Open primaries and crossover voting

Abstract

We examine strategic voting in open primary elections by developing a Poisson voting game. In the model, two parties simultaneously hold primary elections, with two candidates competing in each party. Each voter chooses to vote for one of the four candidates without knowing how many other voters participate in each primary. Analyzing the model, we investigate what types of strategic crossover voting occur in equilibrium and under what circumstances they occur. In particular, we focus on two types of crossover voting: hedging (voting for the moderate candidate of the opposite party) and raiding (voting for the extreme candidate of the opposite party). We show that the pattern of strategic voting in equilibrium critically depends on candidate positions and uncertainty about the outcome in the general election.

Keywords

Crossover voting open primaries Poisson game strategic voting

1. Introduction

Political parties are the main competitors in electoral politics in modern representative democracies. As such, the process of nominating party candidates is one of the most crucial stages in the democratic process. By constraining the choices available to citizens in elections, the method by which parties choose their nominees can potentially influence the identity of policymakers and hence policy outcomes.

The nomination process has become increasingly open to the general public. In the US, since the beginning of the last century, the primary has replaced traditional elite controlled caucuses and is now the chief method of nominating candidates.¹ This opening up of the nomination process has culminated with the introduction of the ‘open primary’ in several states over the last several decades. In open primaries, any eligible voter can participate in the nomination process for various political offices. Currently, about 50% of the US states use the open primary system.

The open primary system provides voters with a unique strategic environment by allowing them to cross the party line with negligible cost. In fact, empirical studies have identified evidence of crossover voting in various open primary elections (Adamany, 1976; Alvarez and Nagler, 2002; Hedlund, 1977; Hedlund and Watts, 1986; Hedlund et al., 1982; Sides et al., 2002; Southwell, 1989, 1991; Wekkin, 1988). Yet insufficient attention has been paid to open primaries in the literature on voting theory. In particular, strategic voting in open primary elections has rarely been a subject of rigorous formal modeling, even though the strategic considerations of voters in open primary elections are so distinct that the existing models of elections cannot be directly applied to open primaries. First, as is the case in any primary election, supporters of a party in open primaries may face a tradeoff between the candidate whose policy position is closest to their own and the candidate who has the best chance of winning the general election. Furthermore, as voters are allowed to cross the party line in open primaries, choosing which primary to participate in is an essential part of strategic considerations.

The possibility of crossover voting has also been an important issue in the normative debate over different primary systems. Proponents of the open primary system are concerned with the representativeness of primary electorates and argue that the inclusive nature of the open primary system has a moderating influence on the electoral outcomes (Geer, 1986; Kaufmann et al., 2003). Yet a recent empirical study on the US state legislature by McGhee et al. (2014) finds that the openness of the primary election has little effect on the ideology of the representatives.² On the other hand, opponents of the open primary system worry that crossover voters dilute the preferences of partisans. In particular, the possibility of ‘raiding’ is often discussed as grounds for opposing the open primary system. The concern is that members of one party intentionally vote for a weak candidate in their rival party’s primary to decrease the chance of the rival party’s winning in the general election.

Both arguments for and against open primaries rest on conjectures about voting behavior. The argument that open primaries promote moderation posits sincere voting, whereas opponents of open primaries worry about the opposite. Still, the direction of strategic voting in open primaries is not so obvious. Although the open primary system provides an opportunity for raiding, raiding may be costly because voters may lose the chance to elect their favorite candidate in their own party. In addition, it may be risky because raiding can result in the worst candidate’s winning in the general election. As such, a rational voter’s choice in open primaries is not self-evident. It may depend on the voter’s expectation of how others will vote, her expectation of who will win each primary absent her vote, and her expectation of who will win the general election given a pair of nominees. Hence, voting behavior in open primaries is a subject on which a rigorous formal model may advance our understanding. An improved understanding will help us identify the conditions under which open primaries induce moderation and the ones under which they may induce extremism.

We develop a Poisson game model of strategic voting in open primary elections. There are two political parties, each of which has two potential nominees. The two parties hold primary elections simultaneously. Each voter chooses to cast her ballot in either one of the two primaries. Voters’ preferences over the candidates are assumed to be derived from their preferences on a unidimensional set of policies. Voters are uncertain how many other voters will participate in the primary elections. To focus on voters’ strategic choices, we take the positions of candidates as exogenous parameters. This allows us to study the relationship between strategic voting and candidate positions. Focusing on voting behavior in large primary elections, our analysis examines limits of equilibria when the expected number of voters becomes large.

Three different types of strategic voting may occur in open primaries:

Compromising: extreme voters support the moderate candidate of their own party because the moderate candidate has a higher chance of winning in the general election than the extreme candidate.

Hedging: partisan voters support the moderate candidate of the opposite party to avoid the worst case in which the extreme candidate of the opposite party wins in the general election.

Raiding: partisan voters support the extreme candidate of the opposite party in order to increase the likelihood of their own party’s winning in the general election.³

By analyzing the model, we investigate what types of strategic voting occur in equilibrium and under what circumstances.

We find that an important factor that shapes the equilibrium pattern of strategic voting is candidate positions relative to the compromise points⁴ of core partisans (i.e. voters whose ideologies are relatively extreme) in each party. The extreme partisans’ compromise points depend on the level of aggregate uncertainty about the outcome in the general election perceived at the time of the primary election. Given the level of uncertainty, on the one hand, if the relatively extreme candidate’s policy position in a party is not sufficiently moderate, the core partisans of that party wish the moderate candidate to be the nominee, as the extreme candidate has too a small chance of winning in the general election. On the other hand, if the extreme candidate’s position is sufficiently moderate, then the core partisans prefer the extreme candidate to the moderate one as the nominee of the party. These two different circumstances lead to different equilibrium outcomes.

When the moderate candidate is more appealing to the core partisans of the party than the extreme candidate in each party, our equilibrium prediction is the following. First, a substantial amount of raiding occurs in equilibrium. The core partisans engage in raiding; that is, they vote for the extreme candidate of the other party instead of voting for the moderate candidate of their own party. However, raiding never makes the extreme candidate of the rival party the winner of the party primary for sure. Such a candidate has a positive probability of winning her primary with the help of raiding partisans from the other party only when she is more moderate than the median of her party’s constituents. Second, compromising may or may not occur depending on specific locations of candidate positions, but hedging never occurs.

On the other hand, when the extreme candidate is more appealing to the core partisans than the moderate candidate in each primary, we predict the following. First, neither raiding nor compromising occurs in equilibrium. Second, a substantial amount of hedging occurs. Third, the voters who engage in hedging are those whose positions are between the two candidates in each party; that is, those who are relatively indifferent between the two candidates in their own party. Thus, the model makes predictions about the relationship between candidate positions and strategic voting as well as about the ideology of voters who engage in different types of strategic voting.

