Paradox of voting (1950)

Also known as Arrow’s theorem.

Developed by American economist Kenneth Arrow (1921- ), paradox of voting states that if there are more than two choices facing voters in a majority democratic selection process, a stalemate will result.

Also see: impossibility theorem

Source:
K J Arrow, ‘A Difficulty in the Concept of Social Welfare’, Journal of Political Economy, vol. LVIII (1950), 328-46;
Social Choice and Individual Values (New York, 1966)

History of scholarship

The issue was noted by Nicolas de Condorcet in 1793 when he stated, “In single-stage elections, where there are a great many voters, each voter’s influence is very small. It is therefore possible that the citizens will not be sufficiently interested [to vote]” and “… we know that this interest [which voters have in an election] must decrease with each individual’s [i.e. voter’s] influence on the election and as the number of voters increases.”^[1]

In 1821, Hegel made a similar observation in his Elements of the Philosophy of Right: “As for popular suffrage, it may be further remarked that especially in large states it leads inevitably to electoral indifference, since the casting of a single vote is of no significance where there is a multitude of electors.”^[2][3]

The mathematician Charles L. Dodgson, better known as Lewis Carroll, published the paper “A Method of Taking Votes on More than Two Issues” in 1876.^[4][5]

This problem in modern public choice theory was analysed by Anthony Downs in 1957.^[6]

Responses

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; that is, rational behavior is restricted to the instrumental as opposed to the intrinsic value of actions.

Some have hypothesized that voting is linked genetically with evolved behaviors such as cooperation. One study of identical and fraternal twins’ voting patterns concluded that 60% of differences in turnout among twins can be accounted for by genetics, but another interpretation of this study put the figure at 40%.^[7]

Another suggestion is that voters are rational but not fully egoistic. In this view voters have some altruism, and perceive a benefit if others (or perhaps only others like them) are benefited. They care about others, even if they care about themselves more. Since an election affects many others, it could still be rational to cast a vote with only a small chance of affecting the outcome. This view makes testable predictions: that close elections will see higher turnout, and that a candidate who made a secret promise to pay a given voter if they win would sway that voter’s vote less in large and/or important elections than in small and/or unimportant ones.

Some argue that the paradox appears to ignore the collateral benefits associated with voting, besides affecting the outcome of the vote. For instance, magnitudes of electoral wins and losses are very closely watched by politicians, their aides, pundits and voters, because they indicate the strength of support for candidates, and tend to be viewed as an inherently more accurate measure of such than mere opinion polls (which have to rely on imperfect sampling). However, these arguments themselves ignore that adherents to the paradox believe their individual vote makes a negligible difference; not only to the electoral outcome but also, by extension, to these supposed “collateral benefits”. Another argument that has been raised, is that researching who or what to vote for may increase the voter’s political knowledge and community awareness, both of which may contribute to a general sense of civic duty, although in such a case the act of voting itself contributes nothing to this.