Condorcet
The Condorcet method is the most sophisticated voting method. Here is an informal description and a few examples. Condorcet method is a pairwise election method where each candidate runs against all the others separately, and one counts the “defeats” X/Y, i.e. the number of times candidates X beats candidates Y versus the number of times X is beaten by Y.
Defeats X/Y are ordered according their magnitude, the magnitude being the number of times candidates X beats candidates Y. For instance in this ballot with three candidates:
We have the following defeats:
Since A beats all its opponents in separate races, it is the clear winner. In more complex situations, no candidate beats all the others. Then the counting proceeds by removing the weakest defeats until an unbeaten candidate is found, or we get a tie. For instance in this case:
we get:
and removing successively the weakest (smallest magnitude) defeat, we remain with the unbeaten candidate B which is the Condorcet winner. This is the algorithm describing the original Condorcet method, but we're actually not going to implement it (since it has some minor defects). We will use a state-of-the-art variation of Condorcet, the Cloneproof or Schwartz Sequential Dropping (SSD) Condorcet method.
Here the winner is A, who beats both B (35/25) and C (60/40). However, if the B voters strategically refuse to rank A while the A voters sincerely rank B, that will let B steal the election from A since
would give the defeats:
Removing the weakest defeat leaves B as the unbeaten candidate. Instant runoff and Borda method do not have this problem: voting C insincerely will go against the interest of B voters, since C will win, not B. However, it may be argued that these methods have even worse problems than Condorcet.
Nobody likes D, whereas three voters like A and three voters like B, but B voters also like A as a second choice, so we would expect A to win. However this is not the case using the Condorcet method, nor the Instant Runoff method. Let us consider first the Condorcet method tallying procedure. The defeats are:
Dropping the weakest defeats we still have a tie between A and B since they are both unbeaten. Still, it is clear that A must be winner, since it gets both first and second choices. The ambiguity is removed by the Borda method:
Condorcet tends to give more ties than other methods. This is a good thing in general, since it is better to return a tie in dubious cases, leaving the decision to a human, than to return a wrong result. Now, let's check if the Instant Runoff method gives a tie too:
Defeats X/Y are ordered according their magnitude, the magnitude being the number of times candidates X beats candidates Y. For instance in this ballot with three candidates:
40: C 35: A B C 25: B A C
We have the following defeats:
B/C:60/40 (B beats C 60 times, C beats B 40 times, the magnitude is 60) A/C:60/40 (A beats C 60 times, C beats A 40 times, the magnitude is 60) A/B:35/25 (A beats B 35 times, B beats A 25 times, the magnitude is 35)
Since A beats all its opponents in separate races, it is the clear winner. In more complex situations, no candidate beats all the others. Then the counting proceeds by removing the weakest defeats until an unbeaten candidate is found, or we get a tie. For instance in this case:
40: C 35: A B C 25: B C
we get:
C/A: 65/35 B/C: 60/40 A/B: 35/25
and removing successively the weakest (smallest magnitude) defeat, we remain with the unbeaten candidate B which is the Condorcet winner. This is the algorithm describing the original Condorcet method, but we're actually not going to implement it (since it has some minor defects). We will use a state-of-the-art variation of Condorcet, the Cloneproof or Schwartz Sequential Dropping (SSD) Condorcet method.
Shortcomings of the Condorcet method
Condorcet is probably the soundest ordinal method available today and it is favoured by many academics. Nevertheless it has its own shortcomings users should be aware of.- The Condorcet method is rather complex, difficult to explain and to grasp. Also, we'll not implement its simplest version, Plain Condorcet, but the SSD Condorcet method, which is even more complex to explain;
- Condorcet results cannot easily be converted in percentages, so are difficult to read for people used to the more common voting methods;
- Condorcet is inferior to the Borda method in the case of sincere voters, since relevant information about the relative rating is lost (but this very fact increases the robustness against insincere votes);
- Still there are cases where Condorcet is not 100% robust against insincere voters. Below is an example of such a case.
A case where Condorcet is not robust
Consider this (sincere) ballot:40: C 35: ABC 25: BAC
Here the winner is A, who beats both B (35/25) and C (60/40). However, if the B voters strategically refuse to rank A while the A voters sincerely rank B, that will let B steal the election from A since
40: C 35: ABC 25: BC
would give the defeats:
C/A: 65/35 B/C : 60/40 A/B: 35/25
Removing the weakest defeat leaves B as the unbeaten candidate. Instant runoff and Borda method do not have this problem: voting C insincerely will go against the interest of B voters, since C will win, not B. However, it may be argued that these methods have even worse problems than Condorcet.
What about ties?
In elections with a small number of voters (typical of decision making process in small committees), ties are relatively frequent and must be managed in some way. There are various ways to solve the ambiguities, for instance the President may decide, or a different voting method can be used to see if the ambiguity disappears. Here is an example with four candidates A, B, C, D and these votes by a six members committee:3: A C 3: B A
Nobody likes D, whereas three voters like A and three voters like B, but B voters also like A as a second choice, so we would expect A to win. However this is not the case using the Condorcet method, nor the Instant Runoff method. Let us consider first the Condorcet method tallying procedure. The defeats are:
A/D: 6/0A / C : 6/0 C/D: 3/0 B/D: 3/0
Dropping the weakest defeats we still have a tie between A and B since they are both unbeaten. Still, it is clear that A must be winner, since it gets both first and second choices. The ambiguity is removed by the Borda method:
A: 15 B: 9 C: 6 D: 0
Condorcet tends to give more ties than other methods. This is a good thing in general, since it is better to return a tie in dubious cases, leaving the decision to a human, than to return a wrong result. Now, let's check if the Instant Runoff method gives a tie too:
This election required 2 rounds.
Round #1: Removing D ... 3: A C - 3: B A - Round #2: Removing C ... TIE EXCEPTION: No winner can be established, since the algorithm cannot decide the weakest candidate between A and B.