In addition, we compare these results with what would happen if the primaries were closed. We find that the nominees under the open primary system are in general more extreme than those under the closed primary system when the moderate candidate is more appealing to the core partisans of the party than the extreme candidate in each party. By contrast, when the core partisans in each party prefer the extreme candidate to the moderate candidate as the nominee of their own party, the equilibrium outcome under the open primary system is more moderate than under the closed primary system. Hence, we provide a more nuanced prediction on the effects of the open primary system on the ideologies of candidates in the general election.

The existing formal models of primary elections mostly focus on the closed primary system. As is well known, primary elections are used to explain divergent positions of political parties. Coleman (1971) and Aranson and Ordeshook (1972) first examine the consequences of primary elections using formal models. While their models are not fully strategic, more recent work has developed models of the closed primary system with strategic partisan voters and strategic candidates (Cadigan and Janeba, 2002; Owen and Grofman, 2006; Serra, 2010). Also, there are models of closed primaries focusing on their informational role (Adams and Merrill, 2008; Meirowitz, 2005).

Only a few formal studies examine the open primary system. Chen and Yang (2002) analyze a model in which one party holds an open primary and voters have an exogenous probabilistic belief about the candidate of the other party. Thus, in their study, voters’ strategic choices between different primaries are not modeled. Oak (2006) develops a model of primary elections and compares equilibrium voting strategies in different primary systems. He shows that, first, the semi-closed primary system induces more moderate outcomes than the closed primary system. Second, the outcome may or may not be more moderate in the open primary system than in the closed primary system. This inconclusiveness is due to the multiplicity of equilibria. His equilibrium concept imposes the rationality assumption only on choosing between candidates in each primary; it does not require any optimality condition for choosing between different primaries. Thus, the very distinct strategic consideration in open primaries is unexamined, and the set of equilibria is large. An important consideration of rational voters in open primaries is which primary election is a closer race, that is, in which primary their votes are more likely to be influential. In any model that captures such strategic participation decisions, calculations of the pivot probabilities are necessarily involved. We employ the Poisson model to minimize the difficulty of these probability computations.

The rest of the paper proceeds as follows. In the next section, we set up the model. In Section 3, we provide the analysis of the model. In Section 4, we present the main findings, in Section 5, we provide an extension, and, in Section 6, we conclude. All formal proofs are contained in Appendix A.

2. Model

Let T = [−1, 1] be the policy space. There are two parties L and R. A party here is simply a collection of its potential candidates for a political office, and a candidate is simply a policy that will be implemented if she takes the office. Each party has two candidates: L = {ℓ_e, ℓ_m} and R = {r_e, r_m}. We assume ℓ_e < ℓ_m < 0 < r_m < r_e. That is, L is the left party, R is the right party, and each party has a relatively extreme candidate and a relatively moderate (centrist) one. For simplicity and tractability, we also assume that the candidate positions are symmetric around zero: ℓ_e + r_e = ℓ_m + r_m = 0. (We will discuss the consequences of dropping this symmetry assumption later.)

The primary elections of the two parties are held simultaneously, and the outcome is a pair of nominees (ℓ, r) ∊ L × R, one per party. Although we do not model a general election explicitly, we assume that voters’ preferences over the outcomes of the primaries are derived from their policy preferences and expectations of the general election outcome. If a policy y ∊ T is implemented, a voter of type t ∊ T receives utility

v (y; t) = - | y - t |

(1)

The types of voters are distributed uniformly on T. Let F denote the cumulative distribution function of the uniform distribution on T.

In the general election where ℓ ∊ L and r ∊ R compete, the vote share of ℓ is assumed to be

F (\frac{ℓ + r}{2}) + ϵ

where $ϵ$ is a uniform random variable on [−β, β].⁵ The justification of the assumption is the following. Since the general election is a binary election, each voter will vote sincerely. Thus, if the number of voters is very large and there is no uncertainty, the vote share of ℓ will be F((ℓ + r)/2). However, a general election is usually held at least a few months after primary elections. Hence, at the time of primaries, there is uncertainty about the aggregate result of the general election, which is captured by the random variable $ϵ$ . The number β > 0 is the precision parameter that measures the level of uncertainty.⁶ Then, ℓ wins the election if and only if

ϵ \geq \frac{1}{2} - F (\frac{ℓ + r}{2})

Thus, the probability that ℓ defeats r is

π (ℓ, r) = \frac{1}{2} + \frac{ℓ + r}{8 β}

(2)

We assume

β > \frac{ℓ_{m} + r_{e}}{4}

This assures that no candidate wins for sure in the general election and that the formula (2) is applicable for every (ℓ, r) ∊ L × R. The winner in the general election implements her policy position. Then, for each outcome (ℓ, r) of the primaries, a type-t voter’s expected utility is

u (ℓ, r; t) = π (ℓ, r) v (ℓ; t) + [1 - π (ℓ, r)] v (r; t)

(3)

We model primary elections as a Poisson game (Myerson, 1998). The players of a Poisson open primary game are a finite number of voters, and the number of voters k is drawn from the Poisson distribution with mean n. Each voter’s type is independently drawn from T according to the uniform distribution. Each voter knows her own type, but does not know the other voters’ types, nor how many voters participate in the game. The distributions of the electorate size and of the voter types are common knowledge.

In the game there is only one stage of simultaneous actions in which each voter votes for one candidate among the four. That is, a voter can participate in either of the parties’ primaries but only one of them. Let C = L ∪ R be the set of available actions. A full play of the game is a profile of votes x ≡ (x(c))_c∊C where, for each c ∊ C, x(c) is the number of votes that candidate c receives. Let $ℤ_{+}$ denote the set of non-negative integers and let

Z (C) = {x \in ℝ^{C} | x (c) \in ℤ_{+} for every c \in C}

denote the set of all vote profiles. In the primary of each party, a candidate who receives more votes than the other becomes the party’s nominee. When a tie occurs, the moderate candidate wins.⁷ Given a vote profile x and a candidate c ∊ C, let ℓ(x, c) and r(x, c) denote the nominees of L and R, respectively, when profile x is played and one additional vote is cast for c. Then,

U (x, c; t) = u (ℓ (x, c), r (x, c); t)

(4)

is the payoff a type-t voter receives when she votes for c and x is the profile of votes chosen by the other players. We let Γ(n) denote the Poisson open primary game with the mean electorate size n, making its dependence on the other parameters implicit.

