[EM] Ranked Preferences

[Juho]

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.)

Send instant messages to your online friends http://uk.messenger.yahoo.com