Abstract
It is commonly assumed that primaries induce candidates to adopt extremist positions. However the empirical evidence is mixed, so a theoretical investigation is warranted. This chapter develops a general model introducing the fundamental elements of primary elections in the well-known spatial voting model by Downs (An economic theory of democracy. Harper and Brothers Publishers, New York, 1957). In spite of significant incentives for candidates to diverge, I find the surprising result that they will all converge to the median voter’s ideal point. The result in this paper suggests that primaries are not sufficient to create polarization by themselves. Rather, for candidates to diverge from the center, other complementary features must be present. An implication is that previous formal results in the literature predicting that primaries lead to polarization probably contain other factors that must be interacting with primaries. Future research should endeavor to disentangle these factors.
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Jones (2012).
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Abjorensen et al. (2012).
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Indriðason and Sigurjónsdóttir (2014).
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Gherghina (2013).
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Field and Siavelis (2009).
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Folke et al. (2013).
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For example Schumer (2014) in the New York Times.
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All the results would hold for any symmetric and single-peaked utility function for voters. The quadratic is used as an illustration.
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This unpublished proof is available upon request.
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All the results would hold for any strictly concave utility function for parties. They would also hold if the parties’ ideal points were not equidistant from the median voter.
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It must be noted that other models of primary elections, some of them withe quite different assumptions, reach similar conclusions. See for example Proposition 2 of Kselman (2014) which finds convergence as a corner solution.
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As I mentioned above, the exact same result would be obtained with any strictly concave utility function for parties and voters. Complete convergence can also be proved to be the only possible outcome with strictly risk loving parties. And the parties’ ideal points could take any value on opposite sides of the median voter.
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We are thus discarding the possibility of flip-flopping during the election season. One way to justify this assumption is that, in this election, flip-flopping would hurt the candidate’s credibility so much that it would never be an optimal strategy.
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This is not an exhaustive list of all the possible configurations. In this section, I only analyze the cases that build an interesting intuition. The proof in the appendix gives the exhaustive list of configurations and determines whether each of them is an equilibrium or not.
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I should note how reminiscent this result is to the one found by Calvert (1985). In his seminal model with two policy-motivated parties with extremist ideal points, he famously proved that both parties will completely converge to the median voter’s ideal point. The logic of his result is similar to the one here, and thus my model can be thought of as a generalization of Calvert (1985) to a situation where a nomination process is added in each party. The fact that a convergence to the median still holds in my model illustrates what Calvert called the “robustness” of the spatial voting model.
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Appendix
Appendix
Without loss of generality, the configurations in Table 1, along with their symmetric counterparts, are an exhaustive list of all the possible configurations of platforms that candidates may adopt. All cases are mutually exclusive.
With this list in mind, I proceed to prove the theorem in this paper.
Proof
The game must be solved by backwards induction. The procedure will be the following: we start by solving the game at its last stage—the general election—and we find the median voter’s strategy profile \(S_{v}^{{\ast}}\) that forms a NE in every situation in which she might be called upon to act. Given \(S_{v}^{{\ast}},\) we consider the reduced game at the second stage—the nominations by each party—and we find the strategies \(S_{L}^{{\ast}}\) and \(S_{R}^{{\ast}}\) that form a NE for the parties in every possible subgame in which they might be called upon to act. Finally, for each \(S_{v}^{{\ast}}\), \(S_{L}^{{\ast}}\) and \(S_{R}^{{\ast}},\) we consider the reduced game at its first stage—the platform adoption—and we find all the strategies \(S_{c}^{{\ast}}\) that form a NE for the candidates. At this stage (the platform adoption), we know that a NE of the reduced game will be a SPNE of the game as a whole.
Third Stage
First we prove that sincere voting is a weakly dominant strategy for voters. When casting her vote, a voter is either pivotal or not. If she is pivotal, then voting other than sincerely will make her worse off (or no better off if she is indifferent between both parties). If her vote is not pivotal then any strategy leads to the same outcome. Therefore, sincere voting is never worse and sometimes better than not voting sincerely. Sincere voting weakly dominates every other strategy for voters. Since we have assumed that a player will never choose a weakly dominated strategy, all voters will vote sincerely. Given that the preferences of voters are symmetric and single peaked, the electorate will behave according to the preferences of the median voter. There are two possible subgames: either \(r_{i} = -l_{j}\) or r i ≠ − l j . In the latter case, the candidate closer to zero will win the election. In the former case, there is a tie between the candidates, and the median voter will decide by flipping a coin.
