The Math Problem That Went to Court
How the gerrymandering puzzle became a trap
Bernard Grofman and Gary King were assigned homework by the Supreme Court of the United States. Naturally they determined to complete the assignment.
Since 1986, the Court had held that partisan gerrymandering could, in theory, violate the Constitution.1 But it had never come close to invalidating a partisan map. The obstacle was the absence of a clear, objective standard. Evaluating a 2003 Texas gerrymander, the Court maintained the same position.
But hidden in the details was a slight shift. Four Justices fixed on a way to measure gerrymandering, a metric called partisan symmetry.
A fifth Justice, the swing vote, remained unconvinced. The problem, wrote Justice Kennedy, was it lacked a legal threshold—how much asymmetry is too much? Second, measuring symmetry requires simulating hypothetical elections. Kennedy would not invalidate a map based on “a hypothetical state of affairs.”
Still, Kennedy did not “altogether discount” symmetry. Solve these problems and he’d reconsider.
Bernard Grofman and Gary King, among the two most prolific political scientists in the country, had helped develop the metric. Victory was one Justice away, and Kennedy had just issued the challenge.
It was their move.
Courts can enforce speed limits because most people agree: speed is distance over time. Anyone who hoped to cure gerrymandering would first need to define it, and Grofman and King had been trying for decades.
The key concept is symmetry, also called bias. It asks whether the same votes produce the same seats for both parties. If Democrats win 7 of 10 seats with 60% of votes, but Republicans would win only 6 seats with 60%, the system is biased.
Grofman proposed measuring bias across a range of plausible outcomes.2 But that required knowing how many seats a party would win under vote shares that had not actually occurred. If Democrats win 55% of the vote and 6 of 10 seats, how do we know how many seats they would win with 54% of votes? Or 40%?
For years, political scientists used a shortcut, shifting every district by the same number of points and seeing which seats flip. This assumes that any statewide shift would hit all districts equally, a “uniform swing.”
Gary King, working with a colleague, relaxed that assumption. Instead of using uniform shifts, they estimated how each district would move under different statewide outcomes.3
For decades they labored in the academic wilderness. Over twenty-five years, Grofman published 10 papers on partisan symmetry; King published 11.
Now, finally, they had the attention of the highest court of the land.
In The Rights Revolution, Charles Epp argues that judicial change depends on people like Grofman and King.
For 125 years after the Bill of Rights, and 50 years after the 14th Amendment, the Court “largely ignored civil liberties.” That changed around 1917, two decades before the Court’s liberal shift. What caused the change, Epp contends, was the infrastructure around the Court.
Before 1915, only large businesses could sustain prolonged constitutional litigation. With the NAACP, ACLU, and AJC, among others, civil-liberties litigants began developing comparable capacity.
Around the same time, legal education shifted from apprenticeship to law schools, fostering a more critical legal culture. Professors developed new constitutional theories, the private bar grew more sophisticated, and civil liberties organizations increasingly drew on these networks.
The U.S. rights revolution, in short, “developed within a broader political economy of litigation.” Scholars helped breathe life into Bill of Rights.
In 2007, Grofman and King turned in the assignment. It was their first article together.
Partisan Symmetry as a Judicial Test for Gerrymandering is not typical scholarship.4 It doesn’t say anything new. It reads like an open letter to the Supreme Court, a memo to a law clerk.
The coauthors make their best case for partisan symmetry. There is, they declare, an academic consensus: “We are aware of no published disagreement or even clear misunderstanding in the scholarly community about partisan symmetry as a standard for partisan fairness.”
They explain concepts patiently, for the quantitatively disinclined, then address Kennedy’s objections one by one.
Addressing the where-to-draw-the-line problem, Grofman and King offer Kennedy five possible thresholds.
On hypotheticals, the authors explain that partisan symmetry requires estimating the relative vote share across districts. Gerrymandering itself depends on the same exercise. Mapmakers pack opposing voters into landslide districts and spread their own voters across winnable ones. This requires knowing the partisan lean of different parts of the state. If this was unknowable, partisan gerrymandering itself would be impossible.
