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

We provide both a theoretical and empirical analysis of the Mean-Median Difference (MM) and Partisan Bias (PB), which are both symmetry metrics intended to detect gerrymandering. We consider vote-share, seat-share pairs $(V, S)$ 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 vote-share, seat share pairs $(V, S)$ for which there is constructed election data with vote share $V$, seat share $S$, and $MM=0$, and see that the corresponding range for PB is the same set of $(V,S)$ pairs. We show how the set of such $(V,S)$ pairs allowing for $MM=0$ (and $PB=0$) changes when turnout in each district is allowed to be different. Although the set of $(V,S)$ pairs for which there is election data with $MM=0$ is the same as the set of $(V,S)$ pairs for which there is election data with $PB=0$, the range of possible values for MM and PB on a fixed $(V, S)$ is different. Additionally, for a fixed constructed election outcome, the values of the Mean-Median Difference and Partisan Bias can theoretically be as large as 0.5. We show empirically that these two metric values can differ by as much as 0.33 in US congressional map data. We use both neutral ensemble analysis and the short-burst method to show that neither the Mean-Median Difference nor the Partisan Bias can reliably detect when a districting map has an extreme number of districts won by a particular party. Finally, we give additional empirical and logical arguments in an attempt to explain why other metrics are better at detecting when a districting map has an extreme number of districts won by a particular party.