User:MarkusSchulze/Schulze method examples

= Example 1 =

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

The graph of pairwise defeats looks as follows:



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.

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 =

Example (30 voters; 4 candidates):


 * 5 ACBD
 * 2 ACDB
 * 3 ADCB
 * 4 BACD
 * 3 CBDA
 * 3 CDBA
 * 1 DACB
 * 5 DBAC
 * 4 DCBA

The graph of pairwise defeats looks as follows:



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.

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 =

Example (30 voters; 5 candidates):


 * 3 ABDEC
 * 5 ADEBC
 * 1 ADECB
 * 2 BADEC
 * 2 BDECA
 * 4 CABDE
 * 6 CBADE
 * 2 DBECA
 * 5 DECAB

The graph of pairwise defeats looks as follows:



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.

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 =

Example (9 voters; 4 candidates):


 * 3 ABCD
 * 2 DABC
 * 2 DBCA
 * 2 CBDA

The graph of pairwise defeats looks as follows:



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.

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.