[maxerickson]

You are stating the efficiency gap incorrectly in the single district case. There can never be one, because the seat goes to the party with more votes (the party with less votes should have 0 seats, no efficiency gap).

The simplest example you can work is with 2 districts.

[andrewla]

The efficiency gap, as described, is the number of "wasted" votes for one party, minus the number of wasted votes for the other party, summed over all districts and divided by the total voting population.

So fundamentally it's just a normalized version of the wasted votes metric. Wasted votes is the thing that concerns me as a metric, and it has the discontinuity noted.

Notably, if you have a circular uniformly populated state that is exactly 50-50 -- let's say the north of the state is 100% Republican and the south is 100% Democrat, and we restrict our districting to straight lines through the center. Then there are two solutions that minimize the efficiency gap, the line going from southwest to northeast, and the one going from southeast to northwest, because both of those result in 75/25 districts (that have 0 net wasted votes). This seems really odd to me.

[maxerickson]

Right, but the efficiency gap isn't obsessed with net 0 districts, it is looking for excess wasted votes. No matter how you draw the line through your circle, the net wasted votes across the districts will be 0, so the state wide efficiency gap will always be 0.

I think it's actually literally impossible to use a straight line there for partisan gerrymanding, so it isn't a real useful scenario.

[andrewla]

That's a fair point. Still seems off to me in terms of sensitivity, and once again putting aside the discussion of what the goal of districting should be.

Simplifying the algebra in the calculation, though, the efficiency gap is actually independent of the inefficiencies of any particular district, but dependent only on the population of the districts -- the numerator simplifies to

D - R + sum_i { I(D_i < R_i) (D_i + R_i) / 2 }

Where D and R are the total Democratic/Republican votes in the state (across all districts) and the I(x) is an indicator function, +1 if true, -1 if false.

If you hold the size of districts to be about even, then this just says that the ratio of the total population of districts that vote R to the total population of districts that vote D should be approximately equal to the ratio of the total state population of R voters to the total state population of D voters.

That seems reasonable, just seems like a roundabout way to get there, and is sensitive to unequal population distributions.

[maxerickson]

An uncontroversial (perhaps the only uncontroversial) goal of redistricting is to make the population of the districts in a state as similar as possible.

Conceptually, I like the idea of having a test for excessive unfairness; states can do whatever they want to deal with their special cases as long as the result is reasonable.