Paradox of Voting

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Paradox of Voting

The most common form of the paradox of voting refers to a situation where the outcome of majority-rule voting over a discrete set of candidates produces no clear winner, even though each individual voter has a clear and transitive rank ordering of preferences over the alternative options. The paradox is that although individual preferences are transitive, the preferences of the majority are cyclical. Thus, although each individual voter has a most preferred candidate, a “reasonable” majority-rule method of voting produces no clear winner.

To see the paradox at work, consider this example. Adam, Bala, and Chen are three candidates for a position on the school committee. There are three voters, whose preferences are as follows:

First Voter: 1. Adam, 2. Bala, 3. Chen;

Second Voter: 1. Chen, 2. Adam, 3. Bala;

Third Voter: 1. Bala, 2. Chen, 3. Adam.

Who should be declared the winner if each voter declares their rankings? Two out of three voters (First Voter and Second Voter) prefer Adam over Bala. Similarly, two out of three voters (Second Voter and Third Voter) prefer Chen over Adam. Should Chen be declared the winner? Not quite. Two out of three voters (First Voter and Third Voter) prefer Bala over Chen, thereby leading to no clear resolution.

The potential for such a paradox was first noted by the marquis de Condorcet (1743–1794), the French mathematician, philosopher, and social scientist, in his Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix ( Essay on the Application of Analysis to the Probability of Majority Decisions, 1785). The voting method used in the example is the so-called Condorcet method, which can be summarized as follows: First, rank each candidate in order of preference (tied ranking is allowed), and then compare each candidate with every other candidate and find a winner for each pair-wise comparison. The candidate that tallies the biggest wins across all pair-wise comparisons wins the election; however, as suggested by the term paradox, there is no guarantee of a winner.

Since Condorcet, other scholars have discussed the paradox and its broader implications, most notably Kenneth Arrow in his seminal work Social Choice and Individual Values (1951). Arrow postulated five “rational” and “ethical” criteria that any social-welfare function must meet, and showed that there is no method of aggregating individual preferences over three or more alternatives that satisfies these criteria and always produces a fair and logical result. Much of the work on social choice theory that has followed Arrow’s results either validates his conclusions or attempts to find a way around them.

Subsequent authors have attempted to resolve the original paradox of voting in various ways, including one that involves using the Condorcet method first, and if it produces no resolution, then using an alternative such as the “Borda count.” In a Borda count, each voter assigns points to candidates in order of his or her preference: If there are n candidates, each voter gives n points to his or her top ranked candidate(s), n – 1 points to the second ranked candidate(s), and so on. There are different formulae for assigning points to each voter’s preferences, with higher points being assigned to higher ranked candidates. The candidate with the highest number of points aggregated across all voters wins.

Other approaches involve taking a multistage approach to finding a winner. In the first stage, if there is no clear winner, a second voting method is used whereby the candidates are restricted in some way, for example with the smallest set such that every candidate in the set beats all candidates not included in this restricted set (the “Smith set”). Other approaches involve the farsightedness of voters. Ariel Rubinstein (1980) introduced the “stability set,” which produces a winner when voters not only make pair-wise comparisons, but also think one step ahead. Yet, Bhaskar Chakravorti (1999) has shown that this notion is itself limited, and if voters do not ignore farsightedness on the part of other voters and are “consistently” farsighted (that is, they can consider comparisons arbitrarily far ahead in the chain), then the paradox returns.

Many alternative voting systems have been proposed to ensure a fair resolution in most practical situations. Common alternatives include run-off elections; approval voting, where voters cast a vote for all the candidates of whom they approve; and the Borda count.

A second version of the paradox of voting is attributed to Anthony Downs (1957). According to Downs’s construct, a rational voter will refrain from voting because the costs of voting usually exceed the expected benefits. The probability of casting an election’s decisive vote is too small to make the benefits worthwhile, whereas the cost of going out to vote and forgoing other activities is positive and quite tangible. The fact that voters do, indeed, participate in elections to vote is paradoxical, given such a rational calculation. Various theories have been put forward to resolve or explain the Downs paradox. Some have suggested that voters consider factors other than the private cost-benefit analysis to decide whether or not to vote. Some vote because they consider it a responsibility and a social duty, whereas others vote to gain satisfaction from the fact they have registered their preferences in some way, even if it is not decisive, and derive utility from participating in a democratic process.

SEE ALSO Arrow Possibility Theorem; Arrow, Kenneth J.; Condorcet, Marquis de; Public Choice Theory; Voting