Ties and Alternative Implementations
When allowing users to have ties in their preferences, the outcome of the Schulze method naturally depends on how these ties are interpreted in defining d. Two natural choices are that d 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.
Although ties in the Schulze ranking are unlikely, they are possible. Schulze's original paper 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 delete all candidates not in the Schwartz set (i.e. any candidate which cannot reach all others) and delete the weakest link
- the winner is the last non-deleted candidate.
Read more about this topic: Schulze Method
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