Schulze Method - Example

Example

In the following example 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

The pairwise preferences have to be computed first. For example, when 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 = 20 and d = 25. The full set of pairwise preferences is:

The cells for d have a light green background if d > d, otherwise the background is light red. There is no undisputed winner by only looking at the pairwise differences here.

Now the strongest paths have to be identified. 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 with d. To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d > d (i.e. the table cells with light green background), omitting the one in the opposite direction (the table cells with light red background).

One example of computing the strongest path strength is p = 33: the strongest path from B to D is the direct path (B, D) which has strength 33. But when computing p, 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.The strength of a path is the strength of its weakest link.

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.

Now the output of the Schulze method can be determined. For example, when comparing A and B, since 28 = p > p = 25, for the Schulze method candidate A is better than candidate B. Another example is that 31 = p > p = 24, so candidate E is better than candidate D. Continuing in this way, the result is that the Schulze ranking is E > A > C > B > D, and E wins. In other words, E wins since p ≥ p for every other candidate X.