I’ve learnt about Arrow’s paradox several years ago and it didn’t show up on my radar since then. This month I ran against references to it twice within a few days via two independent sources:

  1. As an example of the Fallacy of composition (while reading through the full list of fallacies).

  2. In David Deutsh‘s book The Beginning of Infinity.

As a (theoretical) result of social science, to me this “paradox” was not as compelling as the mathematical paradoxes, so I had only looked at its formulation. Several days ago, however, I decided to study it a little bit closer.

The wikipedia article informally summarizes the paradox as follows:

In short, the theorem states that no rank-order electoral 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.

The scope of the theorem is broader than the usage of the terms “voters”, “electoral system” may suggest, since we have to admit that decision making in a single person’s mind (or algorithm) is the outcome of “voting” of multiple factors. Hence was my increased interest in this theorem (as it turned out to be less of a purely social science result).

But after reading the informal proof of this impossibility theorem, I figured out that its essence is that there exist situations where a single actor/voter determines the outcome of the entire election. In such situations that special voter is declared to be the “dictator”. (Note that the superpowered voter becomes such only due to the fixed and unchanging preferences of the rest of the voters - any of them can dethrone the dictator by changing his/her own preference; thus the absence-of-a-dictator requirement in the theorem is not actually violated).

After that revelation my first thought was that the theorem is much weaker than it is made to appear, when it states that

no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting the specified set of criteria: …

roughly claiming that something always fails, whereas it fails only in very specific situations.

But upon reflecting on it somewhat deeper I realized that in reality those special situations that seemed quite artificial may - under certain circumstances - be destined to be the ones where the system ends up after some evolution. Indeed, how come that the Brexit referendum was close to a tie? Why did Republican presidents in the USA occupy the Oval Office almost for the same duration as the Democrats (92 years vs 89)? And though those examples don’t perfectly meet the criteria of Arrow’s theorem they illustrate the important point enabling the “dictatorship of minorities”. In a struggle for the ballots of the voters the competing points of view tend to evolve to a state of equilibrium, whereupon a small minority of voters is empowered to pick the winner.

So I discovered a fallacy in myself. Something that seems to be an edge case not worth too much attention, may turn out to be much more consequential due to the processes governing the system dynamics.

The following analogy from physics came to my mind - in a potential well there may be a single local minimum (of all the infinitely many points in the configuration space), yet it is exactly the (only) point of stable static equilibrium.