3/3/2009
1
Mixed Strategies – Matching Pennies
±
We denote the strategies on the game bimatrix
Guildenstern
q1

q
Heads
Tails
±
Guildenstern’s payoff to
H
: (1)p + 1(1p) =
12p
T
: (1)p + (1)(1p) =
2p1
Rosencrant
zp
Heads
1
±
,1
1 , 1
1p
Tails
1 , 1
1 , 1
Preferences Involving Gambles
±
We revisit our description of payoffs in the
game bimatrix.
±
Consider Tom, who is faced with two
outcomes
±
A: get $1
±
B: get $4
±
Tom likes money, the more the better.
So
Tom thinks that outcome B is better than A
Preferences Involving Gambles
±
Tom’s preferences are summarized by
±
u1(m)=m, so that u1(1)=1, and u1(4)=4
±
u2(m)=m, so that u2(1)=1, and u2(4)=16
±
u3(m)=m,
so that u3(1)=1, and u3(4)=2
any of one these payoff (utility) functions
so long as outcomes are “get some money”
2
½
Preferences Involving Gambles
±
We would like to consider preferences over
gambles (lotteries) involving outcomes
±
Can we come up with a utility function that
ranks preferences over gambles (involving
outcomes) along with outcomes themselves?
±
Example: Suppose G is the gamble
±
Get $1 with probability ½, and
±
Get $4 with probability ½
Preferences Involving Gambles
±
Note that the expected value of G,
E[G] = ½($1) + ½($4) = $2.50
±
Must Tom think that G is just as good as $2.50?
±
No
±
Maybe, Tom thinks G is better than $2.50, or
maybe Tom thinks G is not as good as $2.50
Preferences Involving Gambles
±
Suppose we have a utility function over
outcomes
u:{outcomes}
→
Թ
 the real numbers
that represents a particular set of outcomes
±
It would be nice to have a utility function
v:{gambles over outcomes}
→
Թ
that ranks gambles over outcomes as well as
outcomes
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Preferences Involving Gambles
±
Desirable Property #1: Since outcomes
themselves can be thought of as gambles, v
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 Spring '07
 Emre
 Game Theory, Utility, Daniel Bernoulli, gambles

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