[EM] MinMax definition, Tideman properties
Kevin Venzke
stepjak at yahoo.fr
Wed Dec 31 13:39:01 PST 2003
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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