Erratum Re: [EM] (P1) and monotonicity for single-winner election systems and Condorcet.

[Craig Carey]

At 14:28 20.10.99 , David Catchpole wrote:
>Well, before Craig gets hyped up about the necessity of ignoring voters
>who fail to vote in either "before" (V) or "after" (V') case, in order for
>my definition of monotonicity to operate with meaning, I'll amend the
>following-
>
>> Let Pi(A,B:V) represent the truth of whether voter i prefers candidate A
>> over B in voting schema V.
>> 
>> Let W(V) represent the set of winners of voting schema V
>> 
>> For all V, all V', W(V)<>W(V') implies-
>> 
>> there exists i, A, B such that A is an element of W(V), A is not an
>> element of W(V'), B is an element of W(V'), B is not an element of W(V),
>> not Pi(B,A,V),not Pi(A,B,V')
>
>to read-
>-------------------------------------------------------------------------------
>Let Pi(A,B:V) represent the truth of whether voter i prefers candidate A
>over B in voting schema V.
>
>Let W(V) represent the set of winners of voting schema V
>
>For all V, all V', W(V)<>W(V') implies-
>
>there exists i, A, B such that A is an element of W(V), A is not an
>element of W(V'), B is an element of W(V'), B is not an element of
>W(V), and-
>
>(i) not Pi(B,A,V) and Pi(B,A,V')
>or
>(ii) Pi(A,B,V) and not Pi(A,B,V')
>-------------------------------------------------------------------------------

Pi hasn't been defined. It can't assumed that the words "the truth of" mean
 Condercet because introducing that would be a most arbitrary step and
 eliminate the rule's possible credibility. Mr Catchpole talks about a voter
 and writes Pi(A,B,V): so maybe that is a voter that votes with a few papers
 but not all and not one. 
If Pi means a Condorcet pairwise comparison then there is quite probably
 no need to consider the idea further.

A few problems in the definition need to be fixed:
 The "A" & "B" in some of the Pi's are wrong.
 There is an (Exists) where there ought be an (All).
 The undefined term voter is used. It is unclear what voter is. The idea of
  a voter needn't be fully discarded since a voter that votes for one
  paper can be unable to alter all of the votes for that paper.

Attempt to fix the rule. Let p be a paper, where p does or does not include
 the weight of the paper, but does include the preferences list.
 Let Alt(V,p,p') be the set of elections where a fraction of the papers, p,
 have been altered into p'.

Let "-A:W(V)" mean "(not (A in W(V))", and etc.

Let p be a paper, perhaps a paper and its weight (= count).
Let (c in p) mean: candidate c is in list p. 
   E.g. (A in (ABC...)), not (F in (ABC...))
Let tr(p,c) mean p truncated at c.
If c is not in the set containing the candidates that are in the preference
 list, then is defined to be such that: tr(p,c) = p.
   E.g. tr((ABC...),B) = (AB....), tr((ABC...),D) = (ABC...).

Then the P(A,B,p) = ((A in tr(p,B) or not (B in tr(p,A))).
E.g. P(A,B,(CA..)) = ((A in {C,A}) or -(B in {C,A})) = (true or true)

Using the expressions from below.

(All V)(All p in V)(All p')(All A,B,A<>B)(All V', V' inAlt(V,p,p'))
        [A:W(V).-B:W(V).B:W(V').-A:W(V') .=>. not [P(B,A,p) => P(A,B,p') ]

(-x. => y) = (-y. => x)
-------------------------------------------------------------------------------
                  'Catchy-Monotonicity-version-3a'

(All V)(All p in V)(All p')(All A,B)(All V', V' in Alt(V,p,p'))
        [(P(B,A,p) => P(A,B,p')) => not (A:W(V).-B:W(V).B:W(V').-A:W(V'))]

This is just some monotonicity-like formula.
It says that if only a single paper is changed and the 'rank'... of
 A, w.r.t B, is not reduced, then the outcome does not flip from A wins
 and B loses, to B wins and A loses.

FPTP passes the version 3a formula (so it passes my (MP1) metarule).

The equation could be rewritten to replace P with tr. That would lead
 to a precise definition of monotonicity. Perhaps, given all the
 publications on that particular idea, noone has already defined
 monotonicity. Anything can happen in P.V. theory,

The formula has one paper change into only one other paper.
With (P1) (my (P1)), many papers can be dispersed into many other papers.

End.

-------------------------------------------------------------------------------

"**" = "and" or "or".

X = (All V,V',W(V)<>W(V')(Exists i,A,B). A:W(V). -A:W(V'). B:W(V'). -B:W(V).
               [not Pi(B,A,V) and Pi(B,A,V') ** Pi(A,B,V) and not Pi(A,B,V')]

Let Y be such that
Y = (All V,V',W(V)<>W(V')(Exists i,A,B). A:W(V). -A:W(V'). B:W(V'). -B:W(V).
                                               [not Pi(B,A,V) .and Pi(B,A,V')]

Swap (A,V) with (B,V'):
Y = (All V,V',W(V)<>W(V')(Exists i,A,B). A:W(V). -A:W(V'). B:W(V'). -B:W(V).
                                               [Pi(A,B,V) and. not Pi(A,B,V')]

Then note X = (Y ** Y). (Y or Y) = Y.
(a.-b) = -(-a or b) = -(a=>b)

Y = (All V,V',W(V)<>W(V')(Exists i,A,B). A:W(V). -A:W(V'). B:W(V'). -B:W(V).
                                            not [Pi(A,B,V) => Pi(A,B,V') ]
-------------------------------------------------------------------------------

To Mr David Catchpole: 

This replies to a future message
    (http://www.egroups.com/group/election-methods-list/4547.html?)

>It has now occured to me that it may be possible to demonstrate that
>Condorcet is a necessary condition of monotonicity without assuming
>majority rules ("2-candidate FPTP") but instead assuming the system is
>neutral to candidates (a switch in candidates brings on a corresponding
>change in results). However, this is going to take some effort, because it
>involves 3!=6 "points" and 2^12 possible permutations of results.

Those phrase, "P is a necessary condition for Q" means "Q implies P". In
 other words, Mr Catchpole is saying, wherever a method is neutral to
 candidates and it satisfies monotonicity, then the method is Condorcet. 

 Neutral is wrongly defined: better would be: "a switch ... does not bring
 on a not corresponding change in results").
 On using the fixed definition of neutral and noting it means almost
 monotonicity, the whole statement would become:

         If a method is monotonic then it is Condorcet.

So it seems to be that Mr Catchpole needn't do that research. [The use of the
 word "necessary" ("necessary condition") is now under clearer suspicion of
 leading accidents in making statements].


G. A. Craig Carey
Auckland