Who put you in charge?

Measuring retrospective voter power

“Who put you in charge?” is not just a common rhetorical question. It’s at the heart of political theory. The definition of a democracy is that the leaders are chosen by the people; so one step in telling how democratic a given system is, is measuring who elected whom.

In this article, I’ll give an answer: a way to measure retrospective voter power for each voter in a given election. Using such a measure, you could quantify how electoral systems where voters tend to have mostly equal power are more democratic than systems where voter power is highly unequal.

The idea of measuring voting power is not new. Political scientists have long used this concept, and related ones like “wasted votes” (ballots with no voting power). But definitions and measures for these concepts have, until now, been designed specifically for certain types of voting systems. These definitions break down when you try to compare systems which are truly different.

For instance, in list proportional systems, you can find the power of the votes for a given party by dividing the number of seats that party wins by the number of votes they received. But this definition breaks down if some or all candidates lack clear party labels, and/or if it’s possible for a single ballot to vote for more than one party (as with ranked-choice or score ballots).

The definition I’ll give below, though, directly applies to any neutral electoral system. This allows you to compare systems with different mechanics and/or philosophical underpinnings, asking the question “which system is more democratic? That is, which one tends more to give voters equal power?”

In this article, I’ll define my measure both in words and in math notation. This will get a bit technical and abstract, but I will try to explain what I’m doing and why in simple and understandable terms — as much as I can, anyway. At the end, I’ll talk a little bit about how to calculate my measure in practice, and how to use the results.

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Defining “neutral electoral system”

First off, I said that this measure applies to “any neutral electoral system”, so I have to define what that means.

Formally speaking, an electoral system is made two functions. First, there is a function from a set of candidates to a set of all possible valid ballots for each voter. Second, there is a function from a set of valid ballots for each voter, to a set of winners.

For the purposes of this article, I’ll only consider electoral systems that elect the same number of winners for any set of ballots. Also, I’ll start out by assuming the systems are deterministic (they always give the same winner set for the same ballots), though later I’ll briefly consider ones that have some randomness (the same ballots can lead to different winner sets, with well-defined probabilities).

There are two common baseline criteria for electoral systems to be considered democratic:

• “Anonymity criterion” (symmetry over voters): if you keep the same ballots, but switch which voters cast which ballots, the winners should not change. Note that real-world systems like MMP violate this, because it matters which district the ballots were cast in. Note also that this is to some degree incompatible with deterministic methods which elect a fixed number of candidates; but, since there are reasonable ways to resolve this incompatibility, I’ll ignore it from here on.
• “Neutrality criterion” (symmetry over candidates): If you switch the names of a certain two candidates on all ballots, the outcome should be the same as before, except with the outcomes for those two candidates switched.

The measure I’ll define below requires neutrality, but not anonymity. It also requires an additional criterion:

• “Independence of Unsupported Candidates (IUC) criterion”: If you add a new candidate to an election, and all voters cast ballots which are the same as the original election for all existing candidates and give as little support as possible to the new candidate, the outcome will be unchanged.

Nearly any voting system anybody would seriously propose already meets the IUC criterion. But it is possible to create one that doesn’t. For instance, Borda with mandatory strict ranking (in either single-winner or bloc multi-winner forms) would violate it. For such a system, my voting power measure would not apply.

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What a retrospective voting power measure *should* be

Now that I’ve defined “electoral system”, I want to define what a “measure of retrospective voter power” should look like. Later, I’ll define a specific measure of this type, which I think is the “right” measure. But before I shoot my arrow and claim it was a bullseye, I have to describe the target I’m aiming for.

I want to be able to answer the question in the title of this article: “Who put you in charge?” That is, once an election has occurred, and a given candidate X was elected, I want to be able to dole out responsibility for having elected X among the voters. I want to assign a number to each voter for each winning candidate, so that the total across voters for any given candidate will be V/M, where V is the number of voters and M is the number of candidates. In this way, the average voters’ total voting power (the average across voters of the sum across candidates) will, by construction, be 1.

Once we have such a measure, in a highly-democratic voting system, each voter will tend to have total power close to 1; in a highly-unequal one, total power will vary widely, with many voters having 0 total power (not having helped elect any winners) and others having power of 2 or more.

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What my retrospective voting power (RVP) measure *is*

I’ll start with the definition, then explain it later. For a given election, winner, and ballot, the retrospective voting power is defined as:

How likely is it that voter i was pivotal in electing candidate A?

By “pivotal”, I mean the following proceture:

1. Add an unsupported candidate Z to the election.
2. Put the ballots in some arbitrary order.
3. Go through them, one by one, swapping A with Z. For a ballot which didn’t support A anyway, this might not be a change; for one which did, this will give that support to Z and put A at the bottom.
4. If the outcome changes in a way detrimental to A or favorable to Z when changing ballot i, then voter i was pivotal.

It might appear that this is two separate questions: whether i was pivotal in not-electing A, or whether they were pivotal in electing Z. But by the neutrality criterion, any ballot which is pivotal in the first sense for one ballot order must also be pivotal in the second sense for the opposite ballot order. So asking either of these questions implicitly asks both.

Now I’ve defined “pivotal”, the definition of “likely to be pivotal” is obvious: we just assume a uniformly random distribution for the permutation (order) that we switch the ballots. Thus, the definition becomes:

The RVP for a given voter to have helped elect a given winner in a given election is: the reciprocal of the number of winners, times the expectation, over all orderings of the ballots in the election, of the indicator of whether the given voter is pivotal in the given candidate not winning, if you first add a new unsupported candidate and then go through the ballots in order swapping the given candidate with the new candidate. (If, because of nonmonotonicity, more than one voter is pivotal for a given ordering, then for that ordering divide the indicator function by the number of pivotal voters.)

