The core argument for x^2 is basically just that it's the only function with linear derivative, so for rational voters who calculate marginal benefit of additional votes purchased at cost x^2 (in terms of likelihood of changing the outcome of the "election"), you get a welfare maximizing outcome.

There's some assumptions underlying the theoretical result, but there's also a lot active experimentation in the real world, a few links: https://www.radicalxchange.org/concepts/quadratic-voting/, https://en.wikipedia.org/wiki/Quadratic_voting

Here's what I believe is the key paper on the topic https://www.aeaweb.org/articles?id=10.1257/pandp.20181002 (appendix: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2790624)

Lalley, Steven P., and E. Glen Weyl. 2018. "Quadratic Voting: How Mechanism Design Can Radicalize Democracy." AEA Papers and Proceedings, 108: 33-37.

> rational voters

That's a rather ridiculous assumption, isn't it? I still remember the 3rd year psychology university student, who asked: "What's a square root?"

This is a good objection. The system might be so complicated that people don't understand how to correctly express their preferences.

Definitely a challenge for many mechanism design approaches to intervening in the world. As I understand, the only really solution is to experiment in the real world, see what holds and see what doesn't, and tweak accordingly (e.g. test out different interfaces that communicate how the mechanism works!)

Rationality expectation is good if irrationality of others does not harm rational voter.

In this case, a well-organized and mathematically adept voter can and will harm less rational ones, thus skewing the balance instead of restoring it.

No. Irrational voter harms himself. Rational voter just gets what he wanted.

That's too close to calling them lesser beings for my comfort. They harm themselves, so who cares? Greed is good!

It incentivizes the welfare-maximizing result. I don’t have a short “intuitive” explanation of why handy but the math is covered in section 8.3 of Public Choice III (should be easy to get in college libraries, pdf copies can be found online if needed).

There are a few mathematical reasons. I forget the details, but I remember that a key point is to consider marginal cost: the derivative of a quadratic is linear. I think the Central Limit Theorem is also relevant, as someone else here pointed out. Anyway, if you really want to know, you can read the papers!

Actually it's a good point and IMO for a population that may have a certain distribution of voting credits e.g. shareholders of a stock, adjusting the exponent can make sense to more fairly distribute power.