# [EM] Ranked Preferences

Juho juho4880 at yahoo.co.uk
Fri Nov 17 08:58:28 PST 2006

```Earlier I mentioned that the Ranked Preferences Method may need some
fine tuning to avoid situation where votes of some voters may work
against their interests. Now I found the time to do that.

The Ranked Preferences Method could be characterized by saying that
during the process of eliminating the candidates the votes are
treated at each round in a way that we suppose to be in the best
interest of the voters (no such strategic changes though that would
depend on the progress of the calculation). This means that
situations where the vote works against the voter's interests should
be eliminated - also to avoid generating any need to change the
sincere vote to a strategic one.

I'll explain the new variant of the method starting from how the
individual votes are handled. Capital letters indicate candidates
that are still in the race.

Vote a>>>B>c>>D>>e>F will be handled at this round (A, C and E have
already been eliminated) as if it was a=B=c>D>e=F since >> is the
highest preference relation that still has non-eliminated candidates
at both sides.

Relation >>> will be handled as = since after A has been eliminated B
and C are now her favourites and she has no interest to push them
down. This feature was not included in my previous description of the
method.

In addition to all this "dynamic interpretation" of the ranking based
votes the method also has the tied at top and tied at bottom rules.
Tied at top candidates are all considered to win each others but
never lose to each others (+1 point in all comparisons). Tied at
bottom candidates all lose to each others and never win each others
(-1 point in all comparisons). (These rules are included to eliminate
the need to put the top and bottom candidates in some preference
order for strategic reasons.)

Note that votes that use only one preference strength are calculated
just like in the regular Condorcet methods. Tied at top/bottom rules
however apply if = is used in addition to >.

At each round each vote is thus processed so that first the strongest
preference relation with non-eliminated candidates at both sides is
sought. Then all other preference relations (than preferences of this
strength that have non-eliminated candidates at both sides) are
considered to be =. After this the matrix is calculated as usual in
the Condorcet methods except that also tied at top and tied at bottom
rules apply. Also the already eliminated candidates are included when
counting the comparison results (remaining candidates may thus be
beaten by them).

After this the described method uses simply minmax(margins) to
eliminate the weakest non-eliminated candidate. And sequentially
drops candidates one after another until only the winner is left
(dropping weakest groups would be an option, but a complex one with
no very clear benefits).

The result of the following example changes as a result of the
changed rules.

45: L>>C>R
20: C>>R>L
35: R>>C>L

I use * to mark the use of the tied at top/bottom rules.

L-C = +45 -20 -35* = -10
L-R = +45 -20* -35 = -10
C-L = -45 +20 -35* = -60
C-R = -45* +20 -35 = -60
R-L = -45 -20* +35 = -30
R-C = -45* -20 +35 = -30

C will be eliminated. C would be the Condorcet winner in regular
Condorcet elections where preference strengths are not taken into
account. Top strength of the 20 C supporter votes is now ">".

L-C = +45 -20 -35* = -10
L-R = +45 -20 -35 = -10
R-L = -45 +20 +35 = +10
R-C = -45* +20* +35 = +10 (this value changed in the new variant)

A will be eliminated. B wins.

Juho Laatu

P.S.
In my first description from October 24th two lines should be changed
to get this new version.
- Add one point if x is ranked higher than y (top strength or higher)
and
- Subtract one point if x is ranked lower than y (top strength or
higher)
become
- Add one point if x is ranked higher than y (top strength
preferences with non-eliminated candidates at both sides)
and
- Subtract one point if x is ranked lower than y (top strength
preferences with non-eliminated candidates at both sides)
(Higher strengths need not be mentioned in the new version.)

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