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User:MarkusSchulze/Schulze method examples

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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 matrix of pairwise defeats looks as follows:

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.

... to A ... to B ... to C ... to D ... to E
from A ...
A-(30)-D-(28)-C-(29)-B
A-(30)-D-(28)-C
A-(30)-D
A-(30)-D-(28)-C-(24)-E
from A ...
from B ...
B-(25)-A
B-(33)-D-(28)-C
B-(33)-D
B-(33)-D-(28)-C-(24)-E
from B ...
from C ...
C-(29)-B-(25)-A
C-(29)-B
C-(29)-B-(33)-D
C-(24)-E
from C ...
from D ...
D-(28)-C-(29)-B-(25)-A
D-(28)-C-(29)-B
D-(28)-C
D-(28)-C-(24)-E
from D ...
from E ...
E-(31)-D-(28)-C-(29)-B-(25)-A
E-(31)-D-(28)-C-(29)-B
E-(31)-D-(28)-C
E-(31)-D
from E ...
... to A ... to B ... to C ... to D ... to E
The strongest paths are:
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
The strengths of the strongest paths are:

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 matrix of pairwise defeats looks as follows:

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.

... to A ... to B ... to C ... to D
from A ...
A-(20)-C-(21)-B
A-(20)-C
A-(20)-C-(17)-D
from A ...
from B ...
B-(19)-A
B-(19)-A-(20)-C
B-(19)-A-(20)-C-(17)-D
from B ...
from C ...
C-(21)-B-(19)-A
C-(21)-B
C-(17)-D
from C ...
from D ...
D-(18)-B-(19)-A
D-(18)-B
D-(18)-B-(19)-A-(20)-C
from D ...
... to A ... to B ... to C ... to D
The strongest paths are:
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
The strengths of the strongest paths are:

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 matrix of pairwise defeats looks as follows:

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.

... to A ... to B ... to C ... to D ... to E
from A ...
A-(18)-B
A-(21)-D-(20)-C
A-(21)-D
A-(21)-E
from A ...
from B ...
B-(19)-E-(20)-C-(19)-A
B-(19)-E-(20)-C
B-(19)-E-(20)-C-(19)-A-(21)-D
B-(19)-E
from B ...
from C ...
C-(19)-A
C-(19)-A-(18)-B
C-(19)-A-(21)-D
C-(19)-A-(21)-E
from C ...
from D ...
D-(20)-C-(19)-A
D-(20)-C-(19)-A-(18)-B
D-(20)-C
D-(30)-E
from D ...
from E ...
E-(20)-C-(19)-A
E-(20)-C-(19)-A-(18)-B
E-(20)-C
E-(20)-C-(19)-A-(21)-D
from E ...
... to A ... to B ... to C ... to D ... to E
The strongest paths are:
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
The strengths of the strongest paths are:

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 matrix of pairwise defeats looks as follows:

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.

... to A ... to B ... to C ... to D
from A ...
A-(5)-B
A-(5)-C
A-(5)-C-(5)-D
from A ...
from B ...
B-(5)-D-(6)-A
B-(7)-C
B-(5)-D
from B ...
from C ...
C-(5)-D-(6)-A
C-(5)-D-(6)-A-(5)-B
C-(5)-D
from C ...
from D ...
D-(6)-A
D-(6)-A-(5)-B
D-(6)-A-(5)-C
from D ...
... to A ... to B ... to C ... to D
The strongest paths are:
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
The strengths of the strongest paths are:

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.