Economics 395
Topics in Risk and Uncertainty
Homework Problems
Solutions
Problem 4
Imagine that I hold two envelopes, one in my right hand and one in my left.
In each
envelope there is a check.
You have no idea what the amounts on these checks are, but I
tell you that one check is written for twice the amount of the other.
Then I let you select
one of the envelopes.
You rip open the envelope and find it’s written for $20.
Before you leave with the check, though, I offer you the following opportunity: if you
wish, you may exchange your $20 check for the check in the other envelope.
(a)
If you cared only about the expected value of a lottery, would you swap your $20
check for the check in the other envelope?
Why or why not?
(b) Now repeat this problem assuming that you found $Y in the envelope you chose.
Would you want to swap $Y for the check in the other envelope? Is it true that an
expected value maximizer always wishes to swap her check for the check in the
unopened envelope?
This sounds a lot like the Monty Hall problem.
In the
Monty Hall problem, we deduced that a person would always like to swap the box
initially chosen for the unopened box that remained.
Is a similar result true here?
(c)
Remembering your answer from part (b) (and continuing to assume that you care
only about expected values of lotteries), how much would you be prepared to pay
to play this game?
If this is difficult to answer, explain why.
What kind of
information would help you evaluate this lottery?
The scenario involving the envelopes is another example of a compound lottery:
a lottery
in which the ultimate prize you receive is determined by the resolution of multiple
sources of uncertainty.
In this case, there are two sources of uncertainty.
First, there is
the uncertainty over the values of the two checks.
Then, when the first envelope has been
opened, there still remains the uncertainty over whether the morevaluable or less
valuable check has been revealed.
To make this more concrete, suppose that X is the
value of the lessvaluable check.
The highervalued check is worth 2X.
Again, let Y
represent the value of the check that is revealed.
The values of X and Y are determined
randomly.
The event tree below shows the multitude of ways in which these two sources
of uncertainty might be resolved.
For simplicity, I have drawn only a small handful of
the values of X that can be realized in this problem:
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The probabilities shown as P(X=x1), P(X=x) and P(X=x+1) are all part of the
prior
distribution
over values of X.
The conditional probabilities P(Y = x  X =x) and
P(Y = 2x  X = x) express the likelihoods that the envelope you open contains the low or
the high valued endowment, respectively
(d) Can you “flip” this event tree to depict a random process in which the value of Y
(the value of the check found in the open envelope) is determined first?
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 Fall '08
 MINETTI
 Economics, Conditional Probability

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