# Mixed Condorcet-Plurality

Martin Harper mcnh2 at cam.ac.uk
Wed Apr 11 08:14:59 PDT 2001

```Tom Ruen wrote:

> AC=49, BC=48, CB=3

> is an equally interesting election to challenge the
> Condorcet winner - do we really want a majority support so badly that we are
> willing to elect a candidate with such a small first rank support? I
> reluctantly accept Condorcet in this case, and to me it shows voters need to
> be very careful in lower rankings in Condorcet.
>
> Many voters might vote a centrist second in a purely reactionary way without
> seriously considering if this candidate is superior to the candidate on the
> other side of the political spectrum.

But the election results you detail could equally well occur when the A>C
faction really does consider C (marginally) superior to B, and vica versa with
B>C prefering C to A. This is a case where Condorcet genuinly may have elected a
bad winner - it is possible that all voters would prefer a toin coss between A
and B to C, yet C still be elected. {alternatively, it is possible that all
voters would prefer C to the toin coss: one cannot tell from the Condorcet votes
alone).

The really bad problem is that strategising isn't going to change this: if the
A>C voters change to A>B then they elect B instead of C - and they prefer C to
B. So the A>C voters have no incentive to strategise, and similarly with the B>C
voters. Even if this problem is completely visible from the polls, and all the
pundits warn that this might be the result, there's still no way of avoiding it.

Do other methods suffer from similar bad cases?

> About the plurality process for cycles, I put the question out in hopes
> someone else has already made an evaluation of it. I'm interested to know if
> it can be monotonic for instance, although I expect it isn't. I'm not
> entirely worried about it's failings since we're talking about very strange
> elections (with cycles) anyway.

I would call your suggestion "Smith//Plurality" - the Smith Set is the smallest
possible set of candidates such that all members of the Smith Set pairwise beat
all non-members.

I *think* it's monotonic, but I'm not sure: Plurality itself is monotonic, as is
Smith, so it ought to be. On the other hand, it definately isn't immune to
clones, so there may be issues of vote-splitting. Whether these problems
manifest themselves often enough to be a concern is less certain. If simplicity
and transparency is important, then I might prefer this method over the more
complex Condorcet variants.

```