This is new to me, but apparently the history of this traces back to Lewis Carroll, the mathematician Lionel Penrose and is recently getting attention from political scientists. And spawned a Polish rallying cry, "the square root or death," in 2007.

This is complicated and I do not fully understand the math, but The Economist has a demo that is not behind a paywall that makes this more intuitive.

https://www.economist.com/interactive/2021/12/18/quadratic-votingThey have an article that runs through the history, math and some recent uses of quadratic voting that is interesting, but is behind a paywall.

https://www.economist.com/christmas-specials/2021/12/18/the-mathematical-method-that-could-offer-a-fairer-way-to-voteHere's my best attempt at a summary....

There are a lot of voting schemes. We are most familiar with the scheme where each person gets one vote and the person/issue/item with the most votes takes the day.

The problem with that is let's say 13 of us are voting on paint colors and the choices are tan, gray and pink. Pink gets 5 votes. Tan and grey get 4 each. All the tan and grey voters hate pink. The pink voters are kind of okay with tan and grey, but have a slight preference for pink. The result is a color that five people kind of prefer and eight people hate. That's how US elections work, especially primaries.

There are similar problems with one state one vote (or two votes actually) in the case of the US Senate or one country one vote in the UN where votes are not weighted by population.

Why not weight votes proportionally by population, like the US does in the electoral college in most cases (we don't count the US House because those votes can be split and do not vote as a block on a state-by-state basis)?

It turns out that isn't fair either. The math on this is a bit complicated, but the key point is that the influence of each vote decreases not with the number of votes, but as the square root of the number of votes. This explains why someone can win the popular vote and lose the election in a system like this.

If you have an election with three total voters, it turns out that in 50% of the cases, an individual vote can sway the election. So the "power" of that vote is 50%, meaning that one vote has the power to decide 50% of elections. What if you triple the size of the committee? You might expect that with 3x the number of voters, the power of any given vote would be 1/3 as great, roughly 17%. In fact, because more voters mean more combinations that yield one-vote margins, that ninth vote can decide the election in 140 out of 512 possible combinations, so 27.3%

If you expand the voting pool based on population, the big population gets undue influence. That's because roughly speaking, the power of an additional vote is (# votes)/(sqrt(# votes)). So when you increase the numerator linearly, you're effectively giving more power to that big proportional block.

The Poles got incensed over this in 2007, feeling like their votes would be outweighed by the Germans, and were spouting the slogan, "Square root or death."

Penrose’s idea reappeared when the eu mulled reforms to its voting rules in 2007. Poland worried that big countries like Germany would have too much sway. So it adopted an odd slogan for the summit: “the square root or death”. The phrase “neatly combines obscurity, absurdity and vehemence,” as Gideon Rachman of the Financial Times (and formerly of The Economist) put it, “capturing the spirit of the modern eu”.

I definitely missed that at the time.

So one idea is

**quadratic voting** which gives people an allotment of "vote credits" and each additional vote scales as a square. So if you want cast 1 vote, that's 1 credit. Two votes? That's four credits. Ten votes? That's your whole 100-credit allotment. Fair enough, but that last vote, #10, costs you 100-9^2 = 19 votes. You could, instead, cast 4 votes for your #2 choice and still have three votes to spread around (9^2 + 4^2 + 3 = 81 + 16 + 1 + 1 + 1 = 100).

It turns out that if you reran the Brexit vote this way, you probably end with a "soft Brexit" instead of the "hard Brexit" result that actually came to pass. You also probably end up with John Kasich as the nominee in the 2016 Republican primary.

Most importantly, it ends up being a moderating influence. Let's say nobody picks Kasich as their first choice, but everyone picks him as their second choice. Also, let's assume all Trump voters hate Cruz and Bush. All Cruz voters hate Trump and Bush. And so forth. But they all are kind of okay with Kasich. Don't love him, but don't hate him either. And finally, let's assume that the Trump, Cruz and Bush voters are evenly split.

They all give 9 votes to Trump or Bush or Cruz and they all give 4 votes to Kasich. The result is Kasich wins with 12 and everyone else ties with 9. So in this scenario in our current system, 66.666% get someone they hate and 33.3333% get someone they love. In the quadratic voting system, nobody gets someone they love, nobody gets someone they hate and everyone gets someone they kind of like. It also tends to balance out extremists and lead to a more moderating result.

All very complicated and probably next to impossible to explain to people and sell to them, but if you actually participate in a simulated election, you can see it works very well (or at least it does for me). It does indeed tend to spread my votes out more evenly across priorities than being forced to vote for only one or than getting 100 credits worth 1 vote each.

Anyway, I had never heard of this even though the idea has been kicking around for longer than I have. Found it interesting.