Further thoughts on measuring multiwinner voting system quality
This is #2 in a series of articles about developing quality measures for voting methods — measures that, I argue, will demonstrate that most proportional methods are clearly superior to any nonproportional one. In my recent article, “Who put you in charge?”, I sketched a way to measure an individual voter’s retrospective power to have elected an individual winner. In this article, I’ll:
- Sketch out how to turn that measure for individual voters and candidates into a measure of voter power equality for the entire election.
- Explain how to extend this aggregated equality measure for voting methods that are not deterministic.
- Point out problems with this measure in terms of measuring power to elect individual candidates versus power to elect a party as a whole; suggest some ideas as to how to deal with these.
- Discuss a few things that are missing from this measure, beginning with voter satisfaction. Briefly give one possible rigorous definition for “voter satisfaction”, and suggest how we might begin to estimate it in practical situations, and how we might combine that with the equality measure.
- Mention some other, less-quantifiable qualities of voting methods.
- Briefly touch on why a combined measure of equality and satisfaction is key; that is, how low scores low on this measure connect to undemocratic legislative outcomes.
Combining the individual measure into “percent unwasted votes”
As I said, I’ve already explained a way to divide up the responsibility for having elected a given winner among the voters. With the algorithm I gave, the total across all voters of responsibility for a given winner is a constant; we will arbitrarily scale this so that the average voter has responsibility V/W (the number of voters divided by the number of winners) for a given candidate.
The idea, then, is to measure the total retrospective power of each voter; that is, to add up their retrospective power across all winners. A voting method would be more democratic insofar as that total retrospective power was equal across voters. We’ve scaled this so that, by construction, the average voter will have total responsibility 1; so the optimal situation is that they’re all very close to 1, while the worst case is that there is one voter with responsibility V.
To convert this into an overall measure of “percent unwasted votes”, we’ll take the reciprocal of the average squared total responsibility:
Here, ℬ is the set of ballots (that is, the election); w(ℬ) is the set of winners for those ballots; V is the number of voters; and p(i, C, ℬ) is the retrospective power of voter i to have elected candidate C in election ℬ.
Why do I call this the percent unwasted votes? Well, suppose that there are just two kinds of voters: the “winners” who successfully elected someone, making up a fraction w of the electorate, and those whose votes were wasted, making up a fraction 1-w. The power of an individual “winner” voter will be 1/w; the average squared power will be w(1/w²)=1/w; so the reciprocal of the average squared power will just be w, which is indeed the percent unwasted votes. Obviously, this gets more complicated when the unwasted voters don’t all have equal power, so in that case the name is only an approximate indication of what’s going on; but I think it’s still a good way to talk about this quantity and get an intuition about it.
There are two other ways to look at this which are also instructive:
- Algebraically, the mean square total power is just the variance of voter power plus 1. So its reciprocal will always be 1 or below, and it will approach 1 (aka 100%) only when the variance is zero — when all voters are equal.
- The mean square voting power can also be seen as “the average voter power from the perspective of candidates”. That is, if you choose a random candidate, then choose a random constituent of that candidate (that is, a random voter, weighting the probability of choosing each voter by their power in electing the candidate), then the expected total voting power of that voter is the mean square, because you’re averaging power and also weighting by power. So, in plain English: the higher this number is, the more candidates are accountable to an elite; the lower it is, the more they’re accountable to the average voter. Taking the reciprocal, then, can just be seen as a way to convert this number from “bigger is worse” to “bigger is better”.
Dealing with non-deterministic voting methods
What about non-deterministic voting methods? We should be able to see that sortition or random ballot are actually proportional methods. And after all, even most “deterministic” voting methods are not neutral without some nondeterministic tiebreaker.
(A quick tangent on the distinction between sortition and random ballot: “sortition” is when a body such as a legislature is made up of randomly-selected “voters”, while “random ballot” is when each voter ranks the candidates and the legislature is chosen by randomly choosing ballots and electing their favorite candidate who doesn’t yet have a seat. In other words, sortition can be seen as a special case of random ballot, where each voter is a candidate and votes for themself. Of course, in real historical cases of sortition-like systems, such as ancient Athens or U.S. jury duty, being selected is often seen as a burden; but here, we’re ignoring that aspect and focusing on voting methods in terms of their impact on ideological outcomes, not in terms of the actual work of legislating.)
