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Chapter 8 Multicandidate Elections Although theoretically significant, elections with exactly two candidates are relatively unusual. In general there are at least two candidates and the possibility of multiple electoral competitors raises a variety of substantive and analytic issues, issues that are finessed or largely irrelevant when considering elections with two given candidates seeking a single office. For example, the concept of the "wasted vote" and questions of candidate participation, or the number of candidates, in an election are finessed by assuming a given two-candidate contest. Similarly, proportional representation schemes for determining electoral success are irrelevant when only one office is at stake. As a result, important comparative questions regarding the relative merits of various electoral rules simply cannot be addressed. Furthermore, if the election is for a legislature and legislative policy decisions require, as is typical, majority support of the elected legislators, then rational voters and candidates make their respective electoral decisions taking account of the subsequent legislative bargaining and committee decision-making. Addressing these issues, among others, requires admitting a more general class of electoral rule for multicandidate competition and providing a more complex analysis of voter behavior. In the case of exactly two candidates competing for a single office, we assumed that individuals could vote for at most one candidate with the electoral rule being defined by a simple preference aggregation rule applied to recorded profiles of votes. In effect, the winner of a two-candidate election for one position is defined under a simple electoral rule by the candidate receiving sufficient votes, where "sufficient" is defined by the rule and is not restricted to "most". When there are multiple candidates or parties, however , possibly competing for multiple elected offices or legislative seats, the 333 334 CHAPTER 8. MULTICANDIDATE ELECTIONS restriction to casting at most one vote is too limiting and proportional representation schemes can be important. In this chapter, therefore, the class of electoral rules considered is that of rank scoring rules. Rank scoring rules (defined below) include many of the simple electoral rules of Chapter 7 but in principle allow individuals to cast multiple votes and admit proportional representation where relevant. When a choice involves voting for one of two candidates, individuals'a re either indifferent or have a unique undominated voting decision: vote for the candidate offering the most preferred platform, that is, vote "sincerely". Multiple candidates, however, imply multiple pivot events for a rational voter to consider in completing a ballot and confining attention to undominated strategies often provides relatively little if any analytical purchase. In particular, there is no reason to suppose sincere voting by all individuals constitutes undominated Nash equilibrium behavior at every distribution of candidate platforms. This is illustrated by the following example. Example 8.1 Let the policy space be X = [0,1]; assume N = {1, ..., 11} and that there are three candidates identified with platforms (a, b, c) = (fI-, 1 6 1' fr)· Suppose all individuals have Euclidean preferences on X with ideal points Xi = i/11, all i E N. Finally, assume the election is for a single office and the winner is decided by plurality vote with ties being broken by a fair random device. Then if all i E N vote sincerely, i = 1,2,3 vote for a; i = 4, ..., 7 vote for b; and i = 8, ..., 11 vote for c. This results in a tie between band c, so each of these candidates wins the election with probability 1/2 and a loses for sure. However, individual i = 1 (for example) strictly prefers b to win surely than to have a 50% chance of ending up with c. Consequently, given all i > 1 vote sincerely over {a, b, c}, individual 1's best response is to vote for b; that is, voting sincerely is not an undominated strategy for 1 since to do so here is to "waste" her vote on a sure loser. 0 Empirically, the extent to which voters can be expected to adopt rational , or best response, voting strategies rather than simply vote sincerely in multicandidate elections is unclear. However, as Example 8.1 makes apparent , the equilibrium electoral outcome in such contests can be highly sensitive to the presence of a tiny number of strategically rational voters. Consequently, even if the empirical incidence of strategically rational voting amongst a large electorate is small, an assumption that all voters vote sincerely is not necessarily innocuous. On the other hand, sincere voting behavior is analytically tractable and provides a useful benchmark...

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