Section 3:
von NeumannMorgenstern, Mixed Strategies and Lots of Fun!
Daniel Egel
September 24, 2008
1
von NeumannMorgenstern
von NeumannMorgenstern expected utility
, which I will just call vNM, is a way of dealing with
uncertainty over outcomes. It is discussed in Chapter 27 of Dutta (1999) though I would recom
mend skipping subsection 27.2.2 for the time being. . . and perhaps altogether.
I’ll go through a
quick discussion of the essential concepts and then we’ll look at some examples.
•
Decision under certainty:
So this is not the vNM world. But before we talk about un
certainty it’s probably good to talk about certainty.
–
Certainty
means that we know what the outcomes are going to look like so we don’t
need to deal with probabilities.
–
We are often going to work with utility functions. Utility functions should be an
in
creasing function
of payoffs. i.e. More is better.
For the rest of what follows we will be talking about uncertainty, where a variety of different
outcomes are possible with known probabilities.
•
Expected Value:
This is amount of value that one can expect from playing a game/lottery.
This is very important and not necessarily intuitive. But if we have a lottery with
N
possible
outcomes
X
1
, X
2
. . . X
N
, with probability of
P
1
, P
2
. . . P
N
then the expected value is
EV
=
P
1
×
X
1
+
P
2
×
X
2
+
. . . P
N
×
X
N
Question:
What is the expected value of a game where you win 10 with probability
1
3
, 20
with probability
1
3
and zero with probability
1
3
?
•
Expected Utility:
Given some sort of utility function,
U
(
X
), this is the utility that can be
expected from playing the game. Using the game with
N
outcomes from directly above,
EU
=
P
1
×
U
(
X
1
) +
P
2
×
U
(
X
2
) +
. . . P
N
×
U
(
X
N
)
•
We there are three kinds of attitudes towards risk. Intuitively they are
1.
Risk Averse:
You are worse off from experiencing risk.
EU
<
EV
2.
Risk Neutral:
Risk doesn’t bother you.
EU = EV
1
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3.
Risk Loving:
Risk makes you better off.
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 Fall '10
 Power
 Game Theory, Nash, Mixed Strategies

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