Introduction

In US general elections, ballots cover many different races. In some states, one can circumvent race-by-race voting by ticking a single box at the top of the ballot that automatically registers a vote for every candidate from a particular party in partisan races. This is known as the Straight Ticket Voting Option (STVO), Master Lever or Partisan feature.Footnote 1

To STVO or not to STVO is a controversial question. For example, in the run up to the 2016 general election, Michigan’s GOP-held legislature passed a bill banning STVO—but the Democratic party immediately challenged that decision. In the end, the Supreme Court sided with the Democrats; straight-ticket voting was reinstated just before the election and, unexpectedly, brought more Republicans into power.Footnote 2

Despite heated political debates surrounding STVO and confusion about its consequences, no theoretical model exists that clarifies which party benefits from it, how it impacts candidate selection and the effect it ultimately has on policy; this paper helps fill the gap. Using a pre-election competition model a là Downs (1957), we incorporate the two-principals paradigm into a standard probabilistic voting model (Hinich 1978; Lindbeck and Weibull 1987). For each contested office, parties nominate candidates who maximise vote share discounted by the distance in their positions from party bliss points. The trade-off both parties face is therefore the choice between increasing vote share by fielding candidates who are better aligned with voters and maintaining ideological purity or party unity over the political agenda.

In an election with STVO, voters must first decide whether to use the option or instead go through the ballot and vote in each race individually. At the time of the decision, voters only observe candidates’ political positions and party affiliations. Going through the ballot is costly for the voter, therefore the trade-off he faces is between fine-tuning the choices in every race, on the one hand, and saving his time and effort by using the STVO, on the other.

Specifically, if the voter does not use the STVO and goes through the ballot, he solves a sequence of utility maximisation problems, choosing the candidate who delivers greatest utility in each race. Voters’ utility from electing a candidate has three components: first, a measure of distance between the candidates’ and the voter’s political positions (voters prefer candidates who are closer to them ideologically); second, a bonus for the candidate’s affiliation with a party if the voter is its partisan; third, an idiosyncratic shock that captures the voter’s valuation of the candidate’s quality. The latter is observed only if the voter goes through the ballot. Thus, if a voter uses the STVO, his party choice is based on the expectation of total utility from the party’s candidates, conditional on the voter’s own political positions and partisanship status.

We start by formally exploring STVO’s effect on the position of candidates in each party. Since going through the ballot is costly, voters who are nearly indifferent between voting a straight ticket and making partisan exceptions in a small number of races will be most tempted to use it.Footnote 3 Thus, introducing STVO diverts partisan voters away from positional voting. This impacts politicians’ positions in two different ways. First, because many voters “buy in bulk”, individual candidates’ characteristics such as political positions and quality matter less. Consequently, politicians are more inclined to cater to their party’s political agenda, and not their constituency. We label this STVO effect the party loyalty effect. Second, non-partisan (swing) voters become relatively more important in determining electoral outcomes so politicians align with them in order to win their support.Footnote 4 We call this the swing voter effect.

More specifically, the optimal candidate’s platform on an issue is a convex combination between the party’s bliss point and the position of the average voter in the constituency with a drift proportional to the covariance between the swing voter propensity and political positions. Introducing STVO strengthens the multipliers on the party’s bliss point and covariance and weakens the multiplier on the average voter’s position. Intuitively, STVO leads to an unequivocal increase in partisan votes, meaning both parties can more readily “afford” to put forward candidates who are ideologically closer to their respective bliss points (party loyalty effect). Meanwhile, non-partisan, positional (or swing) voters become more decisive in electoral outcomes, so politicians’ platforms change to accommodate them (swing voter effect).Footnote 5 The STVO’s combined effect is thus determined by the level of partisanship in a state and the distribution of political positions among partisan and swing voters.

Proposition 1 establishes that STVO can have an asymmetric effect across party—e.g., it may make one party’s candidates more moderate in equilibrium, while candidates from the opposing party become more extreme. This result aligns with available data. According to Fig. 1’s right-hand graph, voters in STVO states do not systematically differ from voters in non-STVO states (apart from perhaps a few recent observations). Yet Fig. 1’s left-hand graph suggests STVO correlates with right-wing Republican senators but has no visible relationship to the positions of senators from the Democratic party. (See also Gorelkina et al. 2019 for additional evidence supporting this conclusion.)