A (pure) strategy function is a measurable mapping s : T → C, where s(t) is the candidate whom a voter of type t would vote for. A strategy function, together with the distributions of the electorate size and of the voter types, defines a probability distribution on the set of vote profiles Z(C). Given a strategy function s and a candidate c ∊ C, let

τ (c) = \int_{s^{- 1} (c)} d F (t)

(5)

be candidate c’s expected vote share out of the total votes in the primaries of both parties. Let the vector τ ≡ (τ(c))_c∊C denote the expected vote distribution corresponding to the strategy function s. The independent-actions property of the Poisson game (Myerson, 1998) implies that the number of votes for each candidate c is a Poisson random variable with mean nτ(c) and that it is independent of the number of votes for every other candidate. Thus, the probability that x ∊ Z(C) is the realized vote profile is

P (x | n τ) = \prod_{c \in C} [\frac{\exp (- n τ (c)) {(n τ (c))}^{x (c)}}{x (c)!}]

(6)

Moreover, by the environmental equivalence property of the Poisson game (Myerson, 1998), every single voter assesses the same probability function P(·|nτ) for the vote profile generated by all other voters, counting every vote except her own.

Then, given the expected vote distribution nτ, the expected payoff for a type-t voter from voting for c in the game Γ(n) is

V (c; t | n τ) = \sum_{x \in Z (C)} P (x | n τ) U (x, c; t)

(7)

Our solution concept for the Poisson primary game is as follows.

Definition 1. We say a pair (s, τ) is an equilibrium of Γ(n) if, for every t ∊ T,

s (t) \in \arg \max_{c \in C} V (c; t | n τ)

(8)

and $τ \in ℝ^{C}$ satisfies (5) for every c ∊ C.

That is, in equilibrium, each voter type maximizes the expected payoff given the probability distribution over the profiles of votes which is generated by the strategies of all types. Let Σ_n denote the set of equilibria of Γ(n).

An event is any subset of Z(C). Given the expected vote profile nτ, an event A occurs with probability

P (A | n τ) = \sum_{x \in A} P (x | n τ)

For each c ∊ C, let O(c) be the event that candidate c wins in the primary of her party. Given our tie-breaking assumption, O(ℓ_m) = {x ∊ Z(C)|x(ℓ_m) ≥ x(ℓ_e)} and O(ℓ_e) = {x ∊ Z(C)|x(ℓ_m) < x(ℓ_e)}. Analogously, O(r_m) and O(r_e) are defined. For each party D = {c, c′} ∊{L, R}, let A(c) be the event that c′ wins in the primary of D but one additional vote for c could make c the winner. Again, given the tie-breaking assumption, A(ℓ_m) = {x ∊ Z(C)|x(ℓ_m) + 1 = x(ℓ_e)} and A(ℓ_e) = {x ∊ Z(C)|x(ℓ_m) = x(ℓ_e)}. Analogously, A(r_m) and A(r_e) are defined.

We let $\bar{A} = \cup_{c \in C} A (c)$ , which is interpreted as the event that a single vote can make a pivotal difference in at least one of the primaries. If $\bar{A}$ does not occur, a voter’s choice makes no difference to the outcome. As such, a rational voter cares about her action only in the event that there is at least one close race where her vote matters. Let $\bar{V} (c; t | n τ) = \sum_{x \in \bar{A}} P (x | n τ) U (x, c; t)$ . The following lemma formalizes the discussion.

Lemma 1. Let s be a strategy function and τ be the expected vote distribution corresponding to s. For every $n \in ℕ$ , (s, τ) ∊ Σ_n if and only if $s (t) \in \arg \max_{c \in C} \bar{V} (c; t | n τ)$ for every t ∊ T.

Note that since the size of the electorate can be any non-negative integer with positive probability in the Poisson voting game, $P (\bar{A} | n τ)$ is always positive. Given that, the lemma is so obvious that we omit the proof of it. When n is large, the probability of a close race in general is very small. However, rational voters would choose their ballots as if $\bar{A}$ occurred. Thus, our analysis would be unaltered if all probabilities were replaced by conditional probabilities given $\bar{A}$ . We will use Lemma 1 to prove the results in the next section.

The existence of equilibria is not an issue of our model. Myerson (2000) proves that, in Poisson games with continuous types and finite actions, there exists an equilibrium in distributional strategies⁸ if the payoff function is bounded and continuous in types. This guarantees the existence of mixed strategy equilibria in our model. Moreover, with our specific assumptions of the payoff functions, for any mixed strategy equilibrium, there exists a pure strategy equilibrium that generates the same vote shares for the candidates. Thus, we lose little generality in terms of aggregate predictions by focusing only on pure strategies. From now on, we explore strategic voting in large open primary elections focusing on necessary conditions that must be satisfied in equilibrium.

3. Analysis

3.1. Strategic preferences of voters

In this section, we provide the essential steps that are needed to derive the results in the next section. These steps also help to develop the intuition of strategic incentives in open primary elections. The main distinction of primary elections from general elections is that they determine only the nominees of the parties but not the ultimate policymaker. As such, voters’ strategic considerations that are absent in general elections may affect voting behavior in primaries. For example, an extreme left partisan voter may want her party’s nominee to be centrist in order to avoid the party’s losing in the general election. Or, she may wish an extreme right-wing candidate to be selected in the right party’s primary because such a nomination enhances the prospects of the left party in the general election. To examine these strategic considerations systematically, we investigate voters’ strategic preferences over different positions of one party nominee given the outcome of the primary of the other party.

We first fix the winner of the R primary at r ∊{r_m, r_e} and examine voters’ preferences over policy positions of the nominee of party L. For presentation, it will be convenient if every utility function is multiplied by positive constant 8β. With an abuse of notation but without loss of generality, we assume v(y; t) = −8β|y − t| from now on. Then, from (2) and (3), we obtain

u (ℓ, r; t) = - (4 β + ℓ + r) | ℓ - t | - (4 β - ℓ - r) | r - t |

(9)

Note that, given our assumption, ℓ varies from max{−1, −r−4β} to zero. Simple algebra can show that

u (ℓ, r; t) = {\begin{array}{l} 8 β t - {(ℓ + 2 β)}^{2} + {(r - 2 β)}^{2} & if t < ℓ \\ - 2 (ℓ + r) t + {(ℓ + 2 β)}^{2} + {(r - 2 β)}^{2} + 2 ℓ r - 8 β^{2} & if ℓ \leq t < r \\ - 8 β t + {(ℓ + 2 β)}^{2} - {(r - 2 β)}^{2} & if t \geq r \end{array}

(10)

