投票悖论

示例

假设甲乙丙三人，面对ABC三个备选方案，有如以下的偏好排序：

无论投票结果是什么，都会有至少两人认可的更优方案存在。

如果投票结果是A，则乙和丙都认为C比A更好。当结果是C时，甲和乙都认为B比C好。无论结果如何，多数人都会倾向于另一套方案。

In social choice theory, Condorcet's voting paradox (also called Condorcet's paradox or the Condorcet paradox) is a fundamental discovery by the Marquis de Condorcet that majority rule can be self-contradictory as majority preferences can become intransitive with three or more options. For example, there can be rock-paper-scissors scenarios where a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called Condorcet cycles or cyclic ties.

In such a cycle, every possible choice is rejected by the electorate in favor of another alternative, who is preferred by more than half of all voters. Thus, any attempt to ground social decision-making in majoritarianism must accept such self-contradictions (commonly called spoiler effects). Systems that attempt to do so, while minimizing the rate of such self-contradictions, are called Condorcet methods.

Condorcet's paradox is a special case of Arrow's paradox, which shows that any kind of social decision-making process is either self-contradictory or a dictatorship.

History

Condorcet's paradox was first discovered by Catalan philosopher and theologian Ramon Llull in the 13th century, during his investigations into church governance, but his work was lost until the 21st century. The mathematician and political philosopher Marquis de Condorcet rediscovered the paradox in the late 18th century.

Condorcet's discovery means he arguably identified the key result of Arrow's impossibility theorem, albeit under stronger conditions than required by Arrow: Condorcet cycles create situations where any ranked voting system that respects majorities must have a spoiler effect.

Example

Suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows:

If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. Thus the society's preferences show cycling: A is preferred over B which is preferred over C which is preferred over A.

As a result, any attempt to appeal to the principle of majority rule will lead to logical self-contradiction. Regardless of which alternative we select, we can find another alternative that would be preferred by most voters.

Practical scenario

The voters in Cactus County prefer the incumbent county executive Alex of the Farmers' Party over rival Beatrice of the Solar Panel Party by about a 2-to-1 margin.

Likelihood

It is possible to estimate the probability of the paradox by extrapolating from real election data, or using mathematical models of voter behavior, though the results depend strongly on which model is used.

Impartial culture model

We can calculate the probability of seeing the paradox for the special case where voter preferences are uniformly distributed among the candidates. (This is the "impartial culture" model, which is known to be a "worst-case scenario"—most models show substantially lower probabilities of Condorcet cycles.)

For n voters providing a preference list of three candidates A, B, C, we write Xₙ (resp. Yₙ, Zₙ) the random variable equal to the number of voters who placed A in front of B (respectively B in front of C, C in front of A). The sought probability is pₙ=2P(Xₙ>n/2,Yₙ>n/2,Zₙ>n/2) (we double because there is also the symmetric case A> C> B> A).

Implications

When a Condorcet method is used to determine an election, the voting paradox of cyclical societal preferences implies that the election has no Condorcet winner: no candidate who can win a one-on-one election against each other candidate. There will still be a smallest group of candidates, known as the Smith set, such that each candidate in the group can win a one-on-one election against each of the candidates outside the group. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. The Condorcet methods which always elect someone from the Smith set when there is no Condorcet winner are known as Smith-efficient. Note that using only rankings, there is no fair and deterministic resolution to the trivial example given earlier because each candidate is in an exactly symmetrical situation.