Fig. 1
figure 1

Voters’ and senators’ positions with and without STVO. Note: Left-hand graph shows average senatorial positions by party and STVO status. Right-hand graph displays self-declared average voter positions by STVO presence. Senators’ positions correspond to the first dimension of DW-NOMINATE, a multidimensional scaling application developed by Poole and Rosenthal (2015). Voters’ positions are the first dimension of Enns and Koch (2013)’s dynamic scale of voters’ policy “moods”. Data on the presence of the STVO on state ballots are from Gorelkina et al. (2019). All positional data are projected onto a left (0) right (100) axis. State-level data on voters’ partisanship and positions calculated at the beginning of each Congressional term. See Gorelkina et al. (2019) for a full description of the data used to generate each graph

Proposition 2 examines the impact STVO has on vote shares. In the model, the Republican party’s vote share increases as more voters become Republican partisans, and as the average voter’s views tend to the right; the opposite holds for the Democratic party. The first effect of the STVO is to reinforce the impact of partisanship on vote shares. In contrast, the STVO diminishes the effect of the average voter’s position as it brings the swing (non-partisan) voter to the forefront. When the STVO is available, fewer partisan voters elect by position as they pull the partisan lever instead; thus, the fraction of swing (non-partisan) voters among those who do elect by position increases. Swing voters become more decisive in determining electoral outcomes.

With Proposition 3 we show that the expected position of the election winner is subject to the compound effect of the STVO on candidates’ positions and vote shares. Consistent with Proposition 2 that states partisanship is a more important determinant of vote share when the STVO is present, it also becomes a more important determinant of elected candidates’ platforms with the introduction of STVO. Elected candidates’ platforms hew more closely to the party that has more partisans in the state. The effect of partisanship advantage acts on all issue dimensions and is reinforced by the STVO. Furthermore there are spillover effects between the issues. The STVO induces a spillover effect in the covariance between partisanship and voters’ political positions when that covariance on one issue affects an elected official’s position on another issue. Thus, two constituencies that differ only on the covariance between partisanship and voters’ political positions on a single issue will nevertheless elect politicians that differ across all issue dimensions. Intuitively, issue spillovers are due to the correlation in parties’ bliss points: on each issue, parties are on opposite sides of the origin. This induces correlation across the positions of elected candidates and explains how the prolonged use of STVO has likely contributed to clustering in candidates’ political positions and sorting of the electorate (for empirical evidence, see e.g., Krasa and Polborn 2014a).

Our paper contributes to the literature in several ways. First, we are the first to formally model the link between a common element of ballot design and the positions of elected politicians. This work builds on research in several related contexts, including split-ticket voting and coattail effects. For example, Zudenkova (2011) shows that coattail voting is the outcome of an optimal re-election scheme through which voters incentivise politicians’ efforts; Halberstam and Montagnes (2015) find that the coattails in presidential elections have an adverse effect on ideological polarisation among candidates. Meanwhile, Chari et al. (1997) study split-ticket voting in an environment where the government finances its spending by uniform taxes. Focusing on the interaction between executive and the legislature when choosing policy, Alesina and Rosenthal (1996) show that some voters split the ticket in equilibrium.

Our study further emphasises the relationship between party identification and voters’ positions. Dziubiński and Roy (2011) and Krasa and Polborn (2014b) develop models of vertically differentiated candidates, where voters take into account not only the candidates’ political positions but also their fixed identities—e.g., cultural, religious, or social (partisanship in this paper). In particular, they study the effects of ideological polarisation of voters on the candidates’ positions on economic issues; thereby polarisation results in voters’ party preferences hinging more strongly on cultural issues. This paper offers another insight into issue spillovers (when voters’ views on one issue affect the candidate’s campaign on another issue) by adopting a model where both issues are treated as different dimensions of the candidate’s platform, and the platform is endogenous. A voter’s partisanship status is exogenous but correlated with her political position. We show that issue spillovers may arise in this framework: a party may select a socially conservative candidate running in a socially liberal state, as long as social issues do not dominate the election. Such spillovers become stronger when straight ticket voting is facilitated (for example by the STVO), and by extension, when candidates’ party affiliation becomes more conspicuous or important to voters.

