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

[David Catchpole]

(Further down...)

On Sat, 23 Oct 1999, Craig Carey wrote:

> >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')
> >-------------------------------------------------------------------------------
> 

I would suggest looking at a first year logic textbook. "Pi(A,B,V)
represents the truth of..." is similar to the way logic
textbooks say "S represents the truth of 'Clancy of the overflow is here',
R represents the truth of 'the colt from Old Regret has got away', R->S,"
etc. (More further down)

> 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.

Consider voter i as an element of the set of voters. Pi(A,B,V) is true iff
(if and only if) voter i prefers candidate A over candidate B in scenario
V. (More further down)

> 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.

What do you mean by that last sentence... "...votes for that paper?" I'm
talking about a voter's individual utility, or if you like that utility as
represented in a ballot. If you wish to assume that a voter might have
multiple ballots and wishes to submit more than one type of ballot then
you might like to substitute "ballot" for "voter," though this goes some
way to missing the point of the perpetrator of the ballot remaining the
same.

Nothing's wrong... thorough checks convince me there is nothing to
criticise in this definition. Again, as Humpty says to Alice, "What I say
means precisely what I mean it to say." In other words, QED, mea culpa.

The thing is, my definition incorporates voting systems where the function of
a ballot may vary according to its perpetrator in ways other than just
weighting. What you're trying to get me to change the definition to won't.
(More further down...)

> 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.

Well that's desperately wrong. c will always be incorporated in a
preference list, though she may be considered to be ranked equivalent to
another candidate, or ranked last, or blah blah. However, it is still
there in a voter's full transitive comparison of candidates.

I would say, rather than me changing the definition, it would be more
rewarding for you to either- try to understand it by yourself, or- get
someone like Markus, who might be interested in discussing it without
descending into a mess of vitriol like I do, to discuss it with you. (More
further down)

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

"(A and B) or A or B" is equivalent to "A or B". WTFAYO? (More further
down...)

> 
> 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. 

Got it in one, kiddo.

>  Neutral is wrongly defined: better would be: "a switch ... does not bring
>  on a not corresponding change in results").

Nup. Say you had an election between Roses, Tulips and Stinkweed. If Roses
and Stinkweed were to swap names, we would still assume that Roses, by any
other name, would smell as sweet- that is, were Rose to win the election
before the swap, we would assume the voters would elect the
candidate-now-known-as-Stinkweed after the swap. The rule also extends to
forbidding the system from being based on any consideration of who the
candidates are, rather than how the votes for them go.

You don't like me, Craig, do you?