0 Condorcet winners was: [EM] "No splitting rule"

[Blake Cretney]

>       A note on Condorcet's picking 0 winners when there should be 1

> 
> A note on Condorcet and pairwise comparing. This is relevant to the new
>  rules.
> 
> Let's estimate F, where 10**(-F) equals the probability that the Condorcet
>  method actually returns a single winner when there are 10**99 candidates.
> 
> The approximate derivation here requires the drawing of a graph, and noting
>  or saying that the directions of the arrows are quite random, and that
>  F = log base 10 of approximately this: 10**99 times (1/2)**(10**99 - 1).
> 
> So F = -3*(10**98) (approximately).
> 
> So Condorcet rarely picks a winner for large enough elections.

You are assuming that arrow direction is random.  Even if the
electors vote randomly, the arrows still won't be random.  This is
because actual ballots impose some consistency on the arrows.  For
example, if it is randomly determined that everyone votes A over B,
then it follows that C cannot do better against A than against B.  If
you chose all the pair-wise contests independently, this would not be
the case.  In general, elections corresponding to real ballots (even
randomly filled out ones) will give more consistent pair-wise
contests, and therefore increase the chance of a Condorcet winner.

Even with this consideration, your argument is based on the
assumption that votes are random.  That would seem to be a neutral
assumption, but it is unlikely to be true in real elections.  As well,
the whole principle of Condorcet's method is based on the assumption
that votes are NOT random, and that the results are therefore
meaningful.  Once you assume that votes are random, neither Condorcet,
nor any other electoral method will appear sensible.

> The next step in the argument is to wait for the defender of Condorcet to
>  say that there are ''too many candidates'. That happens.
> It is not trivial to prove that, but a statement will substitute for a
proof.
> 
> At a certain number of candidates, Condorcet becomes a bad method.

A Condorcet winner is possible at any number of candidates.  If the
electors vote randomly, the chance of this occurring will decline with
the number of candidates.

> 
>                      Bounding where Condorcet goes bad
> 
> Maybe Condorcet is OK for 102 candidates but it is not OK for 103.
> Or does it go bad gradually and the only way to save Condorcet is to say
that
>  it gets worse slowly and use ideas from probability theory to save the
day.

It depends what you mean by "goes bad".  If you mean "gives no
conclusion" then this could happen at three candidates or not happen
by 103.

Condorcet's method says that where there is a Condorcet winner, that
candidate should win.  It doesn't say that when there is no Condorcet
winner no one should win.  You just have to have some prepared way to
deal with the situation.  Many have been suggested.

In the same way, most people think that someone who gets a majority
of first-place votes should win.  We could define the "Majority
method" as the procedure that carries this out.  Of course, the
procedure would sometimes not give us a winner, so we would advocate
various majority-completion methods that gave a winner even in these
cases.  Plurality, Condorcet, and plurality-elimination, are all
majority-completion methods.  Anyone using any of these can be said to
be using the Majority method.

I think it clearly makes no sense to talk about the majority method
as "going bad," or criticize the majority completion methods because
there isn't always a majority winner.  

If you disagree with the majority method, the same argument can be
made using a "unanimity method".  It is quite possible to say that
someone who is unanimously chosen as the favourite should win, and
still acknowledge that such a candidate may not exist.  The unanimity
method doesn't "go bad", it just doesn't always give a conclusion.

---

Blake Cretney