Electoral David vs Goliath: How does the Spatial Concentration of Electors affect District-based Elections?

Abstract

Many democratic countries use district-based elections where there is a "seat" for each district in the governing body such as parliament or assembly. In each district, the party whose candidate gets the maximum number of votes is declared the winner of the corresponding seat. The result of the election is decided based on the number of seats won this way by the different parties. The electors (or voters) are assigned to different districts based on their residence, and each elector can vote only in the district in which they are assigned. Thus, locations of the electors and boundaries of the districts may severely affect the election result even if the proportion of popular support (number of electors) of different parties remains unchanged. In this setting, it is also possible that a party with less number of supporters overall, may end up winning more seats if their supporters are suitably distributed. This has led to significant amount of research on the topic of "gerrymandering", i.e. how districts may be redrawn or electors may be moved to ensure maximum seats for a particular party. In this paper, we frame the spatial distribution of electors in a probabilistic setting, and analyze different models to capture the intra-district polarization of electors in favour of a party, or the spatial concentration of supporters of different parties. Our models are inspired by elections in India, where supporters of different parties tend to be concentrated, turning various districts into strongholds of certain parties. We show with extensive simulations that our model can capture different statistical properties of real elections held in India. For this purpose, we frame parameter estimation problems to fit our models to the observed election results. Since analytical calculation of the likelihood functions are infeasible for our complex models, we make use of Likelihood-free Inference methods under the framework of Approximate Bayesian Computation (ABC). Since this approach is highly timeconsuming, we explore how supervised regression using Logistic Regression or Deep Neural Networks can be used to speed it up. We also explore how the election results can change drastically by varying the spatial distributions of the voters, even when the proportions of popular support of the parties remain constant.