Uncertainty and Risk - DARP

Microeconomics CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.4 Suppose you are asked to choose between two lotteries. In one
case the choice is between P 1 and P 2 ;and in the other case the choice o¤ered is
between P 3 and P 4 , as speci…ed below:
P 1 : $1; 000; 000 with probability 1
8
<
P 2 :
P 3 :
P 4 :
$5; 000; 000
$1; 000; 000
:
$0
$5; 000; 000
$0
$1; 000; 000
$0
with probability 0.1
with probability 0.89
with probability 0.01
with probability 0.1
with probability 0.9
with probability 0.11
with probability 0.89
It is often the case that people prefer P 1 to P 2 and then also prefer P 3 to P 4 .
Show that these preferences violate the independence axiom.
Outline Answer:
Let there be only three possible states of the world: red, blue and green,
with probabilities 0.01, 0.10, 0.89 respectively. Then the payo¤s in the four
prospects can be written
red blue green
P 1 1 1 1
P 2 0 5 1
P 3 0 5 0
P 4 1 1 0
where all the entries in the table are in millions of dollars. Note that P 1 and P 2
have the same payo¤ in the green state; P 3 and P 4 form a similar pair, except
that the payo¤ in the green state is 0. Axiom 8.2 states that if P 1 is preferred
to P 2 than any other similar pair of prospects (1; 1; z) and (0; 5; z) ought also
to be ranked in the same order, for arbitrary z: but this would imply that P 4
is preferred to P 3 , the opposite of the preferences as stated.
Note also that if the preferences had been such that P 4 was preferred to P 3
then the independence axiom would imply that P 1 was preferred to P 2 .
cFrank Cowell 2006 118