[EM] hey Markus, can you confirm something?...

[robert bristow-johnson]

On Feb 5, 2010, at 10:24 PM, Warren Smith wrote:

> robert bristow-johnson:
>> In a Condorcet election in which a cycle occurs, if there are only  
>> three
>> candidates in the Smith set, or even if there are more but the defeat
>> path is a simple single loop, is the outcome of the election any
>> different if decided by Schulze rules than if decided by Tideman  
>> ruules
>> (Ranked Pairs)?
>>
>> Of course, this question is open for anyone to answer.
>
> Kristofer Munsterhjelm:
> AFAIK, if there are three candidates in a cycle, RP and Schulze  
> returns
> the same: they break the cycle by its weakest defeat. This is obvious
> for RP, because the weakest defeat (least victory) gets sorted  
> last. For
> Schulze, consider the Schwartz set implementation (CSSD) as  
> described on
> http://en.wikipedia.org/wiki/ 
> Schulze_method#The_Schwartz_set_heuristic .
> Since there's a cycle and all three candidates are involved in it,  
> steps
> one and two can't be done, so 3 is done, which also discards the  
> weakest
> defeat.
>
> Warren D Smith:
> To complete KM's answer, the case where
> "there are more [than three in the Smith set]
> but the defeat  path is a simple single loop"
> does not exist.

that, i didn't expect.

>   That is because if there is a directed N-cycle
> for some N>3, you can always "draw a chord" in the "circle"

<smacking forehead>

> and no matter which way the arrow on that chord points, the result  
> is always
> at least one shorter directed cycle, i.e. with smaller N.    
> Consequently, no
> matter what the value of N>3, there is always a 3-cycle.

so everyone in the smith set is related to everyone else via a 3- 
cycle?  or is that also not general?

> So... do Schulze & Ranked Pairs always return the same winner? No.
> If the Smith set has 3 or fewer members? Yes.


i suspected these two answers but don't mind getting misconceptions  
enlightened.

thanks, Warren.

--

r b-j                  rbj at audioimagination.com

"Imagination is more important than knowledge."