mixed_strategies_handout - Handout on Mixed Strategies Ben...

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Handout on Mixed Strategies Ben Polak, Econ 159a/MGT522a October 4, 2007 The main lesson of the last class was the following: Main Lesson If a mixed strategy is a best response then each of the pure strategies involved in the mix must Before explaining why this must be true, let’s just try to rewrite this lesson formally, using our new notation: More Formal statement of the Same Lesson. If player i ’s mixed strategy p i is a best response to the (mixed) strategies of the other players, p i , then, for each pure strategy s i such that p i ( s i ) > 0 , it must be the case that s i is itself a best response to p i : In particular, Eu i ( s i ; p i ) must be the same for all such strategies. Why is this true? Suppose it were not true. Then there must be at least one pure strategy s i that is p i . (low-yield) pure strategy from my mix, assigning the weight I used to give it to one of the other (higher- batting average on a team must raise the team average). But then the original mixed strategy cannot have been a best response: it does not do as well as the new mixed strategy. This is a contradiction. So what? An immediate implication of this lesson is that if a mixed strategy forms part of a Nash Equilibrium then each pure strategy in the mix must itself be a best response. Hence all the strategies in Finding Mixed-Strategy Nash Equilibria. Let’s look at some examples and use our lesson to ±nd the mixed-strategy NE. Example 1 Battle of the Sexes a b A 2 ; 1 0 ; 0 B 0 ; 0 1 ; 2 In this game, we know that there are two pure-strategy NE at ( A; a ) and ( B; b ). Let’s see if there are any other mixed-strategy NE. Suppose that there was another equilibrium in which the row mixed on both A and B . By our lesson of the day, we know that in this case both A and B must be best responses to whatever the column player is doing. But for them both to be best responses, they must both yield the same expected
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This note was uploaded on 02/13/2012 for the course ECON 159 at Yale.

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mixed_strategies_handout - Handout on Mixed Strategies Ben...

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