User:MarkusSchulze/Schulze method examples
Example 1
[edit]Example (45 voters; 5 candidates):
- 5 ACBED (that is, 5 voters have order of preference: A > C > B > E > D)
- 5 ADECB
- 8 BEDAC
- 3 CABED
- 7 CAEBD
- 2 CBADE
- 7 DCEBA
- 8 EBADC
d[*,A] | d[*,B] | d[*,C] | d[*,D] | d[*,E] | |
---|---|---|---|---|---|
d[A,*] | 20 | 26 | 30 | 22 | |
d[B,*] | 25 | 16 | 33 | 18 | |
d[C,*] | 19 | 29 | 17 | 24 | |
d[D,*] | 15 | 12 | 28 | 14 | |
d[E,*] | 23 | 27 | 21 | 31 |
The graph of pairwise defeats looks as follows:
![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Schulze_method_example1.svg/300px-Schulze_method_example1.svg.png)
The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table lists the strongest path from candidate X to candidate Y. The critical defeats of the strongest paths are underlined.
p[*,A] | p[*,B] | p[*,C] | p[*,D] | p[*,E] | |
---|---|---|---|---|---|
p[A,*] | 28 | 28 | 30 | 24 | |
p[B,*] | 25 | 28 | 33 | 24 | |
p[C,*] | 25 | 29 | 29 | 24 | |
p[D,*] | 25 | 28 | 28 | 24 | |
p[E,*] | 25 | 28 | 28 | 31 |
Candidate E is a potential winner, because p[E,X] ≥ p[X,E] for every other candidate X.
As 25 = p[E,A] > p[A,E] = 24, candidate E is better than candidate A.
As 28 = p[E,B] > p[B,E] = 24, candidate E is better than candidate B.
As 28 = p[E,C] > p[C,E] = 24, candidate E is better than candidate C.
As 31 = p[E,D] > p[D,E] = 24, candidate E is better than candidate D.
As 28 = p[A,B] > p[B,A] = 25, candidate A is better than candidate B.
As 28 = p[A,C] > p[C,A] = 25, candidate A is better than candidate C.
As 30 = p[A,D] > p[D,A] = 25, candidate A is better than candidate D.
As 29 = p[C,B] > p[B,C] = 28, candidate C is better than candidate B.
As 29 = p[C,D] > p[D,C] = 28, candidate C is better than candidate D.
As 33 = p[B,D] > p[D,B] = 28, candidate B is better than candidate D.
Therefore, the Schulze ranking is E > A > C > B > D.
Example 2
[edit]Example (30 voters; 4 candidates):
- 5 ACBD
- 2 ACDB
- 3 ADCB
- 4 BACD
- 3 CBDA
- 3 CDBA
- 1 DACB
- 5 DBAC
- 4 DCBA
d[*,A] | d[*,B] | d[*,C] | d[*,D] | |
---|---|---|---|---|
d[A,*] | 11 | 20 | 14 | |
d[B,*] | 19 | 9 | 12 | |
d[C,*] | 10 | 21 | 17 | |
d[D,*] | 16 | 18 | 13 |
The graph of pairwise defeats looks as follows:
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Schulze_method_example2.svg/210px-Schulze_method_example2.svg.png)
The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table lists the strongest path from candidate X to candidate Y. The critical defeats of the strongest paths are underlined.
p[*,A] | p[*,B] | p[*,C] | p[*,D] | |
---|---|---|---|---|
p[A,*] | 20 | 20 | 17 | |
p[B,*] | 19 | 19 | 17 | |
p[C,*] | 19 | 21 | 17 | |
p[D,*] | 18 | 18 | 18 |
Candidate D is a potential winner, because p[D,X] ≥ p[X,D] for every other candidate X.
As 18 = p[D,A] > p[A,D] = 17, candidate D is better than candidate A.
As 18 = p[D,B] > p[B,D] = 17, candidate D is better than candidate B.
As 18 = p[D,C] > p[C,D] = 17, candidate D is better than candidate C.
As 20 = p[A,B] > p[B,A] = 19, candidate A is better than candidate B.
As 20 = p[A,C] > p[C,A] = 19, candidate A is better than candidate C.