A recent survey article (Dal Bó and Finan 2018) stresses the importance of parties in candidate selection. However, our paper joins only a handful of studies that explore the effect of institutions on intra-party dynamics. Kselman (2017) compares the equilibria of different electoral systems and finds that open list proportional representation avoid the free-riding problem inherent in closed-list proportional representation systems. Buisseret and Prato (2018) focus on the conflict of party and individual politicians’ goals and show that flexible lists in proportional representation systems may weaken politicians’ incentives to cater to voters and focus on toeing the party line instead. Buisseret et al. (2019) study party nomination strategies in list proportional representation systems, focusing on candidate quality (human capital). Motivated by insights from Hix (2002) and Carey (2007), we contribute to this earlier work by exploring a setting where candidates face two principals—the voter and the party—and uncovering how ballot design can have an asymmetric impact on candidate selection that depends on the correlation between voters’ partisanship and political positions.

Our model also sheds a new light on the classic median-voter theorem (Black 1948; Downs 1957) and provides an explanation for the possibly asymmetric effects of STVO. In particular, we show that while candidates chosen by parties are not at the median voter’s position, their platforms depend critically on the non-partisan voter, which is the source of asymmetry. The position of partisan voters—who tend to be more extreme—is less significant to the party’s choice of candidate in STVO states, since it takes partisan votes there for granted. On the one hand, the party’s relative disregard for partisan voters’ positions produces an effect similar to Downs’s original insight where extreme voters matter less to politicians. On the other hand, swing (non-partisan) voters—who are less sensitive to party labels when they vote—play a more decisive role in determining parties’ candidate choices, but their political positions may, in fact, be very far from the median voter. Theoretically, this swing voter effect creates an asymmetry absent from the original Downs model.

Finally, empirical motivation for theoretically exploring the effects of STVO comes from evidence of voter roll off and the importance of ballot design.Footnote 6 Of particular relevance are Schaffner et al. (2001), Hall (1999) and Gorelkina et al. (2019): the first two papers show that roll-off is higher in elections featuring nonpartisan candidates; the third paper demonstrates an empirical link between the STVO’s presence and policy-making. More generally, we believe our findings are also useful for interpreting empirical research on the impact of ballot design, and especially those features facilitating straight-ticket voting.Footnote 7

The rest of the paper is organised as follows. In Sect. 2, we develop a simple probabilistic model of electoral competition. In Sect. 3 we solve the model with and without STVO and derive the impact it has on candidates’ platforms, vote shares and the expected platform of the election winner. Section 4 concludes.

Setup

Fix a US state and an election period and let the offices listed on a ballot be indexed by \(k\in {\mathcal {K}}\equiv \left\{ 1,2,\ldots , K\right\} .\) \(\mu \in \{0,1\}\) indicates the availability of a straight-ticket voting option (STVO), where \(\mu =1\) when the STVO is present and \(\mu =0\) when it isn’t. Our policy space is multi-dimensional and defined as the product of N unit-length intervals:

$$\begin{aligned} {\mathcal {P}}\equiv \left[ -\frac{1}{2},\frac{1}{2}\right] ^{N}\!, \end{aligned}$$

where N is the number of policy issues (e.g., economics or national defense). Three types of actors are positioned within \({\mathcal {P}}\): voters, parties, and candidates.

Each party \(j\in \{R,D\}\) (Republican or Democratic) has a bliss point denoted by a vector of issue positions

$$\begin{aligned} Y_{j}\equiv \left( Y_{j1},Y_{j2},\ldots , Y_{jN}\right) \in {\mathcal {P}}. \end{aligned}$$

Without loss of generality, positions are labelled so that the Democratic party’s bliss point is always to the left of the Republican party, i.e., \(Y_{Dn}<Y_{Rn},\) for all \(n\in \left\{ 1,2, \ldots , N\right\} .\)

Candidates are characterised by an office, k,  the party they represent, j,  and their positions, y. For each office, the pool of candidates is \({\mathcal {P}}\) and each party selects exactly one candidate, \(y_{jk}\in {\mathcal {P}},\) to represent it and run for the open seat (see Eq. (5) below).

