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

[David Catchpole]

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 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."

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

("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.

Take for granted majority rules. 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.

Now- consider a Condorcet situation where a Condorcet winner is known. I'm
going to show that it is a necessary (but not 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 preferences 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 a specific candidate to have their vote
switched with relation to the Condorcet Winner. (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. 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 seems to be a sub-condition of monotonicity.