The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners. The Schulze method is also known as Schwartz Sequential Dropping (SSD), Cloneproof Schwartz Sequential Dropping (CSSD), the Beatpath Method, Beatpath Winner, Path Voting, and Path Winner.
The Schulze method is a Condorcet method, which means the following: if there is a candidate who is preferred over every other candidate in pairwise comparisons, then this candidate will be the winner when the Schulze method is applied.
The output of the Schulze method (defined below) gives an ordering of candidates. Therefore, if several positions are available, the method can be used for this purpose without modification, by letting the k top-ranked candidates win the k available seats. Furthermore, for proportional representation elections, a single transferable vote variant has been proposed.
One typical way for voters to specify their preferences on a ballot (see left) is as follows. Each ballot lists all the candidates, and each voter ranks this list in order of preference using numbers: the voter places a '1' beside the most preferred candidate(s), a '2' beside the second-most preferred, and so forth. Each voter may optionally:
give the same preference to more than one candidate. This indicates that this voter is indifferent between these candidates.
use non-consecutive numbers to express preferences. This has no impact on the result of the elections, since only the order in which the candidates are ranked by the voter matters, and not the absolute numbers of the preferences.
keep candidates unranked. When a voter doesn't rank all candidates, then this is interpreted as if this voter (i) strictly prefers all ranked to all unranked candidates, and (ii) is indifferent among all unranked candidates.
Schulze Method
Let d[V,W] be the number of voters who prefer candidate V to candidate W.
A path from candidate X to candidate Y of strength p is a sequence of candidates C(1),...,C(n) with the following properties:
C(1) = X and C(n) = Y.
For all i = 1,...,(n-1): d[C(i),C(i+1)] > d[C(i+1),C(i)].
For all i = 1,...,(n-1): d[C(i),C(i+1)] ≥ p.
p[A,B], the strength of the strongest path from candidate A to candidate B, is the maximum value such that there is a path from candidate A to candidate B of that strength. If there is no path from candidate A to candidate B at all, then p[A,B] = 0.
Candidate D is better than candidate E if and only if p[D,E] > p[E,D].
Candidate D is a potential winner if and only if p[D,E] ≥ p[E,D] for every other candidate E.
It can be proven that p[X,Y] > p[Y,X] and p[Y,Z] > p[Z,Y] together imply p[X,Z] > p[Z,X].1:§4.1 Therefore, it is guaranteed (1) that the above definition of "better" really defines a transitive relation and (2) that there is always at least one candidate D with p[D,E] ≥ p[E,D] for every other candidate E.
Example
Consider the following example, in which 45 voters rank 5 candidates.
5 ACBED (meaning, 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
First, we compute the pairwise preferences. For example, in comparing A and B pairwise, there are 5+5+3+7=20 voters who prefer A to B, and 8+2+7+8=25 voters who prefer B to A. So d[A, B] = 20 and d[B, A] = 25. The full set of pairwise preferences is:
Directed graph labeled with pairwise preferences d[*, *]
Matrix of pairwise preferences
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
To help visualize the strongest paths, the set of pairwise preferences is depicted in the diagram on the right in the form of a directed graph. An arrow from the node representing a candidate X to the one representing a candidate Y is labelled withl d[X, Y]. To avoid cluttering the diagram, we have only drawn an arrow from X to Y when d[X, Y] > d[Y,X], omitting the one in the opposite direction.
Recall that the strength of a path is the strength of its weakest link. One example of computing the strongest path strength is p[B, D] = 33: the strongest path from B to D is the direct path (B, D) which has strength 33. For contrast, let us also compute p[A, C]. The strongest path from A to C is not the direct path (A, C) of strength 26, rather the strongest path is the indirect path (A, D, C) which has strength min(30, 28) = 28.
For each pair of candidates X and Y, the following table shows the strongest path from candidate X to candidate Y in red, with the weakest link underlined.
Strongest paths
... 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
Strengths of the strongest paths
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
Now we can determine the output of the Schulze method. Comparing A and B for example, since 28 = p[A,B] > p[B,A] = 25, for the Schulze method candidate A is better than candidate B. Another example is that 31 = p[E,D] > p[D,E] = 24, so candidate E is better than candidate D. Continuing in this way we get the Schulze ranking is E > A > C > B > D, and E wins. In other words, E wins since p[E,X] ≥ p[X,E] for every other candidate X.
Implementation
The only difficult step in implementing the Schulze method is computing the strongest path strengths. However, this is a well-known problem in graph theory sometimes called the widest path problem. One simple way to compute the strengths therefore is a variant of the Floyd–Warshall algorithm. The following pseudocode illustrates the algorithm.
# Input: d[i,j], the number of voters who prefer candidate i to candidate j.
# Output: p[i,j], the strength of the strongest path from candidate i to candidate j.
for i from1 to C
for j from1 to C
if(i ≠ j) then
if(di,j > dj,i) then
pi,j := di,j
else
pi,j :=0
for i from1 to C
for j from1 to C
if(i ≠ j) then
for k from1 to C
if(i ≠ k and j ≠ k) then
pj,k :=max( pj,k,min( pj,i, pi,k ))
This algorithm is efficient, and has running time proportional to C3 where C is the number of candidates. (This does not account for the running time of computing the d[*,*] values, which if implemented in the most straightforward way, takes time proportional to C2 times the number of voters.)
Ties and alternative implementations
When we allow users to have ties in their preferences, the outcome of the Schulze method naturally depends on how we interpret these ties in defining d[*,*]. Two natural choices are that d[A, B] represents either the number of voters who strictly prefer A to B (A>B), or the margin of (voters with A>B) minus (voters with B>A). But no matter how the ds are defined, the Schulze ranking has no cycles, and assuming the ds are unique it has no ties.1
Although ties in the Schulze ranking are unlikely,2 they are possible. Schulze's original paper1 proposed breaking ties in accordance with a voter selected at random, and iterating as needed.
An alternative, slower, way to describe the winner of the Schulze method is the following procedure:
draw a complete directed graph with all candidates, and all possible edges between candidates
iteratively [a] delete all candidates not in the Schwartz set (i.e. any candidate which cannot reach all others) and [b] delete the weakest link
the winner is the last non-deleted candidate.
Satisfied and failed criteria
Satisfied criteria
The Schulze method satisfies the following criteria:
The main difference between the Schulze method and the ranked pairs method (both of which have tick the same boxes in the above table) can be seen in this example:
Suppose the MinMax score of a set X of candidates is the strength of the strongest pairwise win of a candidate A ∉ X against a candidate B ∈ X. Then the Schulze method, but not the ranked pairs method, guarantees that the winner is always a candidate of the set with minimum MinMax score.1:§4.8 So, in some sense, the Schulze method minimizes the strongest pairwise win that has to be overturned when determining the winner.
The Schulze method is not currently used in parliamentary elections. However, it has been used for parliamentary primaries in the Swedish Pirate Party. It is also starting to receive support in other public organizations. Organizations which currently use the Schulze method are:
Markus Schulze, Condorect sub-cycle rule, October 1997 (In this message, the Schulze method is mistakenly believed to be identical to the ranked pairs method.)
^ The MKM-IG uses Condorcet with dual dropping. That means: The Schulze ranking and the ranked pairs ranking are calculated and the winner is the top-ranked candidate of that of these two rankings that has the better Kemeny score. See:
