Suppose there are three possible candidates that might run for office, and each has to decide whether or not to enter the race. Assume the electorate’s ideal points can be defined by the Hotel ling line from Chapter 26 — i.e. the ideal points are uniformly distributed on the interval [0, 1].
A: Let πi denote the probability that candidate i will win the election. Suppose that the payoff to a candidate jumping into the race is (πi −c) where c is the cost of running a campaign.
(a) How high must the probability of getting elected be in order for a candidate to get into the race?
(b) Consider the following model: In stage 1, three potential candidates decide simultaneously whether or not to get into the race and pay the cost c. Then, in stage 2, they take positions on the Hotel ling line — with voters then choosing in an election where the candidate who gets the most votes wins. True or False: If there is a Nash equilibrium in stage 2 of the game, it must be that the probability of winning is the same for each candidate that entered the race in stage 1.
(c) Suppose there is a Nash equilibrium in stage 2 regardless of how many of the three candidates entered in stage 1. What determines whether there will be 1, 2 or 3 candidates running in the election?
(d) Suppose that the probability of winning in stage 2 is a function of the number of candidates that are running as well as the amount spent in the campaign, with candidates able to choose different levels of c when they enter in stage 1 but facing an increasing marginal cost p(c) for raising campaign cash. (The payoff for a candidate is therefore now (πi −p(c)).) In particular, suppose the following: Campaign spending matters only in cases where an election run solely on issues would lead to a tie (in the sense that each candidate would win with equal probability). In that case, whoever spent the most wins the election. What might you expect the possible equilibria in stage 1 (where entry and campaign spending are determined simultaneously) to look like?
(e) Suppose the incumbent is one of the potential candidates — and he decides whether to enter the race and how much to spend first. Can you in this case see a role for strategic entry deterrence similar to what we developed for monopolists who are threatened by a potential entrant?
(f) With the marginal cost of raising additional funds to build up a campaign war chest increasing, might the incumbent still allow entry of another candidate?
B: Consider the existence of a Nash equilibrium in stage 2.
(a)What are two possible ways in which 3 candidates might take positions in the second stage of our game such that your conclusion in A (b) holds?
(b) Can either of these be an equilibrium under the conditions specified in part A?
(c) Suppose that, instead of voter ideal points being uniformly distributed on the Hotel ling line, one third of all voters hold the median voter position. How does your conclusion about the existence of a stage 2 Nash equilibrium with three candidates change? Does your conclusion from A (c) still hold?