# Condorect sub-cycle rule

David Marsay djmarsay at dra.hmg.gb
Tue Jan 6 04:04:00 PST 1998

```Dear Markus, and anyone interested in Condorcet/Dodgson/Tideman.

Markus has described two criteria.

1. (attributed to Tideman) 'successively locking resp. skipping the
worst defeats'.

2. beat-path method.

Markus had thought that they were the same, but Mike Ossipoff (off
stage) conjectured that they were different. Markus has shown this,
and questions if (2) meats the twins criteria.

See  refs :
A. T.Tideman, "Independence of clones as a criterion for voting
rules" Social choice and welfare, vol 4, 185-206, 1987

B. T. Zavist & T.Tideman, "Complete independence of clones in the
ranked pairs rule", Social choice and welfare, vol 6, 167-173, 1989

Ref A gives Tideman's version of method 1 and ref B gives method 2.
Tideman says they are the same. His version says "When a
pair-ordering is encountered that cannot be preserved while
also preserving all pair-orderings with greater majorities, ..." In
Markus' example this clause applies to C>A, since one has a greater
majority for A > D > C. However, Markus only takes account of
conflicts with 'locked' rankings.

In terms of Condorcet's notion of evidence, in Markus' example the
evidence for A > C via D outweighs the direct evidence, even though
the evidence for D > C is itself outweighed. I have no intuition
about wether this ought to be expected or not, but note the
equivalence of Tideman and Dodgson.

Tideman shows that his methods 1 and 2 are equivalent.
By comparing his proof with Markus' example one can see the
problem: that a higher contrary majority may have already been
discounted without forcing a ranking on the pair under consideration.

Ref B gives a proof that method 2 satisfies his twin criterion. My
argument is simpler.
If I add a twin (or clone) I introduce new beats-paths, but they may
be matched with old ones of the same size of majority. Hence the
result is unchanged. As Tideman shows, you have to have a fair
tie-breaking criterion to make this work in all cases.

I feel sure that there is a definitive answer to 'How does one
determine the majority preference?', but we await a cogent
treatment.

It's the 100th anniversary of Dodgson's death. Can we expect some
relevant publications?
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Sorry folks, but apparently I have to do this. :-(
The views expressed above are entirely those of the writer
and do not represent the views, policy or understanding of
any other person or official body.

```