Musing on Condorcet’s Paradox

Paradoxes make good brain candy, in my opinion. As a rough approximation, statements can be deemed a paradox when they provoke the reaction “That can’t be right but I also don’t see how it can be wrong.” W. V. Quine once described how paradoxes can be put into three different categories – veridical, falsidical, and antinomy.

A paradox is veridical when it seems impossible or contradictory but turns out to be true when properly understood. Statistics is full of these kinds of paradoxes. One example is Simpson’s Paradox which, as Steven Landsburg has explained, can lead to the seemingly paradoxical situation where you can see “median income shoot up in every demographic sector while the overall median remains nearly unchanged”. Russ Roberts has also skillfully unpacked this paradox as well.

The second category is a falsidical paradox. This is a when the paradox is dissolved by understanding why it is actually false. A classic example is Zeno’s argument for why Achilles can never catch up to a tortoise in a race. We instinctively understand the argument must be mistaken – indeed, we know it’s mistaken because we see the supposedly impossible outcome occur in reality all the time. But for centuries nobody could explain what the error was, until mathematicians proved an infinite series can sum up to a finite total.

The third type of paradox is antinomy, which is the category of “wait, are logic and reality broken?” These are paradoxes that remain unsolved. To resolve a paradox is to move it out of antinomy and into the veridical or falsidical spaces.

Lately, I’ve been pondering Condorcet’s paradox in this framework. Very briefly, this describes when majority rule produces a rock-paper-scissors style of result. Individually, I might prefer waffles over French toast, and French toast over pancakes. These preferences will be transitive – which that since I like waffles more than French toast and I like French toast more than pancakes, I also like waffles more than pancakes. But voting doesn’t necessarily produce transitive outcomes. Among these three options, individuals can rank them in a different order, and voting by majority rule can produce results where waffles beat French toast, French toast beats pancakes, and pancakes beat waffles.

This outcome seems paradoxical. When the majority prefers waffles, the majority prefers pancakes. But when the majority prefers pancakes, the majority prefers French toast. However, when the majority prefers French toast, the majority prefers waffles. How can we use democracy to discern and carry out the will of the majority, when what the majority wants is simultaneously not what the majority wants? We seem stuck in a contradiction.

One way to resolve the seeming contradiction by recognizing there is a fallacy of equivocation. The fallacy of equivocation is when the same word is used throughout an argument but refers to different things at different points in the argument. In this case, we’re using the same word – “the majority” – to refer to partially overlapping but different collections of individuals. Let’s say there are three hundred voters in total. The “majority” that picks waffles is not the same “majority” that picks for French toast, and so on. What we actually have are three distinct “majorities” – with apologies to any monetary economists reading this, we can call them M1, M2, and M3. M1 consists of the first and second hundred voters, M2 is made up of the second and third hundred voters, and M3 is the first and third hundred. Now we can reformat our previously paradoxical statements as “M1 prefers waffles, M2 prefers French toast, and M3 prefers pancakes.”

This approach might seem to resolve the logical contradiction, but there are still practical problems on the table. Democracy, we are often told, is justified because it makes the government accountable to the majority. But M1, M2, and M3 can all with equal legitimacy claim to be “the majority”, and they all want incompatible outcomes. How do we resolve this problem, in a way that preserves democratic legitimacy?

The answer to this, in my opinion, is to recognize that the very question is falsidical. To speak of “the will of the majority” or “what we as a society have decided” is to commit a category error. It makes the mistake of applying concepts like decisions, desires, will, and so forth, to a category where they simply don’t apply. It treats “the majority” is though it were a thing that has an active and independent existence, as though “majorities” or “societies” are the kinds of things that are capable of having desires or making choices. But the statement “M1 prefers waffles” is meaningless.

This is because a majority is not a thing that exists. Or, to phrase the point more precisely, the majority does not exist as a thing. A majority is simply a relative form of measurement, not an entity with a real existence that’s capable of having preferences or making choices. “The majority” is real and exists in the same way and in the same sense that velocity exists. That is, things have velocity, but there’s nothing that is velocity. Similarly, a certain number of people or things in various schemas can be defined as “the majority,” but there’s nothing that is “the majority.” To use phrases like “the will of the majority” or “as a society, we have decided such and such” is as fundamentally incoherent as saying “for this experiment, I’m going to need three pounds of velocity.”

And that’s how I see Condorcet’s Paradox. Instead of wondering how it can be the case that what the majority wants is also not what the majority wants or wondering how a democracy can respect “the will of the majority” when there are multiple, equal majorities with incompatible “wills”, we would do better to recognize that the whole premise is ill-formed. Hayek once said, “We shall not grow wiser before we learn that much that we have done was very foolish.” Endlessly searching under the rainbow for “the will of the majority” is a foolish task – because there never was such a thing to start with.