Mathematically speaking, “one man, one vote” sounds about as exciting as “1 = 1.” Yet it turns out that something so simple can produce a Nobel Prize in economics, not to mention a slew of graduate school statistics homework.

“Our class spent three weeks just on Arrow’s Theorem, looking at it from different angles,” said Jameson Quinn, a Ph.D. candidate in statistics from Harvard who showed up in Concord last week to testify before the House Elections Committee in favor of a bill allowing something called approval voting.

Arrow’s Theorem, key to the aforementioned Nobel prize, is to social choice theory what E=MC2 is to physics. It is usually described as saying that all voting systems are imperfect, a synopsis which misses lots of nuance and isn’t all that helpful to laymen, because most of us don’t even known that other systems exist.

Voting means everybody casts one ballot for one person per race, and the candidate who gets the most votes wins, right?

Not necessarily. Different methods of voting have been proposed since the 13th century, and ever since then they have generated impassioned debate that sees people wrestling with distinctions between ordinality and cardinality in postulates as well as counter-intuitive results like the monotonicity problem, in which voting for a candidate harms their chances.

All such systems would allow voters to choose more than one candidate at a time for any race, which might come in handy this year for indecisive primary voters, in hopes of better reflecting the overall wishes of the electorate. (They are distinct from other voting-related issues such as the presidential electoral system or how to establish voter registration.)

Approval voting, the method being sought in New Hampshire, counts all the resulting votes equally. Other systems, called ranked-choice, let us rank our votes on the ballot and different ranked-choice systems analyze the results in different ways.

That analysis is where the mathematics shows up. If you have fond math-class memories of manipulating matrices – two-dimensional grids of numbers – then social choice theory is for you.

The analysis is also where Arrow’s theorem shows up, as does the related Gibbard–Satterthwaite theorem. These theorems show mathematically that no voting system is perfect, in the sense of optimally reflecting group preferences in all situations, although some may be better than others depending on your definition of “better.”

A few ranked-choice systems actually exist in the wild. Cambridge, Mass., uses instant-runoff to choose its city council, and Burlington, Vt., did the same for its mayoral races for a few years, while in Maine there’s an initiative to allow it in statewide voting.

Approval voting doesn’t seem to exist in any federal races, although it is used in elections for some organizations.

In the New Hampshire State House, alternative voting has been discussed since at least 1979, when Concord native Steven Brams testified in favor of a bill that would allow the state to adopt approval voting. It died in committee.

A few attempts have been made over the years to introduce ranked-choice systems into the state, but none of them have succeeded, either.

Brams, a professor of politics at New York University, is still an advocate, having written a couple of books on the topic. He’s psyched that the issue has come up before the legislature again.

So is the Center for Election Science, a D.C.-based think tank that advocates for approval voting because it “recognizes the peril of our choose-one plurality system” and sent a representative to watch last week’s hearing at the House Elections Committee.

And so is Quinn, who is getting his doctorate in statistics partly because he wanted to understand how to quantify choices among election systems. That’s how he came to be testifying in Concord.

“We need to be able to resolve that question in some rational manner,” he said in a interview after his testimony.

As part of his research Quinn created computer models of election scenarios and ran online tests with 672 volunteers, pitting various voting methods against each other to see what happened.

The work isn’t finished yet, so we can’t grub around in the details, but Quinn says it seems to support one of the main arguments of alternative voting advocates: that they filter out small but very vocal support for fringe candidates, better reflecting the views of the silent, or at least out-shouted, majority.

“This leads me to various sub questions. Within the utilitarian framework we have to figure out: what are the chances of different kinds of election scenarios, how would actual human beings behave in those scenarios and under different election rules, and how does it work out. None of those questions in itself is simple,” said Quinn.

Indeed not. This is why I personally think that alternative voting methods will never catch on. They just seem too complicated.

The current system of plurality voting (whoever gets the plurality of votes wins, even if not the majority) has huge problems and might be shown mathematically to be inferior to other systems, but it’s easy to understand. It feels fair, no matter what the matrices say.

As evidence, look to Lake Champlain. In 2005, Burlington voters installed a system called instant-runoff for their mayoral races. They were perfectly happy as long as the person who would have won with plurality voting also won with instant-runoff. In 2010 that didn’t happen – the second-place finisher by plurality-vote standards ended up winning – and they promptly went back to what most of us think of as “real” voting.

If I was a betting man, I’d bet that the House Elections Committee will once again table the bill for approval voting – and even if it doesn’t, the full House and Senate will never support it. Opposition from the Secretary of State’s elections division almost guarantees that result.

Nonetheless, Quinn and other advocates aren’t giving up.

“The statistical point of view that I’m taking does still work. . . . Even if you can’t prove that’s something is perfect, you can still ask, is it better? And how much better, and is that worth whatever trade-offs you’re making? . . . Those are interesting questions,” he said.

Far more interesting, it turns out, than 1=1.

Science Cafe Concord

On Feb. 2, Science Cafe Concord will hold a discussion titled “Bitcoin and Beyond,” asking whether the digital currency bitcoin is the future of finance or just a geeky trend. It will also look at uses of its underlying technology, called blockchain, which some say is more important than bitcoin itself.

The free session, open to all, starts at 6 p.m. at The Draft Sports Bar, 67 S. Main St., Concord.

For details, check the website

(David Brooks can be reached at 369-3313,, or on Twitter @GraniteGeek.)

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