Events have overtaken me---an eternity ago, when Bush's lead in Florida was 12,000 votes and everyone was talking about how "achingly close" it was and how this really proves that everyone's vote counts, I wrote a brief column about how it actually proves exactly the opposite: Your vote counts when the margin is 1, and 12,000 is a very poor approximation to 1.
More precisely, I calculated as follows: Assume each voter votes for Bush with probability p and Gore with probability 1-p, and each choice is made independently. There are 2N voters. The probability of a tie is then (pretty easily) seen to be about
(4p(1-p))^n / Sqrt(Pi n)
Plugging in the actual value of N (call it 3,000,000) and assuming the final observed percentages to be a good approximation to p, I get a probability of about 1 in 500,000,000. In other words, your vote doesn't count.
But now that Bush's lead is down in the neighborhood of 300, a recalculation gives a probability of about 1 in 3000, which starts to make voting seem worthwhile.
Surely this is the wrong model, in the sense that I should really start with a prior distribution for p, update on the basis of the observed outcome, and then integrate against the updated prior---but I'd guess this oversimplified model gives a pretty good approximation.
Anyway, since the column is no longer relevant, I'll publish in the place most appropriate for an irrelevancy: Usenet! Herewith the column:
Which is stupider---playing the lottery or voting for president? Let's look at the odds.
To make the best possible case for voting, suppose the entire election hinges on your state alone---in other words, suppose you live in Florida. And suppose your state is so evenly divided that the margin of victory is just 12,000 votes out of 6 million cast. Then what's the probabilty that in the next election, with a similarly divided electorate, your vote will actually decide the outcome? The answer turns out to be about 1 in 500 million. (Click here for the calculation.) That's roughly 10 times worse than your chance of winning the jackpot in the New York State Millenium Millions game.
That won't stop Bob Schieffer of CBS News (and presumably other anchors on other networks that I wasn't watching) from telling you that the closeness of the election proves ``every vote can make a difference''. This is of a piece with the lottery propaganda that says ``If you don't play, you can't win''. True enough, but the right question is: what are the odds? And the odds are pitiful.
Moreover, if you'd hit the Millennium Millions jackpot last week, you'd have won $130 million. If you'd hit the jackpot in the voting booth by casting the deciding vote, you'd have won nothing more than the opportunity to choose the next president of the United States. That's not an inconsiderable prize, but I'm willing to bet that most people would rather have the $130 million.
And look at the cost of playing. A Millenium Millions ticket costs $2. Voting, by contrast, costs most people at least 15 minutes (including travel time). If your time is worth at least $8 an hour, then voting is more expensive than buying a lottery ticket. So compared to the lottery, voting offers a smaller chance at a less valuable prize for a higher price.
If you vote to affect the outcome, you're far more foolish than your neighbor who buys a lottery ticket to win the jackpot. On the other hand, maybe your neighbor isn't so foolish after all---maybe he's playing the lottery just for the fun of it, clearly understanding that he has no realistic chance of winning. And maybe that's why you voted, which is fine. I voted too. But I didn't fool myself into thinking my vote might matter.