Lecture11 - Mixed Strategies Matching Pennies In Matching...

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2/17/2009 1 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 so-called “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 rock-paper-scissors, 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 where the probability distribution degenerates all the probability weight is on one pure strategy Mixed Strategies – Matching Pennies Rosencrantz’s strategy set is now a probability
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