[EM] Monotonicity

[MIKE OSSIPOFF]

Yes, my more general short definition could be interpreted in a way so that 
Borda,
and the pairwise-count methods would fail. It might be an impossible 
situation, caused
by the fact that, though what we feel we mean by Monotonicity is 
simple--voting someone
higher shouldn't cause him to lose--we expect something else when we apply 
it.
It's customary to apply Monotonicity in a way that assumes that, if 
possible, we don't
change the order in which we vote the other candidates.

Say you've ranked Smith in 7th place, and Jones in 5th place, and you vote 
Smith higher
by moving him to 5th place. That causes it to be necessary to move Jones. 
Lacking
any rules about where to move him, we could move him to 1st place, and move 
the
previous 1st place candidate down to 7th place. We do that because we have 
to move
Jones, and that's one way to do that. Whatever happens when we move Jones to 
1st
place can be said to have been caused by upranking Smith, since doing so 
caused it
to be necessary to move Jones.

That's the problem that I notice with my brief definition, the one that 
seems to say what
we mean by Monotonicity--till we apply it.

But maybe it isn't to un-general to speak of order. After all, what can a 
ballot do with
a candidate, other than tell how good he is, or tell if he's better than 
someone else?

So my long version, and/or whatever you've written along those lines, might 
be ok
after-all. My long version might be ok with the "voting Smith higher" 
definition that I
posted with it. Probably is, though I haven't studied that yet. At worst, my 
previous
definition of voting Smith higher would be needed.

My long definition doesn't mention rating or ranking, but does mention 
order.
It mentions how candidates are marked on the ballot, but that's general.

I suspect that the problem defining Monotonicity results from what we intend 
for
Monotonicity being different from how we customarily apply it. The 
definition(s) that
speak of order should be the ones used now, since they conform to custom in 
applying
Monotonicity. If eventually someone writes a definition that doesn't have to 
mention order,
and works as expected by custom, then that could then replace the previous 
definition.

Or maybe order just has to be accepted as part of the customary application 
of
Monotonicity.

Mike Ossipoff

Forest had written:

Our attempts were either too strong (eliminating all Condorcet methods,
for example) or, too ambiguous, or lacking in generality.

We were striving to be so general that we would never have to mention
ranking, rating, or order ... just the results of the election method
(i.e. the method being tested for monotonicity) when restricted to various
combinations of ballots.  Such a definition would work on any method
whatsoever, based on any possible ballot type, even those types that have
not yet been envisioned due to our lack of imagination and ingenuity.

We haven't given up, but it's on the back burner for now.

When we have time, we should summerize some of the blind alleys and
partial results so as to prevent unnecessary duplication of effort.
Perhaps some of the list members could then carry it beyond the point
where we left off.




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