Turnout and Polarization Under Alternative Electoral Systems

Abstract

This chapter presents a formal model of electoral competition where parties’ platforms are endogenously chosen and depend on the degree of the electoral rule disproportionality. We first show that proportional electoral systems generate centrifugal forces that increase candidate differentiation. This in turn implies that more proportional systems are associated with lower levels of abstention from indifference. This two-step theoretical prediction of the effect of electoral systems on turnout is then empirically validated even when we jointly control for the prevailing pivotality and party-system size hypotheses. Thus, our work highlights an additional link in the proportionality-turnout nexus.

Appendix

Proof of Proposition 1

The proof of this proposition can be split in to five distinct parts.

Part 1 :

Since the behavior of voters is unambiguous in this model we should focus on understanding the dynamics which determine candidate behavior. Given n we have that party L decides p _L in order to maximize

$$\displaystyle{u_{L}(\hat{p}(p_{L},p_{R},n)) = -(\pi _{L} -\hat{ p}(p_{L},p_{R},n))^{2} = -\hat{p}(p_{ L},p_{R},n)^{2}}$$

while party R decides p _R in order to maximize

$$\displaystyle{u_{R}(\hat{p}(p_{L},p_{R},n)) = -(\pi _{R} -\hat{ p}(p_{L},p_{R},n))^{2} = -(1 -\hat{ p}(p_{ L},p_{R},n))^{2}.}$$

Hence, $u_{L}(\hat{p}(p_{L},p_{R},n))$ is strictly decreasing in $\hat{p}(p_{L},p_{R},n)$ for every $\hat{p}(p_{L},p_{R},n)$ between zero and one and $u_{R}(\hat{p}(p_{L},p_{R},n))$ is strictly decreasing in $\hat{p}(p_{L},p_{R},n)$ for every $\hat{p}(p_{L},p_{R},n)$ between zero and one. Letting aside mixed strategies, this means that our two-player game is strategically equivalent to the two-player zero sum game in which one player (party L) decides p _L in order to minimize $\hat{p}(p_{L},p_{R},n)$ and the other player (party R) decides p _R in order to minimize $\hat{p}(p_{L},p_{R},n)$. Therefore, if we characterize the equilibrium set of this zero-sum game, we will have the equilibrium set of the game we are interested in.

Part 2 :

We notice that in equilibrium it has to be the case that $\hat{p}(p_{L}^{{\ast}},p_{R}^{{\ast}},n) = \frac{1} {2}$. As said our game is strategically equivalent to the described zero-sum game. Therefore, it has to be the case that the equilibrium implemented policy (which coincides with the value of the zero-sum game) is unique; all equilibria should deliver the same implemented policy. Imagine that in this unique equilibrium implemented policy is such that $\hat{p}(p_{L}^{{\ast}},p_{R}^{{\ast}},n) <\frac{1} {2}$. If party R deviates and proposes a platform $1 - p_{L}^{{\ast}}$, it will switch the implemented policy to $\frac{1} {2}$; this is obviously profitable for party R as it will bring the implemented policy nearer to its ideal policy. An equivalent argument rule out possibility of $\hat{p}(p_{L}^{{\ast}},p_{R}^{{\ast}},n)> \frac{1} {2}$. Therefore, in equilibrium it has to be that $\ \hat{p}(p_{L}^{{\ast}},p_{R}^{{\ast}},n) = \frac{1} {2}.$

Part 3 :

