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

[David Catchpole]

What might be rewarding is an initial definition of "candidate" and
"preference," a distinction which is not incomprehensible, as "candidate"
and "option" are synonymous and "preference" refers to a voter's
set of preferences (A>B>C, etc.).

You could ask other people on this list about which (written in English or
in quasi set algebra) is more easy to understand. You may be surprised.
You may be displeased.

On Mon, 18 Oct 1999, Craig Carey wrote:

> 
> 
> 
> Hi David.
> 
> I am sending this to you personally.
> I can't sedn it to the list since the topic is boring, others
>  won't need to pause for a moment to know where error is, and
>  since the IIA issues of incomprehensible English and fundamental
>  ambiguity over what is a "candidate" and what is a "preference"
>  would seem to have to be part of a discussion. At least the
>  word "options" has not been used
> 
> 
> At 19:22 18.10.99 , David Catchpole wrote:
> >On Fri, 15 Oct 1999, Craig Carey wrote:
> >
> >> 
> >> The 18-Sep-99 definition is repeated
> >> (P1) is similar to the the usual "monotonicity" rule
> >> 
> >> : Principle 1 (P1), Sat 18 Sept 1999
> >> : 
> >> : 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'.
> >> : 
> >> : AltAtAfter(V,c) is defined to be the set of all election
> >> :  papers that can be derived from V by altering
> >> :  preferences at and/or after the preference for
> >> :  preference c.
> >
> >Woops! Looks as though my memory of Craig's (P1) was wrong. (Though
> >Craig's (P1) is a sub-criterion of the one I gave).
> 
> No that is not : your (P1) is incomprehensisble, bad [fails FPTP
>  while being a monotonicity like requirement], and it has nothing
>  to do with (P1).

What's the disappeal of failing FPTP? I guess you mean two candidate FPTP.
In that case monotonicity doesn't fail FPTP. The thing is that the (P1)
principle as you restate it is only a sub-condition of monotonicity as
defined by published academics such as Arrow, Sen, most microeconomists,
etc. FPTP (for more than two candidates) is clearly non-monotonic.

> I suggest you get you thoughts clearer. That could be done by
>  saying a voting paper is p, where p is a list
> let p:A be the list trucated at the preference for candidate A.
> B beats A in a Condorcet sense if (B in p:A), and 'in' treats the
>  RHS list as if a set. Whether it is better to have (A in p:A)
>  or not is not obvious.

This itself is unclear. What do you mean? Does (B in p:A) represent the
number of voters who favour B over A? Clearly there's a cultural
difference here which can only be resolved through beginning again from
first principles and _not_ attempting to express principles solely through
a quasi set algebra.

So far, it's not clear that you've been getting to a point over this
issue. I sent you the original e-mail because of the parallels with (P1).
I was expecting you would take it and run with it rather than take such an
obstructionist attitude.

> 
> 
> >
> >> At 15:04 15.10.99 , David Catchpole wrote:
> >> >The statement of (P1) is thus-
> >> >
> >> >"No change in ballots which does not effect preferences between the winner
> >> >and other candidates should not change the outcome"
> >> 
> >> The above may have drafting errors in it. 

Well, yeah, on second thoughts it does have a double negative in there.

Replace the first "No" with "Any".

Consider the amended version in something approximating your algebra,
later on in this message.

> So David may have meant to have the word "preference" mean
>  "preference".

Ala Humpty Dumpty in "Alice in the Looking Glass," what I say means
exactly what I mean it to say.

> 
> That fast runs into a big problem because of the use of the word
>  "between", which a key item in the MP1 reject of Catchpole's
>  2nd (P1) (and it would probably get the 1st too).
> 
> Children know waht "between" means.
> If there are two (or three) children in a row, then child between
>  those two is known, and beyond dispute.

If there are only children, and we're choosing members for a cricket team,
we choose "between" these children. If you like, you could try "amongst,"
etc.  