There is a unit mass of voters, indexed by i,  each with a bliss point given by \(x_{i},\)

$$\begin{aligned} x_{i}\equiv \left( x_{i1},x_{i2},\ldots , x_{iN}\right) \in {\mathcal {P}}. \end{aligned}$$

To obtain the average voter’s position in the state-period, integrate over the mass of voters:Footnote 8

$$\begin{aligned} X\equiv \int _{\left[ 0,1\right] }\!x_{i}\,{\mathrm {d}}i\in {\mathcal {P}}. \end{aligned}$$

(Consequently, the average voter’s position on issue n is given by \(X_n\equiv \int _{\left[ 0,1\right] }\!x_{in}\,{\mathrm {d}}i\in \left[ -\frac{1}{2},\frac{1}{2}\right] .\))

Apart from their political positions, voters are characterised by partisanship status. Let \(p_{i}(j)\) denote the probability that voter i (whose position is \(x_{i}\)) is a partisan of party j.Footnote 9 The realisation of the random variable is denoted by the indicator \(I_{i}^{P},\) where \(I_{i}^{P}=1\) implies that the voter is a partisan, and \(I_{i}^{P}=0\) implies he is a non-partisan, or swing. Assuming i is partisan to at most one party, his total probability of being a partisan voter is defined as

$$\begin{aligned} p_{i}\equiv p_{i}(j)+p_{i}(-j), \end{aligned}$$

where \(-j=\{R,D\}\setminus j\) (e.g., if \(j=R\) then \(-j=D\)). Party j’s partisan advantage in the state is \(p(j)-p(-j),\) where \(p(j)=\int _{[0,1]}p_{i}\!(j)\,{\mathrm {d}}i\) is the mass of party j partisans. By analogy, \(p=\int _{[0,1]}\!p_{i}\,{\mathrm {d}}i=p(j)+p(-j)\) is the share of partisans, irrespective of party affiliation.

We do not assume a specific causal relationship between partisanship and political orientation—their joint distribution can be any. We denote their covariance by \(\sigma _{n}\equiv \int \!(p_{i}-p)(x_{in}-X_{n})\,{\mathrm {d}}i\) but more often refer to the negative of \(\sigma _n,\) namely the covariance between voters’ positions on issue n and their likelihood of being swing (non-partisan):Footnote 10

$$\begin{aligned} \bar{\sigma }_{n}\equiv -\int \!(p_{i}-p)\left( x_{in}-X_{n}\right) \,{\mathrm {d}}i. \end{aligned}$$
(1)

If \(\bar{\sigma }_{n}>0,\) then non-partisan status is associated with a more right-wing position on issue n compared to the rest of the state. Similarly, \(\bar{\sigma }_{n}<0\) implies that swing voters tend to be to the left—and partisans to the right—of the state’s average position on issue n.

Actions, payoffs and timing Our model of an election with STVO is a game between two parties and a mass of voters. The game proceeds according to the following timeline:

\(t=1\):

Party j chooses a candidate, \(y_{jk},\) to compete for seat \(k=1,\ldots ,K.\) The party derives utility from the vote share \(V_{j}\) it wins but incurs a loss increasing in the distance between the candidate’s positions and the party’s bliss points \(Y_{jn}\):

$$\begin{aligned} \max _{y_{jkn}}\left\{ V_{j}-\sum _{n}\gamma _{n}\left( Y_{jn}-y_{jkn}\right) ^{2}\right\} . \end{aligned}$$
\(t=2\):

If the STVO is on the ballot, voter i decides whether to use it. He makes his decision by comparing the cost \(c_{i}\) and the estimated benefit \({\mathbb {E}}[U_{i}^{*}-{\hat{U}}_{i}|x_i, I_{i}^{P}]\) of going through the ballot. (More detail below.)

\(t=3\):

If voter i does not use the STVO, he selects the candidate that maximises \(u_{ik}(j)\) (Eq. (3)) for each of the \(k=1,\ldots ,K\) offices on the ballot.

We solve the game by backward induction.

\(t=3:\) Electing candidates This sub-game is only reached if the voter does not use the STVO. In that case, he votes by solving a sequence of K distinct maximisation problems. His aggregate utility is defined as

$$\begin{aligned} U_{i}^{*}\equiv \sum _{k=1,\ldots, K}\max _{j_k\in \{R,D\}}u_{ik}(j_{k}), \end{aligned}$$
(2)

where

$$\begin{aligned} u_{ik}(j)={\left\{ \begin{array}{ll} -\sum _{n}\omega _{n}\left( x_{in}-y_{jkn}\right) ^{2}+\beta _{k}+\varepsilon _{ij}, &{} \text{if}\, {i}\, \text{is a partisan of}\,\,{j},\\ -\sum _{n}\omega _{n}\left( x_{in}-y_{jkn}\right) ^{2}+\varepsilon _{ij}, &{} \text{otherwise.} \end{array}\right. } \end{aligned}$$
(3)

