University:
Boston University
Course:
CAS PO 501 | Formal Political Theory
Academic year:

2023
Views:

321

Pages:

40
Author:

gylchynarfps

A Theory of Strategic Voting in Runoff Elections Abstract This paper analyzes the properties of runoff electoral systems when voters are strategie. A model of three-candidate rumoff elections is presented, and two new features are included: voters participating in the two rounds may be different, and many types of runoff systems are considered (eg. any threshold for first-round victory). Three main results emerge. First, ranoff elections produce equilibria in which only two candidates receive a positive fraction of the votes. Second, the sincere voting equilibrium does not always exist. Finally, runoff systems with a threshold below 50% produce an Ortega effect that may lead to the systematic victory of the Condorcet loser. 1 Introduction In a runoff election, the candidate with the greatest number of votes wins outright in the first round if she obtains more than a predefined fraction of the votes (called the threshold for first-round victory). If no candidate wins in the first round, then a second round is held between the two candidates with the most first-round votes. The winner of that round wins the election. The runoff electoral system is the single most used electoral system for presidential elections: 61 out of 91 countries that elect a president directly have a runoff provision (Blais et al. 1997), France being a notorious example. Moreover, its popularity has continued to rise over the past decades: about 70% of the presidential elections held in the 90s' were runoff elections, compared to only 30% in the 60s (Golder 2005). The widespread use of the runoff system is also striking in the U.S.: runoff primaries are a trademark in southern states, and most large American cities have a runoff provision (Bullock III and Johnson 1992, Engstrom and Engstrom 2008). The perceived rationale for the worldwide use of runoff systems is twofold: first, runoff elections are expected to be more conducive to preference and information revelation than plurality elections, and second, they are intented to prevent the victory of minority candi- dates. Nonetheless, despite their relative ubiquity, our understanding of their properties and of voters' behavior is limited and mostly informal. The few formal models of runoff elections left important features aside (Cox 1997 and Martinelli 2002). In this paper, I propose a new model of three-candidate runoff elections which challenges the conventional wisdom in several important ways. My model includes two new features. First, the voters participating in the two rounds are not necessarily the same (for empirical evidence, see Wright 1989, Bullock III and Johnson 1992, Morton and Rietz 2006). I show that this leads voters to believe that all candidates participating in the second round have a positive probability of winning. Actually, the U.S. presidential electoral system is a runoff system: if no candidate receives a majority of the electoral votes, the House of Representatives chooses the President among the top three contenders. The latter rationale can be found in the literature in a slightly different form. In particular, it is often argued that runoff electoral systems should be used because they guarantee that the elected president has a mandate of a large part of the population. This is intented to legitimate her position once elected. ballot to be pivotal, candidates have to receive (almost) the same number of votes. This is more likely to happen if the second round is expected to be close. 3.1.2 First Round The first round influences the final result either directly (if one candidate wins outright in the first round) or indirectly (through the identity of the candidates participating to the second round). A ballot can affect the outcome of the election directly in two ways. First, a ballot is threshold pivotal ifij, denoted piv, if candidate i lacks one vote (or less) to pass the threshold for first-round victory and the other candidates are all below that threshold. Thus, without an additional vote in favor of i, a second round opposing i to j is held. The complementary event is the threshold pivotability ij/i, denoted piv, that refers to an event in which any ballot against candidate i, i.e. in favor of either jor k, prevents an outright victory of i in the first round and ensures that a second round opposing i to jis held. Second, a ballot is above-threshold pivotal i/j, denoted piu, if candidates i and j have (almost) the same number of votes and are both above the threshold. An additional vote in favor of candidate i allows her to win outright in the first round but, without any other ballot in favor of i, candidate j wins outright. Since two candidates cannot simultaneously obtain more than 50% of the votes, the above threshold pivotability is possible if and only if the threshold is below 50%. A ballot may also affect the final outcome if it changes the identity of the two candi- dates participating in the second round. This happens when a ballot changes the identity of the candidate who ranks third in the first round. A ballot is second-rank pivotal ki/kj when candidate k ranks first and candidates i and j tie for the second rank. An additional vote in favor of candidate i allows her, instead of j, to participate in the second round with k. Table 1 below summarizes the different first-round pivotal events that influence the first-round voting behavior. 18 Table 1: first-round pivotal events. 181 consider three-way ties as a specific case of two-way ties. For instance, the pivotal event pivl is more likely to occur when (=T2T2 than when>>>. Indeed, the occurrence of the pivotal event requires a "larger deviation with respect to the expected outcome". 3.2 Payoffs The value of each ballot, and thus voters' behavior, depends on its probability of being pivotal. In the first round, it also depends on voters' expectations about the outcome of the second round. 3.2.1 Second Round Let G² (vt) denote the expected gain of playing action in the second round. This gain depends on the voter's preference, summarized by U (It), and on the strategy functions of second-round voters, o². The strategies of other voters determine the expected number of votes received by each candidate in the second round, and thereby the pivot probabilities in that round. For a type t, the expected gain of voting for candidate Pin the first round is: (6) G² (Pit) = Pr (pivo) [U (Pt) - U(Qt)]. This reads as follows: a ballot in favor of candidate P can be pivotal in favor of P against candidate Q. If this happens, then P is elected instead of Q and voter t's payoff is U (Pt)-U(Qit). 3.2.2 First Round Let G¹ (it) denote the expected gain of playing action in the first round. This gain depends on the voter's preference, summarized by U (t), and on the strategy functions of all voters: σ' and σ². First-round strategies determine the expected number of votes received by each candidate in the first round, and thus the pivot probabilities in that round. Second-round strategies allow first round voters to compute their expected utility for the different possible second rounds (A vs. B, A vs. C, and B vs. C). For a type t voter, the expected utility of a second round opposing i to j is given by

A Theory Of Strategic Voting In Runoff Elections

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