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

We consider two symmetry metrics to detect partisan gerrymandering: the Mean-Median Difference (MM) and Partisan Bias (PB). To lay the groundwork for our main results, we first assert that the foundation of a partisan gerrymander is to draw a map so that the preferred party wins an extreme number of seats, and that both the Mean-Median Difference and Partisan Bias have been used to detect partisan gerrymandering. We then provide both a theoretical and empirical analysis of the Mean-Median Difference and Partisan Bias. In our theoretical analysis, 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. By observing the results of this theoretical analysis, we give examples of how these two metrics are unable to detect when a map has an extreme number of districts won. Because these examples are constructed, we follow this with our empirical study, in which we show on 18 different U.S. maps that these two metrics are unable to detect when a map has an extreme number of districts won.