Does one's vote make a difference?

Suppose that there is a simple majority election, with two candidates, and there is a large odd number of voters. Suppose polling data makes the election too close to call. How likely is it that you can decide which candidate wins?

I could look up this stuff, but it’s more fun to figure it out.

A quick and dirty model is this. We have N people other than you voting, each choosing between candidates A and B with probabilities p and 1 − p respectively. You don’t know what p and 1 − p are, but polling data tells you that p is between 1/2 − a and 1/2 + b for some positive numbers a and b. Your vote decides the election provided that exactly N/2 people vote for candidate A. This requires that N be even (if N is odd, at best you can decide between a candidate winning and the election being undecided, so you can’t decide which candidate wins), which has probability 1/2. Given that N = 2n is even, the probability that the other votes are exactly balanced is (a+b)⁻¹ C(2n,n)∫_1/2−a^1/2+bpⁿ(1−p)^n − 1dp, where C(m,n) is the binomial coefficient. Assuming n is large as compared to a and b, the integral can be approximated by replacing its bounds by 0 and 1 respectively, and some work with Mathematica shows that for large n the probability is approximately 1/(N(a+b)).

So what? Well, suppose you think that candidate A will on average make a person in the jurisdiction be u units of flourishing better off than candidate B will, and there are K persons, where K ≥ N + 1 (there are at least as many persons as candidates). So, the expected amount of difference that your voting for A will make is at least Ku/(2N(a+b)). This is at least u/(a+b). Thus, if the polling data gives you a range between 0.48 and 0.52 for the probability of a person’s preferring candidate A, and half of the people in the jurisdiction vote, the expected amount of difference that your vote makes is 25u. This is quite a lot if you think that which candidate wins makes a significant difference u per governed person.

Interestingly, some numerical work with Mathematica also shows that as number of people increases, then the expected amount of difference your vote makes also increases asymptotically, up to the limit of Ku/(2N(a+b)). So for larger jurisdictions, even though the probability of your vote making a difference is smaller, the expected difference from your vote is a bit bigger.

My quick and dirty model is not quite right. Of course, people don’t come to the polls and randomly choose whom to vote for. A more likely source of randomness has to do with who actually makes it to the polls (who gets sick, who has something come up, who decides it’s pointless to vote, etc.). A better model might be this. We have M people eligible to vote, of whom pM want to vote for A and (1−p)M want to vote for B. Some random subset of the M people then votes. My probabilist intuitions say that this is not that different from my model if the number of actual voters is, say, half of the eligible voters. If I had an election that I was eligible to vote in coming, I might try to figure our the more complex model, but I don’t.