Condorcet method

Possible ballot

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

Each voter ranks the candidates by rank, with multiple candidates of the same rank possible. During the evaluation, the data from the votes are used to simulate duels in which every candidate competes against every other candidate. This counts how often a candidate is placed above his opponent. Whoever wins each of these fights is a Condorcet winner.

All Condorcet methods completely agree on the winner if someone is a Condorcet winner. They differ in who they choose to be the winner when there is no Condorcet winner.

The social choice theory examines and compares u. a. different aggregation methods and their problems and advantages.

The possibility of tactical voting behavior of the voters with the aim of achieving the best possible election result is not taken into account. ("I would like candidate A best, but since he has no prospect of winning, I am voting for candidate B, who is second best for me.") Such considerations cannot be ruled out in real votes.

Definitions

Let a set of candidates K = {k ₁ … k _n } be given. Each participating voter x now brings these candidates into a total preference order ≤ _x , i.e. H. indicates which candidates he prefers to which others or which he rates equally.

Preference

A candidate k _i is preferred over a candidate k _j if there are more voters x for whom k _i < _x k _j than voters with k _j < _x k _i .

Condorcet winner

If there is a candidate (or policy) who defeats every other possible candidate in a pair-wise vote, that candidate is called the Condorcet winner . (There does not necessarily have to be one, see below.)

Condorcet losers

If there is one candidate to whom all other candidates are preferred, that candidate is the Condorcet loser . (This does not necessarily have to be there either.)

Condorcet criterion

An election procedure (general) fulfills the Condorcet criterion if, in cases where there is a Condorcet winner, this winner is also chosen.

Condorcet loser criterion

An election procedure (general) fulfills the Condorcet loser criterion if, in cases in which there is a Condorcet loser, this loser is definitely not chosen.

General example with three candidates

There are three candidates or options A, B and C. The voters must now specify a list of preferences. The election result is:

rank	u	v	w	x	y	z
1	A.	A.	B.	B.	C.	C.
2	B.	C.	A.	C.	A.	B.
3	C.	B.	C.	A.	B.	A.

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

(I) u + v + y> w + x + z

and

(II) u + v + w> x + y + z.

The first inequality means that A is preferred over B (because u, v and y rate A before B, the others do not), the second says that A also beats C.

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

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 results .

Paradoxical quirks

It is possible that there will be a majority favoring candidate A over B, as well as 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 electoral method, but that Condorcet only shows real existing, differently composed (and thus not so paradoxical) majorities.

Another aspect that goes against intuition is the low importance of the first choice in comparison with another ranking method, instant runoff voting (IRV). It is entirely possible that the Condorcet winner was not voted first by anyone.

Examples

100 Wähler, 3 Kandidaten
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 C's victory over A is the most unspectacular, it makes sense to ignore it. If you do that, A is the winner.

If a candidate receives over 50% first placements, he will also win every duel. If the voter is allowed to give several candidates the same rank (and Condorcet advocates stand for it) and there are several candidates with over 50% first placements, the winner comes from this same group with over 50%. But then it is not necessarily the one with the most first places, as the following example shows:

100 Wähler, 3 Kandidaten
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 due to the fact that equal placements are in principle counted like abstentions.

If no candidate achieves over 50% first placements, someone without a single first place can become the winner. A particularly drastic example:

100 Wähler, 4 Kandidaten
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 weighting of the first placements compared to IRV means that the voter is exposed to significantly less pressure to compromise with good chances over a favorite with bad chances (low “ spoiler effect ”).

Different Condorcet methods

The currently most widely used Condorcet method is the Schulze method . It is used by the Pirate Party Germany , Wikimedia , Debian , Gentoo , Software in the Public Interest (SPI) and Sender Policy Framework (SPF) , among others .