[Election-Methods] Challenge Problem

[Jobst Heitzig]

Dear Forest,

well - thanks.

Anyway, there is still room for improvement.

Our last version was this: Let x be the highest approval rate (=approval 
score divided by total number of voters). Draw a ballot at random. With 
probability 1/(5-4x), the option with the highest approval score amoung 
those approved on the drawn ballot wins. Otherwise the favourite of that 
ballot wins.

We saw that this method performs well in a large number of situations. 
But it seems to me that, with more than three options, it can be hard to 
find the optimal strategies because approving a non-approval-winner can 
be bad.

For example, consider this case:

   33: A > B >> C=D=E
   34: C > B=D >> A=E
   33: E > D >> A=B=C

Here the C faction can either cooperate with the A faction to give B a 
high probability of winning, or with the E faction to give D a high 
probability of winning. But when the A faction approves of B but the C 
and E factions approve D, it would have been better for the A faction to 
have bullet-voted.

The following even simpler method, however, makes it safe to approve of 
an option which does not turn out the approval winner:


**
** 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.
**


FAWRB is again monotonic and solves the original challenge problem in 
the same way as the other methods we discussed recently. But in the 
above situation it makes it safe for the A and C factions to approve of 
B and D since only one of the two factions will actually partially 
transfer their winning probability from their favourite to the 
compromise option.

I guess it should be possible to analyse FAWRBs strategic implications 
in detail since the method is so extremely simple!

I'm pretty sure already that with FAWRB you will never have an incentive 
to misrepresent your favourite, and seldom or never to approve of one 
option while not approving of all more preferred options as well. With 
the other methods these variations of "order reversal" would occur more 
often I think.


Yours, Jobst


PS: I have not yet thought much about your most recent proposals. Only 
it seems that they won't elect any compromise option that's not the 
favourite of anyone, right?


fsimmons at pcc.edu schrieb:
> Jobst wrote
> 
> ...
> 
>> What do you think about this?
>>
> 
> I think you have the "golden touch!"
> 
> Forest
>