Voting Paradox

A Dictionary of Philosophy, Third Edition

. Let three issues A, B, C, be voted on by three voters whose respective orders of preference are ABC, BCA, CAB. If the first vote is on two issues, and the second vote on the winner and the third issue, the third issue will always win, so that the winner will depend on the order in which the issues are voted on. This example also shows that majority preference is not TRANSITIVE, even when that of each individual is; for a majority prefers A to B, and a majority prefers B to C, but also a majority prefers C to A. This second feature shows that an electorate can be irrational in a way that none of its members need be.

It is sometimes called Arrow’s paradox because he used it to prove his ‘impossibility theorem’, that four plausibly desirable conditions cannot be satisfied together by any voting system.

Another paradox sometimes called the voting paradox, or paradox of democracy, asks how we can consistently hold, when outvoted, both that our favoured policy ought to be enacted and that the policy favoured by the majority ought to be enacted.

K.J.Arrow, ‘Values and collective decision-making’, in P.Laslett et al. (eds), Philosophy, Politics and Society, 3rd series, Blackwell, 1967, § IV.

R.Wollheim, ‘A paradox in the theory of democracy’, in P.Laslett et al. (eds), Philosophy, Politics and Society, 2nd series, Blackwell, 1962.