[EM] Re: twisted prism, etc.

[Jobst Heitzig]

Dear Forest!

Big Sorry for not having read carefully enough. I missed the point about 
public elections not being supposed to show much symmetry. That is a 
very good point I guess which we could even strengthen to this claim:

----
In almost all public elections, there will either be a CW or a 3-cycle 
so that each other candidate is beaten by at least 2 of the 3.
----

Do you agree? When we assume this, you are perfectly right to suggest to 
use a two-step method which first reduces to Dutta's minimal covering 
set (=the above-mentioned CW or 3-cycle) and then defines an efficient 
lottery on the 3 candidates, for example using the wonderfully simple 
Random Ballot method.

So, YES, I think that Random Ballot Dutta is a very good approach -- 
good work!

Now we could even try to include some preference strength information, 
for example like this:
a) Voters specify an approval cutoff. Determine the Dutta set D. Draw a 
random ballot and remove all elements of D which are not approved on 
that ballot, except when no element of D is approved. Repeat drawing 
random ballots until only one element remains.
b) Or like this: Voters specify ratings and defeat strengths s(X>Y) are 
computed as in Cardinal Weighted Pairwise. In the Dutta set X>Y>Z, give 
X a probability proportional to s(X>Y)/s(Z>X).
c) Or any other way :-)

Yours, Jobst

PS:
One minor point: We should justify the "public elections" claim a bit 
further since it might collide with a similar but contradictory 
assumption from tournament theory I referred to yesterday. The claim is 
that in a tournament with a sufficient number of almost equally good 
teams, it can be expected that none of the teams is covered and no set 
of teams is a proper clone set. Or to put it otherwise: If we assume 
that it is equally likely that X>Y and that Y>X no matter how the 
defeats come out between other pairs, then there will most likely be no 
clones and no covered candidates. It would be interesting to know 
whether in such a "random tournament" also the Dutta set must be 
expected to consist of all candidates...

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