[EM] Proportional preferential voting

[Blake Cretney]

Craig Carey wrote:

>                       Principle 1 
> 
>      Alterations of preferences after a preference
>      for a winning candidate never cause that
>      candidate to lose.

You later revised this to a principle that seems equivalent, except
for ties.

> For all c (c is a candidate), all V, all V' (where 
> V and V' are election systems), then if 
>  V' in AltAtAfter(V,c) and c loses V, then c 
>  also loses V'. 

Assuming X' in AltAtAfter(X,c)

it follows that X is in AltAtAfter(X',c) [I think this is obvious]
Then, by above, substituting X' for V and X for V'
if c loses X', then c also loses X. 
so, if c wins X, c must win X' (assuming no ties) [A -> B = ~B -> ~A]

which, I think is the meaning of your original principle.

So, the revised principle follows from the original.  I think it's
obvious that you can reverse this process to show that the original
follows from the revised.  Therefore, they are equivalent assuming no
ties.

--snip--

>                 CONDORCET AND TRAILING PREFERENCES
> 
> I have a question for Mr Catchpole and Mr Markus Schulze, or
>  anybody who wishes to answer:
> 
>  Does Condorcet (1 winner) satisfy the 'principle 1'
>  (given above) ??.
> 
> Please respond with a proof or a counter example.
> 

I have a web site at:
	http://www.fortunecity.com/meltingpot/harrow/124

that lists many common methods, and which criteria they pass and
fail.  I call "Principle 1" the Secret Preferences Criterion.

Anyway, Condorcet doesn't exactly fail SPC, because it is undefined
in some situations.  That is, Condorcet's method is only defined when
there is a Condorcet winner (CW).  So, to violate the criterion, we
would have to be able to arrange candidates on some ballots (lower
than the CW) and cause this candidate not to be the CW.  But changes
between candidates other than the CW cannot give the CW a pair-wise
loss, and therefore cannot cause the candidate to be defeated.

On the other hand, any method that satisfies the Condorcet criterion,
and gives a defined result in all situations, will violate SPC.

Consider a Condorcet criterion method (X) with the following ballots.

45 A B C
25 B C A
30 C A B

if the result of X is A, then alter 45 A B C to 45 A C B.  Now C is
the CW, so must win.  SPC is violated.

if the result of X is B, then alter 25 B C A to 25 B A C.  Now A is
the CW, so must win.  SPC is violated.

if the result of X is C, then alter 30 C A B to 30 C B A.  Now B is
the CW, so must win.  SPC is violated.

To prove the same thing for probabilistic methods, substitute "the
result of X may be" for "the result of X is".  Obviously, at least one
candidate is a possible winner.

---

At first, SPC sounds reasonable.  It's a logical consequence of Independence
of Irrelevant Alternatives, which also sounds reasonable.  But given that we
have to allow the effect of "irrelevant" alternatives, I don't think that the
situation in SPC should be given special treatment.  Although I do not have a
proof, SPC seems to follow inevitably to either monotonicity violations (as in
IRV) or to plurality.

I consider SPC to in fact be a decidedly unreasonable criterion.  If someone
votes
A B C instead of A C B, that's evidence in favour of B, whether or not A
might have won otherwise.

---
Blake Cretney