Goldfish (single-winner method)
David Marsay
djmarsay at dera.gov.uk
Mon Sep 7 05:22:15 PDT 1998
RE: (belatedly)
> From: "Norman Petry" <npetry at sk.sympatico.ca>
> To: <election-methods-list at eskimo.com>
> Subject: Re: Goldfish (single-winner method)
> Date: Sun, 23 Aug 1998 18:08:29 -0600
You asked about the relationship between a method of Tideman and
Shulze's method. I've put the relevant snip at the end.
Tideman defines a 'stack' and allows the use of a stack to break a
tie. Shulze considers 'beats-paths'. I claim the two are sort-of the
same. Tideman allows ties to be broken in a variety of ways. I claim
that X beats-path beats Y if and only if X 'stack' beats Y for every
stack.
I'm going to have to have lots of notation!
Let <(x,y) be the number of ballots in which x beats y.
Let M(x,y) = <(x,y) - <(y,x) be the 'plurality'. (Same as Tideman).
(Note: some people take [number voters - <(y,x)], they can work out
the consequences!)
Following Condorcet:
Let p denote a path from x1 to xn: x1,x2,... xn.
Let M'(p) = min{M(x1,x2),M(x2,x3),...} be the ''plurality along a
path'.
Let M''(x,y) = max {M'(p)} be the maximum over all possible
paths from x to y, a 'best path plurarilty'.
Let M*(x,y) = M''(x,y) - M''(y,x), the differential best path
plurality, a 'weight of evidence'.
Note that M*(x,y) + M*(y,x) = 0 and M* is transitive in the sense
that If M*(x,y) > 0 and M*(y,z) > 0 then M*(x,z) > 0.
Thus the order x > y when M*(x,y)> 0 is a partial ranking.
Shulze et al have suggested that this should be 'the' interpretation
of Condorcet.
Tideman has 'stacks'. These are complete rankings that satisfy:
let x, y be options with x < y. Let p be the path from x to y that
goes from x to the next ranked option, until it reaches y.
Then M'(p) >= M(y,x).
I claim (a) every refinement of the Shulze-order is a stack. (b)
every stack is a refinement of the Shulze order. I claim the proofs
are straightforward, even for a Monday morning.
I hope this clarifies the matter.
>
> In his second paper ("Complete Independence of Clones in the Ranked Pairs
> Rule", Soc Choice Welfare (1989) 6:167-173), Tideman provides two things:
> (1) a special tiebreaker-ballot rule to provide _complete_ clone
> independence, and (2) a new definition of Tideman's method, based on his
> idea of "Stacks". On January 6, 1998, David Marsay claimed ("Re: Condorect
> sub-cycle rule") that this new definition is the same as Markus' beat-path
> method (what we now call the Schulze method). Yet, I don't think this can
> be correct. In the paper, Tideman provides a proof that his two methods are
> identical:
>
> "[...] Hence, if a candidate wins according to the definition of the ranked
> pairs rule in terms of stacks, then it also wins under the algorithmic
> definition. Thus the two definitions are equivalent."
>
> To be honest, I don't entirely follow the "stack" definition of Tideman's
> method, and he gives no examples. Yet, we've seen cases where Tideman's
> (algorithmic) method provides different (and inferior) results to Schulze,
> even when there is no problem with ties. Therefore, rather than try to
> fully decipher Tideman's formulese, I'll just assume the assessment above is
> correct, in which case his stack definition *cannot* be the same as
> Schulze's beat-path method.
>
> I'd be interested in hearing what others on this list (particularly David
> and Markus) have to say about this (who's method is it, anyway?!)
>
>
> Norm Petry
--------------------------------------------------
Sorry folks, but apparently I have to do this. :-(
The views expressed above are entirely those of the writer
and do not represent the views, policy or understanding of
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