041807 - Lecture Recall Finite strategic form games PSNE...

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Lecture ??: 04/18/2007 Recall: Finite strategic form games, PSNE, mixed strategies, Nash equilibrium with possibly mixed strategies, examples. Recall definition of Nash equilibrium : A (possibly mixed) strateg profile (σ 1 * , σ 2 * , …, σ n * )= σ * is a NE for every I and every mixed stratigey σ i є Ai , the following holds: u i 1 * ,…, σ i-1 * , σ i * , σ I+1 * ,…, σ n * )>=u i ( ie. σ i * is a best reply or best response for player i to the “opposition strategy profile” σ -i * = (σ 1 * , σ 2 * ,…, σ i-1 * , σ i+1 * ,…, σ n * ) ie. “σ * is a NE” just means “For all I, σ i * is a best rply to the opposition choices in σ * Defintion: Given σ i є Ai , the support of σ i (supp(σ i )) is the set of pure actions in A i to which σ i assigns positive probability. Fact: If σ * is a NE, then, for every I, every action in supp(σ i * ) is a best reply to the opposition strategy profile (σ 1 * ,…, σ i-1 * , σ i+1 * ,…, σ n * ) (Nb.: actions in A i <--> strategies in Ai assigning all probabilities to one action) Idea of proof: u i * )=a convex combination of terms of the form u i 1 * ,…, σ i-1 * , a i * , σ I+1 * ,…, σ n * ) where a i is a member of supp(σ i * ). If any such a i is not a best reply to σ- i *
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