[EM] MinMax definition, Tideman properties

[Markus Schulze]

Dear Kevin,

Woodall uses the following terminology:

   v is the number of voters.

   g(x,y) is the number of voters who strictly prefer
   candidate x to candidate y.

   n (x,y) := g(x,y) - g(y,x).
   g1(x,y) := 0.5 * [v + n(x,y)].
   g2(x,y) := v - g(y,x).

   mings(x)   := min { g (x,y) : y e C \ {x} }.
   minags(x)  := min { g1(x,y) : y e C \ {x} }.
   mindags(x) := min { g2(x,y) : y e C \ {x} }.
   minns(x)   := min { n (x,y) : y e C \ {x} } = 0.5 * [v + minags(x)].

   MinAGS (Minimum Augmented Gross Score) is also well known as the
   minimax method. It elects the candidate x with the largest minimum
   augmented gross score minags(x), which is the same as the candidate
   with the largest minimum net score minns(x).
   MinDAGS (Minimum Doubly Augmented Gross Score) elects the candidate
   x with the largest minimum doubly augmented gross score mindags(x).

******

So when we compare Woodall's terminology with the terminology used
in this mailing list then "g1" is "margins" and "g2" is "Minus Pairwise
Opposition". "MinAGS" is "MinMax(Margins)" and "MinDAGS" is
"MinMax(Pairwise Opposition)".

You wrote (30 Dec 2003):
> Despite the name "minimax," this definition looks for the maximum
> minimum. This has confused me for some time.  Does the definition
> look right to you?

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".

******

You wrote (30 Dec 2003):
> Also, do you have an opinion as to whether Woodall is aware of
> Tideman(WV)? He defines TidAGS and TidGS which to me appear to be
> Margins and All-Votes respectively (page 14).  TidGS is supposed
> to have equivalent properties to "D min GS," which is charted
> (page 17) as failing Condorcet.

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.

Markus Schulze