Dear Jobst,
Here's my die toss version of your FAWRB method:
1. Draw a ballot at random.
2. If this ballot does not approve the AW, THEN elect its favorite (and STOP).
Else continue to step 3.
3. With the help of dice, coins, spinner, balls and urn, or an icosahedron
with numbered faces,
conduct a Bernoulli experiment with a twenty percent success rate.
4. In case of "success," then elect the AW (and STOP), else repeat from step 1.
Do you like that?
Now, for comparison, here's a method that I call AWFRB:
1. Draw a ballot.
2. If this ballot does not approve the AW, THEN draw another ballot and elect
its favorite,
ELSE continue to step 3.
3. Toss two coins. If they both show heads, then elect the AW, else repeat
from step 1.
It turns out that this method elects the AW with probability x/(4x-3) , where
x is the approval of the AW.
AWFRB doesn't satisfy your "Bullet Proportionality Property," which is
satisfied by FAWRB, but it gives slightly more encouragement for approval
cooperation
in our 33, 33, 33 test case. The compromise A of the two compromising factions
(A1 and A2) is elected with probability 1/3 under AWFRB, but only 2/7 under
FAWRB.
I have some analysis, but no time for it now.
Forest
> ** Method FAWRB (Favourite-or-Approval-Winner Random Ballot):
> ** -------------------------------------------------------------
> ** Everyone marks a favourite and may mark any number of "also
> approved"** options. The approval winner X and her approval rate
> x are
> ** determined. A ballot is drawn at random. If the ballot
> approves of X,
> ** X wins with probability 1/(5-4x). Otherwise, or if the ballot does
> ** not approve of X, its favourite option wins.
> **
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