No Polarization in Spite of Primaries: A Median Voter Theorem with Competitive Nominations

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.

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.

Table 1 Table 1

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.

Table 2 Table 2

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.

Table 3 Table 3

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. □