The Supreme Court Faces a Voting Paradox

There is a chance — not a likelihood, but a chance — that the decision the Supreme Court reaches on abortion in Dobbs v. Jackson Women’s Health Organization will be confusing, inconsistent and even paradoxical. If so, it might not be the fault of the court. It could be because of an inherent problem in voting procedures — all possible voting procedures.

The risk of a paradoxical decision is explained by a branch of economics called social choice theory, for which the economist Kenneth Arrow was awarded a Nobel in economic science in 1972. Arrow’s big idea was that no system for ranking preferences that satisfies basic criteria for consistency and fairness can guarantee that it won’t produce nonsensical results.

In Dobbs, such a voting paradox could occur. David S. Cohen, a professor at Drexel University’s Kline School of Law, recently described for me a hypothetical situation — an unlikely one, he stressed — in which it would.

The justices must answer three questions: Is Mississippi’s ban on almost all abortions after 15 weeks constitutional? Does Roe v. Wade require the ban to be found unconstitutional? And should Roe be upheld?

As Cohen noted, it is possible for most of the justices to conclude that Mississippi’s ban is constitutional and for most of them to conclude that Roe requires the ban to be found unconstitutional — positions that in combination would say that Roe must be overturned — yet for most of them to conclude that Roe must be upheld.

This diagram illustrates what that result could look like.

Credit…The New York Times

So that could be five votes for upholding Roe, even though six justices could say the ban is constitutional (the question at hand) and seven could say that Roe requires the ban to be found unconstitutional.

This outcome would be paradoxical in the sense that no individual justice could consistently hold the three positions that the court as a whole could hold.

The reason this could happen is that each cluster consists of different justices, as the illustration shows. For example, both liberals and conservatives could decide that if Roe holds, the Mississippi law must be unconstitutional. But they could draw opposite conclusions from that. For liberals, it would be an argument for tossing out the Mississippi law. For conservatives, it would be a reason for tossing out Roe.

The point here is not to predict how the court will actually rule in Dobbs. Cohen said he thinks that Roe v. Wade will likely be overturned, so there won’t be any voting paradox. Still, it’s cause for concern that it’s possible for this sort of thing to happen.

“If you got an opinion like we’re describing, you would have deep arguments among people who follow the court about whether Roe had been overturned” or just weakened, Cohen said. “How do you derive rules from Supreme Court decisions when there’s a voting paradox? Frankly, I think you would get people arguing all different sides.”

While a voting paradox in Dobbs may be unlikely, there have been documented voting paradoxes in other Supreme Court decisions, including those involving retired miners’ health benefits, state restrictions on truck length, a claim against a Cuban bank and gun rights. In an article for the Boston University Law Review in 2010, Cohen argued that voting paradoxes can arise whenever questions of precedent are at stake and thus “almost every case can result in a voting paradox.”

This issue comes up in any voting system in which voters rank their choices. One solution is to allow voters to express the strength of their preferences, not just how they rank them, but Arrow warned that it’s impossible to see inside people’s heads to compare what they mean by a given strength of preference. (In markets, as opposed to rank-order voting systems, people can reveal the strength of their preferences through how much they’re willing to pay for something.)

The Marquis de Condorcet, a French mathematician, discovered a version of the voting paradox in the 18th century. Charles Dodgson, better known as Lewis Carroll, the author of “Alice’s Adventures in Wonderland,” proposed a partial solution in the 19th century. Arrow generalized the result to all election systems in his doctoral thesis and a 1951 book, “Social Choice and Individual Values.” It’s now known as Arrow’s impossibility theorem. “Most systems are not going to work badly all of the time,” Arrow told New Scientist magazine in 2008. “All I proved is that all can work badly at times.”

The implications of Arrow’s work have troubled many political thinkers. The Stanford Encyclopedia of Philosophy says:

Maxwell Stearns, a professor at the University of Maryland Carey School of Law who has written on social choice theory and the law, argues that the Stanford entry is a bit strong. (Stearns speaks with some authority: Arrow provided a blurb for one of his books.)

The possibility of paradox “doesn’t mean that these systems aren’t legitimate,” he says. He says lower courts are practiced at interpreting Supreme Court decisions with muddled messages. The losing side, he explains, “might say, ‘I lost this round, but I can’t say that it was unfair. There will be other rounds, and I might win those.’”

The bigger problem, Stearns says, is when the rules are abruptly changed in a way that is clearly partisan. He cites the procedures in Congress that allowed President Donald Trump, who served four years, to replace three justices who died or retired over four years and seven months. It’s hard to devise a theoretical justification for that. “There’s a difference between arbitrariness and a lack of legitimacy,” Stearns says. “The problem is when you have both.”


Tell your kid who plans to become a superhero to study hard anyway, because even superheroes have to pay the bills. The British website money.co.uk calculated, with the help of the job search engine company Adzuna, the annual salaries of various Marvel characters, based on their known skills: Doctor Strange could earn the British pound sterling equivalent of $107,904 as a general practitioner; Black Widow, $76,794 as a head teacher; Hulk, $42,109 as a geneticist; and Wolverine, $33,292 as a translator. Alas, the Marvel character Thor, the website notes, “was excluded from the study due to no evident education or work experience.”

Quote of the day

“Over the past 20 years or so, rising market power has hurt, not helped, investment and output. And it has further hurt workers by giving large firms undue influence in labor markets and rewarding them with extraordinary profits.”

— Kristalina Georgieva, the managing director of the International Monetary Fund, in a speech on Dec. 6

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