Economics 395
Risk and Uncertainty
Problem Set 3
Answer Key
Question 9
Consider a set of monetary prizes, V = {$0, $24000, $27500} and the following four
lotteries over those prizes:
α
1
= (0.01, 0.66, 0.33)
α
2
= (0, 1, 0)
α
3
= (0.67, 0, 0.33)
α
4
= (0.66, 0.34, 0)
where the lottery α = (α
1
, α
2
, α
3
) means that the lottery delivers $0 with probability α
1
,
$24000 with probability α
2
and $27500 with probability α
3
.
(a)
Given a choice between lotteries α
1
and α
2
, which would you choose?
Obviously, this is only for you to decide.
(b) Given a choice between lotteries α
3
and α
4
, which would you choose?
Again, this one is up to you.
(c)
In an experimental setting, many people are shown prefer α
2
to α
1
(i.e. α
2
>
i
α
1
).
Suppose these peoples’ preferences satisfy the independence axiom.
Then how
do you think such a person would choose between α
2
and another lottery, β, where
β = (1/34, 0, 33/34)?
First, observe that α
1
can be written as a compound lottery formed using α
2
and β.
α
1
= (0.66) α
2
+ (0.34) β.
From the independence axiom, then, we know that if
α
2
>
i
β
then
α
2
~
i
(0.66) α
2
+ (0.34) α
2
>
i
[(0.66) α
2
+ (0.34) β] ~
i
α
1
.
But as we know that α
1
>
i
α
2
, it must be the case that β ≥
i
α
2
.