Does VA Schulze violate SEC?

[Blake Cretney]

On Fri, 02 Oct 1998 20:23:41   Markus Schulze wrote:
Dear Markus,
>Dear Blake,
>
>you wrote (02 Oct 1998):
>> I think you're mistaken about Schulze and Tideman because both of
>> these methods are identical to Condorcet(EM) in the 3 candidate
>> case.  So, if you agree that Condorcet(EM) encourages random-fill
>> and violates SEC, then the same must be true of Schulze and Tideman
>> for three candidates.  Because your argument would seem to apply
>> to three candidates examples as well as any others, I have to
>> conclude that it includes an error.  Furthermore, I don't see
>> why adding additional candidates beyond three would fix these
>> problems where they exist.
>
>I don't agree. For example: When I say, that the Copeland method
>can be manipulated by cloning candidates, then this means, that
>you will never have a disadvantage if you present clones and that
>you will sometimes have an advantage if you present clones.
>
What if I proposed a method where clones could sometimes hurt, but
almost always helped.  Wouldn't that encourage more clones too.
So, to prevent the cloning strategy, it would be necessary either that
clones had no effect, or that they were as likely to hurt as help.

>But you say, that already if there is only one situation, where
>an election method can be manipulated using a certain strategy,
>this election method has this strategical problem.

I intend to prove that the strategy works, not in only one case, but on
average.  Only the average can be proven for Condorcet(EM), but
you have in the past been willing to conclude it has the problem.

Let me review what I think we agree on:
1.  In Condorcet(EM), if two candidates are ranked as equal, the
chance is better that one of them will win, than if they had been
differentiated.  So, voters are encouraged to differentiate candidates 
at the end of the ballot and not at the beginning.
2.  These strategies can sometimes back-fire, however, on average they
succeed, and so #1 holds.
3.  To prove that this is true for Schulze, it is only necessary to
show this effect on average, not that it will never back-fire.
4.  Schulze is equivalent to Condorcet (EM) in the 3 candidate example.
And so will have this effect if there are only 3 candidate.

In fact, because you have said that statement #1 also holds for
Smith//Condorcet(EM), and because this is equivalent to Schulze
when there are 3 or fewer candidates in the Smith set, #1 will
have this effect in cases where the Smith set has 3 or fewer
members.

You make a strong intuitive case that if we do not consider problems
of limited Smith set examples, Schulze will not have this effect, but
neither will it have the opposite effect.

>This is not true. By ranking the less favourite candidates,
>the worst beat-path from a more favourite candidate to a
>less favourite candidate could be increased. But it could
>also happen, that -by ranking the less favourite candidates-
>the worst beat-path from a less favourite candidate to a
>more favourite candidate is increased (especially if the
>worst beat-path from this less favourite candidate to that
>more favourite candidate contains other less favourite
>candidates and is limited by the pairwise defeat between
>two less favourite candidates).

So, since under Schulze, for size 3 Smith set, differentiation 
decreases combined chances, and as the Smith set increases in
size this has less effect, moving towards equality, and since
the Smith set could be under size 3, don't we have to conclude
that on average the strategy will work for Schulze.  In the same
sense that it works for Condorcet(EM).

The only way Schulze could recover from the difference with the
size 3 Smith set is if higher Smith set size actually experienced
the opposite effect that could cancel out the effect for the size
3 Smith set case.

Note that I am not saying that just because the strategy holds in the
three candidate example, the strategy holds over all.  However, because
the strategy works well in the three-candidate example, and you yourself
seem to argue that it is not deterred as you increase the number of
candidates, I conclude that it works on average.

>You wrote (17 Sep 1998):
>> Sincere Expectation Standard
>> Given that a voter has no knowledge about how others will
>> vote, a sincere vote must be at least as likely as any
>> insincere vote to give results that are in some way better
>> in the eyes of the voter.
>
>I don't think that this standard is important, because if
>the voters have no knowledge about how others will vote,
>then every election method is strategy-proof.
That's just my point; that isn't true for VA.  Of course, it is
easy to imagine a marginal version of Schulze that wouldn't have
this problem.  That's the system I advocate.

Blake



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