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Suppose that two candidates competing for political office must each choose a position from the set S; {1,2,..., 10}. Voters are equally distributed across these ten positions, and vote for the candidate whose position is closest to their own. If the two candidates are equidistant from a given position, the voters at that position split their votes equally. Each candidate wants to maximize their share of the total vote. Thus, for example, u₁(8,8) = 50 and u₁(7,9) 75. Note that you do not need to write out the payoff matrix to answer either of the following questions.

Suppose that two candidates competing for political office must each choose a position from the set S; {1,2,..., 10}. Voters are equally distributed across these ten positions, and vote for the candidate whose position is closest to their own. If the two candidates are equidistant from a given position, the voters at that position split their votes equally. Each candidate wants to maximize their share of the total vote. Thus, for example, u₁(8,8) = 50 and u₁(7,9) 75. Note that you do not need to write out the payoff matrix to answer either of the following questions.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Suppose that two candidates competing for political office must each choose a position from the set Sį = {1,2,..., 10}. Voters are equally distributed across these ten positions, and vote for the candidate whose position is closest to their own. If the two candidates are equidistant from a given position, the voters at that position split their votes equally. Each candidate wants to maximize their share of the total vote. Thus, for example, u₁(8,8) = 50 and u₁(7,9) = 75. Note that you do not need to write out the payoff matrix to answer either of the following questions. (a) What strategies are strictly dominated for each candidate? Give a full explanation for your answer; in particular, for each dominated strategy, be sure to explicitly identify at least one other strategy that dominates it. Solution: Strategy 1 is strictly dominated by strategy 2 for each candidate. To see this, note first that if the other candidate chooses position 1, then 2 gives a payoff of 90 whereas 1 gives a payoff of 50. If the other candidate chooses position 2, then 2 gives a payoff of 50 whereas 1 gives a payoff of 10. If the other candidate chooses a position 3 or higher, then position 2 always gives a payoff greater by 5 than that from position 1. Since position 2 gives a strictly higher payoff than position 1 against every strategy for the opponent, position 1 is strictly dominated. Similarly, strategy 10 is strictly dominated by strategy 9. No other strategies are strictly dominated since each is a best response to the opponent choosing a position one spot farther from the median. (b) What strategies remain after iterated elimination of strictly dominated strategies? Solution: Once positions 1 and 10 have been eliminated for both players, position 2 is strictly dominated by position 3 and position 9 is strictly dominated by position 8 (the argument is similar to the one given in part (i)). Once positions 2 and 9 have been eliminated, position 3 is strictly dominated by position 4, and 8 by 7. Once 3 and 8 have been eliminated, 4 is strictly dominated by 5, and 7 by 6. We are left with strategies 5 and 6 for each player, neither of which strictly dominates the other in the remaining game.
Transcribed Image Text:Suppose that two candidates competing for political office must each choose a position from the set Sį = {1,2,..., 10}. Voters are equally distributed across these ten positions, and vote for the candidate whose position is closest to their own. If the two candidates are equidistant from a given position, the voters at that position split their votes equally. Each candidate wants to maximize their share of the total vote. Thus, for example, u₁(8,8) = 50 and u₁(7,9) = 75. Note that you do not need to write out the payoff matrix to answer either of the following questions. (a) What strategies are strictly dominated for each candidate? Give a full explanation for your answer; in particular, for each dominated strategy, be sure to explicitly identify at least one other strategy that dominates it. Solution: Strategy 1 is strictly dominated by strategy 2 for each candidate. To see this, note first that if the other candidate chooses position 1, then 2 gives a payoff of 90 whereas 1 gives a payoff of 50. If the other candidate chooses position 2, then 2 gives a payoff of 50 whereas 1 gives a payoff of 10. If the other candidate chooses a position 3 or higher, then position 2 always gives a payoff greater by 5 than that from position 1. Since position 2 gives a strictly higher payoff than position 1 against every strategy for the opponent, position 1 is strictly dominated. Similarly, strategy 10 is strictly dominated by strategy 9. No other strategies are strictly dominated since each is a best response to the opponent choosing a position one spot farther from the median. (b) What strategies remain after iterated elimination of strictly dominated strategies? Solution: Once positions 1 and 10 have been eliminated for both players, position 2 is strictly dominated by position 3 and position 9 is strictly dominated by position 8 (the argument is similar to the one given in part (i)). Once positions 2 and 9 have been eliminated, position 3 is strictly dominated by position 4, and 8 by 7. Once 3 and 8 have been eliminated, 4 is strictly dominated by 5, and 7 by 6. We are left with strategies 5 and 6 for each player, neither of which strictly dominates the other in the remaining game.
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