User:DCary/WIP/Beatpath

(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 used to describe the Condorcet paradox, the Schulze method, and the Schwartz set (draft) (current).

Beatpath definition
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:


 * 1) X = C(0)
 * 2) Y = C(n)
 * 3) for every k = 0,...,n-1: C(k) beats C(k+1) in their pair-wise contest.

Beatpath strength
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 beatpath strength correspond to different measures of winning strength for a pair-wise contest. Winning strength can be expressed quantitatively as a number, or more generally, it can be expressed ordinally as a relation on parameters of a pair-wise contest, such as the number of votes for the winning and losing candidate, as for example with recent specifications of the Schulze method. 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 differ to the extent that:
 * Total votes = votes for the winning candidate + votes for the losing candidate + abstaining votes.

Cycles
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.


 * 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.

Cycle equivalence
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).

Beatpath order
Beatpaths create a partial order on the set of all cycle equivalence classes.

Definition
The beatpath order is defined by:


 * Cec(X) &gt; Cec(Y) if and only if there is a beatpath from X to Y and there is not a beatpath from Y to X,

or correspondingly:
 * Cec(X) &ge; 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) &ge; Cec(Y); but it is not necessarily true that Cec(X) &gt; Cec(Y).


 * If Cec(X) &gt; 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) &gt; Cec(Y) and Cec(X) is next to Cec(Y) [meaning there is no candidate W with Cec(X) &gt; Cec(W) &gt; Cec(Y)], then there is a candidate in Cec(X) that pair-wise beats a candidate B in Cec(Y).

Ties
With an ordering, there are three possible comparisons between two items:
 * Cec(X) = Cec(Y)
 * Cec(X) &gt; Cec(Y)
 * Cec(Y) &gt; 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) &gt; 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.

Alternative formulation
Given 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

then 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 above.

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.

The cycle equivalence classes are the strongly connected components of the graph of Beat.

Examples
Examples of beatpath orders with 3 candidates are here.

An example of a beatpath order with 12 candidates is here.