Problem Set #4

PlSc 349

Due: Tuesday, 28 September 2010

1. [8 points] Problem U5.1
2. [3] Problem U5.5
3. [3] Problem U5.6
4. [8] Problem U5.8 (including part c).
5. [14] [from Hotelling 1929 via Gintis 2000 and Osborne 2004] Suppose two candidates compete by choosing a policy position in a one-dimensional policy space. Voters are distributed uniformly across the policy space, which, for simplicity, goes from 0 to 1. Each candidate independently and simultaneously chooses a position in the policy space. After candidates have chosen positions, voters vote for the candidate whose position is closest. "Thus, for instance, if one [candidate] locates at point x and the second at point y > x, then the first will get a share x+(y-x)/2 and the second will get a share (1-y)+(y-x)/2 of the [voters] each. (Draw a picture to help you see why.)" "The candidate who obtains the most votes wins. Each candidate cares only about winning; no candidate has an ideological attachment to any position. Specifically, each candidate prefers to win than to tie for first place (in which case perhaps the winner is determined randomly), and prefers to tie for first place than to lose."
   • Find each candidate's best-response rules.
   • Find the Nash equilibrium positions.
   • Suppose that voters are distributed normally across the policy space (centered at 1/2). Find the Nash equilibrium positions.
   • Suppose that voters are distributed in any configuration. Find the Nash equilibrium positions.
   • What might this explain about U.S. politics?
   • Suppose there are three parties. For simplicity, assume voters are distributed uniformly. Show that there is no pure strategy Nash equilibrium. What might this explain about U.S. politics?
6. [14] For the previous problem, discuss what you expect to happen from a behavioral game theory perspective (assuming parties can reason like individuals). In other words, reconcile what we observe with a model of behavioral decision-making. Note that the parties may not act according to Nash equilibrium calculations. Connect your reasoning to the concepts discussed in Camerer. Do this for:
   • the U.S. case (two parties).
   • the U.K. case (three parties).
7. [for future planning] Which problem was most useful in learning the concepts? Which problem was least useful?

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