You saw the subtraction fallacy a lot during the election season just past. The subtraction fallacy might be stated as follows: “A vote added to one column equals a vote subtracted from all other columns.” To take an example from the November 8 presidential poll: a vote in Trump’s column equals a vote subtracted from Clinton’s, Johnson’s, and Stein’s column. A vote’s negative weight, by this reasoning, depends on the number of candidates.
That doesn’t seem right, does it? A vote is a vote: a single, positive quantity. It does not have a variable negative weight that depends on the number of options a voter has. Each voter has one vote, and each voter awards that single vote to one person. One voter awards one vote to one person. A voter cannot deprive candidates of votes.
Subtractive theories of vote counting originate with the idea of a two-way race, where a vote entered in one column is not entered in the other column. In a two-way race, vote counting resembles double-entry bookkeeping, where a credit added to one column of the ledger equals a debit subtracted from the other column. Most elections, however, are not two-way races: they involve three or more candidates. The double-entry bookkeeping model does not apply. You cannot subtract a single vote from multiple people.
The correct model for theories of voting, of course, comes from economists’ concept of opportunity cost. A dollar spent on one item is a dollar not spent on every other possibility. Time you spend on one project is time you do not spend on every other possibility. If you try to count all the other possibilities, you run into some high numbers. The same goes for presidential voting, where if you count all the candidates who did not make it onto your state’s ballot, you are looking at a fair number of candidates. When you award your dollar, or a minute of your time, you forgo the opportunity of using your dollar or minute for something else. That does not mean you subtract your dollar or minute from the other possibilities. The other possibilities did not have the award to begin with, so you cannot take it away.
Because a vote is a single, indivisible and positive award, the principle of additive voting applies no matter how many candidates run for an office.
Party number crunchers and prognosticators may want to use double-entry principles before election day – a vote for any other candidate equals a vote we have lost – but that is not how voters regard their votes when they fill out their ballots. They may feel they want to vote against one or more candidates they disfavor, but casting a vote still means awarding a single, positive quantity to one candidate. No subtraction takes place in the voter’s mind, and certainly no subtraction takes place when election officials count the votes.
Because a vote is a single, indivisible and positive award, the principle of additive voting applies no matter how many candidates run for an office. I have used races with three or more candidates to illustrate the fallacy of subtractive vote counting, but the same principle applies if only two candidates appear on the ballot. If the idea of credits and debits does not apply to races with more than two candidates, it does not apply to a pair of candidates either. A vote does not change its nature based on the number of candidates in a race. It is still a positron. Aggregation of all positrons in a race does not include counting negative electrons, because electrons do not exist in elections.
So when someone says, “A vote for Nader is subtracted from Gore’s total: that’s why Gore lost,” your first thought ought to be, “No, a vote for Nader is a vote for Nader, period.” Nader voters did not permit Bush to win. They simply voted for their preferred candidate. Similar reasoning affected the recently completed race: “You won’t give your vote to Clinton? Then you must be for Trump, or at least willing to let him win! How can you consider that possibility?”
That’s not the case at all, of course. Voters have a number of options to express their preference among multiple possibilities, or they can abstain. Nothing in voting theory stipulates anything but a positive award at the moment a voter casts a vote. Moreover, nothing in the vote aggregation process implies any method of subtractive or double-entry bookkeeping. Vote totals are additive. That is all you can say about them.
So far I have focused on additive vote counting as it applies to candidate races. It also applies to the argument, “Why should I vote? My vote doesn’t count anyway.” That argument seems valid, since even races for class president, in a small class, typically do not turn on a single vote. If the outcome is the same without my vote, then why trouble myself?
If you’re a member of a choir, and you decide not to sing because what’s the point – no one will notice one voice missing anyway – why are you standing up there to begin with?
This question has an obvious response, of course: “What if everyone thought that way?” The vote count for every election would be zero to zero. However faulty or valid one’s reasoning, a decision to abstain is still a decision. The option belongs to every voter in every election. To abstain based on the reasoning, “What’s the point? My vote doesn’t count anyway,” points to a serious misapprehension of the vote counting process, because the point of voting is to participate in the process. It is not a trivial point.
If you’re a member of a choir, and you decide not to sing because what’s the point – no one will notice one voice missing anyway – why are you standing up there to begin with? We may be cynical about civic pressure to vote – voting for these jerks just encourages them – but you do have to ask whether you want to be part of the polity’s civic voice at all.
In that way, voting recalls Horton Hears a Who!, where Horton’s main antagonist, Sour Kangaroo, wants to destroy the elephant’s clover, where all the Whos live on a dust speck. Horton fails to persuade anyone of his faith in the unseen, until little Jojo joins the chorus of people pleading for their lives. When she adds her single voice to the total, she saves herself, all her friends, and her whole world.
A person’s a person, no matter how small.