Ties

Abstract

We tackle the question of the role of pivotality in voter turnout decisions by testing for the first time whether the occurrence of a tied election generates information spillovers onto nearby localities’ subsequent elections. First, we develop a model where voters update their beliefs regarding the probability of their vote being decisive upon observing earlier elections’ outcomes. Next, by exploiting Italian mayoral elections ending in close outcomes during the past two decades and the quasi-experimental conditions created by the staggered electoral calendar, we find a substantial impact on voter turnout rates of exposure to spillovers from tied elections.

Appendices

Appendix A

1.1 Proof of Proposition 1

According to Taylor and Yildirim (2010b), the pivotal probability for an L-type voter is:

$$\begin{aligned} \begin{array}{lll} &{} &{} \Pi ^s _{1L}(\tau ^s _{1L},\tau ^s _{1R}, \lambda ) \\ &{} &{}\quad = \sum \limits _{k=0}^{\lfloor \frac{n_{1}-1}{2}\rfloor }\left( {\begin{array}{c} n_{1}-1\\ k,k,n_{1}-1-2k\end{array}}\right) (\lambda \tau ^s _{1L})^{k}\left[ (1-\lambda )\tau ^s _{1R} \right] ^{k}\left[ 1-\lambda \tau ^s _{1L}-(1-\lambda )\tau ^s _{1R}\right] ^{n_{1}-1-2k} \\ &{}&{}\qquad + \sum \limits _{k=0}^{\lfloor \frac{n_{1}-1}{2}\rfloor }\left( {\begin{array}{c} n_{1}-1\\ k,k+1,n_{1}-2-2k\end{array}}\right) (\lambda \tau ^s _{1L})^{k}\left[ (1-\lambda )\tau ^s _{1R}\right] ^{k+1}\left[ 1-\lambda \tau ^s _{1L}-(1-\lambda )\tau ^s _{1R} \right] ^{n_{1}-2-2k},\ \ \end{array} \end{aligned}$$

(A.1)

where $\lfloor \cdot \rfloor$ is the integer part of a number. The first term after the equality in (A.1) represents the event where there is a tie, and the second term represents the event where the L-type candidate would lose by one vote without his vote.

Similarly, the pivotal probability for an R-type voter is:

$$\begin{aligned} \begin{array}{lll} &{} &{} \Pi ^s _{1R}(\tau ^s _{1L},\tau ^s _{1R}, \lambda ) \\ &{} &{}\quad = \sum \limits _{k=0}^{\lfloor \frac{n_{1}-1}{2}\rfloor } \left( {\begin{array}{c}n_{1}-1\\ k,k,n_{1}-1-2k\end{array}}\right) (\lambda \tau ^s _{1L})^{k}\left[ (1-\lambda )\tau ^s _{1R}\right] ^{k}\left[ 1-\lambda \tau ^s _{1L}-(1-\lambda )\tau ^s _{1R} \right] ^{n_{1}-1-2k} \\ &{} &{}\qquad + \sum \limits _{k=0}^{\lfloor \frac{n_{1}-1}{2}\rfloor } \left( {\begin{array}{c}n_{1}-1\\ k,k+1,n_{1}-2-2k\end{array}}\right) (\lambda \tau ^s _{1L})^{k+1}\left[ (1-\lambda )\tau ^s _{1R}\right] ^{k}\left[ 1-\lambda \tau ^s _{1L}-(1-\lambda )\tau ^s _{1R}\right] ^{n_{1}-2-2k}. \end{array} \end{aligned}$$

(A.2)

