[EM] Some chance for consensus (was: Buying Votes)

[Jobst Heitzig]

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



fsimmons at pcc.edu 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
> 
> 
>