The voter’s utility function in Eq. (3) builds upon the probabilistic voting framework of Lindbeck and Weibull (1987). Its first component, \(-\sum _{n}\omega _{n}\left( x_{in}-y_{jkn}\right) ^{2},\) is a weighted function of the distance between candidate j’s positions and i’s bliss points (where every issue n has weight \(\omega _{n}>0,\) \(\sum _{n}\omega _{n}=1\)) and reflects the dis-utility i experiences from electing a candidate whose views do not precisely mirror his own. The second component is a partisanship “bonus” \(0<\beta _{k}<1.\) It represents the extra payoff a voter enjoys if the candidate from his preferred party wins the race for seat k.Footnote 11 Thirdly, \(\varepsilon _{ij}\) is an idiosyncratic shock capturing j’s quality (valence) advantage over his opponent \(-j\) perceived by voter i,  where \(\varepsilon _{ij}=-\varepsilon _{i,-j}\); it is the result of factors such as advertising and endorsements, perceived differences in personality traits and competence, that the voter associates with the candidate’s name. Since the perceived quality differences are voter-specific and centered around zero, candidates do not systematically differ in quality. Voter i draws \(\varepsilon _{ij}\) from a uniform distribution on \(\left[ -\frac{1}{2},\frac{1}{2}\right] \) at \(t=3\); the draw is independent of \(\left( x_{i},p_{i}\right) .\)Footnote 12 Although the realisation of \(\varepsilon _{ij}\) is known only to i,  its distribution is common knowledge so our model contains no aggregate uncertainty.

\(t=2:\) Voter’s choice to use the STVO The time and effort it costs voter i to go through the ballot race-by-race is denoted by \(c_{i}.\) \(c_i\in {\mathbb {R}}_{+}\) is an i.i.d. random draw from a finite set with a associated cumulative distribution function F. We assume that \(c_i\) is orthogonal to i’s politics.

Race-by-race voting benefits the voter by allowing him to fine-tune his selection of candidates. This is equal to the difference between Eq. (2) and the solution to a single maximisation problem under the straight-ticket constraint:Footnote 13

$$\begin{aligned} {\hat{U}}_{i}\equiv \max _{j\in \{R,D\}}\sum _{k=1,\ldots , K}u_{ik}(j). \end{aligned}$$
(4)

The true benefit of going through the ballot, \(U_i^*-{{\hat{U}}}_i,\) isn’t observed until \(t=3.\) At \(t=2,\) the voter only observes the estimated benefit \({\mathbb {E}}[U_{i}^{*}-{\hat{U}}_{i}|x_i, I_{i}^{P}],\) which is conditional on his partisanship status; the expectation is taken over the random variables \(\left( \varepsilon _{ijk}\right) _{j=R,D; k=1,\ldots, K}\) realised in period \(t=3.\) Hence, the voter decides whether to use the STVO by comparing the cost and expected benefit of not using it. He uses the STVO if \({\mathbb {E}}[U_{i}^{*}-{\hat{U}}_{i}|x_i, I_{i}^{P}]\le c_i\); otherwise, he votes race-by-race.

As regards the information structure, note that the uncertainty that is resolved by choosing not to use the STVO relates to the voter’s idiosyncratic valuations of candidates’ quality, and not to their political positions in the election which are observed.

\(t=1:\) Party’s choice of candidate The party’s problem is a tradeoff between attracting votes and satisfying its own policy agenda (ideological purity).Footnote 14 We assume it is separable across offices, meaning the other \(K-1\) races on the ballot only impact a party’s choice of candidate in race k by changing the tradeoff to the voter of going through the ballot and using the STVO. Thus, the party solves the following optimisation problem for each office:Footnote 15

$$\begin{aligned} \underset{y_{jk}\in {\mathcal {P}}}{\max }\,\left\{ V_{j}-\sum _{n}\gamma _{n}\left( Y_{jn}-y_{jn}\right) ^{2}\right\} , \end{aligned}$$
(5)

where \(V_{j}\) is the share of votes earned by party j’s candidate, the office subscript k has been dropped. The weighting j puts on issue n is represented by \(\gamma _{n}>0\); it satisfies in a given state \(\gamma _{n}/\omega _{n}\ge {1}/{2}\) for all n—i.e., parties cannot put too little weight on issues that are important to voters. (See Online Appendix A for further detail.)

