The most-comprehensive argument that motivates its existence, I've found in Glen Weyl and Eric Posner's book "Radical Markets."
(And by "any reason" I guess I mean "any reason besides lack of comfort with non-whole numbers").
From here it seems like the general principle of social cost still applies / doesn't have anything to do with the step-function nature of things.
In Weyl and Posner's book "Radical Markets" they motivate quadratic voting as outlined as a way of preference voting where a voter's desire intensity is represented in the vote.
Their argument is practically outlined in the pollution + city example I make in the original blog post and where, with the sloppy figure I have attached [1], they reason geometrically that "Nils" should "pay" for reducing the electricity plant's pollution. If you look closely at that figure (which is nicer in the book), you can see that the plane where the cost functions intersect is a triangle.
I can't fully reproduce the mathematical reasoning here, but how I understood it is that dependent on Nils's choice, the externality he imposes has quadratic cost given that the triangle's change is affected quadratically. In an isosceles triangle, where volume V is defined dependent on two same length sides, I can see that work out. But I've never worked through that.
More intuitively, an example, I'm just making up is if you're annoyingly snorting during a visit to the cinema.
If you're the only person snorting and you think that's OK, but nobody else can understand the dialogue - you're not only wasting your money but everybody else's. It's quadratic because of the opportunity cost of everybody wasting 2 hours but having all paid for the film. So both the cost of the cinema running the movie for two hours and everybody paying but not understanding makes your snorting square the cost. Does that make sense? Please correct me if I'm wrong.
So essentially, if Nils is picky with anti-pollution, his pickiness costs the town's society quadratically - which is why he should pay the quadratic cost of his pickiness during governance.
references:
- 1: https://timdaub.github.io/assets/images/cost-of-externality....
I'm curious but skeptical. I would have thought that if there is a good and straightforward justification, it would be in the wikipedia page, but it's not. It simply says:
Quadratic voting is a variant of cumulative voting in the class of cardinal voting. It differs from cumulative voting by altering "the cost" and "the vote" relation from linear to quadratic.
Apart from that there's no comparison to cumulative voting, which seems a lot simpler and more robust against concerns like collusion and sybil attacks. The only other mention of "linear" is in a rather opaque paragraph marked "citation needed".
If it's purely a way to allow people to buy votes without having rich people completely corner the market, I guess it makes some sense, but it still seems rather wrongheaded. If you're going to give everybody the same number of virtual tokens, there are no rich people to begin with. If you insist on using an actual currency, there are other possible mechanisms like a universal basic income.
Edit to add: Aha, I've found a (the?) paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2343956
Rational agents maximize their utility by setting marginal cost equal to marginal benefit. This means that if John values being able to incrementally move the outcome in his favor twice as much as Sue values being able to incrementally move the outcome in her favor, John will pay twice as much at the margin as Sue does. For example, John buys 16 votes while Sue buys 8 votes. The exact number of votes that John and Sue buy depends on their estimates of how likely they will be pivotal voters, as explained below, so if John buys 16 votes for $256 (162 ), this does not mean that he values the project at $256. But it does mean that he values the project twice as much as Sue, who buys 8 votes.
This emphasis on marginal cost seems to ignore the fact that John and Sue will be working with finite budgets -- the overall real cost of the vote is important too! Unless I misunderstand, the implication is that if the cost were linear, John would buy an unbounded number of votes (as the marginal cost never goes up) which seems absurd.
Particularly in the cases where quadratic voting is actually being used in government, when e.g. lawmakers are given 100 tokens to split among different possibilities, I don't think any of the above analysis applies at all, and it's not clear that quadratic voting has any advantages over cumulative voting.
https://www.economist.com/interactive/2021/12/18/quadratic-v...
I don't think that quadratic voting is relevant for the case of "Many people need to answer a few shared questions" as it happens it elections. I believe it is for the case "one person needs to answer many questions" as it happens in polling.
Of course it can be used in elections as a sort of preference voting, but its effect is to punish extreme opinions (you might be extremely opposed to Statement A but might also want to keep some credits to express your milder opinion about B, C, and D.
(of course this also opens more venues for manipulating polling by adding duplicated similar questions...)
Unfortunately the linked article is paywalled.
The motivating example in the QV papers I've just skimmed is more about people voting on a sequence of yes-or-no questions, buying some number of votes and being compensated afterwards each time. That's totally different from this one-shot, multiple-questions, fixed-budget example.
references:
- 1: https://www.aeaweb.org/articles?id=10.1257/pandp.20181002
The use of QV in voting scenarios where you have 100 tokens to split among alternatives seems completely motiveless, beyond “QV is great because there’s a mathematical proof.” But a proof of what? Not a proof of anything relating to that voting scheme, as far as I can see.