They end in hope. “The Supreme Court has seeded the clouds after this 20 year drought. We think that, at last, there can now be a non-trivial probability of rain, and as a result to a new flowering of American democracy.” Non-trivial probabilities aside, it’s a nice line.
Then the academic consensus cracked.
There was precedent for what Grofman and King wanted—a generation earlier, the Supreme Court had remade American democracy.
Until 1964, legislative districts were hugely unequal in population. A 38-person town in Vermont had the same representation as Burlington.
Within days of Reynolds, Michigan drew new districts. More than half the state’s senators retired or lost their primaries. Within two years, nearly every state had followed. Los Angeles went from one senator to fourteen.
The Court accomplished this with what now looks like a simple rule: legislative districts must be “as nearly of equal population as is practicable.” One person, one vote.
But the simplicity is retrospective.
For one, population equality competes with other theories of representation. American institutions, Alexander Bickel argued, historically represented “not only people, but interests, groups, and regions.”5
Even if votes should carry equal weight, one-person-one-vote does not guarantee fair representation. States might respond with at-large elections, allowing a bare statewide majority to capture every seat.6
Scholars canvassed many possible standards. Courts could cap the ratio between the largest and smallest district, the maximum or average deviation from an ideal district, or the percent of the population able to elect a majority.7
One person, one vote—population equality—was the most radical possible outcome. Even supporters thought it unlikely the court would constitutionalize the principle.8
But that’s exactly what the court did.
Grofman and King were not the only scholars who rose to the challenge. Eric McGhee and Nicholas Stephanopoulos fashioned a new metric, custom-built to meet Kennedy’s criteria.
McGhee criticized partisan symmetry for putting weight on implausible outcomes. If Democrats earn 60% of votes and every seat in San Francisco, it doesn’t matter that Republicans would theoretically achieve the same outcome if they somehow won 60%. They never would.9
Stephanopoulos and McGhee introduced the efficiency gap. It compares the votes “wasted” by each party. A “wasted” vote is one either cast for a losing candidate, or every surplus vote, over 50%, cast for a winning candidate. In a 60-40 district, the winner wastes 10% of the total vote and the loser wastes 40%. The efficiency gap is intuitive, requires no hypotheticals, and looks administrable.
Then came the objections. Two mathematicians observed that the metric “fetishizes” 75-25 districts but penalizes proportional outcomes.10 Three economists predicted EG would reduce competition and increase polarization.11 A political scientist argued that surplus winning votes are not equivalent to losing votes.12
The efficiency gap was one of many metrics that emerged around this time. A pair of political scientists proposed repurposing an old statistical measure of skew.13 A mathematician suggested borrowing from angular geometry.14 A redistricting workshop for mathematicians attracted 1,000 applicants.15
Each new metric arrived as a possible solution to Kennedy’s challenge, and also underscored his point; there is no singular answer.
Can courts bring about social change? If not, there would be little point to the whole circus. But there’s a growing band of skeptics.
Gerald Rosenberg posed the question in The Hollow Hope, and answers a faintly qualified “no.” His central target is Brown, his thesis captured by a two-word epigraph: “nothing happened.”16 Ten years after Brown, only 1.2% of Black children in the South were attending desegregated schools.17 Southern states effectively resisted, closing public schools and funding private segregated schools.18
“Brown and its progeny,” Rosenberg concludes, “stand for the proposition that courts are impotent to produce significant social reform.”19
Michael Klarman expands the argument—courts rarely stand up for rights when it’s unpopular.20 Together, Klarman and Rosenberg question the Court’s capacity to act against prevailing opinion.
