Bounds and Bugs: The Limits of Symmetry Metrics to Detect Partisan Gerrymandering

We consider two symmetry metrics commonly used to analyze partisan gerrymandering: the mean-median difference (MM) and partisan bias (PB). Our main results compare, for combinations of seats and votes achievable in districted elections, the number of districts won by each party to the extent of potential deviation from the ideal metric values, taking into account the political geography of the state. These comparisons are motivated by examples where the MM and PB have been used in efforts to detect when a districting plan awards extreme number of districts won by some party. These examples include expert testimony, public-facing apps, recommendations by experts to redistricting commissions, and public policy proposals. To achieve this goal we perform both theoretical and empirical analyses of the MM and PB. In our theoretical analysis, we consider vote-share, seat-share pairs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> for which one can construct election data having vote share V and seat share S , and turnout is equal in each district. We calculate the range of values that MM and PB can achieve on that constructed election data. In the process, we find the range of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> pairs that achieve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , and see that the corresponding range for PB is the same set of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> pairs. We show how the set of such <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> pairs allowing for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> (and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) changes when turnout in each district is allowed to vary. By observing the results of this theoretical analysis, we can show that the values taken on by these metrics do not necessarily attain more extreme values in plans with more extreme numbers of districts won. We also analyze specific example elections, showing how these metrics can return unintuitive results. We follow this with an empirical study, where we show that on 18 different US maps these metrics can fail to detect extreme seats outcomes.