Electoral math  

One of the big problems in game theory (or maybe in people's heads) is the stark fact that people vote at all, given the fact that there's so little chance of casting the deciding vote. The probability of casting the deciding vote in any election is around 1/n where n is the number of voters. For most elections, that 1/n term is infinitessimal, but for voting to pay off in the game theoretical sense, the ratio of the value of the time one expends voting to the value of seeing one's candidate elected would actually have to be smaller than the 1/n term. As much as I hate George Bush, I can't say that seeing him out of office was 100 million times more valuable to me than the 20 min I spent voting (especially since my life expectancy isn't even close to 100 million minutes long). Still, I somehow found the time to pull the lever.

John Quiggen suggests here that some kind of "social" preferences (the quotes are his), beyond the usual sort of rational individual preferences, might account for the fact that people vote. There are a couple of ways of thinking about this. The social benefit perceived by the voter could be cumulative, that is, equal to the sum total of individual benefits in the entire country. This would easily offset the 1/n likelihood of casting the deciding vote because the multiplier would be n or greater. Alternatively, the perceived social benefit could be derived in some other way; however, it would still have to be the same order of magnitude as n in order to overcome the unlikelihood of an individual voter casting the deciding vote.

For me the obvious problem with this is that we don't see a lot of other spontaneous political activity on the part of those who vote. As a rational calculation, if these social preferences are actually large enough to get us to the polls, then shouldn't everybody (or at least, everybody who votes) be out there working night and day for political causes, writing letters, volunteering, etc? There are some who do these things, but they number far fewer than those who actually go out and vote.

A couple ideas might temper this problem somewhat. Quiggen has as interesting sugestion that the social benefit of victory in an election doesn't just take into account the winner, but also the size of the win; this might temper the effect somewhat. Another interesting possibility has to do with our perception of odds; I believe it's Richard Thaler who talks about the way extremely small or large probabilities tend to be perceived as less extreme; certainly this would be relevant to the questions above.

Ultimately though I think the most interesting answers to the question of why people vote are going to be those that look at tradition, habit, socialization, morality... these kinds of inputs don't have to counteract the extreme probabilities involved because both their origins and their payoffs are internal. This doesn't mean the idea of "social" preferences -- even the cumulative sort described above -- aren't relevant; an elegant solution might involve involve these very preferences, but restricted (through democratic tradition or ethics) to the voting booth.

UPDATE: I should have been clearer about the probability I mentioned above about the chances of an individual casting the decisive vote. Quiggen links to this paper (PDF) which explains (in the appendix):

If n individuals vote in an election, then the probability of a vote being decisive is roughly proportional to 1/n (see Good and Meyer, 1975, and Chamberlain and Rothchild, 1981). This result is derived based on the empirical fact that elections are unpredictable. Let f(d) be the predictive or forecast uncertainty distribution of the vote differential d (the difference in the vote proportions received by the two leading candidates). If n is not tiny, f(d) can be written, in practice, as a continuous distribution (e.g., a normal distribution with mean 0.04 and standard deviation 0.03). The probability of a decisive vote is then half the probability that a single vote can make or break an exact tie, or f(0)/n.

For example, if a Democrat is running against a Republican, and the difference between the two candidates' vote shares is expected to be in the range +/-10%, then the probability is about 1/(0.2n) = 5/n that a single added vote could create or break a tie. The exact probability of decisiveness depends on the election and one's knowledge about it, but even if an election is expected ahead of time to be close it is hard to imagine a forecast vote differential more precise than +/-2%, in which case the probability of a decisive vote is still at most 1/(0.04n) = 25/n. In practice, we see 10/n as a reasonable approximate probability of decisiveness in close elections, with lower probabilities for elections not anticipated to be close. Gelman, King, and Boscardin (1998), Mulligan and Hunter (2002), and Gelman, Katz, and Bafumi (2004) estimate these probabilities in more detail for elections for Presidential, Congressional, and other elections.

Some game-theoretic models have been proposed that suggest instrumental benefits for voter turnout (e.g., Feddersen and Pesendorfer, 1996), but these models also imply that large elections will be extremely close, and so they are not appropriate for real elections where the margin of victory varies by several percentage points from year to year. Under a coin-flipping model of voting, the probability of decisiveness is proportional to 1/(n^.5), but this model once again implies elections that are much closer than actually occur (see Mulligan and Hunter, 2002, and Gelman, Katz, and Bafumi, 2004).

MORE: Hei Lun explores this issue further at BTD.