Mirror Universe Voter Power
How Evil Spock can teach us something important about real-world elections
One of the most basic principles of democracy is: one person, one vote. That is to say, as much as possible, each voter’s power should be equal. But what do we mean by voting power? To maximize its equality, we need to define it and measure it.
By the end of this article, I hope to convince you that, for a representative democracy, the best philosophical and mathematical definition of voter power—the best way to measure this thing we want to equalize—uses the trope of a Star-Trek-like Mirror Universe. You know, the parallel reality which has copies of each person from this reality, but they’re all evil—and Spock even has a goatee to prove it.
OK, fine, maybe that name and picture are a little clickbait-y. But I’m entirely serious about the underlying idea. Imagining a mirror universe turns out to be the best compromise to get around the contrasting problems in the two fundamental perspectives on voter power.
In 2005, Felsenthal and Machover clearly laid out these two perspectives, showing how they’d both been reinvented various times. In this and earlier articles, they named these two ideas I-power (for “influence”— that is, the ability to change the outcome) and P-power (for “purse” or “pie”— the idea of dividing up a fixed resource).
I-power leads to the unnormalized Banzhaf power index. To calculate this index for a given voter in a simple two-outcome voting game, simply count the minimal winning coalitions they form part of. This is proportional to the probability that the given voter will be pivotal if all others vote at random. (The constant of the proportionality is the number of ways everyone can vote—2 to the number of voters in the simplest case. Since this constant is huge in large elections, the probability of being pivotal in such an election is tiny.)
P-power leads to the Shapley-Shubik power index. To calculate this index for a given voter as above, for each minimal winning coalition they are part of, give them points equal to the reciprocal of the number of voters in that coalition. This is proportional to the probability that, if the voters cast random ballots in a random order, the given voter would be the last in a minimal winning coalition of voters on the same side. (The constant of proportionality is just the number of voters — exponentially smaller than the corresponding constant above.)
In the 2005 article I linked above, Felsenthal and Machover clearly disfavor P-power. They convincingly argue how, if you’re focused on outcomes — which, in general, aren’t necessarily zero-sum—the inherently fixed-sum P-power leads to contradictions. (They also show that the normalized Banzhaf power index, unlike the un-normalized form I explained above, is an unprincipled hybrid between the two ideas which has the downsides of both without all the upsides of either.)
But I-power also has its problems. As soon as you abandon perfectly-symmetrical toy examples in which all voters have an equal probability of casting any legal ballot, any reasonable extension of the Banzhaf index ends up approaching zero — because unless candidates are very close to equally-balanced a priori, the chance that any one voter will be pivotal becomes negligible.
(The normalized Banzhaf index, in which you divide by the total power of all voters, avoids this problem by construction—but, as Felsenthal and Machover showed, this is an unprincipled hybrid between the I-power and P-power which has the downsides of both without the rigor of either. Basically, the normalized version is the ratio of two exponentially-tiny quantities: the chance that a given voter is pivotal, and the chance that any voter is. Unless candidates are near-perfectly balanced, both of these numbers are so small that their ratio is as practical as counting angels on pinheads.)
Obviously, I’m writing this article because I think I have a way out of this dilemma. The trick is to consider a representative democracy: a system where first voters elect representatives, and then representatives vote on outcomes. Let’s assume the process in the second step can be complex — think, Robert’s Rules—but at least satisfies some minimal fairness axioms such as “anonymity”.
Consider such a representative democracy from the point of view of the moment after representatives have been elected, but before they have met to vote on outcomes. I’d argue that from this point of view, the best way to measure an individual voter’s prospective power to affect the as-yet-undecided outcomes, is to measure their retrospective power—in the first-step election as it actually happened—to choose their own representative.
If this is true, then it offers a way out of the dilemma. We can get the outcome-relevance of I-power, because we’re measuring expected power in the second step. But we also get the share-of-the-pie normalization of P-power, because in practice we’re measuring backwards across the first step.
