12_6 - Pic 1 An alternative way of thinking about Nash...

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Pic 1: An alternative way of thinking about Nash equilibria in two- player Pic2: example: The game in Tables 28.2&29.1 Column Left Right Row Top 2,1 0,0 Bottom 0,0 1,2 For a two-player game: Suppose you are one of the players For any choice that the other player can make, a choice that maximizes your payoff (given the other player’s choice) is called a “best response” or best reply A Nash Equilibrium is a pair of strategies which are mutually consistent (in the sense that each player’s strategy is a best reply to the other player’s strategy) Best responses for row: - If column chooses Left. Row’s best response is Top - If Column chooses Right, row’s best response is Bottom Best response for Column: - If row chooses Top, Column’s best response is Left - If row chooses bottom, Column’s best response is right (top, left) is mutually consistent - if row thinks column will play left, row wants to play top - if column thinks row will play top, column wants to play left - (top, left) is a Nash Equilibrium (Bottom, Right) is mutually consistent - If row thinks column will play right, row wants to play bottom, - if column thinks row will play bottom, column wants to play right - (bottom, right) is a Nash Equilibrium When there is always a unique best response: - -For each choice r that row can make, let (r) =column’s best response to row’s choice r - For each choice c that column can make, let ( c) = row’s best response to column’s choice c - A Nash Equilibrium is a pair of strategies (r*, c*) such that C*= (r*) r*= (c*) For the game in Table 28.2&29.1 -
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