By taking the derivatives of the pivotal probabilities in (A.1) and (A.2) with respect to $\lambda$, and fixing at the equilibrium $(\tau ^{s*}_{1L}, \tau ^{s*}_{1R})$, we have:

$$\begin{aligned}{} & {} \frac{\partial \Pi ^s_{1L}(\tau ^{s*}_{1L}, \tau ^{s*}_{1R}, \lambda )}{\partial \lambda }\nonumber \\{} & {} \quad =\sum \limits _{k=0}^{\lfloor \frac{n_1-1}{2}\rfloor } \left( {\begin{array}{c}n_1 -1\\ k, k, n_1-2-2k\end{array}}\right) \lambda ^{k-1} (\tau ^{s*}_{1L})^k (1-\lambda )^{k-1} (\tau ^{s*}_{1R})^k \left[ 1-\lambda \tau ^{s*}_{1L}-(1-\lambda )\tau ^{s*}_{1R}\right] ^{n_1-2-2k}\nonumber \\{} & {} \qquad \times \left\{ \frac{k(1-2\lambda )\left[ 1-\lambda \tau ^{s*}_{1L}-(1-\lambda )\tau ^{s*}_{1R}\right] }{(n_1-1-2k)} +\frac{k(1-2\lambda )(1-\lambda )\tau ^{s*}_{1R}}{(k+1)}-\frac{\lambda (1-\lambda )\tau ^{s*}_{1R}}{(k+1)}\right. \nonumber \\{} & {} \qquad + \left. \lambda (1-\lambda )(\tau ^{s*}_{1R}-\tau ^{s*}_{1L}) \left[ 1+\frac{(n_1-2-2k)(1-\lambda )\tau ^{s*}_{1R}}{(k+1)\left[ 1-\lambda \tau ^{s*}_{1L}-(1-\lambda )\tau ^{s*}_{1R}\right] }\right] \right\} ,\quad \textrm{and}\nonumber \\{} & {} \frac{\partial \Pi ^s_{1R}(\tau ^{s*}_{1L}, \tau ^{s*}_{1R}, \lambda )}{\partial \lambda } \end{aligned}$$

(A.3)

$$\begin{aligned}{} & {} \quad =\sum \limits _{k=0}^{\lfloor \frac{n_1-1}{2}\rfloor } \left( {\begin{array}{c}n_1-1\\ k, k, n_1-2-2k\end{array}}\right) \lambda ^{k-1} (\tau ^{s*}_{1L})^k (1-\lambda )^{k-1} (\tau ^{s*}_{1R})^k \left[ 1-\lambda \tau ^{s*}_{1L}-(1-\lambda )\tau ^{s*}_{1R}\right] ^{n_1-2-2k}\nonumber \\{} & {} \qquad \times \left\{ \frac{k(1-2\lambda )\left[ 1-\lambda \tau ^{s*}_{1L}-(1-\lambda )\tau ^{s*}_{1R}\right] }{(n_1-1-2k)} +\frac{k(1-2\lambda )\lambda \tau ^{s*}_{1L}}{(k+1)}+\frac{\lambda (1-\lambda )\tau ^{s*}_{1L}}{(k+1)}\right. \nonumber \\{} & {} \qquad + \left. \lambda (1-\lambda )(\tau ^{s*}_{1R}-\tau ^{s*}_{1L}) \left[ 1+\frac{(n_1-2-2k)\lambda \tau ^{s*}_{1L}}{(k+1)\left[ 1-\lambda \tau ^{s*}_{1L}-(1-\lambda )\tau ^{s*}_{1R}\right] }\right] \right\} . \end{aligned}$$

(A.4)

There are no definite signs in general. We then focus on some special cases to see the effect of $\lambda$. Consider a very close election where $\lambda \rightarrow 1/2$ in $s=0$. In this case, we have $\tau ^{0*}_{1R}\rightarrow \tau ^{0*}_{1L}$, according to Lemma 1. Therefore, the terms in the braces in (A.3) and (A.4) approach to zero except the third one. Therefore,

$$\begin{aligned} \frac{\partial \Pi ^0_{1L}(\tau ^{0*}_{1L}, \tau ^{0*}_{1R}, \lambda )}{\partial \lambda }|_{\lambda \rightarrow 1/2}<0 \ \ \textrm{and} \ \ \frac{\partial \Pi ^0_{1R}(\tau ^{0*}_{1L}, \tau ^{0*}_{1R}, \lambda )}{\partial \lambda }|_{\lambda \rightarrow 1/2}>0. \end{aligned}$$

