[EM] MinMax definition, Tideman properties

[Kevin Venzke]

Markus,

Thank you, that was helpful.

 --- Markus Schulze <markus.schulze at alumni.tu-berlin.de> a écrit : 
> Dear Kevin,
> 
> Woodall uses the following terminology:
>    g(x,y) is the number of voters who strictly prefer
>    candidate x to candidate y.
> 
>    mings(x)   := min { g (x,y) : y e C \ {x} }.

"mings" seems to be a rather useless figure.  If you clone a candidate
X to get X and Y, such that no voter distinguishes between X and Y,
mings(x) drops suddenly to 0.

> When "margins" is being used then it is the same whether you use
> the minimum maximum or the maximum minimum.
> 
> The reason why Woodall uses the maximum minimum in the definition
> of "MinDAGS" is that he defines "mindags(x)" in such a manner that
> mindags(x) decreases with increasing pairwise opposition because
> of his definition of "g2".

Ok.  I hadn't even attempted to figure out what MinDAGS was.

> There is no need to define Tideman(WV). Already the fact that the
> g(x,y) are sorted according to their strengths and that each g(x,y)
> is taken in turn until you have a complete ranking of all candidates,
> guarantees that those g(i,j) with g(i,j) < g(j,i) will never be
> taken into consideration. I don't see yet why TidGS and TidDAGS
> fail Condorcet(net) in table 2.

Ok.  So WV and All-Votes are equivalent with Tideman, but not Schulze.

I believe Woodall made a mistake.  It's clear that "D min GS" doesn't
meet Condorcet(net).  It is (page 18):

"DminGS is the set-intersection method in which each set X is given its
minimum gross score mings(X), defined by mings(X):=min{g(x,y) : x e X,
y e C \ X}."

Unlike TidGS, losing opposition votes can affect the winner.  So perhaps
Woodall didn't realize this is not the case with TidGS.

Kevin Venzke
stepjak at yahoo.fr


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