The following is a guest post from Jeff Barton:

When considering a district map, the term “partisan advantage” as introduced by Jon Eguia and advocated by Alec Ramsay refers to the difference between a party’s ideal and actual seat shares. As Nicholas Stephanopoulos discusses, the use of such a measure is a key component of recently proposed voting rights legislation. Two commonly cited standards for what constitutes a fair proportion of seats are proportionality (a party with proportion V of the votes deserves proportion V of the seats), and the efficiency gap (a party with proportion V of the votes deserves proportion 2V-0.5 of the seats).  

Proportionality is an appealing ideal, but it has been excluded by the courts as a requirement for fairness. Moreover, real election results from single-member plurality, or first-past-the-post (FPTP), systems like ours demonstrate that proportionality is not a reasonable expectation. The reason for this disparity is that the winner-take-all nature of our elections causes a lot of votes to be wasted, and this results in an inherent majoritarian bonus that is absent in proportionality.  

In contrast to proportionality, the efficiency gap standard does incorporate the winner-take-all character of single-member elections in requiring that both parties waste equal numbers of votes. It is also a demonstrably better fit for observed election data. Unfortunately, the way that the efficiency gap counts wasted votes is overfriendly to the majority party and hence produces a standard with a majoritarian bonus that is too large. This can be inferred from the fact that the standard 2V-0.5 produces negative seat shares for minority parties whose vote shares are less than 0.25. 

In my paper, “Fairness in Plurality Systems with Implications for Detecting Partisan Gerrymandering,” I deduce a simple, FPTP-specific standard for fairness from first principles of fairness and a careful accounting of wasted votes. The result is that a minority party with proportion V of the votes deserves proportion 2V² of the seats. This FPTP standard, which falls between that of proportionality and the efficiency gap, has several appealing properties.  

First, it handles edge cases appropriately: a party with 50% of the votes should win 50% of the seats, and only a party with 0% of the votes should win 0% of the seats. Second, the new standard is a good fit for the curvature seen in plots of real election data; furthermore, unlike other “good fit” curves such as the cube law, it explains why the curvature exists irrespective of political geography or gerrymandering. Third, when compared to results from Eguia’s jurisdictional method and certain computer simulations, the fit of the new standardimproves when sources of bias are removed from real election results. Fourth, the new standard has an interesting relationship to the efficiency gap. Namely, the efficiency gap standard is the best straight-line approximation to the new standard at parity. This may help explain the reasonableness of the efficiency gap for a fairly wide range of vote proportions. Finally, the new standard has a natural extension to multiparty elections, such as for the Canadian Parliament, where again it is a demonstrably better fit to actual results than proportionality. 

As a simple, deducible, and empirically reasonable baseline for fairness in the context of our election system, the new standard supplies a sensible and necessary way to determine the extent to which a proposed map causes actual harm to either party.