Consider a new $\lambda '$ which is close to $\lambda$ and $1/2<\lambda ' < \lambda$. Then under this $\lambda '$, since $\frac{\textrm{d} F}{\textrm{d} c} > 0$, we know

$$\begin{aligned} \tau ^{0*}_{1L} < F\left( \frac{1}{2} \Pi ^0_{1L}(\tau ^{0*}_{1L}, \tau ^{0*}_{1R}, \lambda ')\right) \quad \textrm{and} \quad \tau ^{0*}_{1R} > F\left( \frac{1}{2} \Pi ^0_{1R}(\tau ^{0*}_{1L}, \tau ^{0*}_{1R}, \lambda ')\right) . \end{aligned}$$

(A.5)

Thus, in order to maintain the equality, the new equilibrium $(\tau ^{0\prime }_{1L}, \tau ^{0\prime }_{1R})$ under $\lambda ^\prime$ must be the case where

$$\begin{aligned} \tau ^{0\prime }_{1L} > \tau ^{0*}_{1L} \quad \textrm{and}\quad \tau ^{0\prime }_{1R} < \tau ^{0*}_{1R}. \end{aligned}$$

(A.6)

That is, when a very close election becomes even closer, the pivotal probability for a supporter of the leading (underdog) candidate will be larger (smaller), so that it is more (less) likely for that voter to turn out and vote in the later election.

Another extreme case is $\lambda \rightarrow 1$, where the election is dominated by the leading party. Then the terms in the braces in (A.3) and (A.4) approach to zero except the first one. Thus, we have

$$\begin{aligned} \frac{\partial \Pi ^0_{1L}(\tau ^{0*}_{1L}, \tau ^{0*}_{1R}, \lambda )}{\partial \lambda }|_{\lambda \rightarrow 1}<0 \quad \textrm{and} \quad \frac{\partial \Pi ^0_{1R}(\tau ^{0*}_{1L}, \tau ^{0*}_{1R}, \lambda )}{\partial \lambda }|_{\lambda \rightarrow 1} < 0. \end{aligned}$$

Consider a new $\lambda '$ which is close to $\lambda$ and $\lambda '<\lambda$. Then under this $\lambda '$,

$$\begin{aligned} \tau ^{0*}_{1L}< F\left( \frac{1}{2} \Pi ^0_{1L}(\tau ^{0*}_{1L}, \tau ^{0*}_{1R}, \lambda ')\right) \quad \textrm{and} \quad \tau ^{0*}_{1R} < F\left( \frac{1}{2} \Pi ^0_{1R}(\tau ^{0*}_{1L}, \tau ^{0*}_{1R}, \lambda ')\right) . \end{aligned}$$

(A.7)

Thus, in order to maintain the equality, the new equilibrium $(\tau ^{0\prime }_{1L}, \tau ^{0\prime }_{1R})$ under $\lambda '$ is such that

$$\begin{aligned} \tau ^{0\prime }_{1L}> \tau ^{0*}_{1L} \quad \textrm{and} \quad \tau ^{0\prime }_{1R} > \tau ^{0*}_{1R}. \end{aligned}$$

(A.8)

That is, when the election becomes less lopsided, the pivotal probability for a supporter of either party increases, so that it is more likely for a voter to vote. Thus, the total turnout rate increases. $\square$

Appendix B

See Tables 7, 8, 9.

Table 7 Summary statistics

Table 8 Ties

Table 9 Elections decided by one vote

Appendix C

See Tables 10, 11, 12, 13, 14, 15 and 16.

Table 10 Spillovers: mayor candidates

Table 11 Spillovers: income tax rate

Table 12 Spillovers from close outcomes

Table 13 Spillovers: turnout (electorate < 5000)

Table 14 Spillovers: turnout (electorate < 2500)

Table 15 Spillovers: turnout (electorate < 1000)

Table 16 Spillovers: turnout leads