Hi,
Alex Small's definition of monotonicity (below) matches one that's
commonly used. (It's the one I use in my website about the Maximize
Affirmed Majorities voting method. MAM happens to be monotonic.)
A couple of years ago someone posted here in EM the web addresses of
two papers by Douglas R Woodall. In a separate message I'll send to
Alex the Woodall paper ("Monotonicity and Single-Seat Election Rules"
1996) that he'll likely find most relevant.
What I don't understand is why people care so much about monotonicity!
It's been written about many times as a criterion that a voting
method ought to satisfy. But I believe it's relatively unimportant.
I've never seen a strong justification for requiring its satisfaction.
It's merely one of many reasonable-seeming "consistency" criteria,
and no voting method can satisfy all such consistency criteria.
For instance, the Reinforcement criterion satisfied by Borda (and
other point scoring methods) and the Clone Independence criterion
failed by Borda (and other point scoring methods) are both consistency
criteria. Reinforcement is unimportant: since it's trivial to add a
rule that prevents a minority from choosing how to partition the
voters into districts, failure to satisfy reinforcement cannot in
practice be exploited by a minority. Clone Independence is very
important, though: since we want small minorities to be able to
nominate candidates, we cannot add a rule to prevent exploitation of
clone dependence by small minorities.
The only reason I care whether an otherwise-good voting method (such
as MAM) is monotonic is that other people might care about monotonicity.
--Steve
-------------------
Alex Small wrote:
> Hi everybody. I know I haven't posted much for quite some time.
> Right now I'm working on a paper on a result in strategic voting.
> I'll post the result once I've written the paper and checked the
> logic of the proof carefully. For now, I'm hoping somebody can
> help me with a reference: At one point in my proof I invoke the
> concept of monotonicity. I define a method as monotonic if a voter
> (or group of voters casting identical ballots) cannot cause the
> winning candidate to lose by ranking him higher (while leaving the
> relative order of the other candidates unchanged), nor can they
> cause a losing candidate to win by ranking him lower (while leaving
> the relative order of the other candidates unchanged). I've heard
> that there are multiple monotonicity definitions out there. Can
> somebody point me to a reference that discusses the definition that
> I'm using, or perhaps one that carefully distinguishes the
> definitions (so that I can sharpen the definition in the paper)?
>
> Thanks for any references that you guys can provide.
>
> Alex Small
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