The Paradox

The criteria listed on the previous page seem like perfectly reasonable goals, but it is surprising to realize that even the most basic criterion, majority rule, is impossible to implement in a reasonable way. This is often referred to as Condorcet’s paradox.

Majority Rule Cycle

The situation in which paradox can arise is what is called a majority rule cycle. This is when, according to the result of the election, there is a cycle of candidates, each beating the next in a runoff. Imagine the simplest case: there are three candidates: A, B, and C. A majority rule cycle occurs when A defeats B, B defeats C, and C defeats A. This is a paradox because to install any one of these three candidates defies majority rule, since there exists another candidate which the majority prefers.

Is this Possible?

The first impression to this idea is that it appears bogus, but it is in fact a legitimate concern. What exactly it means for the basic notions of democracy is discussed below. To convince you of why it is not altogether impossible, consider the following thought experiment.

• Every citizen is given a card with “rock,” “paper,” or “scissors” written on it. Exactly one third of the citizens have each type.
• The government conducts will choose to "play" one of the three moves.
• Anyone voter who is beaten (rock beats scissors, scissors beats paper, paper beats rock) by the government must pay extra taxes; anyone who wins receives a tax rebate; anyone with the same move as the government is unaffected.

Notice that if the question of what move the government should play is put to the public, there is a majority rule cycle, in the following sense: two thirds of voters prefer rock to paper (those who have rock and those who have paper), and likewise two thirds prefer scissors to rock and paper to scissors. Thus if majority rule is taken to be the will of the people, the people contradict themselves.

Obviously this example is artificial, but it is not hard to imagine that something close to this could occur in some electoral decision process: there might exist three categories of people who would be harmed or benefitted by certain decisions in this cyclical way.

True Ambiguity?

Situations along these lines are literally ambiguous. There is no choice that majority rule prefers, thus the will of the people, if this term has any meaning, is ambiguous. The critical observation, however, is that this ambiguity is not an artifact of a Condorcet system. The ambiguity is present, regardless of the electoral system, but every other system simply hides it. For a particular example of how current electoral systems fail to consider the intricacy of this problem, consider the example of the California recall election of 2003 (we intend to create a page on this topic, and how it relates to Condorcet, in good time).

What Can be Done?

The solution is to acknowledge this possibility, recognize it when it occurs, and debate the best way to resolve it. The solution is not to sweep it under the rug as PluralityThis is the voting system currently used for most American elections. Each voter chooses only one candidate, and the winner is the candidate with the most votes. elections do, but rather to endorse CondorcetThe general term for any election method that uses ranked ballots and has, as it's first princple, the Condorcet Criterion: any candidate which beats every other candidate individually must win the election. Any Condorcet method must come along with an ambiguity resolution procedure for cases in which there is no winner by this first criterion., and bring this skeleton out of the closet. The only solution is to decide on a procedure, accepted by the citizenry, to reach a compromise and choose one of the competing choices.

In the rock-paper-scissors example, there is obviously no right answer, since the citizenry is divided exactly into thirds, but if this division is not exact, there could be a majority rule cycle, but still a clear best choice; suppose for example that 40% have rock, 30% scissors, 30% paper. In this case, it is clear that scissors is the best choice on the whole since it is best for the greatest number of people, yet fully 60% of the populace would still vote for paper over scissors.

The problem of resolving these ambiguities is discussed under Condorcet Systems.