Second Stage
Without loss of generality, the configurations in Table 2, along with their symmetric counterparts, are an exhaustive list of all the possible subgames that parties may face, along with their corresponding NE (considering only the NE in pure strategies and non-weakly dominated strategies). In this list, the pair of strategies \(\left (l_{i},r_{j}\right )\) refers to the decision of party L to nominate l i in conjunction with the decision of party R to nominate r j . The strategy “randomize” stands for the decision of the party to randomize equally between its two candidates.
To be part of a SPNE, any strategy profile \(S_{L}^{{\ast}}\) and \(S_{R}^{{\ast}}\) must induce these NE in the corresponding subgames. Note that subgames 3, 9, 10, 15 and 16 allow two NE in pure strategies, while all the other subgames allow a unique NE. To illustrate how this table was derived, I will prove the NE in subgame 3. Party R does not have a real choice since both of its candidates have adopted indistinguishable platforms. Its unique available strategy is to randomize between r 1 and r 2. On the other hand, party L has a choice between l 1 = 0 and l 2 > 0. If L nominates l 1 it will tie with R and the policy implemented will be 0 for sure. If L nominates l 2 it will lose against R and the policy implemented will be 0 for sure. Hence, both nominations lead to the same policy outcome and give L the same utility. Therefore, L is indifferent between l 1 and l 2 and the Nash equilibria are (l 1, randomize) and (l 2, randomize). Analysis of the other 27 subgames follows a similar logic.
First Stage
Without loss of generality, the configurations in Table 3, along with their symmetric counterparts, are an exhaustive list of all the possible configurations of platforms that candidates may adopt, along with a profitable deviation, if any. Below, \(\varepsilon\) is some small positive number.
I will prove why configuration 1 is a NE for the candidates. Suppose none of the candidates deviated. Then parties would face subgame 1, and we can see from Table 2 that each party randomizes between their candidates. Each candidate has a probability of \(\frac{1} {4}\) of winning the election (\(\frac{1} {2}\) probability to be nominated times \(\frac{1} {2}\) probability to win the election conditional on being nominated). Suppose, on the other hand, that one of the candidates deviated unilaterally. Then parties would face subgame 3 or its symmetrical counterpart, and we can see that the candidate who deviated would either lose the nomination or win the nomination but lose the election for sure, depending on which of the two equilibria in subgame 3 was selected. Such a deviation is therefore not profitable, and the configuration is a NE.
Now I prove why configuration 2 is not a NE. Suppose none of the candidates deviated. Then the parties would face subgame 2, and we can see from Table 2 that each party randomizes between their candidates. Each candidate has a probability of \(\frac{1} {4}\) of winning the election. Suppose, on the other hand, that one of the candidates, say r 1, deviated unilaterally to zero. Parties would face subgame 7 and r 1 would win both the nomination and the election. Since this is a profitable deviation for r 1 this configuration is not a NE.
In a similar way it can be proved that configurations 3 to 28 are not NE (see the profitable deviations in each case). Thus configuration 1 is the unique NE of the reduced game, and it is the unique strategy profile of candidates that can be part of a SPNE. Therefore in any strategy profile \(S_{c}^{{\ast}},\) \(S_{L}^{{\ast}}\) and \(S_{R}^{{\ast}},\) \(S_{v}^{{\ast}}\) that forms a SPNE, the outcome will be the same: candidates adopt the platforms in configuration 1, which are \(0 = r_{1} = r_{2} = -l_{1} = -l_{2},\) parties have no choice but to select the strategies (randomize, randomize), and voters have no choice but to randomize between the two parties. This is exactly what the theorem says. □
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Serra, G. (2015). No Polarization in Spite of Primaries: A Median Voter Theorem with Competitive Nominations. In: Schofield, N., Caballero, G. (eds) The Political Economy of Governance. Studies in Political Economy. Springer, Cham. https://doi.org/10.1007/978-3-319-15551-7_11
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