(Electoral competition between candidates who care only about the winning position) Consider the variant of Hotelling's model in which the candidates (like the citizens) care about the winner's position, and not at all about winning per se. There are two candidates. Each candidate has a favorite position; her dislike for other positions increases with their distance from her favorite position. Assume that the favorite position of one candidate is less than m and the favorite position of the other candidate is greater than m. Assume also that if the candidates tie when they take the positions x1 and x2, then the outcome is the compromise policy 1/2(x1 + x2). Find the set of Nash equilibria of the strategic game that models this situation. (First consider pairs (x1, x2) of positions for which either x1 < m and x2 < m, or x1 > m and x2 > m. Next consider pairs (x1, x2) for which either x1 < m < x2, or x2 < m < x1, then those for which x1 = m and x2 ? m, or x1 ? m and x2 = m. Finally consider the pair (m, m).) The set of candidates in Hotelling's model is given. In the next exercise, this set is generated by an equilibrium.
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Qudsiya A.
Each of n players determines whether to be a political candidate, and if so, which position to take. There is a continuum of citizens, each of whom has a favorite position, which is distributed in the range [0, 1] with a probability density function f(). A candidate attracts those citizens whose favorite positions are closer to hers than to other candidates. If k candidates choose the same position, then each receives 1/k shares of the votes they attract. The unique winner gets Z > 0. If k candidates out of n get a tie, each of them gets Z/k. The cost of the campaign is c > 0 for each candidate. Suppose that Z/n = a. Suppose that candidates can only choose one of the following positions: m, where m is an even number greater than 2. What strategies survive IESDS? What strategies survive IEWDS? What strategies survive IESDS? Suppose that n = 2 and candidate C can choose any position in the range [0, 1]. What are the Nash Equilibria in this case? Suppose that n = 3 and candidates can choose any position in the range [0, 1]. What are the Nash Equilibria in this case?
Sri K.
Q1 Consider a population of voters uniformly distributed along the ideological spectrum from left (x = 0) to right (x = 1). Each of the candidates for a single office simultaneously chooses a campaign platform (i.e., a point on the line between x = 0 and x = 1). The voters observe the candidates' choices, and then each voter votes for the candidate whose platform is closest to the voter's position on the spectrum. If there are two candidates and they choose platforms x1 = .3 and x2 = .6, for example, then all voters to the left of x = .45 vote for candidate 1, all those to the right vote for candidate 2, and candidate 2 wins the election with 55 percent of the vote. Suppose that the candidates care only about being elected -they do not really care about their platforms at all! A) If there are two candidates, what is the pure- strategy Nash equilibrium? B) If there are three candidates, exhibit a pure-strategy Nash equilibrium. (Assume that any candidates who choose the same platform equally split the votes cast for that platform, and that ties among the leading vote-getters are resolved by coin flips.)
Supreeta N.
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