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2/17/2009
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Mixed Strategies – Matching Pennies
±
In Matching Pennies, we found there was no
Nash Equilibrium
Guildenstern
Heads
Tails
±
We expand the strategy set to allow for
“mixed strategies” randomizations over
Heads and Tales, the socalled “pure
strategies”
Rosencrantz Heads
1, 1
1, 1
Tails
1, 1
1,1
Why Mixed Strategies?
±
If you were to play the same opponent
repeatedly, you would like to be
unpredictable
.
±
In rockpaperscissors, if you behave
predictably, your opponent will be able to beat
you
you.
±
In a tennis match, you do not want your
serves to be predicted by the opponent.
Why Mixed Strategies?
±
Maybe your opponent is picked from a
large population of players…
±
You play many times, each time with new
opponent
±
Each opponent does the same thing each time
he plays
±
Different opponents do different things
±
You don’t know which type of opponent you
get
±
Model your opponent as randomly picking a
strategy
Why Mixed Strategies?
±
Technical reason: Not all games have Nash
equilibria in pure strategies.
All games have
Nash equilibria when allowing for mixed
strategies.
±
We are generalizing our notion of strategy
±
pure strategies are a kind of mixed strategy
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where the probability distribution degenerates
±
all the probability weight is on one pure strategy
Mixed Strategies – Matching Pennies
±
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 Spring '07
 Emre

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