Mathematics of Gerrymandering Identification with Linear Metrics

32 Pages Posted: 11 Apr 2022

Date Written: March 3, 2022


The identification of bias in election results due to partisan gerrymandering and redistricting manipulation is important to preserve functioning democracy. Many methods have been proposed but none have been proven superior. After an overview of existing methods, the paper looks at the mathematical foundation of gerrymandering metrics. It is shown that often hidden variables, average party percent vote in district races won and lost, are the basis of most metrics. Using these variables it is shown that seats-vote ratio based Partisan Bias and Efficiency Gap are closely related and are described by a common formula, which does not entirely conform to the principle of packing and cracking. A new simple to calculate linear metric (named Partisan Disparity) based on packing asymmetry is introduced and shown to have non-linear seats vs votes neutral value curves which better account for vote distribution. This metric is shown to have over 0.9 correlation with Declination based on US Congress 1976-2020 elections. A generalized model of linear metrics is introduced as a 3-dimensional parametric space of functions which allows to create and study gerrymandering metrics. A metric based on statistical skew is also introduced and compared with other metrics. Finally a comparative summary of the extent of gerrymandering over last 50 years using average results of 3 metrics is provided that shows 2012-2020 as being the decade with highest level of gerrymandering in US Congress elections.

Keywords: Gerrymandering, Partisan advantage, Redistricting, Functional Analysis

Suggested Citation

Leibzon, William, Mathematics of Gerrymandering Identification with Linear Metrics (March 3, 2022). Available at SSRN: or

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