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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

ｐ２
ｘ２

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).
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  .
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.
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
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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.