Figure 1 graphs u(ℓ,r;t) for different values of t when r < min{4β,1 − 4β}. A few notable features are in order. First, for every voter whose type is less than −2β, the function u(ℓ, r; t) is maximized at ℓ = −2β. That is, the extreme leftist voters would choose −2β if they could decide the position of the left party nominee. They would not choose any point less than −2β because the winning probability is too small. Nor would they choose any point greater than −2β because the policy then would be too rightist. Given that all voters whose ideal points are less than −2β share the same strategic ideal point and that these voters are most extreme leftist voters, we call these voters the core leftists or the core left partisans. We also refer to −2β as the compromise point of the core leftists. Second, for every voter whose type is between −2β and zero, the function u(ℓ, r; t) is maximized at ℓ = t. Thus, for each moderate leftist, her ideal point is the best position of the left party’s candidate. Third, for every voter whose type is between zero and r, the function u(ℓ, r; t) is maximized at zero. That is, moderate rightists like the left candidate’s position to be as moderate as possible. However, this is not the case for extreme rightists. Fourth, for the voters whose type is greater than r, the best position of the left party’s candidate is either the most extreme position (max{−1, −r −4β}) or the most moderate position, zero, depending on r. Thus, there may be an incentive for them to choose the extreme leftist as the candidate of the left party. A symmetric argument holds for u(ℓ, r; t) as a function of r. Thus, we use the term the core rightists to refer to the voters whose ideal point is greater than 2β and the term the compromise point of the core rightists to refer to the point 2β. Note that β measures the degree of uncertainty about the general election outcome. As the uncertainty is small, more moderate positions have greater strength in the general election, and thus the core partisans are willing to choose more centrist positions.

Figure 1.

Graphs of u(ℓ, r|t) as a function of ℓ.

Next, we explore how voters’ preferences between the two candidates of one party vary across their types. For each given r ∊ R, let Δ^L(t|r) = u(ℓ_m, r; t) − u(ℓ_e,r; t), and for each given ℓ ∊ L, let Δ^R(t|ℓ) = u(ℓ, r_m; t) − u(ℓ, r_e; t). Let

\bar{ℓ} = \frac{ℓ_{e} + ℓ_{m}}{2}

and

\bar{r} = \frac{r_{e} + r_{m}}{2}

be the midpoints of the candidate positions of party L and of party R, respectively. From (10), we derive the following: for each r ∊ R,

Δ^{L} (t | r) = {\begin{array}{l} - 2 (\bar{ℓ} + 2 β) (ℓ_{m} - ℓ_{e}) & if - 1 \leq t < ℓ_{e} \\ 2 (4 β + ℓ_{e} + r) (t - ℓ_{e}) - 2 (\bar{ℓ} + 2 β) (ℓ_{m} - ℓ_{e}) & if ℓ_{e} \leq t < ℓ_{m} \\ - 2 (ℓ_{m} - ℓ_{e}) (t - r) + 2 (\bar{ℓ} + 2 β) (ℓ_{m} - ℓ_{e}) & if ℓ_{m} \leq t < r \\ 2 (\bar{ℓ} + 2 β) (ℓ_{m} - ℓ_{e}) & if r \leq t \leq 1 \end{array}

(11)

Figure 2 graphs Δ^L(t|r) as a function of t.⁹ There are a few notable features.

Figure 2.

Graphs of Δ^L(t|r).

First, for every t ∊ [−1, ℓ_e], Δ^L(t|ℓ) > 0 if and only if $\bar{ℓ} < - 2 β$ . That is, the core leftists prefer the candidate of the left party whose position is closer to their compromise point −2β, as expected. On the other hand, the core rightists’ preferences over the candidates of the left party are exactly the opposite. They prefer the candidate of party L whose position is farther from the compromise point of the core leftists. Except for the non-generic case in which ℓ_e and ℓ_m are equidistant from −2β, the preferences of the core leftists and the core rightists over the two candidates are divergent. This will play a major role in our analysis.

Second, Δ^L(t|r) is strictly increasing on the interval [ℓ_e, ℓ_m]. For the intuition, let $\tilde{t}, \hat{t} \in [ℓ_{e}, ℓ_{m}]$ and $\tilde{t} < \hat{t}$ . Note that, given the right party’s nominee r, the chances of ℓ_e and ℓ_m winning in the general election are independent of the voters’ preferences. Then, since voter $\hat{t}$ likes ℓ_m more (or dislikes it less) than $\tilde{t}$ does, $Δ^{L} (\hat{t} | r) > Δ^{L} (\tilde{t} | r)$ .

Third, Δ^L(t|r) is strictly decreasing on the interval [ℓ_m, r]. Given our assumption of linear utilities, the utility difference v(ℓ_m; t) − v(ℓ_e; t) is constant across different ideal points in the interval [ℓ_m, r]. Let $ℓ_{m} \leq \tilde{t} < \hat{t} \leq r$ . Voter $\hat{t}$ likes the right party’s nominee r more than voter $\tilde{t}$ does. Then, the relatively high winning chance of ℓ_m in the general election is less valuable for $\hat{t}$ than for $\tilde{t}$ . Hence, Δ^L(t|r) is strictly decreasing on the interval.

Lastly, unless the two candidates in the L primary are equidistant from −2β, there exists a unique cutpoint voter t_L who is indifferent between ℓ_e and ℓ_m. When $\bar{ℓ} < - 2 β$ , the cutpoint t_L is greater than zero, as the voter with ideal point zero always prefers ℓ_m to ℓ_e. Thus, when the moderate left candidate is more appealing than the extreme left candidate to the core leftists in terms of the voters’ strategic preferences, all leftist voters (t < 0) unanimously prefer the moderate candidate to the extreme one. On the other hand, when $\bar{ℓ} > - 2 β$ , $t_{L} \in (ℓ_{e}, \bar{ℓ})$ . Note that voter $\bar{ℓ}$ who is indifferent between ℓ_e and ℓ_m in terms of her naive policy preference always prefers the moderate candidate to the extreme one since the former is more likely to win in the general election.

3.2. Logic of rational voting in large open primaries

As our study is primarily motivated by primary elections with large electorates, we concentrate our inquiry on the properties of the limits of equilibria as n goes to infinity. We define the concept of large equilibrium by the following.

Definition 2. We say a pair (s, τ) is a large equilibrium of the Poisson open primary model if there exists a sequence ${(s_{n}, τ_{n})}_{n = 1}^{\infty}$ such that (s_n, τ_n) ∊ Σ_n for every $n \in ℕ$ , s = lim_n→∞ s_n, and τ = lim_n→∞ τ_n.

In a large equilibrium (s, τ), the strategy function s partitions the type space T into a finite number of intervals in each of which all voters vote for the same candidate. Generally, it is possible that there are multiple large equilibria. When the multiplicity of equilibria is inconsequential, we select only one large equilibrium. This occurs when every voter in an interval, say $[\underline{t}, \bar{t}]$ , is indifferent between two candidates, say c and c′. In this case, if a certain partition of this interval into those who vote for c and those who vote for c′ is a part of a large equilibrium, then any other partition that generates the same τ(c) and τ(c′) is also an equilibrium. However, all of the equilibria are essentially equivalent in the sense that they generate the same outcome. Hence, in this case, we choose the simplest equilibrium, that is, the one with the minimum number of cutpoints in the partition of T. From now on, when we say ‘large equilibrium’, we mean a large equilibrium that satisfies this criterion of refinement. Let Σ denote the set of large equilibria.

