[EM] Hybrid/generalized ranked/approval ballots

[Juho]

On May 9, 2010, at 11:22 PM, Dave Ketchum wrote:

> 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.

I think that should lead to the same conclusion. In a three candidate  
cycle there are three cyclic defeats. Ranked Pairs should keep the two  
strongest ones. Schulze should eliminate the weakest one and keep the  
remaining two.

The philosophy of Minmax is a bit different since it does not aim at  
approving or ignoring defeats (until there is a transitive order) but  
just finds the candidate that has smallest opposition against her.

Juho







-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20100510/b80589e3/attachment-0004.htm>