Or, equivalently:

The reciprocal of the number of winners, times thez expectation, over all orderings of the ballots in the election, of the indicator of whether the given voter is pivotal in the new candidate winning (with the same procedure and caveats as above).

By “indicator” here, I mean the indicator function; that is, 1 if the given voter is pivotal for the given candidate and order, or 0 if they are not.

Illustrative examples

To illustrate why this makes sense, imagine a simple 1-winner plurality election with three candidates, X, Y, and Z, who get 45, 40.5, and 15.25 votes respectively (the fractional votes come from a special randomly-allocated tiebreaker ballot). X is the only winner. An X voter is pivotal in the first sense if they are the 5th X voter in the order (41st from last); they are pivotal in the second sense if they are the 41st X voter in the order (5th from last). Note that every time the first is true, the second is true for the reverse order; this is always the case, regardless of electoral system. In this simple case, it’s clear that each X voter is pivotal the same number of times, so we can see that each of them must have retrospective voting power to elect A of 100/45 or 2.222, while the other voters have 0 retrospective voting power (the candidates they voted for lost).

(Side note: in this scenario, candidate Z might be called a “spoiler”, as Z voters could have gotten better voting power by choosing between X and Y. There is an different definition of retrospective voting power which would have given each Z voter exactly half as much voting power to elect X as an X voter had. I won’t discuss that further here, but may do so in another article.)

Let’s consider a slightly more-complex scenario: the electoral system is multi-winner reweighted range voting, the candidate we care (A) about is the first to get a seat, and there is another similar candidate (B) who will get the first seat if A does not, causing A’s ballots to be deweighted so that A does not get a seat. In that situation, it can be shown that a given ballot’s RVP to elect A is proportional to that ballot’s score for A (up to some “rounding error” which is negligible in large, diverse elections).

This second example illustrates a few things:

• In at least some circumstances, RVP conforms well to our naive intuitions about “who was responsible for getting A to win”.
• A given voter’s voting power to elect a given candidate in a certain election relates to many aspects of that entire election, not just to the voter’s support for that one candidate. That’s why I had to assume there was some candidate B who would win if A didn’t, preventing A from being a contender after the first seat, and thus preventing the complex math that would be involved in evaluating RVP in the context of such second- or third- chances. If B dropped out, the outcome of the election would be unchanged, but RVP to elect A would change. This might not be ideal, but it allows defining RVP without considering the effects of changing ballots for more than one candidate at a time.

Formal definition

To write this in math notation, we need to start with some definitions:

• M: the number of winners
• V: the number of voters
• ℬ: the set of ballots for the election we are retrospectively analyzing
• w(ℬ): the set of winners for ballot set ℬ.
• p(i, A, ℬ): The retrospective power that voter i had to elect candidate A in an election with ballots ℬ. This assumes that A∈w(ℬ).
• 𝒫: The set of orderings (permutations) of the ballots in ℬ.
• {j:p(j)<p(i)}: the set of voters who come before i in permutation p.
• {j:p(j)≤p(i)}: the same set as above, but also including voter i.
• Z: A new candidate, not in ballots ℬ.
• r(ℬ,{…},X,Z): The result of taking ballots ℬ, then adding candidate Z with no support on any ballot, then taking the set of ballots {…} and swapping candidate X with candidate Z on them.
• I[]: an indicator function; 1 when its argument is true, and 0 when it is false.
• c(ℬ, p, A): The number of times A drops out of the winner set as you go through ℬ in order of p, replacing A with Z. This is always at least 1, and only more than that if there are nonmonotonicities where moving A to the bottom of a ballot causes them to win.

With all of that defined, the formula is:

Or, equivalently:

(In LaTeX, the first equation above is $p(i, A, \mathcal{B}) = \frac{1}{V!}\sum_{p\in\mathcal{P}}\frac{1}{c(\mathcal{B}, p, A)}I[w(r(\mathcal{B},\{j:p(j)<p(i)\},X,Z))=A)I[w(r(\mathcal{B},\{j:p(j)\leq p(i)\},X,Z))\neq A)$)

Calculating RVP

The definition above involves a sum over all V! possible ballot ordering. If you had to calculate each of these individually, it would take eons even on the fastest computer. Luckily, as we saw in the simple plurality case above, in most voting methods, reasoning using symmetry allows you to calculate the answer much faster. Even if this is doesn’t get you all the way to a final answer, when combined with random or systematic sampling, it allows estimating the answer reasonably quickly.

Implications of RVP

I’ll be returning to these ideas in later articles, but for now, here are a few notes on what I think RVP can teach us:

• Once you can measure RVP for individual voters and candidates, you can measure each voter’s total RVP across all candidates. Insofar as this total is close to equal across all voters — for instance, insofar as it has a relatively-low variance empirically—we can say the voters have roughly equal power and the method is democratic.
• Not all proportional voting methods will always have low variance of total RVP, but all disproportional results will have high variance of total RVP. You can’t address inequality of voter power without proportionality.
• Equality of total RVP is of course not the only desirable characteristic of voting systems. For instance, closed list proportional systems will tend to have lower variance in total RVP than open-list systems (at least, for existing implementations of the latter idea); but while closed list might be said to represent voters more equally, open-list does so more faithfully.