"One person, one vote" is wrong
How a common fallacy has kept us using a terrible voting system for centuries
No, this essay is not about disenfranchising any group of people; quite the opposite, as you will see. It’s about the mathematics of voting systems — a topic that may sound dry, but that critically affects the politics of any democracy.
In this essay I will show you how the familiar slogan “one person, one vote” has led us astray. Its heart is in the right place — we absolutely do want every voter to have the same amount of influence over the result — but taking it literally as a guide to voting system design is a terrible mistake. “Make everything as simple as possible, but no simpler” is the operative dictum here: it turns out that “one person, one vote” is a little too simple. A system based on it works fine for two-candidate races, but fails, for three or more candidates, to live up to its aspiration of giving every voter the same amount of influence. Yes, I mean exactly that: it calls for fairness, yet leads to unfairness. I will explain exactly why, and how to repair it.
The Problem
It’s obvious how voting should work in a two-candidate race. Each candidate votes for Candidate A or Candidate B, and whoever gets the most votes wins. Easy peasy. But what should happen when we add a third candidate to the ballot? Well… “one person, one vote”, right? So each voter should have to pick one of the three to vote for. That’s just as obvious, right? If we let voters vote for two candidates if they wanted to, that wouldn’t be fair to those who voted for just one, would it?
Let’s turn that around: why would it be unfair? The intuition we all have is that voting for a candidate is a vote, but failing to vote for a candidate is nothing, a non-vote, the absence of a vote. As I will now demonstrate, not to vote for a candidate is actually to vote against them; and so there is actually no such thing, in our system1, as a non-vote. (I will show how a different voting system could permit non-votes, which will make it clearer why ours doesn’t.)
One way to see this is to consider the probability of each candidate winning, and the way that this probability changes as the votes are cast. Suppose we have three candidates, A, B, and C. When a vote is cast for A, clearly their chances of winning improve. But the probabilities must always total to 100% — there will be exactly one winner — so however much the probability of A winning increases, the sum of the probabilities of B and C winning must decrease by the same amount.
Here’s another way to see it. We can refer to the usual way of counting votes as a {+1, 0} system, because it assigns a value of +1 to a “yes” vote on a candidate, and 0 to a “no” vote. But what if we instead assigned a value of -1 to a “no” vote, yielding a {+1, -1} system? A candidate’s total, then, would be the number of “yes” votes they received, minus the number of “no” votes. In this system, it’s clearer that not voting for a candidate is equivalent to voting against them. But it’s easy to see that it must produce the same winner as the usual {+1, 0} system.2
So then let’s suppose, hypothetically, we give the voters another choice. Along with the options of casting a +1 vote for a candidate or a -1 vote against them, let’s give them a 0 option. This would be a neutral vote, an abstention, that wouldn’t change the candidate’s total when counted. Now we have what can fairly be called a non-vote option, right? But notice how the value for that option is not equal to either the “yes” value or the “no” value, but is the average of the two. That means that if we wanted to add this option to the {+1, 0} system, we would give it value +1/2. I call {+1, 0} and {+1, -1} two-valued systems; and {+1, 0, -1} and {+1, +1/2, 0} three-valued systems.
The point of my hypothetical is this. The argument that “one person, one vote” requires us to restrict voters to voting for a single candidate on a 3-or-more-candidate ballot contains a fundamental confusion, between the zero of {+1, 0} and the zero of {+1, 0, -1}. Although those are obviously both numerically zero, they’re zeros in different systems and so aren’t the same thing. So the zero of {+1, 0}, corresponding as it does to the -1 of {+1, 0, -1}, is not a non-vote; it’s a negative vote, a vote-against.
The plain fact is, once you get three candidates on a ballot with no abstention option, “one person, one vote” isn’t just wrong — it’s incoherent: there’s no two-valued system you could design that could be fairly described that way. The only way to make “one person, one vote” work with three or more candidates would be to use a three-valued system, and allow voters to cast either a “yes” (+1) or a “no” (-1) vote on one candidate, leaving all the rest at neutral (0). I’m not seriously proposing such a system, but it would satisfy a literal reading of “one person, one vote”.
And the only reason “one person, one vote” seems to work to describe a two-candidate election is that we can ignore the above argument. It doesn’t matter whether we count a ballot that votes for A as also voting against B, because there are only two possible valid ballots anyway: one must either vote for A and against B, or for B and against A. Once one has decided to vote for one of the candidates, there are no more choices to make.
With three candidates, that’s no longer true. A voter who decides to vote for A could still desire to vote for B and against C; against B and for C; or against both of them. In any case, it’s clear that three votes are being cast, even if we require two of them to be negative votes. This brings us to the essence of the problem with the usual system: by allowing a voter to vote for only one candidate, we force them to vote against all the others.