Although all decisions are simultaneous in the model, it turns out that the equilibrium voting decisions in large open primary elections can be well understood as hypothetical two-step decision-making. First, a voter has to decide which party’s primary she will participate in. Second, once a voter determines the primary she will participate in, she then needs to choose between the two candidates in the primary. Let us begin with the second hypothetical step. Once a voter decides which party primary she participates in, her voting decision within a party is binary. Our intuition suggests that a rational voter will vote according to her strategic preference. That is, she will vote for the candidate that she believes the better nominee given the expected outcome of the other party primary.

Given a large equilibrium (s, τ) with the corresponding sequence of equilibria ${(s_{n}, τ_{n})}_{n = 1}^{\infty}$ , let p_L(τ) = lim_n→∞ P(O(ℓ_m)|nτ_n). That is, p_L(τ) is the limit probability that ℓ_m wins in the L primary. Similarly, let p_R(τ) = lim_n→∞ P(O(r_m)|nτ_n). The next lemma formalizes the intuition in the previous paragraph.

Lemma 2. Let (s, τ) ∊ Σ. For every t ∊ T, the following is true.

If s(t) ∊ L, then s(t) ∊ arg max_ℓ∊L [p_R(τ)u(ℓ, r_m; t) + (1 − p_R(τ))u(ℓ, r_e; t)].

If s(t) ∊ R, then s(t) ∊ arg max_r∊R [p_L(τ)u(ℓ_m, r; t) + (1 − p_L(τ))u(ℓ_e, r; t)].

One notable characteristic of large equilibria is that the limit probabilities p_L(τ) and p_R(τ) can take only three values: 0, 1 or 1/2. When the number of voters is large, a small difference between the candidates’ expected vote shares almost surely decides the winner in an election. For example, suppose τ(ℓ_m) ≠ τ(ℓ_e). In the sequence of equilibria {(s_n, τ_n)}, the expected numbers of votes for ℓ_m and ℓ_e are the Poisson random variables with means nτ_n(ℓ_m) and nτ_n(ℓ_e), respectively. Note that the standard deviation of a Poisson random variable is the square root of its mean. As n goes to infinity, the difference between the expected numbers of votes for the two candidates, n|τ_n(ℓ_m) − τ_n(ℓ_e)|, goes to infinity. Moreover, both of the standard deviations, $\sqrt{n τ_{n} (ℓ_{m})}$ and $\sqrt{n τ_{n} (ℓ_{e})}$ , become infinitesimally small relative to n|τ_n(ℓ_m) − τ_n(ℓ_e)|. This implies that if τ(ℓ_m) > τ(ℓ_e), then the probability of ℓ_m being the nominee of party L converges to one and vice versa. The only case where there is no sole expected winner in a large equilibrium is where the two candidates’ expected vote share in the limit is equal: τ(ℓ_m) = τ(ℓ_e). In that case, each candidate’s winning probability converges to 1/2. Hence, in any large equilibrium, for any party D ∊ {L, R}, p_D(τ) ∊ {0, 1∕2, 1}. This, together with Lemma 2, substantially reduces the number of potential candidates of large equilibria.

We next consider how rational voters decide which primary they participate in. Given a large equilibrium (s, τ), |Δ^L(t|τ)| is the utility difference for a t-type voter between electing the more preferred candidate and electing the less preferred candidate in the L primary. Thus, we can regard |Δ^L(t|τ)| as the benefit for a t-type voter from electing her favorite candidate of the L primary. Similarly, |Δ^R(t|τ)| is the benefit for a t-type voter from electing her favorite candidate of the R primary in the large equilibrium τ. All else being equal, if the former value is greater than the latter, the voter will vote in the L primary.

However, this is not the whole story. Although |Δ^L(t|τ)| > |Δ^R(t|τ)|, the voter may vote in the R primary rather than in the L primary if she believes that the R primary is a much closer election than the L primary. That is, a rational voter considers not just the benefits but also the effectiveness of her ballot. Thus, the relative magnitude of the pivot probabilities in the two races does matter. Given a large equilibrium (s, τ), let

Θ (τ) = {t \in T | Δ^{L} (t | τ) Δ^{R} (t | τ) \neq 0}

That is, Θ(τ) is the set of voter types that are not indifferent in either of the primaries in the large equilibrium. Then, for each type t ∊ Θ (τ) and each party D ∊ {L, R}, the candidate a t-type voter would vote for if she participated in the D primary is uniquely determined by Lemma 2. Let c_D(t) denote this candidate. Then, the pivot probability for a type-t voter in the D primary in equilibrium (s_n, τ_n) for large enough n is q_D(t|nτ_n) ≡ P(A(c_D(t))|nτ_n).

We consider the ratio of the pivot probabilities

\frac{q_{R} (t | n τ_{n})}{q_{L} (t | n τ_{n})}

Note that, for every n, q_R(t|nτ_n) > 0 and q_L(t|nτ_n) > 0 since the number of actual voters is uncertain. Yet, as n goes to infinity, both of the pivot probabilities converge to zero. The sequence

{\frac{q_{R} (t | n τ_{n})}{q_{L} (t | n τ_{n})}}

either converges to a finite non-negative limit or diverges to the positive infinity. Define the limit ratio of the pivot probabilities for t by

η (t | τ) = {\begin{array}{l} \lim_{n \to \infty} \frac{q_{R} (t | n τ_{n})}{q_{L} (t | n τ_{n})} & if \frac{q_{R} (t | n τ_{n})}{q_{L} (t | n τ_{n})} is convergent \\ + \infty & otherwise \end{array}

Our next lemma shows how rational voters choose between the primaries.

Lemma 3. For every (s, τ) ∊ Σ and every t ∊ Θ(τ), the following is true:

η (t | τ) < \frac{| Δ^{L} (t | τ) |}{| Δ^{R} (t | τ) |}

then s(t) ∊ L.

η (t | τ) > \frac{| Δ^{L} (t | τ) |}{| Δ^{R} (t | τ) |}

then s(t) ∊ R.

Lemma 3 is consistent with our decision-theoretic intuition that a voter participates in an election when the benefit of doing so discounted by her pivot probability exceeds the opportunity cost. In the model, the only opportunity cost is not to participate in the other primary. Note that although the pivot probabilities in both elections go to zero as the number of voters becomes very large, their relative magnitudes still matter.

