[EM] Hybrid/generalized ranked/approval ballots

[Dave Ketchum]

On May 9, 2010, at 2:24 PM, Juho wrote:
> On May 9, 2010, at 6:49 PM, Peter Zbornik wrote:
>> an alternative way to count wins against equally ranked candidates,
>> would be to give both 0.5 wins per vote and not 0 as I did in my  
>> previous mail.
>>
>> Thus the current state: A>B gives 1:0. A=B gives ?:? (A=B is only  
>> allowed at the end of the ballot)
>> The proposal above: A>B gives 2:0. A=B gives 1:1 (linear  
>> transformation from A>B - 1:0 and A=B - 0.5:0.5).
>> The soccer proposal  A>B gives 3:0, A=B gives 1:1 (linear  
>> transformation from A>B - 1:0 and A=B -1/3:1/3. This proposal would  
>> (as in soccer) encourage voters to differenciate between  
>> candidates, and candidates to try to gain more support than the  
>> other candidates. The soccer ranking is purely speculative though.
>
> 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. These differences (and their naturalness with  
> sincere vote and their impact on strategies) have been discussed a  
> lot. I will not analyze the differences in detail here. In margins  
> victory 50:40 has the same strength as victory 10:0. In winning  
> votes victory 50:40 has the same strength as victory 50:0.
>
> Juho

"smallest defeat"?  Oops - I will quote from wikipedia:
> Ranked Pairs and Schulze are procedurally in some sense opposite  
> approaches (although they very frequently give the same results):
> Ranked Pairs (and its variants) starts with the strongest defeats  
> and uses as much information as it can without creating ambiguity.
> Schulze repeatedly removes the weakest defeat until ambiguity is  
> removed.
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