A Test of the Marginalist Defense of the Rational Voter Hypothesis Using Quantile Regression

Abstract

This chapter uses quantile regression to uncover variations in the strength of the relationship between the expected closeness of the outcome, size of the electorate and voter turnout in Norwegian school language referendums. Referendums with a low turnout show a weak positive effect of closeness and a strong negative effect of size, the opposite being true of referendums with a high turnout. The results substantiate the marginalist defense of the Downsian rational voter hypothesis, which asserts that, while closeness and size cannot explain the absolute level of turnout, they can account for change at the margin.

An earlier version of this chapter has been published in HOMO OECONOMICUS 28(1/2): “Essays in Honor of Hannu Nurmi, Volume II,” edited by Manfred J. Holler, Andreas Nohn, and Hannu Vartiainen.

Appendix Probability that a Vote is Decisive in a Two-Way Election

When a voter faces two alternatives her vote becomes decisive either when all other votes would have tied the outcome (Event 1), or when her preferred alternative would lose by a single vote if she abstained (Event 2). The two events are mutually exclusive, as $N$ is odd in the former case and even in the latter. Let $p$ be a prior probability of a vote being cast for the voter’s preferred alternative. Event 1 occurs with probability $P_o$, which is the probability of $\frac{N-1}{2}$ successes in $N-1$ Bernoulli trials with the probability of success $p$:

$$\begin{aligned} P_o=\frac{(N-1)!}{(\frac{N-1}{2}!)^2}p^{\frac{N-1}{2}}(1-p)^{\frac{N-1}{2}}. \end{aligned}$$

(6)

Since $N$ is odd, substitute $N=2k+1$ for $k=0,1,\dots$

$$\begin{aligned} P_o=\frac{(2k)!}{(k!)^2}p^k(1-p)^k. \end{aligned}$$

(7)

By the Stirling’s approximation $x!\cong \sqrt{2\pi }(x^{x+0.5}e^{-x})$, where $\cong$ means that the ratio of the right hand side to the left hand side approaches unity as $x\rightarrow \infty$,

$$\begin{aligned} P_o\cong \frac{\sqrt{2\pi }(2k)^{2k + 0.5}e^{-2k}}{(2\pi )(k^{k + 0.5}e^{-k})^2}p^k(1-p)^k=\frac{2^{2k + 0.5}}{\sqrt{2\pi k}}p^k(1-p)^k. \end{aligned}$$

(8)

Substituting back $k=(N-1)/2$ yields, after some simplification,

$$\begin{aligned} P_o\approx \frac{2[1-(2p-1)^2]^{\frac{N-1}{2}}}{2\sqrt{\pi (N-1)}}. \end{aligned}$$

(9)

Note that in $[0,1]$ both $x(1-x)$ and $1-(2x-1)^2$ attain their maxima at $x=0.5$, so that the approximation preserves $P_o$’s essential property of being highest at $p=0.5$. Using the fact that $1+x\approx e^x$ for small $|x|$ and $1-(2p-1)^2\approx e^{-(2p-1)^2}=e^{-4(p-0.5)^2}$, for all $p$ close to 0.5 the above expression can written as

$$\begin{aligned} P_o\approx \frac{2e^{-2(N-1)(p-0.5)^2}}{\sqrt{2\pi (N-1)}}. \end{aligned}$$

(10)

This formula leads to the convenient log-linear specification with an interaction term between the quadratic measure of closeness $(p-0.5)^2$ and size $N$.

Event 2 occurs with probability $P_e$, which is the probability of $\frac{N}{2}$ successes in $N-1$ Bernoulli trials with the probability of success $p$. By a similar argument using the parity of $N$, for all $p$ close to 0.5,

$$\begin{aligned} P_e\approx \frac{2e^{-2N(p-0.5)^2}}{\sqrt{2\pi N}}. \end{aligned}$$

(11)

Good and Mayer (1975) discuss the magnitude of error in $P_o$ and $P_e$ due to $p$ deviating from 0.5, which can be substantial (Fig. 2). See, also Chamberlain and Rothschild (1981), and in the context of voting power, Grofman (1981). Kaniovski (2008) computes the probability of casting a decisive vote when votes are neither equally probable to be for or against, nor independent. Departures from either assumption induce a substantial bias in this probability compared to the baseline case of equally probable and independent votes. The bias incurred by the probability deviating from one-half is larger than that incurred by the Pearson product-moment correlation coefficient deviating from zero.

Fig. 2

Approximation to the probability of casting a decisive vote. In an election with two alternatives, the Probability of being decisive decreases with the size of electorate $N$ and with $|p-0.5|$. For a fixed $N$, the probability is the highest when $p=0.5$ and decreases rapidly as $p$ deviates from 0.5. The approximation is valid for $p\approx 0.5$