Equation (5)’s first term, \(V_{j},\) reflects the driving force of political competition, namely the party’s desire to capture more votes. The second term corresponds to the party’s loss from disagreeing with the candidate on policy issues—for example, politicians with views that diverge from \(Y_{j}\) may be less determined to pass bills supporting the party’s agenda. Generally speaking, it captures those forces that deter parties from achieving policy convergence.

Lastly, whether the candidate actually implements \(y_{j}\) is not relevant for what follows. The key is that voters consider \(y_{j}\) to be a candidate’s true position, e.g., because it is observed (as in our setup) or the politician is able to credibly campaign on it. We therefore use the terms platforms and positions interchangeably throughout the paper.

Results

In this section, we study the model’s solutions with and without STVO and deduce its effects on three outcomes: candidates’ platforms (Proposition 1), their vote shares (Proposition 2), and the expected platform of the election winner (Proposition 3).

Candidates’ platforms

We start by characterising the optimal platform \(y_{jn}^*\) derived as a solution to the three-stage game. Since the choice set of candidates is unconstrained (i.e., it is the whole policy space, \({\mathcal {P}}\)), \(y_{jn}^*\) also corresponds to the party’s optimal choice of candidate.Footnote 16

Proposition 1

The optimal position for the candidate of party j on issue n is a convex combination of the average voter’s position \(X_{n}\) and the party’s bliss point \(Y_{jn},\) with a drift proportional to the swing-position covariance \(\bar{\sigma }_{n}{:}\)

$$\begin{aligned} y_{jn}^{*}=\frac{1-\mu \,p}{1-\mu \,p+\alpha _{n}}X_{n}+\frac{\alpha _{n}}{1-\mu \,p+\alpha _{n}}Y_{jn} +\frac{\mu \left( \lambda ^{p}-\lambda ^{s}\right) \bar{\sigma }_{n}}{1-\mu \,\lambda +\alpha _{n}}, \end{aligned}$$
(6)

where \(\alpha _{n}\equiv 2\,\gamma _{n}/\omega _{n}\) and \(\lambda ^{p}\) and \(\lambda ^{s}\) are the probabilities that partisan and swing voters use the STVO, respectively.

Absent STVO (\(\mu =0\)), Proposition 1 implies that the optimal candidate’s position on issue n lies between the average voter’s position and the party’s bliss point on that issue. Moreover, as shown in Lemma C1 (Online Appendix C), partisans are more likely than swing voters to use the STVO (i.e., \(\lambda ^{p}-\lambda ^{s}>0\)). This implies the following.

Corollary 1

Introducing STVO increases the weight of the party’s bliss point, \(Y_{jn},\) and the effect of the swing-position covariance, \(\bar{\sigma }_{n}.\)

STVO influences candidates’ positions by diverting their partisan voters away from positional voting. On the one hand, this means that candidates’ positions have less of an impact on voters’ behaviour so parties can nominate more “loyal” candidates (party loyalty effect). On the other hand, since swing voters are weighted more heavily among positional voters, parties will pay more attention to their particular preferences (swing voter effect). We discuss these effects as they appear in Eq. (6); a short derivation of the STVO’s total effect as a sum of both components is shown in Lemma C4 (Online Appendix C).

Party loyalty effect:

To pin down the first effect, we focus on a state in which voters’ partisanship status and positions on issue n are uncorrelated (\(\bar{\sigma }_{n}=0\)).Footnote 17 In this case, the candidate’s optimal political position is a convex combination of the average voter and party bliss points. In the presence of STVO, the party can afford to choose a candidate whose views on the issue are closer to those of the party.

