Condorcet method

Condorcet methods ( by Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet ) are preferential elections in which a candidate wins at least then, if it is preferred to any other candidate in a direct comparison.

Each voter ranks the candidates according to rank, with several candidates on the same rank are possible. When evaluating duels are simulated from the data of voting, in which each candidate competes against each other candidate. For this purpose, it is counted how often a candidate is placed over his opponent. Who will win any of these battles, is a Condorcet winner.

All Condorcet methods are in complete agreement about the winner, if someone Condorcet winner. They differ in who they define as the winner, if there is no Condorcet winner.

The social choice theory examines and compares, inter alia, different aggregation methods and their advantages and problems.

The possibility of tactical voting behavior of the voters is not considered with the aim to enforce the best possible choice for an even result. ( "While I would prefer candidate A, but since he has no chance to win, I vote for Candidate B, who is the second best for me." ) Such considerations can not be ruled out in the real polls.

• 5.1 See also

Definitions

Given a set of candidates K = { k1 ... kn }. Each participating voters x now brings these candidates in a preference - total ordering ≤ x, ie specifies which candidates he which other preferred over or which it considers to equal.

Preference

A candidate is a candidate ki kj preferred over x if there are more voters for the ki

Condorcet winner

If there is a candidate who is over all other candidates preferred, this is the Condorcet winner. ( One such it does not necessarily provide, see below. )

Condorcet loser

If there is a candidate to be the opposite of all the other candidates preferred, this is the Condorcet loser. ( Also, this does not necessarily give it. )

Condorcet criterion

A choice procedure ( general) satisfies the Condorcet criterion if, and this is chosen in cases where there is a Condorcet winner.

Condorcet loser criterion

A choice procedure ( general) satisfies the Condorcet loser criterion if, this will certainly not be elected to the cases where there is a Condorcet loser.

General Example for three candidates

There were three candidates or options A, B and C. The voters now have to specify a preference list. The election result was:

So u people wanted A rather than B and B rather than C, v people have the preference list ACB, BAC w people want and so on. Then A is then winner if:

(I), u v y > w x z, and

(II), u v w> x y z

The first inequality is, A is compared with B is preferably (for u, v and y values ​​A before B, the other does not ), the second is that A C also suggests.

For example, if u = 5, v = 3, w = 2 and x, y and z = 1 would be, would be A winner, for

I: 9> 4

(9 people see A before B, 4 see B before A ) and

II: 10 > 3

(10 people see A before C, only 3 see C before A).

For the case that u = x = y and v = w = z = 0, the Condorcet paradox arises.

Paradoxical peculiarities

It is possible that there is both a majority, the candidate A over B preferred, and B over C and C over A. This is called the Condorcet paradox. Condorcet defenders argue that this contradiction does not result from a defect in the method of election, but that Condorcet only really existing, be different composing (and thus not as paradoxical ) shows majorities.

Another intuition contradictory aspect is the low importance of the first-choice compared with another ranking method, instant - runoff voting. It is quite possible that the Condorcet winner was chosen by anyone in the first place.

Examples

100 voters, 3 candidates 40 A > B> C 35 B> C> A 25 C> A > B A vs. B 65 35 B vs. C 75 25 C vs A 60 40 A Condorcet paradox: A beats B, B beats C and C beats A. Since the victory of C on A is the least spectacular, lends itself to ignore this. If you do that, A is the winner.

If a candidate receives more than 50 % first-place finishes, this also wins every tackle. If the voter is allowed to give multiple candidates the same rank ( and Condorcet advocate advocate ) and there are several candidates with over 50% of first-place finishes, the winner comes from this same group with over 50 %. But then it is not necessarily the one with the most first-place finishes, as the following example shows:

100 voters, 3 candidates 60 A = B> C 39 C > B> A   1 A > C> B A vs. C 61 39 B vs. C 60 40 B vs A 39 1 B wins. This is because direct placements are considered in principle as abstentions.

If no candidate achieves over 50% of first-place finishes, and anyone can be a winner without a single initial placement. A particularly dramatic example:

100 voters, 4 candidates 49 A> B> C > D 26 C > B> D> A 25 D > B> C> A A vs. B 49 51 A vs. C 49 51 A vs. D 49 51 B vs. C 74 26 B vs. D 75 25 C vs. D 75 25 B wins every duel. A loses every duel.

This very low compared to IRV weighting of the first places means that the voter is exposed to a much lower pressure, a compromise with good opportunities to ask about a favorite with poor prospects ( less a "spoiler effect").

Various Condorcet methods