4. Results

In presenting our results, we focus on strategic voting in large equilibria and its consequences. Although all voters are assumed to be strategic in terms of their motivation in our model, we use the term ‘strategic voting’ as indicating any observed voting behavior that misrepresents a voter’s direct preference over the candidates. This usage of the term is consistent with the empirical literature on voting behavior. Formally, we say a voting strategy s(t) is sincere if s(t) ∊ arg max{v(c; t)|c ∊ C}. A voting strategy s(t) is strategic voting if it is not sincere. Among possible types of strategic voting, we are most interested in crossover voting. In order to define crossover voting, we simply consider all negative types of voters as the partisans of the left party and all positive types of voters as the partisans of the right party.¹⁰ We then say s(t) is crossover voting if either t < 0 and s(t) ∊ R, or t > 0 and s(t) ∊ L.

There are different types of crossover voting. A partisan voter may vote for the moderate candidate of the opposite party to avoid the worst case in which the extreme candidate of the opposite party becomes the winner of the general election. We call this hedging. By contrast, a partisan voter may vote for the extreme candidate of the opposite party to lower the likelihood that the opposite party wins the general election. We call this raiding. Formally, s(t) is hedging if either t < 0 and s(t) = r_m, or t > 0 and s(t) = ℓ_m. And s(t) is raiding if either t < 0 and s(t) = r_e, or t > 0 and s(t) = ℓ_e. Loosely speaking, hedging is a defensive crossover voting strategy and raiding is an aggressive crossover voting strategy.

By definition, both types of crossover voting are strategic voting.¹¹ There is another type of strategic voting, which is not crossover voting. A voter can vote for one of her party candidates who is not her top choice. It is easy to see that the voters whose top choice is the moderate candidate of their party will never vote for the extreme candidate of their party. The moderate candidate has a higher chance of winning in the general election against any given candidate of the opposite party than the extreme candidate does. Moreover, the policy position of the moderate candidate is better for the moderate type of voters than that of the extreme candidate. Therefore, if a voting strategy s(t) is strategic voting but not crossover voting, then it is the case that an extreme partisan voter votes for the moderate candidate of her own party who has a better chance of winning in the general election than her favorite candidate. We call such voting behavior compromising. Formally, s(t) is compromising if either $t < \bar{ℓ}$ and s(t) = ℓ_m, or $t > \bar{r}$ and s(t) = r_m.

Now we present our first equilibrium characterization result. From the discussion in the previous section, we expect that equilibrium voting strategies depend on the locations of the candidates’ positions relative to the core partisans’ compromise points. Thus, we divide the parameter space into two cases: the case in which $\bar{ℓ} < - 2 β$ (or, equivalently, $\bar{r} > 2 β$ ), in other words, in each party, the position of the moderate candidate is closer to the compromise point of the core partisans than that of the extreme candidate is; and the case in which $\bar{ℓ} > - 2 β$ (or, equivalently, $\bar{r} < 2 β$ ), in other words, in each party, the position of the extreme candidate is closer to the compromise point of the core partisans than that of the moderate candidate is. (We rule out the non-generic case in which $\bar{ℓ} = - 2 β$ .) We first consider the former.

Proposition 1. Let $\bar{ℓ} < - 2 β$ and let (s, τ) ∊ Σ. Then, there exists $\hat{t} > 0$ such that

s (t) = {\begin{array}{l} r_{e} & i f t \in [- 1, - \hat{t}) \\ ℓ_{m} & i f t \in (- \hat{t}, 0) \\ r_{m} & i f t \in (0, \hat{t}) \\ ℓ_{e} & i f t \in (\hat{t}, 1] \end{array}

(12)

Moreover:

If ℓ_e < −1/2, then $\hat{t} \in (1 / 2, r_{e})$ . Thus, the moderate candidates wins in each primary election.

If ℓ_e ≥ −1/2, then $\hat{t} = 1 / 2$ . Thus, each candidate wins with probability 1/2 in each primary election.

The proposition has several interesting implications. First, a large proportion of voters do not vote sincerely. In the equilibrium, voters whose ideal points are less than $- \hat{t}$ and those whose ideal points are greater than $\hat{t}$ do vote strategically. Second, the type of strategic voting they engage in is raiding. The extreme leftist voters vote for the extreme right candidate, and the extreme rightist voters vote for the extreme left candidate. Thus, the model predicts that those who raid in open primary elections will be extremists, not centrists.

The intuition is as follows. Given the configuration of candidate positions, the extreme left voter whose ideal point is less than ℓ_e prefers ℓ_m to ℓ_e in the L primary and r_e to r_m in the R primary. As seen in (11), Δ^L(t|τ) is the same for all of these voters. Moreover, given the symmetry of candidate positions, |Δ^L(t|τ)| = |Δ^R(t|τ)|; the benefits of participating in the two primaries are equal. Given the symmetric strategies in Proposition 1, the two primaries are equally close races.¹² Yet, the key here is that a vote for the candidate who is behind in a race is slightly more likely to influence the outcome than a vote for the candidate who is ahead. (This does not rely on our assumption of asymmetric tie-breaking.) Suppose the moderate candidate is the expected winner in each primary election. Then, for the extreme left voters, their preferred candidate in the L primary is winning, and their preferred candidate in the R primary is losing while the two primaries are equally close elections. Thus, it is rational for them to cross over the party line by voting for the extreme right candidate of party R. Hence, every voter whose type is less than ℓ_e will vote for r_e if in each party the moderate candidate is the expected winner. Next, the benefit of participating in the L primary is strictly increasing as t increases from ℓ_e. On the other hand, the benefit of participating in the R primary is weakly decreasing. Hence, the cutpoint $- \hat{t}$ is slightly above ℓ_e. If ℓ_e < −1/2, the cutpoint is below −1/2, which is consistent with the supposition that the moderate candidate wins in each election. However, if ℓ_e ≥ 1/2, then the extreme candidate wins in each primary. But then, the extreme left voters want to vote for ℓ_m, who is behind in the L primary. Thus, the only possible large equilibrium for the case that ℓ_e ≥ 1/2 is such that each candidate wins with probability 1/2 in each primary. In that case, the extreme left voters are completely indifferent between participating in the two primaries. The only rational voting strategies that generate the ties in both of the primaries are such that exactly half of partisans in each party engage in raiding the opposite party’s primary.

It is informative to compare this result with what would happen if the primaries were closed. Let us consider the following simple model of closed primaries. Suppose that, for all negative types of voters, the choice set is {ℓ_m, ℓ_e}, and, for all positive types of voters, it is {r_m, r_e}; that is, crossover voting is not allowed. Suppose that all voters are strategic in this closed primary game. Recall that we are considering the case in which the moderate candidate of each party is more appealing even to the extreme voters of the party in terms of the voters’ strategic preferences. Then, the unique equilibrium in this closed primary game is such that all voters in each party vote for the moderate candidate of the party. Thus, when ℓ_e < −1/2, the outcomes of the open and closed primary games are the same in terms of the expected nominees. Raiding occurs in our open primary game, but it does not change the winners of the primaries. However, the results are different when ℓ_e ≥ −1/2. In our open primary model, each candidate in each party is equally likely to be the nominee of the party, whereas the moderate candidate wins for sure in each primary of the closed primary game. Thus, crossover voting is consequential in this case; raiding tends to bring more extreme candidates into the general election.