As 21 = p[C,B] > p[B,C] = 19, candidate C is better than candidate B.
Therefore, the Schulze ranking is D > A > C > B.
Example 3
[edit]Example (30 voters; 5 candidates):
- 3 ABDEC
- 5 ADEBC
- 1 ADECB
- 2 BADEC
- 2 BDECA
- 4 CABDE
- 6 CBADE
- 2 DBECA
- 5 DECAB
d[*,A] | d[*,B] | d[*,C] | d[*,D] | d[*,E] | |
---|---|---|---|---|---|
d[A,*] | 18 | 11 | 21 | 21 | |
d[B,*] | 12 | 14 | 17 | 19 | |
d[C,*] | 19 | 16 | 10 | 10 | |
d[D,*] | 9 | 13 | 20 | 30 | |
d[E,*] | 9 | 11 | 20 | 0 |
The graph of pairwise defeats looks as follows:
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Schulze_method_example3.svg/300px-Schulze_method_example3.svg.png)
The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table lists the strongest path from candidate X to candidate Y. The critical defeats of the strongest paths are underlined.
p[*,A] | p[*,B] | p[*,C] | p[*,D] | p[*,E] | |
---|---|---|---|---|---|
p[A,*] | 18 | 20 | 21 | 21 | |
p[B,*] | 19 | 19 | 19 | 19 | |
p[C,*] | 19 | 18 | 19 | 19 | |
p[D,*] | 19 | 18 | 20 | 30 | |
p[E,*] | 19 | 18 | 20 | 19 |
Candidate B is a potential winner, because p[B,X] ≥ p[X,B] for every other candidate X.
As 19 = p[B,A] > p[A,B] = 18, candidate B is better than candidate A.
As 19 = p[B,C] > p[C,B] = 18, candidate B is better than candidate C.
As 19 = p[B,D] > p[D,B] = 18, candidate B is better than candidate D.
As 19 = p[B,E] > p[E,B] = 18, candidate B is better than candidate E.
As 20 = p[A,C] > p[C,A] = 19, candidate A is better than candidate C.
As 21 = p[A,D] > p[D,A] = 19, candidate A is better than candidate D.
As 21 = p[A,E] > p[E,A] = 19, candidate A is better than candidate E.
As 20 = p[D,C] > p[C,D] = 19, candidate D is better than candidate C.
As 30 = p[D,E] > p[E,D] = 19, candidate D is better than candidate E.
As 20 = p[E,C] > p[C,E] = 19, candidate E is better than candidate C.
Therefore, the Schulze ranking is B > A > D > E > C.
Example 4
[edit]Example (9 voters; 4 candidates):
- 3 ABCD
- 2 DABC
- 2 DBCA
- 2 CBDA
d[*,A] | d[*,B] | d[*,C] | d[*,D] | |
---|---|---|---|---|
d[A,*] | 5 | 5 | 3 | |
d[B,*] | 4 | 7 | 5 | |
d[C,*] | 4 | 2 | 5 | |
d[D,*] | 6 | 4 | 4 |
The graph of pairwise defeats looks as follows:
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/88/Schulze_method_example4.svg/210px-Schulze_method_example4.svg.png)
The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table lists the strongest path from candidate X to candidate Y. The critical defeats of the strongest paths are underlined.
p[*,A] | p[*,B] | p[*,C] | p[*,D] | |
---|---|---|---|---|
p[A,*] | 5 | 5 | 5 | |
p[B,*] | 5 | 7 | 5 | |
p[C,*] | 5 | 5 | 5 | |
p[D,*] | 6 | 5 | 5 |
Candidate B and candidate D are potential winners, because p[B,X] ≥ p[X,B] for every other candidate X and p[D,Y] ≥ p[Y,D] for every other candidate Y.
As 7 = p[B,C] > p[C,B] = 5, candidate B is better than candidate C.
As 6 = p[D,A] > p[A,D] = 5, candidate D is better than candidate A.
Possible Schulze rankings are B > C > D > A, B > D > A > C, B > D > C > A, D > A > B > C, D > B > A > C, and D > B > C > A.