06.MixedStrategy - MIXED STRATEGIES A PURE STRATEGY is a...

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MIXED STRATEGIES A PURE STRATEGY is a (sure) complete, contingent plan of action (which is to be followed with probability one). A MIXED STRATEGY is a probability distribution over the set of pure strategies (assigns a probability to each pure strategy). Example : if the set of pure strategies is { s 1 , s 2 , s 3 , s 4 }, a mixed strategy could be { (0.25)s 1 , (0.25)s 2 , (0.5)s 3 , (0)s 4 }. This player plans to play s 1 with probability 0.25, s 2 with probability 0.25 and s 3 with probability 0.5. He or she, however, will eventually be choosing s 1 , or s 2 or s 3 . HOW TO FIND NASH EQUILIBRIA IN MIXED STRATEGIES? Example 1: The inspector and the thief The inspector and the thief play a simultaneous-move game where the inspector chooses between “ inspect ” and “ don’t inspect ,” and the thief chooses between “ rob ” and “ don’t rob .” Thief Inspector Rob don’t inspect 10 , 0 0 , 5 don’t 0 , 20 5 , 5 1
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This game has no Nash equilibrium in Pure Strategies. A mixed strategy for the Inspector assigns a probability distribution over his set of pure strategies , {inspect, don’t inspect} (0.25) Inspect Mixed strategy (0.75) Don’t inspect Play “ inspect ” with 25%, “ don’t inspect ” with 75% probability. CALCULATION OF EXPECTED PAYOFFS: Suppose there is no “outside ” uncertainty (no “moves” by Nature) 1. Given a PURE strategy profile {s 1 , s 2 }, the outcome is completely determined, hence also the payoffs: Player 1 gets U 1 (s 1 , s 2 ), and Player 2 gets U 2 (s 1 , s 2 ). Example: U Inspector (Inspect , Rob) = 10, U Thief (Inspect , Rob) = 0 2. Let “ p ” denote the probability with which Inspector plays the pure strategy “Inspect” (so “1-p” is the probability with which he plays “don’t inspect”) , and let “ q ” denote the probability with which the Thief plays “Rob” (so “1-q” is the probability with which he plays “don’t rob”) . Notice: Play (inspect) with probability p, play (don’t inspect) with probability 1-p “ IS A MIXED STRATEGY for the Inspector. 2
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Play (Rob) with probability q, play (don’t rob) with probability 1-q “ IS A MIXED STRATEGY for the Thief. The Inspector ’s expected payoff under these mixed strategies is: E U Inspector = (probability that “inspect,rob” is played). 10 + (probability that “inspect,don’t” is played). 0 + (probability that “don’t inspect,rob” is played). 0 + (probability that “don’t inspect, don’t rob” is played). 5 = = p [q].10 + [ p .(1-q)].0 + [ (1-p) q].0 + [ (1-p) (1-q)].5 = 10 p q + (1-p) [5(1-q)] The Thief ’s expected payoff under these mixed strategies is: EU Thief = (probability that “inspect,rob” is played). 0 + (probability that “inspect,don’t” is played).
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06.MixedStrategy - MIXED STRATEGIES A PURE STRATEGY is a...

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