[EM] Strategic voting in Condorcet & Range N-canddt elections

[Warren Smith]

We shall assume N>=3 and N is fixed (N = #candidates).

1. In general, your vote (in an election with a large #voters) is
going to have no effect -- you cannot create or break a tie because
the election is too far away from being a tie. In this case, how you
vote is irrelevant.

2. In the unlikely event your vote CAN have an effect, it is most
likely to be because you have a chance to create or break a tie
between TWO candidates -- call them A & B -- the two "frontrunners."
More precisely, you have an opportunity to make either win.
Wlog we shall assume you prefer A over B.

3. Much less likely still is the situation where your vote actually
has the power to make one of THREE or more candidates win (you choose
which).  Let us NEGLECT that as just too incredibly unlikely.

4. If in range voting the two frontrunners are A & B, your
strategically best vote is
score A max, B min.  Now your scores for the others are irrelevant, although
to help yourself in the ultra-unlikely 3-way-tie case in (3), you'd want to
(semi-honestly) score everybody you like better than A, also max,
everybody you hate worse then B, also min, and everybody you think lie
between A & B, ordered in some non-dishonest manner.  This strategic
vote then is "semi-honest," that is, never
orders X>Y for any X,Y for which you honestly feel Y>X.

5. In a monotone Condorcet method (such as Schulze, Tideman ranked
pairs, etc) you cannot go wrong by ranking A top and B bottom (both of
which, in general, will be dishonest, but this is always strategically
correct).   How you then rank the others, may be important.  (In range
voting, those are irrelevant if 3-way ties are neglected, but with
Schulze they could be crucial.) It also can be highly mysterious.
There might be
some mysterious magic strategic ordering of the others that does the
job for you to make A win.   It might also be that you can get away
with ranking some coequal;
you may not need to use a strict rank-ordering.   However, I can prove
at least a constant fraction of the time (in random elections), it
will be strategically forced for you to dishonestly order X>Y for some
X,Y where you honestly feel Y>X; ordering X=Y will not work. So in
that sense. all Condorcet methods are inferior in terms of
strategy to range voting -- they a constant fraction of time force
full-dishonest ordering.

6. My old (1999-2000) Bayesian Regret simulations, when considering
strategic voters,
made as their first move, the decision to rank the two frontrunners
top and bottom.
As we've seen, that decision was wholy justifiable... though it what
it then did about the other candidates may have been less
justifiable...

Then

THEOREM:  If all voters rank the two frontrunners unique-top and unique-bottom
(and the identity of the two frontrunners is publicly known before
election starts)
then in any Condorcet method (ignoring exact ties) one of these two frontrunners
will always win, and it will be the same as the strategic
plurality-voting winner.

Thus in this sense, Condorcet is no improvement over plurality voting
(for strategic
voters who behave this way). With voters of this ilk we'd expect to develop
two-party domination, same as with plurality voting. Which is rather pathetic.

7. However, it might be that voters would, after ranking A top, B bottom, then
rank the others in a manner involving lots of equalities with A or B.
(The theorem above was assuming NO equalities with A or B were
used in the strategic vote.)   The ultimate limit of this would
be casting an approval-style vote.

But if every voter did THAT then, by another THEOREM stated in a
recent post by me,
the Condorcet election ssytem would degenerate to become identical to
approval voting.

Now, heuristically, these are two strategic voter-behavior "endpoints":
Endpoint I. strategic voters rank A>others>B and B>others>A only.
Result: Condorcet degenerates to strategic Plurality voting.

Endpoint II.  strategic voters rank A=others1>others2=B and
B=others1>others2=A (approval style only).  Result: Condorcet
degenerates to strategic approval voting.

The "intermediate points" would be a MIXTURE of voter behaviors of the
two forms in I and II.   Remember, only these are strategic votes;
there are no other possibilities.

Heuristically speaking, it makes sense to conjecture now that the result of
"intermediate" stategic voting, will be intermediate between the two
endpoint results,
i.e. roughly equivalent in quality to what we would get with a mixture
of approval voters and plurality voters in an approval-voting
election.

But in my Bayesian Regret work, it was found that you get the most quality
(least regret) using pure approval voting, in mixtures of this sort.

This suggests (heuristically only - I am not claiming this is a
rigorous result) that
strategic Condorcet voters will ALWAYS produce results which are no better
(in terms of Bayesian Regret) than you would get with pure approval voting.

Here 1-6 were all rigorous results, but 7 is merely a heuristic argument.
To the extent it is valid, though, it is rather important -- it means
anybody who believes
strategic voters are all that matters,  should discard every Condorcet
voting system and prefer Approval voting (as simpler and at least as good)

Further, anybody who believes honest voters are all, should prefer range voting
both over every Condorcet system and over approval (except that if you
regard range voting as the same thing as approval for strategic
voters, this is not a distinction,
we could just use range for both honest & strategic voters, getting
superiority over Condorcet in all cases).

Which would lead to the conclusion range is superior to all Condorcet
systems including
if rank-equalities are permitted in them.

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