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 chance 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.
Thanks to Ben for the related Wiki entry on Arrow’s Impossibility Theorem:
In voting systems, Arrowâ€™s impossibility theorem, or Arrowâ€™s paradox, demonstrates that no voting system based on ranked preferences can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. These criteria are called unrestricted domain, non-imposition, non-dictatorship, monotonicity, and independence of irrelevant alternatives.
The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his Ph.D. thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was entitled “A Difficulty in the Concept of Social Welfare”.  Arrow was a co-recipient of the 1972 Nobel Prize in Economics.