Most people assume there is a duty to vote. Philosophers generally agree. (In fact, there’s even some evidence that they rate voting more morally good than donating to charity or giving blood, which I find bizarre.) Despite the fact that most philosophers believe in a duty to vote, the arguments on behalf of this position are surprisingly weak. Or, perhaps because philosophers tend to believe there’s a duty to vote, the arguments are weak.

In chapter 1 of The Ethics of Voting (which is only $24 hardcover and the most fun you'll  ever have in a serious philosophy book), I try to look for or come up with the very best arguments on behalf of there being a duty to vote. The best arguments on behalf of a duty to vote rely upon ideas of collective responsibility and civic virtue rather than on the value of individual votes. In chapter 2, I repudiate those arguments.

However, along the way, in chapter 1, I consider and rebut a number of weaker arguments for a duty to vote. Among these are a class of arguments that we might call instrumental arguments. Instrumental arguments (attempt to) ground a duty to vote by showing that individual votes produce a lot of good. They try to show that in some way or other, individual votes can be expected to make a difference and lead to good consequences. 

Or, more precisely, instrumental arguments  try to show that the expected utility of voting, in light of its expected consequences, is sufficiently high that a citizen has compelling moral grounds to vote.  For many philosophers, the search to prove voting is morally obligatory amounts to a search for the right consequences.

Here are some of the valuable consequences they have proposed:

1. My vote might change the outcome of the election.
2. My vote might prevent democracy from collapsing.
3. My vote might improve the quality of government simply because the quality of government positively correlates with the number of voters.
4. My vote might sufficiently increase the mandate my preferred candidate enjoys and thus help the candidate govern more effectively.

Alas, arguments based on 1-4 all fail.

Today let's talk about one instrumental argument. Consider the following argument for a duty to vote:

            The Beneficence Argument:

1. All things being equal, if you can perform an action that has a large expected benefit to the public good, you should do so.
2. Voting the right way has a large expected benefit to the public good, because there is some chance that your vote will decide the election.
3. Therefore, you should vote the right way.

Notice that this argument does not imply that citizens should vote however they wish rather than abstain.  Rather, the argument, if successful, would show citizens should vote well rather than abstain. It leaves open whether it is better for a citizen to abstain than to vote badly.

Premise 1 of the Beneficence Argument is questionable. Even if my vote were to do a lot of good for others and didn’t cost me much to cast, that doesn’t automatically imply I would have a duty to vote. Voting might instead be supererogatory—above and beyond the call of duty. Or it might just be one of many equally good ways of acting beneficently. There are some other problems with premise 1. However, I’ll just grant premise 1 here.

Premise 2 is deeply problematic. Let’s take a look at it here.

There’s not much chance that your vote will decide the election. However, just how small the chance is will make a big difference for the Beneficence Argument. Suppose I had only a one in a million chance of changing the outcome. Even on that small probability, individual votes could have a lot of expected value. Imagine that one candidate were worth $10 billion more to the common good (whatever that is) than the other. If so, then a one in a million chance of changing the election from the worse candidate to the better would have an expected utility of $1000. Spending 10 minutes voting for the better candidate would seem like a swell thing to do. That 10 minutes has an expected value of $1000 for the rest of society. Voting is like donating $1000 to a good charity.

However, most economists and political scientists who study this issue would say that a 1 in a 1,000,000 chance of changing an outcome is much too high. In this post, I’ll discuss Geoffrey Brennan and Loren Lomasky’s method (from Democracy and Decision) for calculating the expected utility of voting and probability of being decisive.

The expected utility of a vote (UV_i), in terms of its ability to change the outcome of an election, is given by formula 1:

            (1)  UV_i = p[V(G) – V(E)]

            where

            p is the probability a vote will be decisive

            V(G) is the expected value of the candidate voted for

            V(E) is the expected value of the candidate voted against.

Equation (1) works regardless of whether you have altruistic or selfish concerns. You can render V(G) and V(E) in terms of candidates' value to you, or in terms of candidates' value to the common good.

Calculating the probability a vote will be decisive is trickier. Here’s a shorthand formula, formula 2:

            (2)  p =  f(N, m)

            where

            N is the number of voters

            m is the “anticipated proportional majority”

The degree to which one candidate has an edge—the degree to which people tend to favor that candidate over the other—is the degree to which the leading candidate has an anticipated proportional majority.  Once you do the mathematics, it turns out that the probability that an individual vote will be decisive decreases slowly as the number of voters increases, but it decreases dramatically when there’s even a slight anticipated proportional majority.

Let’s now do a calculation using their formulae. (You’ll need to check out Democracy and Decision, pages 56-57, 118-119 if you want to see the full version of formula 2.) 

Suppose I want to promote the common good. Suppose the election is held between two candidates, Good and Evil. Suppose that I can easily calculate how much better it is for Good to win instead of Evil, and this value can be expressed in monetary terms: $10 trillion. Though it’s worth $10 trillion for Good to win, my vote for Good isn’t worth $10 trillion. If my goal is to change the outcome of the election, my vote has value only if it’s decisive. Most methods of trying to calculate the probability of my vote’s being decisive (and the related expected utility of my vote) indicate that in typical large-scale elections, unless voters are expected to almost perfectly evenly split between candidates, my vote has only a vanishingly small chance of changing the outcome, and thus has only a vanishingly small expected utility. On the most commonly used formula, if Good has a mere 0.5% lead over Evil in the polls, and if the number of voters will be the same as in the 2004 US presidential election, the my vote for Good is worth only $1.45 x 10^-2647, 2645 orders of magnitude below a penny. So, while it’s worth $10 trillion for Good to win over Evil, so far it looks like it’s worth nothing to vote for Good over Evil.

Brennan and Lomasky’s method for calculating the probability of being decisive seems to be the most widely accepted. There are some critics. Most alternative formulae lead to similar results—individual votes have a vanishingly small chance of being decisive in any realistic election (including, for example, the US presidential race in Florida in 2000). Edlin, Gelman, and Kaplan use a rather different method from everybody else for estimating the probability of being decisive, and if their work is correct, the Beneficence Argument is on firmer ground. (However, chapter 2 of my book ends up doing quite a bit to undermine premise 1 of the Beneficence Argument.)