Mark Tushnet makes the normative leap: the Supreme Court should not have the final say in what the Constitution means. Political institutions should settle their own boundaries.21
The cost of litigation-based reform, Tushnet says, is deeper than popular backlash. Public debate adopts legal vocabulary. Discussion of principle becomes circuitous, doctrinal. Resources—money, elite attention—funnel into litigation, at the expense of other channels of change.
The moment arrives.
A typical story, a partisan gerrymander. Wisconsin.
The case is brought on behalf of twelve voters, represented by the Campaign Legal Center and co-counsel, including one Nicholas Stephanopoulos, co-framer of the efficiency gap.
The district court declares the map unconstitutional. The efficiency gap is central to its analysis.
The Supreme Court agrees to review the case. Now everyone is paying attention.
Eric McGhee files an amicus brief for the efficiency gap. Gary King urges the Court to rely on partisan symmetry. Bernard Grofman files separately, urging a framework of symmetry, responsiveness, and causation.
They are few among many. King’s brief is joined by the soon-to-be Dean of Yale Law School. Eight political scientists, in two briefs, assess whether the bias could be explained by geography. It cannot. Historians argue gerrymandering has long been denounced in constitutional language.
John McCain and Sheldon Whitehouse file a brief. John Danforth, Bob Dole, John Kasich, Richard Lugar, and Arnold Schwarzenegger file a brief.
In all, thirty-six political scientists, ninety-five law professors, fifteen historians, and one-hundred-fifty-four politicians join or file briefs.
It’s thirty-five years after Grofman published his first article on symmetry, eleven years after Grofman and King predicted a non-trivial probability of rain, four years after Stephanopoulos and McGhee introduced the efficiency gap.
The support structure has assembled.
It’s a punt.
Chief Justice Roberts writes for a unanimous court. The plaintiffs have not shown the right type of injury. The Court does not consider whether gerrymandering is constitutional, whether any metric can prove it.
It has been the greatest honor and privilege to serve our nation in the federal judiciary for 43 years.22 With these words, Justice Kennedy announced his retirement from the Supreme Court of the United States, effective July 31, 2018. Twelve days later, President Trump nominated his replacement, Brett Kavanaugh.
If Grofman, King, McGhee, Stephanopoulos, and all the others were wondering Kavanaugh’s position on gerrymandering, they were not left guessing for long. Rucho was decided in Kavanaugh’s first Term.
The door was slammed shut.
“There are no legal standards discernible in the Constitution for making such judgments,” wrote the Chief Justice, “let alone limited and precise standards that are clear, manageable, and politically neutral.”
And that, as they say, was that.
Reviewing studies on the effects of Reynolds—one person, one vote—Gerald Rosenberg, in Hollow Hope, concludes “any [policy] effects that can be traced to reapportionment are small.”23
More recent scholarship suggests otherwise. When suburbs and cities received more representation, they secured significantly more government funding.24 More importantly still, Reynolds weakened the Southern Democratic party and helped create space for Southern Republicans.25 It was among the key ingredients in America’s partisan realignment.
Fixing gerrymandering through courts was not inevitably doomed to fail.
Had the Court chosen to intervene, it would probably have been effective. Gerrymandering is more similar to Reynolds than Brown; enforcement would not require mass cooperation or social change.
But aren’t metrics flawed? Yes; certainly, partisan symmetry and the efficiency gap both carry defects. Other metrics, especially those that rely on simulations, are probably better. Regardless, any metric would restrict the degrees of freedom partisan mapmakers can exercise.
Wouldn’t the Court have to draw an arbitrary line? Courts draw them for a living. “Beyond a reasonable doubt,” “probable cause,” “undue burden”—as arbitrary as they are ambiguous. “Where do you draw the line?” is so commonly asked by law professors that one of them termed an acronym: WDYDTL.26
How could Grofman and King have known that the Court would punt, that Kennedy would then be replaced by a less-receptive Justice?
Who could blame them for trying?
And yet I wonder if Grofman and King, if McGhee and Stephanopoulos, would find some truth in Tushnet’s frame. That in judicializing rights, the academic dialogue came to fixate on the perfect metric.
