but violates

independence. Interpret this DM’s preference.

Reference

Andreu Mas-Colell, Michael D. Whinston and Jerry R. Green (1995), Microeconomic

Theory, Oxford University Press.

Ariel Rubinstein (2006), Lecture Notes in Microeconomic Theory: the economic agent,

Princeton University Press.

…

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