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

[David Catchpole]

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). 

> 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. 
> 
> >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"

> Definition:         Metarule (MP1)
> 
>         A rule satisfies (MP1) if it passes the
>            1 winner 2 candidate FPTP method.
> 
>                                
> 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. 

> >I brought it up because there are interesting parallels with Craig Carey's
> >(P1) condition, which seems to be a sub-condition of monotonicity.
> 
> 
> (P1) implies monotonicity. The two are different.

Actually, monotonicity implies (P1) (The version I expressed above), not
the other way round, which is why we define (P1) as a sub-condition of
montonicity.

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