“Reweighted” voting methods are second-rate.

Warning: this article is very, very wonky about voting theory/mechanism design. If you’re not already familiar with this field, perhaps starting somewhere else would be best. I have written many other Medium articles that are less technical, or you can find such resources at electology.org.

Current voting methods in most English-speaking countries (US, Canada, UK, mostly India, etc.) are horrible, and are at the root of a lot of the current political dysfunction in these countries. I’ve made this argument many times, and I’m not going to repeat it here.

Fixing this, especially in the US case, involves two different kinds of elections: single-winner, such as for President/Governor/Mayor, and multi-winner, such as for national or regional legislatures.

For single-winner reform, there are various groups of advocates with different proposals. Personally, I’m in favor of approval voting as a first step, eventually moving to a rated runoff method such as Star Voting or 3–2–1. Another method that has a lot of advantages is score voting, similar to many online review systems. What all of these methods have in common is that they start with expressive ballots (unlike the current choose-one ballots) and use all the data on all the ballots to reach an outcome (unlike IRV/RCV methods). This means they all have good voter satisfaction efficiency (VSE), a measure of how happy they make voters across millions of simulated elections.

For multi-winner reform, most people agree that the goal is some form of proportional representation. But there are many voting methods that are proportional; which is best? Approval/score advocates often look for methods that have the same advantages as those single-winner methods, and thus often favor “reweighted” methods.

This is, in my opinion, a mistake.

In order to explain why, I have to explain how reweighted methods work. I’ll take a typical version of reweighted score voting as my example. In this method, voters are first divided into multiseat districts. Say your district had 5 seats — so it would be 5 times as large as a current single-seat district. Your ballot would list all the candidates running in this district. That’s probably a total of 15–25 candidates (4–5 each from 2 larger parties, 2–3 each from 2–4 smaller parties, and a smattering of independent candidates). You’d rate each of these candidates on some fixed point scale, say 0–5 stars.

Then, to find the winners, election administrators would repeatedly choose the candidate with the top total score, then “reweight” ballots so that those that supported this winner would “use up” some of their voting power. In theory, this ensures that all ballots end up using about the same amount of voting power, generally to help the candidates they like the best to win. Specifically, one formula would weight each ballot by the maximum score, divided by the maximum score plus that ballot’s total score for each winner so far. So in a 5-star system, if you’d given the 4 winners so far ratings of 5, 3, 2, and 0, your ballot would be weghted at 1/3 strength (which is 5/[5+5+3+2]=5/15). So if you gave some other candidate a rating of 3, your ballot would be contributing just 1 point to their current total when choosing the 5th seat.

(Note, there are various ways to modify this system and keep proportionality. For instance, the above is a “D’Hondt”-type method, but if you double ballot ratings when calculating weighting factors, you’d get a “Sainte-Lagüe”-type method. Or you could count the weighted points a ballot actually contributed to each winner, instead of the raw scores. These modifications probably improve the method marginally but they make it more complex and don’t change the essential flaws I’ll discuss below.)

Why is this a proportional method? Well, imagine voters fall into two parties: about 40% “yellow” and 60% “purple”. Each party has 4 candidates for a 4-seat district, and each rates all their candidates at 5 and the opponents at 0. After 2 purples and 1 yellow have been selected, the yellow ballots would be weighted at 1/2, and thus have a voting strength of 40%/2=20% of the original electorate. Similarly, the purples would be at 60%/3=20%. So the last seat would go to whichever group had a slight advantage over these numbers (or by coinflip, in the unlikely case these numbers were exact). In general, this would guarantee the result can’t be more than 1 seat away from proportionality in this kind of purely-partisan election.

But of course, in the real world, voters are motivated by more than just pure partisanship (as well they should be). I might prefer purples over yellows in general, but think some yellows are not quite as bad—perhaps worth 1 star instead of 0—and some purples are not quite as good—perhaps worth only 4 stars instead of 5. If I vote honestly, my ballot will end up getting slightly de-weighted every time a 1-star yellow candidate wins. When it comes down to the last seat, this slight loss of my purple ballot weight could let a yellow beat a purple.

So strategically, as a purple voter, I should make sure I give a 0 rating to all yellow candidates, even the less-horrible ones. Whichever party can make sure its voters follow this kind of strategy more faithfully will have a small but key advantage; arguably, an unfair one.