Swing voter effect:

Now drop the assumption of zero covariance and suppose we are in a state with few partisan voters, so that the party loyalty effect is small. In this case, introducing STVO forces both parties to follow the direction of the swing voter. The reasoning is as follows. Assume that \(\bar{\sigma }_{n}>0\) so that holding more left-wing views on issue n is associated with being a partisan and, as a result, use of the straight-ticket option. In this case, STVO attracts left-wing voters, so the average position of those who go through the ballot—and judge the candidates by their political positions—shifts to the right. Hence, the candidate’s optimal position must satisfy the more right-wing swing voters when STVO is introduced. More generally, STVO makes swing voters more decisive in electoral outcomes so when \(\bar{\sigma }_{n}>0\) the swing voter effect is also positive (more extreme Republican candidate, more moderate Democrat), and vice versa for \(\bar{\sigma }_{n}<0.\)

Due to the separability of our model across dimensions, the STVO’s impact may vary by issue. Consider for instance a state that is left wing in social issues and right wing in economics. Introducing STVO will induce the state’s Democratic party to choose candidates who are more extreme on social issues while possibly opting for candidates with moderate positions on economic issues; conversely, the Republican party will likely put forward candidates who are more extreme on the economy but moderate with respect to social issues.

Along with the effects of introducing STVO in an election (i.e., the effect of binary variable \(\mu \)), Eq. (6) allows us to study the local effects of model variables: p\(X_n,\) and \(\bar{\sigma }_{n}.\) For example, when \(X_n\) marginally increases, so does the optimal candidate’s position on the same issue, whether STVO is available or not. Observe that we restrict our attention to marginal changes in model parameters; effects of non-marginal changes cannot be inferred due to the potential equilibria multiplicity. Similarly, the following Propositions 2 and 3 focus on marginal effects and local equilibrium dynamics.

Vote share

While both parties choose candidates to maximise vote share, one may be at a disadvantage due to the average partisanship and distribution of voters’ positions in a state. The STVO differentially impacts the relative importance of these determinants of election success.

Proposition 2

For all nthe Republican (Democratic) vote share increases (decreases) in the: (i) Republican partisan advantage, \(p(R)-p(D);\) (ii) swing-position covariance \(\bar{\sigma }_{n}\) (only with STVO); and (iii) average voter bliss point \(X_{n}.\) STVO increases effect (i) and decreases effect (iii) on the distribution of votes between parties.

Voters’ positions and partisanship have the anticipated effect on parties’ electoral success: the greater the party support in the state and the closer the party is to the average voter the higher its vote share ((i) and (iii) of the proposition, respectively). (ii) is also straightforward: \(\bar{\sigma }_{n}\) determines electoral outcomes only with STVO present, in which case it benefits the party that follows the direction of swing voters (Republicans if positive; Democrats if negative).

Proposition 2’s most intriguing result is that STVO has a differential effect on (i), (ii) and (iii). While it reinforces the role of partisanship and swing-position covariance, it diminishes the importance of being positionally proximate to the average voter. To illustrate, suppose an exogenous shock causes a uniform rightward shift in all voters’ positions but has no effect on their partisanship status. According to Proposition 2, the shock benefits Republicans most when STVO is absent since without it, the average voter’s position is a more important determinant of vote share.

Note that voters’ positions across issues are substitutes with respect to party vote share. Suppose that in a given state the average voter becomes more right-wing in economic issues \(X_{econ},\) but more left-wing in social issues \(X_{soc}.\) If shifts are inversely proportional to the weights of the issues then a Democratic (Republican) candidate’s probability of winning will not change.

Expected positions of elected politician

Knowing the optimal positions of candidates and their corresponding vote shares, we can evaluate the expected position of the election winner:

$$\begin{aligned} y_{n}^{**}=V_{j}y_{jn}^{*}+\left( 1-V_{j}\right) y_{-jn}^{*}. \end{aligned}$$
(7)

\(y^{**}_n\) is a convex combination of the endogenous positions of the Republican and Democratic candidates, where weights are determined by each party’s respective vote share.

Proposition 3

An elected politician’s expected position on issue n\(y_{n}^{**},\) tends to the right as the following increase: (i) the Republican partisan advantage, \(p(R)-p(D)\); (ii) the swing-position covariance in all issues, \(\bar{\sigma }_{m},\) for all monly with STVO present; (iii) the average voter bliss point in all issues, \(X_{m}\) for all m. The STVO increases the effect (i); its effect on (iii) depends on the relative importance of issues, \(\{\alpha _{n}\}_{n=1,2\ldots , N}.\)

