[EM] Hybrid/generalized ranked/approval ballots

[Juho]

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