The Efficiency Gap, Voter Turnout, and the Efficiency Principle

Recently, scholars from law and political science have introduced metrics which use only election outcomes (and not district geometry) to assess the presence of partisan gerrymandering. The most high-profile example of such a tool is the efficiency gap. Some scholars have suggested that such tools should be sensitive enough to alert us when two election outcomes have the same percentage of votes going to political party $A$, but one of the two awards party $A$ more seats. When a metric is able to distinguish election outcomes in this way, that metric is said to satisfy the efficiency principle. In this article, we show that the efficiency gap fails to satisfy the efficiency principle. We show precisely how the efficiency principle breaks down in the presence of unequal voter turnout. To do this, we first present a construction that, given any rationals $1/4< V<3/4$ and $0<S<1$, constructs an election outcome with vote share $V$, seat share $S$, and EG = 0. (For instance, one party can get 26% of the vote and anywhere from 1% to 99% of the seats while the efficiency gap remains zero.) Then, for any election with vote share $1/4<V<3/4$, seat share $S$, and EG= 0, we express the ratio $\rho$ of average turnout in districts party $A$ lost to average turnout in districts party $A$ won as a function in only $V$ and $S$. It is well known that when all districts have equal turnout, EG can be expressed as a simple formula in $V$ and $S$; we express the efficiency gap of any election as an equation only in $V, S,$ and $\rho$. We also report on the values of $\rho$ that can be observed in actual elections.