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

[David Catchpole]

Starting again, because I made some mistakes and managed to baffle
and anger Craig in the process. (We'll try "relative ranking," instead of
"preference," shall we?) - 

(P1 as I remembered it) a necessary condition for a change in winner is
that some voters' relative rankings of the old winner and another
candidate change (i.e.-
(i)(a) from OW>AC to AC>OW or;
(ii)(a) from OW>AC to OW=AC or;
(iii)(a) from OW=AC to AC>OW or;
(ii)(b) from OW=AC to OW>AC or;
(i)(b) from AC>OW to OW>AC or;
(iii)(b) from AC>OW to OW=AC). 

("M") a necessary condition for a change in winner is that some
voter's relative rankings of the old winner and the new winner change to
favour the new winner (i.e.-
(i) from OW>NW to NW>OW or;
(ii) from OW>NW to OW=NW or;
(iii) from OW=NW to NW>OW).

(P1 as Craig most recently expressed it) a necessary condition for a
change in winner is that some voter's relative rankings of two different
candidates, one of them being (in that voter's rankings) ranked below
the new winner either before or after the change, change (i.e.-
(i)(a) from AC>NW to NW>AC or;
(ii)(a) from AC=NW to NW>AC or;
(i)(b) from NW>AC to AC>NW or;
(ii)(b) from NW>AC to AC=NW or;
(iii)(a) from AC1>AC2, NW>AC1 to AC1=AC2 or;
(iv)(a) from AC1>AC2, NW>AC1 to AC2>AC1 or;
(v)(a) from AC1>AC2, NW>AC2 to AC1=AC2 or;
(vi)(a) from AC1>AC2, NW>AC2 to AC2>AC1 or;
(iii)(b) from AC1=AC2 to AC1>AC2, NW>AC1 or;
(iv)(b) from AC2>AC1 to AC1>AC2, NW>AC1 or;
(v)(b) from AC1=AC2 to AC1>AC2, NW>AC2 or;
(vi)(b) from AC2>AC1 to AC1>AC2, NW>AC2).

As can be seen, both (P1) as I incorrectly remembered it and (P1) as Craig
has now expressed it are implied by monotonicity. They do not imply
monotonicity, and (P1) as I remembered it and (P1) as Craig has now
expressed it do not imply each of the other.

Now, take for granted majority rules (what Craig has defined as 2-member
FPTP rule). That is, we already have a space of voting schema for which we
know the answer- those in which one candidate recieves more than half of
first preference votes (the candidate is most highly ranked by more than
half of voters who express some non-indifference between candidates).

Also, assume that the system is neutral to ballots ("one vote one value")-
elections with identical aggregates of ballots elect the same candidate.

Now- consider a Condorcet situation where a Condorcet winner is known. I'm
going to show that it is a necessary (but not that it is a sufficient)
condition on "monotonicity" that where a Condorcet winner is known to
exist then the Condorcet winner must win.

Consider a situation for which we know the Condorcet winner.

(1) For any non-Condorcet candidate, in order to attain an absolute
majority of the votes some voters must switch their relative
rankings between the n-C candidate and the C candidate to favour the n-C
candidate. This is because more than half of the voters (who express a
preference between the two) prefer the C candidate to the n-C candidate,
and in order for the n-C to recieve an absolute majority the n-C candidate
must win votes from this majority.
 
(2) Any change of votes which leads to the Condorcet winner recieving an
absolute majority does not require of a specific candidate that
some voters must change their relative rankings of the Condorcet Winner
and the specific candidate (e.g. Consider all the voters who prefer A (the
CW) to C change their votes so that A is their first preference. Because
more than half of the voters prefer A to C, this immediately means A has
an absolute majority without any relative rankings between A and C
changing. Therefore, C is ruled out as a monotonic solution). This rules
out any candidate being the monotonic choice except for the
Condorcet winner.
 
Therefore, a necessary condition for monotonicity is that Condorcet
winners get in.
 
However, unfortunately, monotonicity can't be completed- the thing gets
more complicated by the fact that these Condorcet solutions themselves
rule out the existence of _any_ monotonic solution for some non-Condorcet
examples.
 
There was a discussion I had with Blake Cretney in the dim distant past
(i.e. I can't find it in the egroups search) about this (I was trying to
"fill in the gaps" but Blake discovered the paradox and I was left lost).
I brought it up because there are interesting parallels with Craig Carey's
(P1) condition, which is a sub-condition of monotonicity. However, I don't
have time to work out explicitly what the implications for (P1) are. I
suggest Craig tries to work out whether Condorcet is a necessary condition
of (P1).

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.