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7. Two-Candidate Elections
- University of Michigan Press
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Chapter 7 Two-Candidate Elections Defining exactly what it is that makes a political system democratic is peculiarly difficult. A common feature of those systems typically labeled democratic , however, is the prevalence of elected officials. There are many different electoral systems in use and many more can be imagined. The institutions differ in who can vote and how often they can vote; in the number of votes any individual can cast; in the way votes are aggregated to arrive at an outcome; in the number of candidates or political parties seeking office ; in the number of elected officials associated with any given geographic district or electoral constituency; and so on. In this chapter, we look at one particular sort of election: two-candidate competition for a single office under a given electoral rule. The introduction of more than two candidates and multiple offices is left for the next chapter. In Chapter 3 (section 3.4), an electoral rule is described abstractly as a mechanism consisting of a set of admissible vote profiles on a set of possible outcomes, and an outcome function mapping vote profiles into outcomes. In what follows the focus is on two-candidate electoral competition and it is useful to be more explicit about such mechanisms. Given two candidates, an individual typically decides either to vote for one or other candidate, or to abstain. An admissible vote profile, therefore, is a list of such decisions, one for each member of the electorate, and the election is decided according to a rule for aggregating the votes. And because individuals can in general abstain or vote "strategically" for a candidate other than the one they most prefer, there is no presumption that an individual's vote invariably reflects the individual's underlying preferences over candidates. Example 7.1 Suppose there are five individuals and two alternatives, a, b. The preference profile over a, b has individuals 1,2 and 3 strictly prefer a 253 254 CHAPTER 7. TWO-CANDIDATE ELECTIONS to b, and individuals 4 and 5 strictly prefer b to a. If a collective choice is made under plurality preference aggregation, then a is chosen. However, if individuals have to record their preferences by voting and individuals 1 and 2 abstain, plurality vote aggregation results in b defeating a. 0 Despite the fact that recorded preferences, votes, need not coincide with individuals' true preferences, Example 7.1 indicates that, at least from a formal perspective, vote aggregation procedures are equivalent to collective choice rules, associating electoral winners with realized vote profiles. In particular, with only two candidates for an office, a vote for one of them is treated exactly as a strict preference for that candidate and abstention is treated as indifference. In this chapter, therefore, we assume for the most part that votes cast in a two-candidate election are aggregated by a simple rule (Definition 1.3). Although, among others, the class of simple rules includes majority rule and the q-rules, it excludes plurality rule and weighted q-rules. However, as we argue later, at least when individuals have continuous and strictly convex preferences over policy alternatives, the existence and characterization results for two-candidate electoral equilibria under a simple rule generalize easily to a broader class of rules ("voting rules") that includes these latter aggregation procedures along with many others. 7.1 Electoral equilibrium and the core The benchmark model of electoral competition is a spatial voting model in which two candidates seek a single office by offering policy platforms freely chosen from some given set of alternatives. Eligible voters have given preferences over the set of alternatives and vote on the basis of the platforms offered by candidates. Let the set of candidates be {0:,,B} and let N = {I, ... , n} be the electorate. The policy space is a (typically, convex and compact) set X C ~k. Each voter i E N has policy preferences on X representable by a strictly quasi-concave utility function Ui : X -> R As in earlier chapters, we abuse notation slightly and occasionally write the utility profile U E R~s when U = (Ul, ... ,un) represents the profile p E R~s' Candidates too have preferences although they need not be defined directly on the policy space, X; candidates, for example, might plausibly be more interested in winning office than in policy per se, or in some combination of winning and the policy eventually implemented, irrespective of who wins. Although the implications of adopting various assumptions for candidates...