A Probabilistic Model for Detecting Gerrymandering in Partially-Contested Multiparty Elections

Abstract

Classic methods for detecting gerrymandering fail in multiparty partially-contested elections, such as the Polish local election of 2014. A new method for detecting electoral bias, based on the assumption that voting is a stochastic process described by Pólya’s urn model, is devised to overcome these difficulties. Since the partially-contested character of the election makes it difficult to estimate parameters of the urn model, an ad-hoc procedure for estimating those parameters in a manner untainted by potential gerrymandering is proposed.

Supported by the Polish National Science Center (NCN) under grant no. 2014/13/B/HS5/00862, Scale of gerrymandering in 2014 Polish township council elections.

A Appendix

We have developed two proof-of-concept tests to evaluate the correctness of the proposed method for detecting gerrymandering. First, for any set of jurisdictions to which we seek to apply the proposed method we can test whether the empirical marginal distribution of $X_{m}$, $m=1,\dots ,11$, agrees with the theoretical marginal distribution expressed by (). We have run such test for our primary data set of interest, Polish local election of 2014, producing, for the following values of parameters $\kappa _{m}$ and $\theta _{m}$ (fitted in accordance with the procedure described in Sect. 3.5), the following total variation distances $\varvec{d}_{\mathbf {TV}}$ between the theoretical density function and the kernel density estimator of the empirical density (Fig. 3):

Fig. 3.

The theoretical (red) and empirical (black) densities of $v_{i}^{k}$ for different values of m. The empirical density is a kernel density estimate with the number of points and the bandwidth given below each plot. (Color figure online)

Even if the probabilistic model underlying the test is correct, it remains to be seen if the method is sensitive enough to detect actual gerrymandering (or other instances of electoral bias) and specific enough to keep the level of false positives low. Ideally, we would test the above using an empirical dataset that includes some known instances of gerrymandering, but our main dataset included none. Therefore, we have tested the sensitivity and specificity of the method against an artifical dataset, although one based on empirical data. We have algorithmically created a sample of 1024 districting plans for our home city of Kraków, each having 43 seats (as is the case in reality). Of those, 1020 were generated randomly using a Markov chain Monte Carlo districting algorithm developed by [25] and implemented in R package redist. The remaining four have been generated using two algorithms from [70], designed to produce districting plans gerrymandered in favor of one of the two largest parties (the first algorithm has also been designed to try to keep the districts relatively compact, while the second has been freed of all compactness constraints). Under each of those districting plans, we have calculated simulated election results using the 2014 precinct-level data. We treat each of such simulated elections as a single jurisdiction for which we carry out the procedure described in the article to obtain $\pi$. There are 12 distinct outcomes arising in simulated elections. As under all 1024 plans, all seats are won by the two largest parties, those outcomes are uniquely characterized by $s_{1}$ (or $s_{2}$). We list all of them, with corresponding $\pi$ values, in the table below (the outcomes arising under the four intentionally gerrymandered plans are identified by the bold font):

$s_{1}$	$s_{2}$	No. of plans	% of plans	$\pi$
$\mathbf {36}$	$\mathbf {7}$	$\mathbf {1}$	$\mathbf {0.10\%}$	$\mathbf {7.8E-06}$
$\mathbf {31}$	$\mathbf {12}$	$\mathbf {1}$	$\mathbf {0.10\%}$	$\mathbf {0.39\%}$
26	17	1	$0.10\%$	$13.67\%$
25	18	3	$0.29\%$	$21.53\%$
24	19	43	$4.20\%$	$31.47\%$
23	20	226	$22.07\%$	$42.94\%$
22	21	433	$42.29\%$	$55.04\%$
21	22	262	$25.59\%$	$44.96\%$
20	23	48	$4.69\%$	$33.32\%$
19	24	4	$0.39\%$	$23.09\%$
$\mathbf {13}$	$\mathbf {30}$	$\mathbf {1}$	$\mathbf {0.10\%}$	$\mathbf {0.48\%}$
$\mathbf {6}$	$\mathbf {37}$	$\mathbf {1}$	$\mathbf {0.10\%}$	$\mathbf {4.4E-07}$

As can be seen from the above table, for all four intentionally gerrymandered plans the value of $\pi$ are small enough to identify them as suspect, while none of the unbiased plans are so identified.