Why Democracy Is Mathematically Impossible [video]

The examples are always artificially constrained, like 3 candidate supermarket locations around a lake, with inhabitants distributed around it.

consider 3 groups objectively each preffering:

1) A over B over C

2) B over C over A

3) C over A over B

Then 2 thirds of the population agrees A is better than B, but also that B is better than C yet also that C is better than A!

In fact the world and human incentives are much more high dimensional than the problem statement. (people could propose a rent reduction when further away from the supermarket, if not democratically then economically, so instead of a democratic issue, its housing owners bickering over the coming supermarket location, which just reveals housing is not very democratically distributed, in a democracy the prices could be agreed to depend on supermarket distance).

In fact there is only good news for democracy: by detecting preference cycles, we KNOW the kind of thing that is going on.

And we know that A,B,C is a false trilemma, and formal statements depicting rules or laws can be mathematically much much richer in structure than a simple limited list of "choices".

I value the contribution of Arrow, but at the same time one shouldn't try to understand the concept of democracy by reading the Arrow impossibility literature. The moment democracy has been reduced to simply a predetermined multiple choice form, is the moment democracy is no longer understood.

As we know, people don't have access to same set of facts and even if they do, they don't have time to reason over them. I believe if each person could stop the world, spend time on going over all the knowledge accumulated by humanity and then make a decision, democracy is possible in such a scenario. But that is not true, we are made to play many timed chess matches where we have only few microseconds to make a decision before some other issue (another chess match) calls for our decision.

Still, my guess is that these types of edge cases could be handled by using some randomness and low probability arguments. For example, claiming that such dictators exist with negligible probability given a small-ish fraction of voters that are considered "random."

While you would never want to inject random votes into an election, it's probably reasonable to assume that some fraction of voters behave naturally random (e.g., people get sick and don't vote, or are just politically fickle). This probably conflicts with the property of "unrestricted domain" but it seems like a reasonable concession in practice.

Tldr; While the mathematics do prove the impracticality of guaranteeing the socially optimal candidate in ALL situations, realistically in the majority of cases plurality can achieve a candidate with a high social utility even if they aren't the best. In fact in most average situations it can outperform Harmonic or Borda voting procceses because of the lower spread on total scores leading to more deterministic outcomes for close candidates.

When it's messy people complain that it's failing. But the messiness is really a reflection of society and/or the media coverage.

[0] https://www.youtube.com/watch?v=GVhFBujPlVo

"It is democratic for state offices to be filled by lot, and oligarchic for them to be filled by election" (Aristotle, Politics)

This idea was the majority view until the French revolution:

"The decision by lot corresponds to the nature of democracy, the decision by election to that of aristocracy." (Montesquieu, The Spirit of Law)

So, just because voting systems cannot be perfect doesn't mean democracy is mathematically impossible. In fact mathematically and statistically democracy is only possible without any elections, simply by choosing candidates by lot, see https://en.wikipedia.org/wiki/Sortition .