Proposition 3 shows how Propositions 1 and 2 interact with each other and describes the STVO’s impact on policy implementation. Consider first point (i): \(y_n^{**}\) tends to the right as the Republican partisan advantage increases. Recall that our model contains two parties, one of which is (by construction) consistently more right-wing than the other (i.e., \(Y_{Dn}<Y_{Rn},\) for all \(n\in \{ 1,2,\ldots ,N\} \)). This implies that an increase in the vote share of the Democratic party shifts all dimensions of every candidate’s political platform to the left, while an increase in the vote share of the Republican party shifts them all to the right. For example, suppose the number of Republican partisans increases in a state (ceteris paribus). Because partisan voters are more likely to elect candidates from the parties they identify with, elected politicians will become more right-wing on every issue. The STVO then reinforces this effect since partisan voters are more likely to use the option. That is, adding STVO to the ballot will cause partisan voters to cast even more straight tickets and increase the uniform shift in the positions of elected politicians.

Point (ii) highlights that the positions of swing voters relative to the general population becomes an important determinant of elected officials’ platforms when the STVO is present. Some partisans use the STVO to vote a straight party line although they would have voted positionally had the option not been made available. Thus, STVO increases the proportion of swing voters—and reduces the proportion of partisan voters—who vote positionally. Since candidates from both parties cater to positional voters, \(y_n^{**}\) shifts towards swings.

Point (iii) describes the direct and spillover effects, respectively, of voters’ positions on elected politicians’ platforms regardless of STVO status. To understand both effects, consider a state’s electorate becoming more right-wing on only one issue dimension, e.g., social issues (i.e., \(X_{soc}\) goes up but, say, \(X_{econ}\) remains unchanged, where soc and econ stand for social and economic issues, respectively). From Proposition 3(iii), elected politicians will now be further to the right not only on social issues (i.e., \(y_{soc}\) goes up), but also on economic issues (i.e., \(y_{econ}^{**}\) goes up). While the effect of the average voter position on the vote share and the average optimal candidate’s position declines when STVO is introduced, the option’s effect on \(y_n^{**}\) is generally ambiguous. However, in the special ‘symmetric’ case where parties and voters assign the same weights to issues (i.e., \(\gamma _{n}=\omega _{n}\) for all n) STVO decreases the impact of \(X_m\) on \(y_n^{**}\) for all m and n. To continue with the example of two states that only differ in \(X_{soc},\) the implication is that removing the STVO in both states will drive their elected politicians further apart on both social and economic issues.

To conclude this section, we have shown that the straight-ticket voting option changes the importance of political positions relative to partisanship in a state and thus affects the types of voters targeted by candidates. When the STVO is present, partisanship becomes more significant in that it is a more important determinant of vote shares and may thus allow candidates to offer platforms closer to the parties they represent. In terms of the political positions of the electorate, the average voter loses significance, whereas swing voters and their positions become more decisive.

Conclusion

This paper explores how STVO impacts candidate selection, vote share and the expected positions of elected politicians. Introducing STVO induces more partisan voters to cast straight-party ballots, meaning fewer of them vote by position. This grants candidates extra flexibility in appealing to the party (party loyalty effect) and remaining positional voters (swing voter effect). As a result, partisanship status and the positions of swing voters become more decisive determinants of vote share and the expected positions of election winners. Meanwhile, the average voter’s position becomes less decisive for vote share; the direction of its impact on the expected positions of election winners depends on the relative importance of issues to parties and voters.

Our model is specific to STVO, but it also speaks to a broader question about the impact of ballot design on candidates’ platforms. Figure 2 maps a range of electoral systems according to the degree to which their ballots facilitate straight-ticket voting: on the one end, it’s mandatory; on the other, elections and their ballots eschew party affiliation entirely.Footnote 18 Introducing each should result in a change in the type of voters targeted by candidates—and therefore a change in platforms they run on—that is similar to the STVO’s effect but proportionate to the extent to which the electorate is encouraged to vote on a straight-party line.

Fig. 2
figure 2

Party-candidate association in elections. Note: Figure shows the availability and ease of voting a straight ticket in different electoral systems and the implied strength of association between parties and their candidates, from no association (left) to full association (right)

By exploring the consequences of STVO, we also address how ballot design affects party influence through candidate selection. In electoral systems where straight-ticket voting is enforced, the party has full control over the politicians who represent it, since it is impossible to vote for individual candidates. In systems that have abandoned party allegiance, however, voters do not associate candidates with party labels and parties have no control over the electoral process. Each of these systems shapes party influence via candidate selection—as occurs with STVO—but again in a manner that relates to how much the system facilitates straight-ticket voting.