> >> >The statement of "monotonicity" for the purposes of the exploration below
> >> >is-
> >> >
> >> >"Any change in ballots that results in a change of winner must involve
> >> >someone changing their ballot to rank the new winner over the old
> >> >winner."
> >
> >Well, if we include the loss of voters (I assume a non-vote simply
> >represents A=B=C=...) and other cases, I could state a new "monotonicity"
> >as-
> >
> >"Any change in ballots that results in a change of winner must involve
> >someone changing their ballot either so their preferences go-
> >
> >(i) From old winner over new winner to indifference;
> >(ii) From old winner over new winner to new winner over old winner;
> >(iii) From indifference to new winner over old winner"
> >

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')

Thar we go. No misplaced negatives any more.

> >> Theorem: The (MP1) metarule fails the "(P1)" rule defined just above.
> >> 
> >> Proof: Here are two FPTP examples where preferences were shifted:
> >> 
> >>   1 A B        FPTP winner = A
> >> 
> >>   1 B A        FPTP winner = B
> >> 
> >>  Let the old winner be A and the other candidate be B. Then none of the
> >>  preferences "between" those two have changed. However the winner did
> >>  change, so the the new "(P1)" fails FPTP, and the new "(P1)" method is
> >>  therefor failed by (MP1). 
> >
> >I don't catch on... It seems to me that the example above does involve an
> >inversion of preferences between A and B. 
> 
> "A" means either the candidate or the preference. Why not write
>  "preferences between the two preferences for A and B"?. The words
>  A and B refer to candidates, not preferences. There are no preferences
>  between A and B since A and B are are called, "candidates". I can't
>  understand why you would want to be ambiguous.

While I've described what I mean by "preferences" (individual
voter preferences between candidates) what you've just written here is
tres ambiguous and now I don't catch on to what _you_ mean by preference.

> 
> The word (and idea) "inversion" is not present in the wording of the
>  definition. So that's that.

Well, not really, I'm saying "BOUNCE BOUNCE" and you're saying "BAH BAH."

> >Actually, monotonicity implies (P1) (The version I expressed above), not
> >the other way round, which is why we define (P1) as a sub-condition
> montonicity.
> 
> One of us made a valid statement and the other made an invalid
>  statement. What is that "why we". Surely you have absolutely no persons
>  agreeing with your 'new' ideas?.

Umm... this is where I give up and direct you to printed work.

Admittedly, this is the "Royal" "we," but it is also the "mathematical"
"we."

> If you can improve on my (P1) then I'd be interested. 
> I suspect (P1) implies 1 man one vote (and of course, monotonicity
>  does not):

What you mean is one value one vote. Actually, this condition is
independent of (P1) and monotonocity, but combined with them it has far
reaching implications (check out Arrow's theorem).


> Note: (P1) needs to be applied repeatedly before it can be seen that
>  monotonicity is a corollary of (P1).
> 
> Monotoniciy does not imply (P1), because (P1) allows preferences to be
>  disordered as the preference that has to remain a winner, is shifted
>  over them.

Where monotonicity applies in a situation, (P1) is implied to also be true
(give me an example where (P1) applies but monotonicity does not).
Monotonicity is more stringent than (P1), so the reverse is not true.

> 
> 
> >> (1) http://www.ccrc.wustl.edu/~lorracks/dsv/diss/node4.html
> >> 
> >>     A voting system is monotonic if when a voter raises the valuation
> >>     for a winning alternative it remains a winning alternative,
> >>     when a voter lowers the valuation for a losing alternative it
> >>     remains a losing alternative. [a false statement is omitted]
> >
> >The new, improved statement of monotonicity implies this. However, this
> 
> Is that your (P1) or mine?. It has to be my (P1) if it has to do with
>  monotonicity.

No, new improved monotonicity, which has had my original errata removed.
An new expression in something approximating your formalism resides above.

> 
> >statement does not imply the new improved statement of monotonicity
> >because of the fact that changes of preference between a third and fourth
> 
> "Changes of preference"?
> 
> >candidate will satisfy Lorrie's monotonicity but not the wow! look! new
> >improved! statement of monotonicity.
> 
> What is the "wow! look! new improved!" statement of monotonicity?.

Again, see above.

> So a case has been made out for you to not devalue my ideas by
>  naming them (P1).

I weren't devaluing your ideas. I was trying to give you an example of a
precedent on this list. Ungrateful bloody sod.