The solution is to take the expectation of each voter’s voting power, over the randomness inherent in the voting method. That is, if w(ℬ) (the set of winners) is seen as a random variable, the above formula becomes:
This is a very simple change to the formula. But a couple of notes are in order:
- Note that this averages the squares of the expectations of voters’ power, not the expectation of the squares. That is to say: it’s OK if, as in sortition, any given election has wildly unequal voting power, as long as on average each voter can expect equal power.
- By the same token, this measures your expected retrospective power to have elected some candidate. The winners themselves are not taken as fixed, so unlike in the deterministic case above, this cannot be seen as a measure of the accountability of the actual winning set of candidate.
- The idea of “retrospective expectation” here is perfectly well-defined mathematically, but intuitively it sounds like an oxymoron, so it’s worth looking at what that means. It’s “retrospective” because it’s holding the ballots themselves fixed; it takes no account of the prospective uncertainty that you would have when voting about how other people would vote. But it’s an “expectation” because it is averaging over any randomness in the counting process. So if we want an intuition for what it means in terms of subjective probability, it’s what you’d expect at the moment when the ballots were all tallied (thus, “retrospective” in regards to the election) but before you started any random procedures (thus “expectation” in regards to randmness).
- In principle, this is still a flawed measure, because it doesn’t distinguish between something like “choose three ballots randomly and elect the favorite on each ballot” and something like “choose one ballot randomly and elect the three favorites on that ballot”. It’s possible to fix the definition so that it would make this distinction and correctly say that the former voting method is better; but that would add needless complexity for ordinary cases. For simplicity’s sake, we’ll just assume that nobody would seriously propose anything like that second method as “democratic”.
Power to elect an individual candidate versus power to elect a party
The individual voter power measure defined so far, p(i, A, ℬ), measures the retrospective power of voter i to have elected candidate A. But it doesn’t do a great job measuring the quality of “open party list”-type methods, where your voting power is split between helping elect an individual winner A and a party (that is, set of candidates) 𝒫. In practice, if a party gets, say, 3 seats, this will assign voting power for their first seat almost exclusively to the voters who chose the individual winner who got that seat, but voting power for the last seat will be close-to-evenly divided among the party’s voters. On the whole, this unfairly exaggerates the inequalities in voting power between that party’s voters. Such inequalities are real, but this overstates them.
Conceptually, t’s not hard to generalize this measure so that p(i, 𝒫, ℬ)—voter i’s retrospective power to have given more total seats to party 𝒫—is well-defined. The notation is finicky so I won’t write that out here, but I trust that readers who have followed my definitions so far will be able to see how that would work.
But then, we’d have two different measures. Looking at voting power at the individual candidate level would unfairly exaggerate the inequality in voting power in open party list methods, making such methods look worse than they really are; but looking at voting power at the party level would do the opposite, making them look unfairly good. I have some ideas about ways to clearly define the “right” compromise between these two extremes, but for the purposes of this article, I’m just pointing out that it’s an issue.
Voter Satisfaction
(To be completed: Explain what “voter satisfaction” is; why it’s missing from the above measure(s); one idea on how to define it if you were omniscient; and some rough ideas on how to estimate that in a real-world case.)
Non-quantifiable desirable characteristics
(To be completed: Point out that, while equality and satisfaction are important, there are other things that are desirable. List a few such aspects, such as ease of voting; ease of tallying; transparent accountability; partisan/coalition incentives; and, possibly, keeping out intransigent extremists.)
Combining Voter Equality with Voter Satisfaction
(To be completed: Briefly touch on why a combined measure of equality and satisfaction connects to a separate philosophy of ideal democracy; that is, how low scores low on this measure connect to undemocratic legislative outcomes.)