Ending gerrymandering doesn’t require a metric.
The Supreme Court could adopt a presumption that any partisan map—any map drawn by one party, without the consent of the other—is unconstitutional. Or the Court could forbid any “self-interested insiders” in participating in the districting process.27 Neither requires a metric.
Congress could address the problem in all sorts of ways. It could require maps to be drawn by nonpartisan commissions, as did a bill the House passed in 2019. It could require multi-member districts with proportional elections, or simply statewide proportional representation.28
There is something seductive about Kennedy’s challenge. How do we measure gerrymandering without hypotheticals? How do we flatten a many-dimensional object into something easy to understand and apply?
It’s a neat little puzzle. Trivial to understand, excruciating to solve. On the other side, a fairer electoral system.
A hard math problem that matters—or so they hoped.
Davis v. Bandemer (1986)
Bernard Grofman, Measures of Bias and Proportionality in Seats-Votes Relationships (1983)
Andrew Gelman & Gary King, A Unified Method of Evaluating Electoral Systems and Redistricting Plans (1994)
Grofman & King, The Future of Partisan Symmetry as a Judicial Test for Partisan Gerrymandering after LULAC v. Perry (2007)
Alexander Bickel, The Durability of Colegrove v. Green (1962).
Phil C. Neal, Baker v. Carr: Politics in Search of Law (1962); Colegrove v. Green, 328 U.S. 549, 551-556 (1946) (Rutledge concurring)
Arthur L. Goldberg, The Statistics of Malapportionment (1962)
Robert B. McKay, Political Thickets and Crazy Quilts: Reapportionment and Equal Protection (1963)
McGhee, Measuring Partisan Bias in Single-Member District Electoral Systems (2014)
Bernstein & Duchin, A Formula Goes to Court (2017). More generally, EG demands a particular relationship between seats and votes that a proportional map does not satisfy.
Chambers, Miller & Sobel, Flaws in the Efficiency Gap (2017)
Cho, Measuring Partisan Fairness (2017)
McDonald & Best, A Diagnostic Applied to Six Cases (2015). The Mean-Median Difference is the gap median and mean vote share. If median >> mean, Party A is losing a few districts by large margins, which is what you would expect from a successful gerrymander. E.g., if Party A’s mean vote share is 40% and its median district vote share is 50%, Party A is winning around half of districts with two-fifths of the vote.
Warrington, Quantifying Gerrymandering Using the Vote Distribution (2018). A disadvantaged party may lose many close races and win a few landslides. The metric captures this by sorting a party’s district vote shares, splitting wins and losses, finding the center of each group, and comparing the angles from those centers to the 50% threshold line. If the losing center is close to 50% but the winning center is close to 100%, the party is losing narrowly while winning overwhelmingly, which suggests systemic disadvantage.
https://www.quantamagazine.org/the-mathematics-behind-gerrymandering-20170404/
Gerald Rosenberg, The Hollow Hope 52 (1991). Emphasis original.
Id. at 52.
Id. at 79 – 80.
Id. at 71.
Michael Klarman, Rethinking the Civil Rights and Civil Liberties Revolutions 13 (1996)
Mark Tushnet, Taking the Constitution Away from the Courts 99-102 (1999).
https://www.supremecourt.gov/publicinfo/press/pressreleases/pr_06-27-18
Gerald Rosenberg, The Hollow Hope 297 (1991).
Stephen Ansolabehere, Alan Gerber & James M. Snyder Jr., Equal Votes, Equal Money: Court-Ordered Redistricting and Public Expenditures in the American States (2002).
Stephen Ansolabehere & James M. Snyder Jr., Reapportionment and Party Realignment in the American States (2004).
Orin Kerr, Line-Drawing (2020)
Samuel Issacharoff, Gerrymandering and Political Cartels (2002)
Lee Drutman & Scott Mainwaring, PR and Presidentialism: Yes, We Can (2023)