And these two are united by the observation that the prospective second-step power is roughly equally divided between the winning representatives. This holds over the second step because, from a prospective point of view, symmetry between representatives is not yet broken; so the pie of power has a fixed size, with one slice per winner. And then, retrospectively, we can use a P-power-type index to divide that pie between the original voters.
And to do that — to measure P-power in an arbitrary multi-winner voting system—you should use the mirror universe.
If the mirror universe existed, then for every election, there would be a mirror-universe reflection. Each voter and each candidate in the election would have an “evil” counterpart. Say that the candidates are the bridge crew on the original Enterprise: Kirk, Spock, Dr. McCoy, Scotty, Uhura, Chekov, and Sulu. I’m supposing that however you vote in the real world—say you cast a ranked ballot for Spock, Uhura, and Chekov, in that order—your mirror counterpart would cast the corresponding ballot in the mirror universe—so, in this case, Evil Spock, Evil Uhura, and Evil Chekov.
Suppose that both sets of candidates (Real and Evil versions of each) are up for election together, but the voters are all the real voters. None of the real voters voted for Evil candidates, so the winners would all be Real.
But now, begin to replace the normal-universe voters, one by one, with their mirror-universe counterparts. When you finish this process, replacing the last voter, all the voters will be mirror-universe denizens, voting for only Evil candidates, so winners will all be Evil.
In order for this story to work, we need to fix the voting method beforehand. And that voting system must have a couple of simple characteristics.
First, the voting method must be Neutral —usually stated as “the names of the candidates do not matter”. That is to say that, when you replace all the normal-universe ballots with mirror-universe candidates, this is the equivalent of just changing each candidate X’s name to “Evil X”, so the final winners will be the couterparts of the initial winners (though strange things could happen in the middle.)
Second, the voting method must allow equal-bottom ranks/ratings, and never elect a candidate who was ranked thus by all voters. That allows all normal-universe voters to rank or rate all mirror-universe candidates at the bottom, without distinguishing between them; and vice versa.
But with these minimal assumptions — which essentially all voting methods worthy of the name meet—we can now define a measure of the your individual retrospective voting power. It’s the probability (over random orders for replacing all voters with mirror-universe twins) that, when you are replaced by your twin, a real-world winning candidate is replaced by a mirror-universe winning candidate. And, if there are M winners in this election, then summed over all voters, that voting power should naturally add up to around M — because, by the neutrality of the voting method and the symmetry between the normal and mirror universes, by the time all the voters are replaced, all the candidates should be too.
(Yes, I said “around M”, not “exactly M”, because it’s possible that some seats switch back and forth more than once between normal-universe and mirror-universe winners. But that can only happen insofar as the voting method is setwise-non-monotonic. Few voting methods are in all cases, but most good ones are in most cases. So, it’s possible to adjust the above definition of voting power so that it adds up to exactly M — for instance, by counting only the first time a seat switches hands between universes, or by allowing “negative voting power” when it goes backwards. And, more importantly, the exact adjustment shouldn’t make much difference for mostly-monotonic voting methods.)
So (with a little handwaving in the last paragraph) I’ve defined a measure of retrospective voting power of an individual voter in an election with an arbitrary number of winners and an arbitrary (neutral and equal-bottom-allowing) voting method. What good is that?
Plenty! In the next article in this series, I’ll show that:
- This measure accords well with, and indeed can help strengthen, intuitions about democratic ideals. In particular, it helps show why and how proportional multi-winner methods are better than non-proportional ones — and which proportional ones are better than others.
- Even though it relies on averaging over the space of all permutations of voters, this measure is computationally tractable in practice.
- This measure leads naturally to other meaningful and useful ones, such as “percent unwasted votes” and “power overlap” / “specificity of representation”.
- Of course, there are limits to what this line of thinking can help measure. But even then, defining a precise measure helps clarify what those limits are, and thus what lies beyond them.
