Paradox of voting
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The paradox of voting, also referred to as Downs paradox is a reference to the fact that for a rational, self-interested voter, the costs of voting will normally exceed the expected benefits. Because the chance of exercising a decisive vote (i.e. the change of a tied election) is tiny compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the costs. The fact that people do vote is a major problem for public choice theory, first observed by Anthony Downs [1].
[edit] Illustration
As a first approximation, it seems reasonable to argue that the likelihood of voter x being decisive, that is, of the vote being tied in the absence of voter x is of the order of 1/n where n is the number of voters. In a typical national election in the United States, n will be of the order of 100 million. On the other hand, the benefit to an individual voter of one party winning, rather than the other, is presumably smaller than the individual's personal wealth which for most people is less than $1 million. So, the expected benefit of voting is less than 1 cent for any given person, much less than the cost of turning out to vote.
On the assumption of rational egoism, the number of participants in an election should decline to the point where the expected benefit of voting is equal to the cost. In the example above, if the cost of voting is $1, this point would be achieved if turnout fell to 1 per cent.
[edit] Responses
One possible response to the paradox of voting is to argue that the probabiltiy of being decisive is much larger than 1/n. It has been argued that if each individual voter is equally likely to vote either way then the binomial theorem yields a probability of a tie of the order of 1/n^1/2, where n is the number of voters . Unfortunately, this approach also implies that the result of, say, a US presidential electioncan be known, with virtual certainty, to within a few thousand votes. More realistic models yields a probability that is somewhat larger than 1/n, but still very small.
Alternative responses modify the postulate of egoistic rationality in various ways. For example, Brennan and Lomasky suggest that voters derive 'expressive' benefits from supporting particular candidates. However, this implies that voting choices are unlikely to reflect the self-interest of voters, as is normally assumed in public choice theory.
