# 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

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