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

[Craig Carey]

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.
: 
: For example, 
: 
: V:
:   10 ABC
:   11 B
:   S
: 
: One system in AltAtAfter(V,'B') is this:
:  10 A
:   2 AB
:   2 ACB
:   4 AD
:   1 B
:   1 C
:   S
: 
: No deletion of the ABC papers was possible, so
:  since A has 18, 8 papers did come from the alteration
:  of the 11 B preferential voting papers.
:  1 paper that was for B was discarded.
: 

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

I test some of some news rules against a new (MP1[-provisional])
 metarule, which fails or passes other rules.


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."

What about discarding of votes?: say the First Tyrarkh, who normally gets to
 cast 1,000 votes (wich is permitted under the constitution), decided to not
 vote just in the 2nd of the two elections. He was the single winner of the
 first election, but he lost the 2nd election. The Theoblimp won the 2nd
 election.

The rule would hold if the change was from the 2nd to the 1st, but votes
 can't be created by a process of "changing the ballots".

When going from the 1st election to the 2nd, the "ballots" certainly "changed"
 because the 2nd election had 1,000 votes less. 
David: Who changed their ballot to rank the Theoblimp over the Tyrarkh?.


>The two are more or less the same-
>
>(P1) a necessary condition for a change is that some preferences between
>the old winner and another candidate change

----------------------------------------------------------------------------
(A => B) = (a necessary condition for A is B) = (-B => -A) = (if A then B)

a necessary condition for (a change in the winners) is that (some preferences
 between the old winner and another candidate change)
=
(a change in the winners) implies (some preferences between the old winner
 and another candidate change)
=
[not (some preferences between the old winner and another candidate change)]
  implies [not (a change in the winners)]
=
(none of the preferences between the old winner and another candidate change)
  implies (no change in the winners)
=
if none of the preferences between the old winner and another candidate change
then there is no change in the set of winners
----------------------------------------------------------------------------


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). 


>("M") a necessary condition for a change is that some preferences between
>the old winner and the new winner change to favour the new winner.
>
....

>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.


(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, and
    when a voter lowers the valuation for a losing alternative it
    remains a losing alternative. [a false statement is omitted]

    If is a system is not monotonic, a situation may arise in which
    voters could prevent their favorite alternative from winning by
    voting for that alternative. Obviously this is contrary to the
    purposes of a voting system -- voters should not be penalized for
    supporting their favorite alternative. 

(From "Declared-Strategy Voting: An Instrument for Group Decision-Making
by Lorrie Faith Cranor, Prepared under the direction of Ron K. Cytron")