However, it should be noted that this occurs only when the extreme candidate of each party is sufficiently moderate. A necessary condition is that the extreme candidate of each party is more moderate than the median voter in the party’s primary. Under the range of parameters we consider here, the extreme partisans prefer the moderate candidate of their party to the extreme candidate even though the extreme candidate is substantially moderate. The reason is that the parameter β that captures aggregate uncertainty about the general election result is small relative to the positions of the candidates and thus the moderate candidate is much more likely to win in the general election than the extreme candidate. Hence, the open primary system may extremize the positions of the candidates in general elections only when the candidates’ positions are moderate overall but uncertainty about the general election is small.

The next proposition concerns the case in which, in each party, the extreme candidate is more appealing to the core partisans than the moderate candidate is.

Proposition 2. Let $\bar{ℓ} > - 2 β$ and let (s, τ) ∊ Σ. Then, there exist $\bar{t}$ and $\tilde{t}$ with $0 < \bar{t} < \tilde{t}$ such that

s (t) = {\begin{array}{l} ℓ_{e} & i f t \in [- 1, - \tilde{t}) \\ ℓ_{m} & i f t \in (- \bar{t}, 0) \cup (\bar{t}, \tilde{t}) \\ r_{m} & i f t \in (- \tilde{t}, - \bar{t}) \cup (0, \bar{t}) \\ r_{e} & i f t \in (\tilde{t}, 1] \end{array}

(13)

Moreover:

If ℓ_e < −1/2, then $\tilde{t} \in (1 / 2, r_{e})$ . Thus, the moderate candidates wins in each primary election.

If ℓ_e ≥ −1/2, then $\tilde{t} = 1 / 2$ . Thus, each candidate wins with probability 1/2 in each primary election.

In contrast to the previous case, the core partisans in both parties vote sincerely. Given the level of uncertainty β, the core leftists prefer ℓ_e to ℓ_m in their own party’s primary since ℓ_e also has a fair chance of winning in the general election. As in the previous case, their preferences over the candidates in the right party’s primary are exactly the opposite. Thus, their choices are effectively between ℓ_e and r_m. Suppose the moderate candidates win in both of the primaries. As before, the two races are equally close under the symmetric strategies. Since ℓ_e is behind in the L primary and r_m is ahead in the R primary, a vote for ℓ_e is more likely to be pivotal. Hence, all voters whose ideal points are less than ℓ_e vote for ℓ_e, their favorite candidate in their sincere preferences. A symmetric argument applies to the voters whose ideal points are greater than r_e; they sincerely vote for r_e. When ℓ_e < −1/2, the vote share of the extreme candidate in each party is less than 1/2, and, thus, the moderate candidate wins in each primary, which is consistent with the supposition. On the other hand, when ℓ_e ≥ −1/2, the only strategies that can be consistent with the pivot probabilities are those that lead to a tie in each election, by the logic applied to the previous case.

The type of crossover voting that occurs in this case is hedging. The voters whose ideal points are in $(- \tilde{t}, - \bar{t})$ vote for r_m, and those whose ideal points are in $(\bar{t}, \tilde{t})$ vote for ℓ_m. Consider the voters in $(- \tilde{t}, - \bar{t})$ . Note that Δ^L(ℓ_e|τ) < 0, that Δ^L(ℓ_m|τ) > 0, and that Δ^L(t|τ) is strictly increasing on [ℓ_e, ℓ_m]. On the other hand, Δ^R(t|τ) ≥ |Δ^L(ℓ_e|τ)| for every t ∊ [ℓ_e, ℓ_m] since it is symmetric to Δ^L(t|τ). Then, there are some types of voters such that Δ^L(t|τ) is close to zero and thus participating in their own party’s primary is of little value. Hence, these voters vote for their preferred candidate in the opposite party’s primary, r_m. The same incentive is present for those voters in $(\bar{t}, \tilde{t})$ . Intuitively, the voters in these intervals are relatively indifferent between the two candidates in their own party but are strongly opposed to the extreme candidate of the opposite party, so they hedge. Thus, an implication of Proposition 2 is that the voters who engage in hedging in open primary elections are those who would be swing voters in closed primary elections.

Lastly, we compare this result with the closed primary case. As seen in Figure 2(b), the cutpoint type t_L that is indifferent between ℓ_m and ℓ_e belongs to the interval $(ℓ_{e}, \bar{ℓ})$ . The set of voters who vote for the extreme candidate in the left party in the closed primary game is exactly [−1, t_L]. Note that voters who are in a close neighborhood of t_L will participate in the R primary since they are almost indifferent between the two candidates in the L primary. This implies that $- \tilde{t} < t_{L}$ . Hence, the expected vote share of ℓ_e in the open primary game is less than that in the closed primary game. By the symmetry, this also holds for the vote share of r_e. Hence, if the moderate candidates are the nominees in the closed primary game, they are so in the open primary game as well. On the other hand, whenever t_L > −1/2, the nominee of each party in the closed primary game is the extreme candidate while this never occurs in the open primary game. Thus, the open primary institution has moderating effects on the general election in contrast to the previous case.

Overall, the effects of the open primary system on the ideologies of the party nominees depend on the positions of candidates relative to the core partisans’ strategic ideal points, which in turn depend on the level of uncertainty in the general election. For the comparison of the two primary systems, we so far have defined partisan voters so that every voter belongs to one party. One might wonder whether this definition is problematic since participants in closed primaries may have more extreme ideologies in general than those in open primaries. Alternatively, we could define partisan voters of one party as those voters who will vote for the party in the general election regardless of the outcomes of the primaries. That is, a voter t is L-partisan if

t < \frac{ℓ_{e} + r_{m}}{2}

and a voter t is R-partisan if

t > \frac{ℓ_{m} + r_{e}}{2}

We then compare the outcomes of the closed primary system to our results assuming only partisan voters participate in the closed primaries. However, recall that in the case of Proposition 1, all negative types prefer ℓ_m to ℓ_e and all positive types prefer r_m to r_e. Thus, even when only the narrowly defined partisans participate in the closed primaries, the moderate candidates are still the winners of both primary elections. Hence, the effect of the open system is the same as what we discussed previously in the case of Proposition 1. Also, clearly, in the case of Proposition 2, the open system has a moderating effect even with this new definition of partisans, as is the case with our broad definition of partisans.

