The Math of Gerrymandering
In 2015, two researchers proposed a single number meant to fit on one line of a legal opinion. Count every vote that did not help elect the winner in its district; subtract the two parties' totals; divide by total votes cast. If the gap exceeds about 7%, the map is suspect. Four years later the Supreme Court ruled federal courts cannot police such maps at all. The equation kept working. The courtroom door closed.
How the efficiency gap works
Nicholas Stephanopoulos and Eric McGhee defined a "wasted vote" as any vote cast for a losing candidate, plus any vote for the winner beyond the 50%+1 needed to win. A gerrymander works by packing one party's voters into a few districts they win in landslides (wasting their surplus) and cracking the rest across districts they narrowly lose (wasting all of them).
$$ EG = \frac{W_A - W_B}{\text{total votes}} $$
A fair map under this measure has equal waste on both sides, EG ≈ 0. The 7% threshold was not pulled from the air: Stephanopoulos and McGhee found that historically, gaps above that level rarely self-corrected within a decade.
Where it showed up
| Map | Year drawn | EG (approx) | Outcome |
|---|---|---|---|
| Wisconsin Assembly | 2011 | ~13% R | Struck down by district court (Gill v. Whitford), reversed on standing 2018 |
| Pennsylvania congressional | 2011 | ~15% R | Struck down 2018 under state constitution, redrawn |
| Maryland 6th congressional | 2011 | severe D | Plaintiffs in Benisek, consolidated into Rucho |
| North Carolina congressional | 2016 | ~20% R | Lead case in Rucho v. Common Cause |
The North Carolina map was a clean test: the state legislator who chaired the redistricting committee said on record they drew 10 Republican and 3 Democratic districts "because I do not believe it's possible to draw a map with 11 Republicans and 2 Democrats." Intent was admitted. The math agreed. The Court still declined.
What Rucho actually held
Chief Justice Roberts, for a 5-4 majority in Rucho v. Common Cause (2019), did not rule that partisan gerrymandering is constitutional. He ruled that federal courts have no "judicially discoverable and manageable standards" for deciding when a map crosses the line. Partisan gerrymandering claims are a political question, non-justiciable. State courts and state constitutions can still act; Pennsylvania's supreme court already had. Federal doors closed.
Justice Kagan's dissent argued the efficiency gap, along with two other measures the lower courts had used, were precisely the manageable standards the majority claimed did not exist. The disagreement is methodological, not factual: both sides accepted the math worked. They disagreed about whether courts should use it.
What's contested
The efficiency gap is one of three competing numerical tests. The mean-median difference asks whether the median district's vote share lags the statewide mean by an unusual amount. The partisan symmetry test asks whether the two parties would win the same seat share if their vote totals were swapped. Each captures something real; none captures everything. A map can score well on EG and poorly on mean-median, or vice versa, especially when one party's voters cluster naturally in cities, the "unintentional gerrymander" problem that political geographers like Jonathan Rodden have documented.
Critics on the right argue EG penalises geographic reality: Democrats pack themselves into urban districts without anyone drawing a line. Critics on the left argue any threshold becomes a safe harbour: draw to 6.9% and you are immune. A deeper critique is structural. Single-member districts plus first-past-the-post will always allow some gerrymandering. Proportional systems do not have this problem because they do not have this geometry.
The simulation alternative
After Rucho, the action moved to outlier analysis. Generate thousands of computer-drawn maps that satisfy traditional criteria (contiguous, compact, equal population, respect counties). Compare the enacted map's partisan outcome to the distribution. If the enacted map sits in the 99.9th percentile of partisan advantage, the legislature did something the geography alone does not explain. Jonathan Mattingly's group at Duke used this approach in North Carolina; Wendy Cho at Illinois built the PEAR algorithm. State courts in North Carolina and Ohio have accepted simulation evidence. The federal door stays shut.
Why this has to do with other realms
Gerrymandering is a problem in concept combinatorial optimization dressed in legal clothing. The space of valid district maps for a state with 100 precincts is astronomical, which is exactly why Markov Chain Monte Carlo became the working tool: you cannot enumerate the maps, but you can random-walk through them and ask where the enacted map sits in the distribution. The same mathematical machinery shows up in concept statistical mechanics and in protein folding. A redistricting plan is a low-energy state in a constrained configuration space; the question is whether a particular state was reached by chance or by design.
An open question
If federal courts will not enforce any standard, do state-level efficiency-gap rulings drift apart enough to create fifty different definitions of a fair map? And does that fragmentation make the underlying math more useful or less?
Key Sources
- Stephanopoulos & McGhee, Partisan Gerrymandering and the Efficiency Gap, University of Chicago Law Review (2015) — the paper that defined EG and proposed the 7% threshold.
- Rucho v. Common Cause, 588 U.S. ___ (2019) — the opinion (and Kagan's dissent) that closed the federal door.
- Mattingly et al., Quantifying Gerrymandering group, Duke University — the canonical ensemble/outlier approach.
- Jonathan Rodden, Why Cities Lose (Basic Books, 2019) — the geographic-clustering argument for why Democrats waste votes without anyone drawing a line.
- Moon Duchin's Metric Geometry and Gerrymandering Group (MGGG) — bridges discrete geometry, MCMC, and redistricting litigation.
Further Reading
- Gerrymandering in America by Anthony McGann et al. — book-length treatment of the empirical record before Rucho.
- Moon Duchin's public lectures on metric geometry and political districts — accessible entry point to the MCMC approach.
- The DRA 2020 platform (districtr.org) — interactive tools the same researchers use; useful for building intuition by drawing your own map.
- Justice Kagan's Rucho dissent — short, sharp, and the cleanest summary of why the manageable-standards argument was unconvincing.
See Also
- concept combinatorial optimization — the configuration-space framing that simulation evidence relies on.
- concept arrow impossibility (why every voting system makes some unfair trade) — the deeper reason single-member districts produce these pathologies.
- concept mcmc — the algorithmic engine behind ensemble redistricting evidence.
- event rucho v common cause — the case file in more detail.
- concept proportional representation (the geometry-free alternative) — what countries that avoid this problem do instead.
Abhishek's take
What I find sharp about the EG story is that the math did its job and the institution refused to use it. Most "evidence-based policy" failures are the other way around: we have the institution but not the metric. Here, two academics handed the Court a number Roberts could have written into a footnote, and the majority chose unmanageability as a doctrine rather than a fact. Reading Rucho alongside the Mattingly ensembles, I think the next decade of American districting is going to be decided in state courts using simulation evidence the federal bench declined to touch, and the resulting patchwork is itself a fairly clean natural experiment.
Tags: #gerrymandering #redistricting #voting-theory #supreme-court #applied-math