(For information about the Beatpath method, see Schulze method).
In an election, a beatpath is a sequence of candidates such that each candidate in the sequence beats the next one in a pair-wise contest.
Beatpaths can be defined for any election in which candidates are evaluated in pair-wise contests.
A beatpath from candidate X to candidate Y is a series of candidates, C(0),...,C(n), for some n >= 1, where:
- X = C(0)
- Y = C(n)
- for every k = 0,...,n-1: C(k) beats C(k+1) in their pair-wise contest.
Note: The same candidate can occur more than once in the sequence. Also, the definition excludes zero-length paths, where n=0.
The strength of a beatpath is the minimum winning strength of each of the pair-wise victories of C(k) over C(k+1) in the beatpath. Different measures of winning strength for a pair-wise contest result in different measures of beatpath strength.
Winning strength can be expressed quantitatively as a number. Common quantitative measures of winning strength include:
- margins: the difference between the number of votes for the winning candidate
- winning votes: the number of votes for the winning candidate
- losing votes: the number of total votes minus the number of votes for the losing candidate
- ratio: the number of votes for the losing candidate divided by the number of votes for the winning candidate
These measures of winning strength can produce different effects to the extent that:
- Total votes = votes for the winning candidate + votes for the losing candidate + abstaining votes.
More generally, winning strength can be expressed comparatively as a relation on parameters of a pair-wise contest. The parameters can include the number of votes for the winning and losing candidate, as for example with recent specifications of the Schulze method.
A cycle is a beatpath that starts and ends at the the same candidate.
Two candidates X and Y are in a cycle when there is a cycle C(0),...,C(n) with X=C(i) and Y=C(j) for some i and j, i ≠ j.
- Two candidates are in a cycle if and only if there is a beatpath from X to Y and there is a beatpath from Y to X.
An equivalence relation can be defined on the set of candidates by:
- Candidate X is cycle equivalent to candidate Y if and only if X and Y are in a cycle or X = Y.
- For every candidate X, the cycle equivalence class of X, Cec(X), is the set consisting of X and all candidates that are in a cycle with X.
- For any candidates X and Y, either Cec(X) = Cec(Y) or Cec(X) is disjoint from Cec(Y).
Beatpaths create a partial order on the set of all cycle equivalence classes.
The beatpath order is defined by:
- Cec(X) > Cec(Y) if and only if there is a beatpath from X to Y and there is not a beatpath from Y to X,
- Cec(X) ≥ Cec(Y) if and only if X = Y or there is a beatpath from X to Y.
Compared to pair-wise wins
- If X beats Y, then Cec(X) ≥ Cec(Y); but it is not necessarily true that Cec(X) > Cec(Y).
- If Cec(X) > Cec(Y), then every candidate in Cec(X) pair-wise beats or ties every candidate in Cec(Y), and it is possible that every candidate in Cec(X) pair-wise ties with every candidate in Cec(Y).
- If Cec(X) > Cec(Y) and Cec(X) is next to Cec(Y) [meaning there is no candidate W with Cec(X) > Cec(W) > Cec(Y)], then there is a candidate in Cec(X) that pair-wise beats a candidate B in Cec(Y).
With an ordering, there are three possible comparisons between two items:
- Cec(X) = Cec(Y)
- Cec(X) > Cec(Y)
- Cec(Y) > Cec(X)
The beatpath ordering is a partial ordering, because there may be candidates X and Y such that Cec(X) and Cec(Y) are not comparable, meaning none of the three comparisons are true.
- If Cec(X) and Cec(Y) are not comparable, then every candidate in Cec(X) pair-wise ties with every candidate in Cec(Y).
- If there are no pair-wise ties among any of the candidates, then the beatpath order becomes a linear order.
Maximal cycle equivalence classes
A cycle equivalence class Cec(X) is maximal in the beatpath order if and only if there is no other candidate Y with Cec(Y) > Cec(X).
Because the beatpath order may only be a partial order, there can be more than one maximal cycle equivalence class. But if Cec(X) and Cec(Y) are different and both are maximal, then they are not comparable, and hence only have pair-wise ties between them.
If there are no pair-wise ties, the beatpath order is linear and there is exactly one cycle equivalence class set that is maximal.
The Schwartz set is the union of the maximal cycle equivalence classes in the beatpath order.
An alternative but equivalent formulation of the beatpath order is based on the binary relation Beat over the set of candidates defined by:
- (X,Y) is in Beat if and only if candidate X pair-wise beats candidate Y
The transitive-reflexive closure of Beat is a preorder on the set of candidates, and the natural partial order generated from that preorder is the beatpath order defined previously.
Typically, Beat is irreflexive and anti-symmetric, but this formulation does not depend on those properties. When Beat is irreflexive and anti-symmetric, the cycle equivalence classes can have 1, 3, or more candidates, but never exactly 2 candidates.
This alternative formulation highlights that beatpaths are used in the first definition of the beatpath order mostly as a way of expressing the transitive closure of the pair-wise victories.
A parallel concept is the beat-or-tie path, differing from beatpaths in that each candidate in the sequence is only required to beat or tie the next candidate in the sequence in their pair-wise contest. Beat-or-tie paths create beat-or-tie cycles, beat-or-tie cycle equivalence classes, and a beat-or-tie order on those beat-or-tie cycle equivalence classes.
Alternatively but equivalently, the beat-or-tie order is generated from the Beat-or-tie relation defined by:
- (X,Y) is in Beat-or-tie if and only if candidate X pair-wise beats or ties candidate Y
The transitive-reflexive closure of Beat-or-tie is a preorder on the set of candidates, and the natural partial order generated from that preorder is the beat-or-tie order defined by beat-or-tie paths.
If the election requires every candidate to be evaluated against every other candidate in a pair-wise contest which always results in one of the candidates beating the other, or in a tie, then the beat-or-tie order is a linear order. This is because for any two candidates X and Y, X beats-or-ties Y or Y beats-or-ties-X, so the beat-or-tie cycle equivalence classes of X and Y are always comparable. When the beat-or-tie order is linear, it has a single maximal cycle equivalence class that is the Smith set.
Given either the Beat relation or the Beat-or-tie relation as an N × N matrix, Kosaraju's algorithm can be used to partition the set of candidates into cycle equivalence classes (the strongly connected components of the graph of the relation). Kosaraju's algorithm runs in Θ(N2) time. The corresponding order and/or the maximal elements can also be determined in Θ(N2) time.
A version of the Floyd-Warshall algorithm can be used to identify the candidates in the maximal elements of the beatpath order or the beat-or-tie order. The Floyd-Warshall algorithm runs in Θ(N3) time.