[EM] Strategic voting in Condorcet & Range N-canddt elections - boiled down (& Ketchum reply)

[Warren Smith]

> I would go for a different strategy - whatever leads toward long-term
> strength.
>
> Winning the current election is usually pretty good.
>
> A third party, looking ahead, can think of what will make a stronger
> future for it, even though perhaps not having a chance to win the
> current election.

--my strategy analysis was only about winning current election.  If you do not
care about current election, then it's a whole different ballgame.
Still, if third parties always lose, then I don't see how they get
long term strength!

--Thinking some more about this whole argument, it can be boiled down further.
Here is the result:

THEOREM:
Consider a single-winner voting method that inputs rank-order ballots
with rank-equalities permitted.  Assume the method is
1. monotone
2. always elects a candidate ranked unique-top by a majority (if+when
one exists).

[EXAMPLES of methods satisfying these conditions:
Plurality, Black=Condorcet with Borda as fallback, Schulze beatpaths Condorcet,
Tideman ranked pairs Condorcet.   Where note Borda has to be defined
in such a way it is monotonic even if equalities are permitted in
rankings.
Examples of methods NOT satisfying these conditions:
IRV, Coombs, Nanson-Condorcet, WBS-IRV-Condorcet: not monotone.
Borda: fails condition 2.]

Fix the number N>=3 of candidates.
Assume the number of voters is large and your vote is
1. unlikely to be capable of altering the winner
2. but if it can, it can only choose between TWO winners A & B;
we assume elections in which your vote can choose between THREE OR MORE
winners are neglectibly unlikely.

Then: a maximally-strategic vote for you (in hindsight, knowing all ballots) is
always of the form ranking A unique top, B unique bottom, and the rest
in between in
some manner (assuming you prefer A>B).

Further, if everybody votes this strategic way, then A or B (one of
the two "frontrunners") will always win except if there is a 3-way
perfect tie (which we regard as neglectibly unlikely) and it will be
the same winner as Plurality.

COROLLARY 1:
Under any circumstances where Approval is a better voting system than
plurality for
strategic voters, Approval also is a better voting system than every
system meeting the criteria of the theorem (and with voters of the
maximally-strategic ilk in the theorem).

COROLLARY 2:
Under any circumstances where Approval is a better voting system than
plurality for
strategic voters, and if "strategic range voters" mean the same thing as
"strategic approval voters", range is a better voting system than
every system meeting the criteria of the theorem (and with voters of
the maximally-strategic ilk in the theorem).

REMARK 1:
Range is known to be better than every rank-order ballot voting system
for honest voters under the "RNEM" probability model, where "better"
is measured by Bayesian Regret.  This proven by WD Smith in recent
papers for each N=3,4,5,...,31.
Permitting rank-order ballots to include equalities has no effect on
this because in the RNEM exactly-equal utilities occur with
probability=0, hence honest rank-order
voters would not use them.

So in these senses, range is better for both honest & strategic voters
than every
rank-order method meeting the criteria in the theorem, measured by
Bayesian Regret.

REMARK 2:
Approval also is known to be better than or equal to every rank-order
ballot voting system for honest voters under the "RNEM" probability
model, where "better" is measured by Bayesian Regret, if N=3.

REMARK 3:
If N=4 and N=5 however, there exist rank-order systems superior to
approval voting with
honest voters (under RNEM).   Indeed, Borda is thus-superior, and
certain Condorcet methods also appear to do this job.

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html