[EM] Juho's idea

[Juho]

I generated one example to demonstrate the difference between basic  
mimax and the proposed method that seeks the candidate with "the  
smallest number of ballots on which some alternative that beats A  
pairwise is ranked higher than A". All the votes in this example rank  
all the candidates to eliminate the impact of margins vs. winning votes.

12: a>b>c>d
13: a>c>d>b
12: b>a>c>d
13: b>c>d>a
12: c>d>a>b
13: c>d>b>a
12: d>a>b>c
13: d>b>a>c

The basic idea of the example is that there is a cycle A&B > C > D >  
A&B. B also beats A with a small margin. The difference between the  
two methods comes from the difference of one bad defeat vs. wide  
front of defeats (to any candidate).

With minmax the largest defeats (margins) are: A:26, B:26, C:24, D: 
50. C wins since it loses to A and B only with margin 24.

With the proposed method the results are: A:75, B:63, C:75, D:75. C's  
result is now worse than in minmax since there are 75 voters that  
think that A or B or both are better than C. The opposition against C  
is thus wide (in number of voters who have "justified" reasons to  
object C) although C doesn't lose that much to any single candidate.  
Note also that although A and B lose to D in the same number of  
ballots, there are also some additional voters that feel that B  
should win A, and this makes B the winner.

Minmax and the proposed new method both have a reasonably justifiable  
utility function (with sincere votes) that is just different -  
looking for either the worst defeat or the worst number of opposing  
voters with well justified opinions (=someone would beat the  
candidate also in pairwise comparison).

Juho Laatu


On Jan 13, 2007, at 14:46 , Juho wrote:

> On Jan 9, 2007, at 2:25 , Simmons, Forest wrote:
>
>>> Juho wrote:
>>>
>>>> How about "the smallest number of ballots on which some
>>>> alternative  that beats A pairwise is ranked higher than A"?
>>>>
>>>> Juho
>>
>>
>>
>> If I am not mistaken, this idea is equivalent to electing the
>> alternative A with the greatest number of ballots on which A is
>> ranked higher than any alternative that beats A pairwise.
>
> Yes, seems so, if we don't care about the ties.
>
> I'll explain a bit more my interest in this particular method. The
> vote of a voter in a way says: "I object the election of alternative
> A since voters prefer B to A (and I do so too)". We thus do not try
> to seek the alternative that beats A most (directly or in chain) (as
> typical in the Condorcet methods), but we try to count the number of
> "justifiable" objections (that may be based on losing to any of the
> other alternatives). This reasoning is mainly based on trying to
> achieve good performance with sincere votes using an alternative
> utility function. The strategic defence properties may or may not be
> there.
>
>> Here's a variant that could be used with cardinal ratings/range
>> ballots:
>>
>> Elect the alternative A with the greatest number of ballots on each
>> of which A is rated above the expected rating of the alternatives
>> that beat A pairwise.
>
> Aha, that's an interesting approach. And the same with the
> replication of names on the ballot. I'll also explain a bit more the
> thoughts I had on how this method could be developed further (to
> better or worse :-).
>
> Instead of seeking the candidate with "the smallest number of ballots
> on which some alternative that beats A pairwise is ranked higher than
> A" one could sum up the margin comparison values (=> instead of just
> adding 1 per each vote). This would make the largest possible result
> n*n where n is the number of votes (10000 in the typical EM elections
> with 100 voters). The idea is to go closer to margin style results
> instead of the quite winning vote style results of the method where
> just the number of votes is counted. (Also (WinningVotes-LosingVotes)/
> WinningVotes is an option.)
>
> Juho Laatu
>
>
>
> Send instant messages to your online friends http:// 
> uk.messenger.yahoo.com
> ----
> election-methods mailing list - see http://electorama.com/em for  
> list info

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