lec4_17.pdf - Lectures 4 Mixed Strategies 1 Intro...

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Lectures 4: Mixed Strategies 1 Intro Previously, we de°ned a strategy for each player as simply the action he planned to play, and a NE as a vector of strategies=actions s.t. (i) each player±s action is a best response (maximizes his payo/) to his belief about the opponents±strategies, and (ii) this belief about the opponents± strategies=planned actions is correct , namely, equal to their equilibrium strategies. In some games, there is no pure strategy equilibrium. For example, 1.1 Motivating Example: Matching Pennies H T H 1,-1 -1,1 T -1,1 1,-1 ° no equilibrium where P1±s strategy is to play H: in such a NE, P2 would have to play his best-response strategy T; but then if P2±s strategy is to play T, in fact P1 should not be playing H ° no equilibrium where P1±s strategy is to play T: in such a NE, P2 would have to play his best-response strategy H; but this (P1 plays T, P2 plays H) cannot be NE, since P1 is not best-responding to P2±s strategy (he should play H instead) 1.2 Solution: ° expand set of allowable strategies to include all probability distributions over actions: rather than picking one speci°c action that he plans to play, each player is allowed to choose a ²randomized strategy², promising to play each action with a speci°ed probability ° the set of available strategies for player i is now °( S i ) ± the set of all possible probability distributions over the set of available actions S i : a typical strategy ° i 2 °( S i ) speci°es the probability ° i ( s i ) with which player i plans to play each action s i 2 S i (and for ° i to be a probability distribution over S i ; each probability ° i ( s i ) must be a number between 0 and 1, and probabilities must sum to 1: P s i 2 S i ° i ( s i ) = 1) ° correspondingly assume that players care about
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