Is Democracy Illusory? Of Kenneth Arrow and the Marquis de Condorcet

Kenneth Arrow. Creative Commons.

Kenneth Arrow died last week. 

Professor Kenneth Joseph Arrow of Stanford, naturally the son of immigrants (in this case Romanian Jewish ones), in his lifetime won the John Bates Clark Medal for best economist under 40, the John von Neumann Theory Prize, and, oh yes, the Nobel Prize for Economics. In fact, he remains the youngest person ever to win that particular award.

Arrow was 95.

I discovered Arrow in college, as so many other did as well. It was hard to study social science, any social science (political in my case), without coming across Arrow’s Impossibility Theorem. And for a college sophomore coming across the Theorem for the first time, the idea of it is deeply disturbing, like tremors beneath your feet where you didn’t realize there was ever a fault line.

Here’s Wikipedia’s summary of Arrow’s Impossibility Theorem:

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In short, the theorem states that no rank-order voting system can be designed that always satisfies these three “fairness” criteria:

• If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
• If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged (even if voters’ preferences between other pairs like X and Z, Y and Z, or Z and W change).
• There is no “dictator”: no single voter possesses the power to always determine the group's preference.

That’s without the mathematical derivations that Arrow went through to prove that his postulate was true and always true. The implications of the theorem are immense if not immediate obvious to a non-technical mind. What it means is that given an electorate with more than two choices and a simple preference between them, it is impossible to design a voting procedure that is “fair” and that accurately reflect “the will of the people.” Because of “the will of the people,” that vaunted concept of democracy, never exists as an ascertainable variable that cannot be arbitrarily engineered by the voting process.

The Marquis de Condorcet had actually suggested nearly as much in Enlightenment France, right about the time when James Madison was drafting the U.S. Constitution. The illustration of the Condorcet Paradox was basically this: Assume three voters, A, B, and C, and three candidates, X, Y, and Z. Now suppose A prefers X to Y to Z, B prefers Y to Z to X, and C prefers Z to X to Y. In that scenario, no procedure can fairly pick out a candidate who represents the will of the majority, because there is no will of the majority. Real life is more complicated, but not that much more complicated. In 2016, for example, presumably a subset of voters preferred Sanders to Clinton to Johnson to Trump, while another preferred Johnson to Trump to Sanders to Clinton, and so on. There never was a consensus candidate in 2016 and never could have been.

That was a disturbing thought for a 19-year-old enamored with American democracy. And it’s a disturbing thought today, when democracy worldwide seems on the retreat, when many of our fellow citizens seem not to mind very much if democracy ends, as long as their policy agenda gets enacted. For that matter, the advancements of human thought made possible by the Enlightenment of Condorcet itself seems endangered.

Without those advancements of human reason, we could never have figured out that, mathematically, democracy might be a sham.

And I fall back on Churchill. Mathematics aside, democracy remains the worst form of government excepting all of the others. And that, even without more and in the flimsiness of our lives, is worth fighting for.