Majority Cycle

A majority cycle is a majority voting system's bias leading to a circular result, namely, every alternative loses to at least one other alternative. From both rational and ethical perspectives, majority cycles are undesirable because they undermine the principle of transitivity and are unsuitable for reaching consistent decisions.

The discovery of this phenomenon lies in the works of M. J. Condorcet, illustrated by his famous voting paradox. Drawing on the works of J. C. Borda, who exposed in 1781 that the result of an election largely depended on the voting rules—majority voting or individual ranking of candidates by voters—Condorcet demonstrated in 1785 that the method of simple majority voting may yield a majority cycle. In the simplest case of a poll composed by three voters and three candidates:

• 1.
  A (two votes) defeats B (one vote)
• 2.
  B (two votes) defeats C (one vote)
• 3.
  C (two votes) defeats A (one vote)

This election outcome is not stable. In this case, there is no Condorcet winner—a candidate who is undefeated by any other feasible alternative—because a majority cycle occurs (A>B>C>A). This result is paradoxical because it violates rationality to maintain the moral principle of majority.

The heuristic potential of majority cycles was rediscovered in 1951 by K. J. Arrow through his investigations on collective decision-making systems. In 1963, Arrow acknowledged that, from a formal viewpoint, a decision-making system based on the aggregation of individual preferences must ensure their transitivity (if A>B and B>C, then A>C) and completeness (if A and B are candidates, the only alternatives are A>B and B>A). Such ideal systems also should comply with four moral axioms. First, whatever the individual preference orderings are, there should be defined a collective preference ordering. Second, if an individual prefers A to B and the other individuals' preferences remain the same, the social welfare function should ensure that society still prefers A to B. Third, collective preferences made from any set of available candidates should depend only on individual preferences with respect to those candidates. Fourth, collective preferences should not depend on one individual whose preferences overcome the preferences of the other individuals.

As Arrow pointed out, these rational and moral conditions are mutually incompatible. As a consequence, no voting system is able to avoid the formation of majority cycles and also be legitimate. This theorem has inspired an important literature on various solutions to this impossibility by reducing, multiplying, or reformulating Arrow's postulates.