Hi folks,
I think I know what the problem is with the idea of somehow
automatically match pairs or larger groups of voters who will all
benefit from a probability transfer: It cannot be monotonic when it
requires that the ballots of all members of the matched group indicate
that the respective voter profits from the transfer.
Look at the simplest version where we have only two voters who submit
favourite and approved information:
Situation I:
Voter 1: A favourite, C also approved
Voter 2: B favourite, C also approved
If we interpret the approval information as an indication that the
voters like C better than tossing a coin between A and B, we would be
tempted to let the method match these voters and transfer both their
winning probabilities from their favourites to C. So C will win with
certainty.
But if we want monotonicity also, C must still win with certainty in the
following situation:
Situation II:
Voter 1: C favourite, A also approved
Voter 2: B favourite, C also approved
But in this situation, a matching algorithm would *not* match the voters
since voter 2 obviously does not seem to profit from such a transfer.
D2MAC and FAWRB don't have this problem: they are not based on matching
and *do* elect C with certainty in situation II. For this reason, voter
2 would have incentive *not* to approve of C in situation II when D2MAC
or FAWRB is used. It seems the monotonicity is paid for by a need for a
bit more of information in order to vote strategically efficient.
A similar argument shows why it is so difficult to solve the following
situation:
Situation III:
Voter 1: A1 favourite, A also approved
Voter 2: A2 favourite, A also approved
Voter 3: B favourite
Suppose we want our method to give A a winning probability of 2/3 in
this situation. Then we have a problem in the following situation:
Situation IV:
Voter 1: A favourite, D also approved
Voter 2: B favourite, D also approved
Voter 3: C favourite, D also approved
Here each of the three voters would have an incentive to change her
ballot and *not* approve of D, since that would move 1/3 of the winning
probability from D to her favourite. So, the strategic equilibria in
situation IV will be
Voter 1: A favourite
Voter 2: B favourite, D also approved
Voter 3: C favourite, D also approved
or
Voter 1: A favourite, D also approved
Voter 2: B favourite
Voter 3: C favourite, D also approved
or
Voter 1: A favourite, D also approved
Voter 2: B favourite, D also approved
Voter 3: C favourite
each of which won't result in D being elected with certainty.
So, it seems we can't have efficient cooperation in both situations III
and IV!
Situation IV seems to be the more important, and D2MAC and FAWRB both
make sure that in situation IV full cooperation is both an equilibrium
and efficient. But for this they need to give A less than 2/3 in
situation III, however.
Yours, Jobst
[EMAIL PROTECTED] schrieb:
What do I think? All of these ideas are better than what I have come up
with, and have great potential, whether or not they might need some
tweaking or even major over haul.
I'll try to digest them more in the mean time, to get a better feel for
their strengths and potential weaknesses.
Marriage and matching procedures certainly seem natural in this setting.
Thanks,
Forest
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