[EM] RP ranking criteria

Blake Cretney bcretney at postmark.net
Mon Apr 23 09:22:32 PDT 2001

Some of us have recently been referring to criteria that suggest the
ranking given by Ranked Pairs is in some way more internally
consistent than other methods.  This is going to be an overview of
those criteria.

The first, I'll call the Ranking Reinforcement criterion.  What this
says is, if the method gives you a ranking of the candidates where X
is ranked above Y, then on any ballot where Y is ranked immediately
above X, and both are ranked alone, you can switch the two, without
affecting the result.  The idea is that the switch only gives more
information in favour of the conclusion already reached, so it
shouldn't change the conclusion.

I can already imagine Demorep tapping away about how changing votes is
election fraud, so I'll just say this.  The idea of this criterion, is
not that ballots will actually be changed.  It is simply to suggest
that some methods behave in ways that don't make sense, and we can see
this in how they respond to additional information.  This is also the
reasoning behind monotonicity.

Also, it could be argued that a criterion based on ranking doesn't
matter unless we want a complete ranking.  But if a method has an
obvious interpretation in terms of giving a complete ranking, and that
ranking doesn't make sense in some way, then we might not want to
trust it when it comes to picking a winner either, unless there is
some good reason to.

The second property, I can't think of a name for.  This is that if you
take an RP ranking, and you eliminate the top ranked candidate, the
new RP ranking places the remaining candidates in the same order as
they had in the old ranking.  The same is true if you eliminate the
bottom ranked candidate.

This is certainly what intuition says should happen, although
intuition is often wrong.  For example, you still can't remove any
arbitrary candidate, and expect the ranking to hold.  You can only
remove the topmost or bottommost candidate.  Of course, since you can
do this repeatedly, you could actually remove any number of topmost
and bottommost candidates, and expect the ranking to hold for the
remaining candidates.

In general, we have tended to favour methods that meet independence of
irrelevant alternatives as much as possible (given ranked ballots). 
This is another kind of independence, beyond the local independence
and independence of clones criteria.  The result is independent of
dropping any number of last ranked candidates.

Also, it has a modest practical benefit.  It would often be nice when
electing two people, to hold one election and pick the top two ranked,
rather than hold two elections.  Unless voter's opinions change
between elections, it doesn't matter which you do in Ranked Pairs, so
the decision can be made on the basis of practicality.  In other
methods, it can affect the outcome, so the choice might made to
achieve a particular result.

The third is the following, lets say a method gives a complete ranking
A>B>C>D, but B pairwise loses to C.  We can change the ranking in
agreement with the B>C victory without affecting agreement with any
other majority.  That is, we could have A>C>B>D just as easily, from
this perspective.  So, the third property of the RP ranking is that
each candidate pairwise beats the next lowest ranked candidate. 

Blake Cretney   http://www.fortunecity.com/meltingpot/harrow/124/path

Ranked Pairs gives the ranking of the options that always reflects 
the majority preference between any two options, except in order to
reflect majority preferences with greater margins. 
(B. T. Zavist & T. Tideman, "Complete independence  of clones in the 
ranked pairs rule", Social choice and welfare, vol 6, 167-173, 1989)

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