5. Extension

In this section, we discuss how our findings in the previous section would be altered if we allow asymmetric candidate positions. Dropping the symmetry assumption (i.e. allowing that ℓ_m + r_m ≠ 0 or ℓ_e + r_e ≠ 0)¹³ certainly makes the analysis more complicated, and so we leave the task of complete characterization of large equilibria for the future. However, we can still establish some necessary conditions that must be satisfied in all large equilibria. In the following discussion, we assume that $\bar{ℓ} \neq - 2 β$ and $\bar{r} \neq 2 β$ . That is, we only rule out the non-generic case in which the core partisans of a party are exactly indifferent between the two candidates in the party primary election. It turns out that most of our results on the pattern of crossover voting in relation to candidate positions and voter ideology carry over to the generalized model.

First, partisans of a party engage in raiding only when, in the other party primary, the moderate candidate is more appealing to the core partisans than the extreme candidate.¹⁴ To see this, note that the shape of Δ^L in Figure 2 does not rely on the symmetry assumption. Also, recall that voter t prefers ℓ_e to ℓ_m as the nominee of party L if and only if the graph is below the horizontal line at t. Then, it is easy to see that there are members of party R who prefer ℓ_e to ℓ_m as the nominee of party L only when $\bar{ℓ} < - 2 β$ , that is, when the core leftists prefer ℓ_m to ℓ_e. The intuition is the following. Suppose, to the contrary, the core leftists prefer ℓ_e to ℓ_m in terms of their strategic preference. This means that the extreme left candidate has a relatively good chance of winning in the general election given the level of the electoral uncertainty. Then, the partisans of the right party have no incentive to vote for the extreme left candidate because she is not sufficiently weak in the general election and the worst scenario is realized if she becomes the final winner.

Second, if raiding occurs in equilibrium, then those who engage in raiding are relatively extreme partisans. Observe that in Figure 2(a) Δ^L is weakly decreasing in t on the interval [ℓ_m, 1]. Thus, if a moderate rightist prefers ℓ_e to ℓ_m, then an extreme rightist does so even (weakly) more. Since the ratio of the pivot probabilities in the two elections is independent of voter ideologies, an extreme rightist has at least as strong an incentive to raid as a moderate rightist does.

Third, if the moderate candidate is more appealing to the core partisans than the extreme candidate in each party (i.e. $\bar{ℓ} < - 2 β$ and $\bar{r} > 2 β$ ), then, in every large equilibrium, the core partisans of both parties engage in raiding as in Proposition 1. Note that moderate partisans of each party always prefer the moderate candidate to the extreme candidate in the primary of their party. Hence, in this case, all members of each party prefer the moderate candidate to the extreme candidate in the primary of their party. This implies that the extreme candidate of each party can receive votes only from the partisans of the other party. Lemma 4 in Appendix A proves that, in every large equilibrium, every pivot probability ratio is a finite positive number, without relying on the symmetry assumption. An implication is that every candidate expects to receive a positive vote share in every large equilibrium. Therefore, it must be the case that some core partisans of both parties engage in raiding in equilibrium.

Lastly, if the extreme candidate is more appealing to the core partisans than the moderate candidate in each party (i.e. $\bar{ℓ} > - 2 β$ and $\bar{r} < 2 β$ ), then, in every large equilibrium, some moderate partisans engage in hedging and some core partisans vote sincerely as in Proposition 2. Observe in Figure 2(b) that, for every voter t in a close neighborhood around t_L, the benefit of participating in the L primary, |Δ^L(t|τ)|, is close to zero. Recall that $t_{L} \in (ℓ_{e}, \bar{ℓ})$ . Then, for every t in this interval Δ^R(t|τ) ≥ Δ^R(−1|τ) > 0. Again, by Lemma 4, for any voter t, the ratio of pivot probabilities η(t|τ) is a finite positive number. Then, if t is sufficiently close to t_L, then

η (t | τ) > | \frac{Δ^{L} (t | τ)}{Δ^{R} (t | τ)} |

Hence, there are some moderate partisans who engage in hedging. The reason that some core partisans in both parties vote sincerely is similar to the argument in the previous paragraph. In the configuration of candidate positions currently considered, the only voters that may vote for the extreme candidate of a party are the core partisans of the party. Thus, if no core partisan in the party votes sincerely, then the extreme candidate of the party gets no votes, which contradicts that every pivot probability ratio is finite and positive.

Overall, the pattern of strategic voting as a function of candidate positions and uncertainty about the general election, which is found in the last section, is extended to the general model allowing asymmetry, to some degree.

6. Conclusion

Who are the voters that cross over the party line? When do they engage in crossover voting? These important questions about open primaries have been unanswered theoretically. Analyzing a Poisson model of open primary elections, this study has provided partial answers to these questions. We have shown that the equilibrium pattern of strategic voting critically depends on candidate positions relative to the compromise points of core partisans, which in turn are shaped by the level of uncertainty about the general election. We have also demonstrated that voters with different policy preferences engage in different types of strategic voting. Hence, our study makes testable predictions about the relationship between candidate positions and strategic voting, as well as the relationship between voter ideology and crossover voting.

Specifically, our model predicts the following under the assumption that the candidates’ positions are symmetrically located. First, raiding occurs when the moderate candidate in each party is closer to the core partisans’ compromise point than the extreme candidate. Second, the voters who engage in raiding are extreme partisans. Third, hedging occurs when the extreme candidate in each party is closer to the core partisans’ strategic ideal point than the moderate candidate. Fourth, the voters who engage in hedging are relatively moderate partisans.

One important question is whether the open primary system leads to more moderate outcomes than the closed primary system. Our analysis has identified the conditions under which this may or may not be the case. Given a symmetric configuration of candidate positions, if uncertainty about the general election results at the time of the primary election is high, then the open primary system always leads to an outcome that is at least as moderate as the outcome under the closed primary system. Even when uncertainty is low, if the extreme candidate in each party is sufficiently extreme, then the outcome in the primary election is never more extreme under the open system than under the closed system. However, if there is low uncertainty and the extreme candidate in each party is sufficiently moderate, the party nominees under the open primary system can be more extreme in expectation than those under the closed primary system.

The applicability of our predictions to real-world open primaries is limited to some extent because we take candidate positions as exogenous in this study. In this sense, our model amounts to a partial equilibrium analysis. However, one cannot implement a full equilibrium analysis that includes the endogenous choices of candidate positions without first characterizing voting equilibria under different combinations of candidate positions. Hence, this study takes a progressive step toward a comprehensive understanding of open primaries.

Footnotes

Appendix A

Acknowledgements

The authors thank Jane Bang, Mark Fey, Justin Fox, John Patty, Maggie Penn and seminar participants at Yale University for their valuable comments.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Notes

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