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Fall 2007 G AME T HEORY IN THE S OCIAL S CIENCES Problem Set 2 (Due in Lecture Tuesday, October 2) 1. In lecture we studied how political parties or candidates choose their positons when all they care about is winning. This question explores what happens when political candidates care not only about winning but also about the policies they espouse. The U.S. Congress is up for grabs in the election this November. Bipartisanship has again broken down, and the parties are very polarized. The more liberal faction of the Democratic Party now dominates that party and a more conservation faction dominates the Republican Pary. As the election appoaches, these parties are trying to stake out positions that reflect their own policy preferences and will attract enough voters to win. To simplify matters, suppose the parties have to choose a position along a left-right spectrum and can adopt one of the following positions: Liberal (L), Liberal leaning centrist (LC), Middle of the Road (M), Conservative Centrist, (CC), Conservative (C). These positions are represented on the line below where the distance between any two neighboring positions is the same. LL CMC CC Since the voter’s ideal points are evenly distributed along the political spectrum, the party whose position is closer to the middle of the road (M) wins. If, for example, the Republicans, R , choose a position of CC and the Democrates, D , announces, L, then R would win because CC is closer to M. If the parties run on platforms that are equally distant from M, each party is equally likely to win. Finally, each party chooses its platform secretly. As noted above, R and D care about policies as well as winning. D ’s von Neumann-Morgenstern payoffs are: 5 for winning with L; 4 for winning with LC; 3 for winning with M; 2 for winning with CC; 1 for winning with C; -1 for losing with L; -2 for losing with LC; -3 for losing with M; -4 for losing with CC; and –5 for losing with C. R ’s von Neuman-Morgenstern payoffs are the opposite: 5 for winning with C; 4 for winning with CC; 3 for winning with M; 2 for winning with LC; 1 for winning with L; -1 for losing with C; -2 for losing with CC; -3 for losing with M; -4 for losing with LC; and –5 for losing with L.
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(a) Suppose R adopts position C and D chooses LC. What are their payoffs? (b) Suppose R adopts C and D chooses the more extreme position L. What are the parities payoffs? (c) Specify the strategic form of this game. (d) Solve the game by iterated deletion of dominated strategies. Be sure to indicate the order of deletion, what dominates what, and whether this is strict or weak dominance. (e) What is/are the pure strategy Nash equilibria of this game? (f) Finally, suppose that there are three parties instead of two. Call the third party S (for spoiler) and assume that S has the same preferences that D does. Consider the three-player game in which each player selects his position secretly; each voter votes for the party whose position is closest to her ideal point; and the party that receives the most votes wins. If each candidate chooses M, is this a Nash equilibrium of the game?
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This note was uploaded on 12/25/2010 for the course PO 137 taught by Professor Power during the Fall '10 term at University of California, Berkeley.

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