Furthermore, if I care not only that purple wins the most seats, but which purple candidate wins, I should make sure to give 0 to any purple candidate I don’t like, or who I like but who can win without my vote, so that my vote will keep full strength in favoring the purple candidates I like who need my vote the most. But unlike the above strategy, this hurts my party overall; if all purples vote like that, the yellows will have a huge advantage.

Thus, there are two different problems with strategic voting in reweighted voting methods. And in fact, both can be traced to the philosophy at the heart of such methods—the very same philosophy that makes score- and approval-type methods so good for single-winner elections!

When you’re choosing just one winner, it’s helpful when the voting method does its best to pay attention to all information on all the ballots. For an example of how failing to do this is bad, look at IRV, a single-winner method that counts each ballot for only one candidate at a time, ignoring all the lower rankings on that ballot until the top candidate is permanently eliminated from consideration. This can lead to prematurely eliminating centrist candidates—“center squeeze”, as happened in Burlington, 2009.

But for a proportional voting method, paying attention to the whole of each ballot is exactly what leads to the “free-rider” strategic problems discussed above. Any proportional voting method has some degree of “free-rider” strategy in certain edge cases; but in reweighted methods, such strategies appear not just to marginal candidates, but to your entire ballot, all the time.

And this isn’t just a superficial problem, but one that goes to the philosophical heart of single-winner versus multi-winner voting methods. Philosophically, I’d say that the goal of single-winner methods is to find the best compromise: the one option that maximizes total satisfaction of all voters. That’s why the utilitarian simulations in VSE are a good measure of quality for single-winner voting methods.

Shouldn’t this same utilitarian philosophy work for multi-winner voting, too? On some level, yes; but due to the fundamental paradox of mechanism design, in a more concrete sense the answer is no.

What is this fundamental paradox of voting theory/mechanism design? Here goes:

Or, more simply:

When you’re designing a voting method, don’t push too hard in one direction, or voters will push back harder in the opposite direction.

I’m far from the first person to note this paradox, but as far as I know I am the first to state it as a general law, so I guess you can call it “Quinn’s law” if you want. A typical example is when a voting method subject to spoiled elections which lead to relative extremists winning (such as choose-one or, to a lesser extent, IRV) in practice leads to two-party domination by relatively “moderate” parties.

In the case of multi-winner voting methods, it means that designing a method to take advantage of more information on each ballot can backfire, leading to strategic ballots where voters deliberately minimize the information they give about their preferences, revealing only those preferences where they think their marginal impact will be greatest. Pushing to get more information from each ballot leads strategically to getting less information from each ballot.

So even if your underlying philosophy of multi-winner mechanism design is utilitarian, I’d argue that your surface philosophy should be in terms of representation: choose winners so that each one represents a group of voters that’s as large, as equal to other groups, and as enthusiastic about that winner as possible. This leads to “allocational” proportional methods, where each time a winner is chosen, a certain fixed quota of ballots is “used up” entirely.

I’d argue that this representational philosophy ends up preserving the information from voter more faithfully than methods like reweighted score which are designed superficially to maximize information use. A good representational method will, like sortition or stratified-sortition or asset voting,¹ preserve the information in the voting population at large, even though it pays attention to only one choice per ballot.

Are there representational/allocational methods that use score or approval ballots? Indeed there are. The basic idea for this has been called “Bucklin transferrable voting”, and it’s used in voting methods such as PAD. Essentially, as with reweighted methods, you start out by choosing the winner with the highest total score, but then instead of partially deweighting all the ballots insofar as they helped that winner, you completely use up the 1 quota of ballots that helped them the most.

Reweighted methods are still proportional, and thus far better than choose-one voting (aka plurality or FPTP). But in my opinion, for the above reasons, they’re still second-rate; most other proportional methods, especially allocational methods with score ballots (discussed above) or delegated biproportional methods (not discussed above), are superior.

¹Sortition and asset voting are methods of selecting a legislature that give “perfect representation” (in some sense) by departing from certain traditional ideas of how voting and legislatures should work. Sortition means choosing legislators at random, relying on the law of large numbers to ensure that the resulting legislature will tend to have all the same preferences (that is, preserve the information) as the citizens at large. Stratified sortition means sortition that ensures certain groups (geographical, gender, ethnic, and/or partisan) are represented proportionally, with zero random deviations; it’s basically the same as sortition, with slightly less randomness. Asset voting means that each person can choose their own representative freely, and then the representatives can choose their representatives if they want; when that’s done, each rep gets a number of votes equal to the number of people they represent, so that some might have thousands of times the votes as others.

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