[EM] Hybrid/generalized ranked/approval ballots

[Juho]

On May 9, 2010, at 9:51 PM, robert bristow-johnson wrote:

>
> thanks, Juho, for this summary.
>
> On May 9, 2010, at 2:24 PM, Juho wrote:
>>
>> All classical Condorcet methods can handle equal rankings and their  
>> impact has been analyzed quite well.
>>
>> Usually the discussion focuses on how to measure the strength of  
>> the pairwise preferences. This is the next step after the matrix  
>> has been populated. The two most common approaches are margins and  
>> winning votes.
>>
>> - Let's define AB = number of votes that rank A above B
>> - In margins the strength of pairwise comparison of a A against B  
>> is:  AB - BA
>> - In winning votes the strength of pairwise comparison of a A  
>> against B is:  AB if AB>BA and 0 otherwise
>>
>> - Note that in margins ties could be measured either as 0:0 or as  
>> 0.5:0.5 since the strength of the pairwise comparison will stay the  
>> same in both approaches
>>
>> The most common (/classical) Condorcet methods give always the same  
>> winner if there are three candidates and all votes are fully  
>> ranked. If there is no Condorcet winner then the candidate with  
>> smallest defeat will win. The strengths of the defeats (and  
>> therefore also the end result) may differ in margins and winning  
>> votes if there are equal rankings.
>
> do you mean that marginal defeat strength can differ if equal  
> rankings are allowed compared to if equal rankings are not allowed?   
> that, of course is true.  but if you mean that marginal defeat  
> strength is different where equal ranks are counted (as votes for  
> both candidates) vs. if they are not counted (for both candidates),  
> then i think i disagree.  that's one reason why i think that  
> marginal defeat strength is a more salient measure than simply the  
> number of winning votes in each pairing (as a metric to be used in  
> resolving a cycle).

I was thinking about results given by margins vs. given by winning  
votes with the same ballots. Here's one example that has no Condorcet  
winner.

49: A>B>C
48: B>C>A
03: C>A>B

Both approaches to measuring the strength of pairwise comparisons  
agree (as always with full rankings wthout equalities) that A wins (in  
typical Condorcet methods that "ignore" the weakest defeat in the case  
of a three candidate loop).

But if the A supporters use equal rankings...

49: A>B=C
48: B>C>A
03: C>A>B

... then winning votes gives victory to C while in margins A still wins.

There is a preference loop in the opinions of the society where A>B,  
B>C and C>A. According to margins the strengths of the defeats of A, B  
and C changed from (2, 4, 94)  (= (48+3-49, 49+3-48, 49+48-3) ) to (2,  
4, 45) and according to winning votes from (51, 52, 97) (= (48+3,  
49+3, 49+48) ) to (51, 52, 48).

(I agree that margins is a more natural measure of preference strength  
than winning votes.)

Juho


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