Forcing voters to vote against all but one candidate is what gives rise to the “spoiler effect”. In the 2000 US Presidential election, for example, one had a choice of voting against George W. Bush and Ralph Nader; voting against Nader and Al Gore; or voting against Bush and Gore. So someone whose first priority was voting for Nader had to vote against both Bush and Gore; the option of voting only against Bush was not available. Since Nader’s politics were closer to Gore’s than Bush’s, it’s likely that most Nader supporters would have preferred not to have had to vote against Gore in order to vote for Nader. (If your politics are such that you’re glad Nader drew votes from Gore, consider instead the 1992 election, in which Ross Perot drew votes from George H. W. Bush, helping to elect Bill Clinton. The spoiler effect is nonpartisan.)
The spoiler effect is why our voting system is actually unfair. It’s unfair because it favors voters who want to vote for one candidate and against the rest, and disfavors those who would like to vote for more than one. The latter group have no way to express their true opinions, and thus actually have less influence over the election results. Allowing people to vote for two candidates and against one wouldn’t give them more influence than those who voted for one and against two: it would give them equal influence. In both cases, three votes are being cast.
So not only is “one person, one vote” a fiction — an approximation that works for two candidates, but makes no sense once there are three — but it’s a destructive fiction, because it has locked us into this idea that a voter must vote against all candidates but one.
“One person, one vote” has its heart in the right place, but its math is off. It’s not bad as a slogan to chant at rallies or to remind us that fairness in voting systems is critically important, but as a guide to actually designing such systems, it leads us in the wrong direction.
The Solution
I propose a different motto: “one person, one candidate, one vote”. That is, each voter gets to vote once, either up or down, on each candidate. That’s still fair, right? In fact, as I’ve argued, it’s more fair than requiring voters to vote against all but one. Let’s see how it would play out.
All we have to do is to make a small change to our voting system: we simply remove the rule that you have to vote against all but one candidate, and let you vote for or against each candidate independently. Otherwise, it’s the same: ballots are tabulated by counting the votes for each candidate, and the candidate with the highest total wins. This system is called Approval Voting, because the voter just votes for each candidate they approve of, and against the ones they don’t.
Approval Voting (AV for short) clearly gets rid of the spoiler effect. Consider the elections we were discussing above, where a third-party candidate C entered the race who was ideologically much closer to one major-party candidate, A, than to the other, B. Under AV, the presence of C won’t cause B to win; if anything, it will slightly improve A’s chances, if some B voters decide to vote for A also, to help prevent C from winning. (I’ll go into more depth on this possibility in a later post.) The voters who approve of both C and A aren’t disenfranchised.
But you may be wondering: is there any bad news? Is there any strong reason not to use AV as the standard voting system for elections with more than two candidates? Having studied voting systems at some length, I believe that there is not. While no voting system is perfect, AV is very good, and what flaws it does have are, in my opinion, not fatal. In fact, I strongly suspect that the major reason it isn’t generally used is that it seems to violate “one person, one vote”. I wrote this post in the hope of demolishing that objection.
In case you’re still not convinced, here’s one more point. Let’s ask: what does “one person, one vote” really mean in practice? Does it mean you can cast only one vote in your entire lifetime? Of course not — that’s too literal a reading. Well, does it mean you can cast only one vote per election? No; most elections involve multiple races, and you’re allowed to vote once in each race. So is it “One person, one race, one vote”? That’s closer, but not exactly right either; it’s common, when the race is for more than one seat (such as for a county Board of Supervisors), to allow a voter to vote for as many candidates as there are open seats. So what we’ve been doing is better described as “One person, one seat, one vote.”
That’s still inaccurate, as I’ve been arguing, because it doesn’t count all the candidates you don’t vote for as votes-against. But let’s leave that aside for a moment. The point is that “One person, one vote” isn’t an accurate description of our standard practice even on its own terms, in which votes-against aren’t counted.
Mathematically, is there any reason, in a multi-seat race, to limit voters to one positive vote per seat? No, there isn’t. Such a limit requires voters to cast at least c - s negative votes, where c is the number of candidates and s is the number of seats, and so disenfranchises people who don’t want to cast that many — perhaps they have a particular candidate or two they want to vote against, and they think all the others are acceptable. We don’t tend to have multi-seat races at the state or national level, so this issue seems less important than that of single-seat races, but the principle is the same.
To sum up, then, the rule we’ve been following, completely correctly stated, is not “one person, one vote,” but “one person, one seat, one positive vote, with the rest negative”. In that light, does “one person, one candidate, one vote” still seem all that strange? It’s actually simpler.
Voting system design is a large topic, and one I’ll be writing more on. For the moment, let me leave you with a link to more resources on AV:
The Center for Election Science
I am speaking primarily of elections in the US, but many other places use the same system, and so the same argument is applicable.
If t_old is a candidate’s total under the {+1, 0} system, t_new is their total under the {+1, -1} system, and b is the total number of ballots cast, then: t_new = 2 ⋅ t_old - b (since b is the sum of the number of “yes” votes, which is t_old, and the number of “no” votes). This is a linear relationship with a positive slope, so it doesn’t change ordering; a series of t_new values will be in the same order as the corresponding t_old values.