[EM] Maybe a new thread is in order ...

[David Catchpole]

On Fri, 29 Oct 1999, Craig Carey wrote:

> At 19:52 28.10.99 , David Catchpole wrote:
> >On Thu, 28 Oct 1999, Craig Carey wrote:
> ...
> >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...
> 
> Proof of what?.

Hopefully next week some time, I'll be sending a formal version of
the proof all of this started with (you know- Neutrality to voters +
Majority rules + Monotonicity -> Condorcet) in microsoft equation form
(i.e. as a word document) to this list. At the same time, I'll attempt to
use the same method to establish ramifications for (P1).

> >> 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-
> 
> FPTP is monotonic method. There need not be misunderstanding over that.
> Mr Catchpole is redefining the word monotonic. I suggest it be called
>  David-Catchpole-Monotonicity-version-0.1.

Finding Arrow...(more further down)

> 
> >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)
> 
> 
> First example:  (1)
>     3  A
>     2  AC        A wins
>     4  B
>     1  C
>   FPTP,STV,IFPP: A wins
> 
> The Second example:  (2)
>     3  A
>     4  B
>     2  CA        B wins
>     1  C
>   FPTP,IFPP: B wins  
>   STV : draw between A and B
> 
> Mr Catchpole probably wants A to win election (2)


> So A wins the first if and only if A wins this:  (3)
>     5  A
>     4  B
>     1  C


It's not whether A _has_ to win the election (remember, A could lose the
first, and that too would eliminate the problem for this particular case)!
It's whether voters C>A>B will vote insincerely- you can plainly see that
with FPTP it benefits them to do so.
>    0.9  A
>    2.1  AC
>     4   B
>     2   CA
>     1   C

	I	II	III	IV	|	I	II	III	IV
A>B	5	5	5	5	|C>A	1	3	1	1
B>A	4	4	4	4	|B>C	4	4	4	4
A>C	5	3	5	5	|C>B	3	3	1	1

	VI		VI
A>B	5	|C>A	3
B>A	4	|B>C	4
A>C	3	|C>B	5.1


The possible winners are therefore-

I	II	|I	III	|II	III	
A	C	|C	B	|C	A	
A	A	|A	A	|C	B	
B	B	|B	B	|A	A	
C	C	|C	C	|B	B	
		|		|C	C	
-----------------------------------------
I	IV	|II	IV	|III	IV
C	B	|C	A	|A	A
A	A	|C	B	|B	B
B	B	|A	A	|C	C
C	C	|B	B	|
		|C	C	|
---------------------------------------------------------
I	VI	|II	VI	|III	VI	|IV	VI
A	C	|B	C	|A	C	|A	C
B	C	|A	A	|B	C	|B	C
A	A	|B	B	|A	A	|A	A
B	B	|C	C	|B	B	|B	B
C	C	|		|C	C	|C	C



Logically, the possible winners are syphoned down to-

I	II	III	IV	VI
A	A	A	A	A
A	C	A	A	C
B	B	B	B	B
B	B	B	B	C
C	C	B	B	C
C	C	C	C	C


So there you go, these are the options which are not ruled out outright
for these 5 schema. REMEMBER- I can prove that monotonicity may imply _no_
answer! (This was discussed in the dim distant past with Blake Cretney)
However, where we expect monotonicity to apply for _most_ transitions,
this case-by-case method demonstrates some not-so-noxious options.

(more further down)