Why it’s rational to vote

This October 2004 post on a Columbia University blog deals with the seemingly simple question of whether voting in a presidential election is a rational act. They present a utility function U=p*B - C, where p is the proabbility that your vote will change the outcome of the election, B is your benefit from your candidate winning, and C is the cost you incur by going to vote. They begin by throwing out a value of p=1/10,000,000 as a reasonable value for p. This makes it seem as if voting should be irrational (negative utility) because even for B=$1,000,000, the cost of voting would have to be less than 10 cents. They present 2 explanations for why people still go out and vote. First, people feel happy performing a “civic duty”. Second, they claim that you should consider the potential gain per person, instead of just for yourself. Their example numbers are B=$100/person and p=10,000,000, which means voting is rational if your cost C is less than $30. They support this thinking by citing a study in which British citizens said that they vote in order to obtain “benefits for groups that people care about”.

While their argument reaches a nice conclusion (we want people to go out and vote, in theory), their analysis seems far to simple. The most glaring problem with their reasoning is their value of p=10,000,000. This roughly represents a value of 1/n (n = number of voters) that is seen in other articles on this topic. I would propose a different model for determining p. Call x the fraction of voters who prefer your candidate. If x = 0.5, you expect their will be roughly the same number of votes for each candidate. In this case, the probability that your vote will be the deciding vote is found by the binomial distribution n C (n/2) * (0.5)^(n/2) * (1-0.5)^(n/2). Using Stirling’s approximation, this gives p=1/3963 if there are 10,000,000 voters. That’s good news for showing voting is rational, but there is a downside. If you aren’t sure the election is extremely close to a tie, the p values get reduced dramatically. For x=0.495, p is roughly 10^-221 already. This makes it surely an irrational prospect to go vote according to our utility formula. In states where 60% of the vote is expected for 1 candidate, going to vote could surely never change the outcome of the vote.

How else can we rationalize voting? Attacking the problem from a game theoretic perspective, each player has 2 choices, vote or don’t vote. It would appear that there is no pure strategy equilibrium, because if everyone chose “don’t vote”, then a single person could deviate to “vote”, win the election, and get his nice payoff. If everyone else is voting, then a person would chose “don’t vote” because there is essentially no chance of swinging the outcome of the election. There will be some mixed strategy Nash equilibrium where each person goes and votes with some probability p.

I believe the key to rationalizing voting is to realize that the outcome is not some binary feature based on who won the election. The percentages that each candidate receives are important for many reasons. For example, the outcome of one election surely has an impact on the next election 4 years later. States like Massachusetts are considered “blue states” because they have a history of overwhelming democratic majorities, so it would seem that voting is illogical and it will always stay a democratic state. However, republicans consistently go out and vote, they will pick up larger and larger fractions of the vote each year, and eventually the state will be contested again. However, I don’t feel that this reasoning will get far enough to actually make it a financially rational decision to go vote. Let’s hope that people are happy with it being their “civic duty”.