If the equilibrium set is non-empty, then an equilibrium should exist such that $p_{L}^{{\ast}} <\frac{1} {2} <p_{R}^{{\ast}} = 1 - p_{ L}^{{\ast}}$. If an equilibrium exists and $p_{L}^{{\ast}}\geq \frac{1} {2}$ then $\hat{p}(p_{L}^{{\ast}},p_{R}^{{\ast}},n) = \frac{1} {2}$ suggests that $p_{R}^{{\ast}}\leq \frac{1} {2}$. This implies that (a) if $p_{R}^{{\ast}} = \frac{1} {2}$ party L can deviate to the policy p _L  = 0 and, given our parameters restrictions, receive some votes and thus induce $\hat{p}(0, \frac{1} {2},n) <\frac{1} {2}$ and that (b) if $p_{R}^{{\ast}} <\frac{1} {2}$ party L can deviate to the policy $p_{L} = p_{R}^{{\ast}}$ and thus induce $\hat{p}(p_{R}^{{\ast}},p_{R}^{{\ast}},n) <\frac{1} {2}$. This rules out possibility of an equilibrium such that $p_{R}^{{\ast}}\leq \frac{1} {2}$ too. Moreover, since our game is strategically equivalent to a zero-sum game it has to be the case that every equilibrium strategy is a minimaximizer strategy and the other way round. This along with the fact that our game also satisfies a symmetry notion ($\hat{p}(\mu,\nu,n) = 1 -\hat{ p}(1-\nu,1-\mu,n)$) suggests that if $p_{L}^{{\ast}}$ is a minimaximizer strategy for party L then $1 - p_{L:}^{{\ast}}$ must be a minimaximizer strategy of party R.

Part 4 :

We observe that when $p_{R}> \frac{1+v} {2}$ there exists $\varepsilon> 0$ such that $\hat{p}(p_{L},p_{R},n)$ is differentiable in $p_{L} \in (1 - p_{R}-\varepsilon,1 - p_{R}+\varepsilon )$. Routine algebraic manipulations show that $\frac{\partial u_{L}(\hat{p}(p_{L},p_{R},n))} {\partial p_{L}} \vert _{p_{R}=1-p_{L}} = 0$ if and only if $p_{L}^{{\ast}} = \frac{-1+2n-\sqrt{1-4vn}} {4n} <\frac{1-v} {2}$ (in which case $p_{R}^{{\ast}} = 1 -\frac{-1+2n-\sqrt{1-4vn}} {4n} = \frac{1+2n+\sqrt{1-4vn}} {4n}$). That is, $(p_{L}^{{\ast}},p_{R}^{{\ast}}) = (\frac{-1+2n-\sqrt{1-4vn}} {4n}, \frac{1+2n+\sqrt{1-4vn}} {4n} )$ is candidate for an equilibrium. By plotting $\hat{p}(p_{L}, \frac{1+2n+\sqrt{1-4vn}} {4n},n)$ we notice that it admits a unique minimum for any admissible pair of parameter values (see Figs. 3, 4 and 5) and hence $u_{L}(\hat{p}(p_{L}, \frac{1+2n+\sqrt{1-4vn}} {4n},n))$ admits a unique maximum too. That is party L has a unique minimaximizer strategy $p_{L}^{{\ast}} = \frac{-1+2n-\sqrt{1-4vn}} {4n}$ which suggests that our game admits the unique equilibrium $(p_{L}^{{\ast}},p_{R}^{{\ast}}) = (\frac{-1+2n-\sqrt{1-4vn}} {4n}, \frac{1+2n+\sqrt{1-4vn}} {4n} )$.

Fig. 3

Implemented policy—height—as a function of the policy platform of party L,$p_{L} \in [0,1/2)$—length—and abstention parameter $v \in [0,0.05]$—width—when $p_{R} = \frac{1+2n+\sqrt{1-4vn}} {4n}$ and the electoral rule is purely proportional (n = 1)

Fig. 4

Implemented policy—height—as a function of the policy platform of party L,p _L  ∈ [0, 1∕2)—length—and abstention parameter v ∈ [0, 0. 05]—width—when $p_{R} = \frac{1+2n+\sqrt{1-4vn}} {4n}$ and the electoral rule is mixed (n = 2)

Fig. 5

Implemented policy—height—as a function of the policy platform of party L,$p_{L} \in [0,1/2)$—length—and abstention parameter $v \in [0,0.05]$—width—when $p_{R} = \frac{1+2n+\sqrt{1-4vn}} {4n}$ and the electoral rule is majoritarian (n = 3)

Part 5 :

It is straightforward that $\frac{\partial (p_{R}^{{\ast}}-p_{ L}^{{\ast}})} {\partial n} <0$ and hence the distance between the equilibrium platforms is decreasing in the level of disproportionality of the electoral rule.

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Proof of Proposition 2

This is straightforward as $\frac{\partial T^{{\ast}}(n)} {\partial n} <0$ for every strictly positive and admissible value of the parameters. □