Assume that Uv = the expected utility of voting; Cv = the cost of voting; and D = the expected difference in welfare per person if the voter’s preferred candidate defeats her opponent. Let us further assume that this is a presidential election in a nation with 300 million people; that the voter’s ballot has only a 1 in 100 million chance of being decisive (Riker and Ordeshook 1968); and that the voter values the welfare of his fellow citizens an average of 1000 times less than his own. Thus, we get the following equation:
D*(300 million/1000)/(100 million) – Cv = Uv.
If we assume that Cv is $10 (a reasonable proxy for the cost of voting) and that D is $5000 (this can incorporate monetary equivalents of noneconomic benefits as well as actual income increases), then Uv equals $5, a small but real positive expected utility.
If you care about the well-being of others, even a little bit, you should vote, despite the cost of voting. The reason is that the cost of voting is very low, while the benefit is not as low as you might think. Although your chance of breaking a tie is very low, the benefit from breaking a tie is very high—it’s felt by 300 million people. This multiplier effect offsets, to some extent, the very small chance that your vote will make a difference.
However, breaking a tie is beneficial only if your vote is more likely correct than not—that is, you actually vote for the better candidate. Surely your vote is more likely to be correct than not? After all, you have some information, and that means you are doing better than flipping a coin. However, you need to reflect on your own ignorance with some humility. If, by hypothesis, your vote breaks a tie, then it means that (putting aside the vagaries of the electoral system) half the country prefers one candidate and the other half prefers the other. If all of these people have enough information that their votes are not random, the existence of a tie (aside from your vote) indicates that the two candidates are almost exactly equal in quality. The probability that your own puny knowledge (elsewhere in the same article Ilya discusses the problem of rational ignorance—people have weak incentives to inform themselves about the candidates and policy in general) will distinguish the infinitesimally better candidate is itself infinitesimal.
In other words, D, the expected difference in welfare per person if the voter’s preferred candidate defeats her opponent, is not realistically $5,000; more realistically it is in the range of $0.000000005. Using the equation above, your expected utility from voting is a shade higher than negative ten dollars. Ilya, stay home!