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1 E CONOMICS S-1050 H ARVARD U NIVERSITY S UMMER 2009 S UMMER S CHOOL P ROBLEM S ET 2 A NSWERS due July 9 , in class Show all calculations, explain all answers (up to 5 lines), and fully label graphs. Also: (a) staple sheets, (b) acknowledge assistance, (c) highlight final answers, and (d) write neatly! 1. Consider the following game: - 2.5 - L M R U 3,4 2,2 0,1 M 2,2 4,3 1,2 D 1,3 1,2 3,2 a) Is there a dominant or dominated strategy for either player? Reduce the game by iterated elimination of dominated strategies (IEDS), if possible. Find the best-response of each player to the other’s strategies: L M R U 3,4 2,2 0,1 M 2,2 4,3 1,2 D 1,3 1,2 3,2 row player = player-1 , having strategies = {U,M,D}, denoted by red circles column player = player-2 , having strategies = {L,M,R}, denoted by blue diamonds (These terms and depictions will be used in every assignment in this course.) A dominant strategy for a player is the best strategy to follow no matter what the other player does. Here, if there is a dominant strategy for the row player, all the circles would line up in that row; if there is a dominant strategy for the column player, all the diamonds would line up in that column. This is not the case here, so neither player has a dominant strategy - 0.1 -

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2 A strictly dominated strategy for a player is the worst strategy to follow no matter what the other player does. If there is a dominated strategy for the row player, none of the circles would line up in that row; that is not the case here. If there is a dominated strategy for the column player, none of the diamonds would line up in that column; here we see strategy R is a dominated strategy for the column player . But if strategy R is dominated and so the column player never plays it, the row player has no reason to play D which he would play only if the column player chose R. Hence, strategy D is a dominated strategy for the row player - 0.3 - . This process of sequentially eliminating strictly dominated strategies from a game matrix is called iterated elimination of dominated strategies (IEDS) ,* which results in: L M U 3,4 2,2 M 2,2 4,3 - 0.3 - [*Some texts (e.g. Dutta, p.58-9) consider this “strong IEDS” as opposed to regular IEDS.] If there is a dominant strategy then all other strategies are dominated, but a dominated strategy does not mean all other are dominant, unless there are only 2 strategies. b) What are the Nash Equilibria (NE) of the game? Sketch the best response (BR) functions for both players. A best response (BR) strategy for a player is one that offers her the highest payoff, given the opponent’s strategy. Here, the best response for the row player, if the column player chooses L, is to play U because her payoff from U (3) is greater than her payoff from M (2). Similarly, choosing M is best for her if her opponent chooses M. The BR strategies for the row player (in red circles) and the column player (blue diamonds) are shown. When it is the best response of both players to choose the same strategy pair (the circles
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