[EM] Maybe a new thread is in order: Cross-Tasman Catfight

[David Catchpole]

On Thu, 28 Oct 1999, Craig Carey wrote:
> That last sentence is lengthy given the information it holds.
> A "proposition". Two types of preferences, one on paper known to the
>  formula, and another in the thoughts of voters. There is still no
>  definition of "Pi". There is no presumption in me at all that you have
>  ever defined Pi. You did write that you had thoroughly tested the
>  rule. No one else can: for example, what if a Catchpole-"voter" actually
>  voted strategically in a 150 candidate election, but did not "express"

If you wil, we make the assumption that the ballot reflects a full
ranking of candidates. Obviously, a rational voter will vote strategically
in order to optimise her utility- the ranking reflected by the ballot
will correspond with this. What monotonicity actually asks is whether the
optimum behaviour for a voter is to tell the truth on her ballot, and in
the "reflective ranking." Whether she expresses or not is totally up to
her and not up to supermarket policy on breastfeeding...

>  a "preference". I want to prove your rule is too strong or too weak, if
>  possible, but you have not stated what the entire rule is. Does rule
>  reject FPTP with 3,000 candidates?. I don't know.

Yes it does.

In case you didn't notice- the relevant message contained a proof. If you
couldn't work it out from the plain English, I'll have a go at putting it
in a formal structure of proof. Later, though...

> FPTP is a monotonic preferential voting method, and it isn't Condorcet.
> So therefore the and rule is finally rejected.

It's not! Consider the following example I gave to Markus Schultze-

Say two voters have preferences A>C>B in an election where A has 5
votes (including that of this voter), B has 4 votes and C has 1. A
wins. Say that these two voters change their preferences to C>A>B. B now
wins, even though nobody changed their preferences between A and B. While
some definitions of monotonicity fail to rule on such a scenario, the one
I gave in my message does- and rules FPTP out. (more further down)

> --------------------------------
> At 14:39 27.10.99 , David Catchpole wrote:
> >On Wed, 27 Oct 1999, Craig Carey wrote:
> ...
> >
> >Um... what do you mean by "Condorcet picks the wrong number of winners?"
> >I've been talking about single-winner election systems, SF-dammit!
> >
> >Again, Pi is not Condorcet pairwise comparison. It's an individual
> >comparison with respect to voter i, not an aggregate comparison. ...
> 
> Saying what Pi is not, does not say what it is.
> I have a symbolic algebra program, and on the line describing "Pi",
>  I had to enter a best guess of Pi. Mr Catchpole said I got that guess
>  wrong.

Assuming neutrality between voters, the rule for monotonicity becomes-
-------------------------------------------------------------------------------
A necessary condition for a change in winner is that the number of voters,
whose relative rankings of the old winner and the new winner favour the
new winner, increase, or the number of voters, whose relative rankings of
the old winner and the new winner favour the old winner, decrease
[i.e. N'(NW>OW)>N(NW>OW) or N'(OW>NW)<N(OW>NW)]
-------------------------------------------------------------------------------

or, in Navaho,
-------------------------------------------------------------------------------
For all V, for all V',

if W(V)<>W'(V') then for all A, all B,

-A is an element of W(V) and A is not an element of W(V') and B is an
element of W(V') and B is not an element of W(V)-

implies-

-N(B>A,V')>N(B>A,V) or N(A>B,V')<N(A>B,V)
-------------------------------------------------------------------------------
> --------
> Also, Condorcet does picks the wrong number of winners.

What do you mean? Again, my Navaho's not that good.