[Election-Methods] Challenge Problem

[Jobst Heitzig]

Hi again.

There is still another slight improvement which might be useful in 
practice: Instead of using the function 1/(5-4x), use the function
   (1 + 3x + 3x^7 + x^8) / 8.
This is only slightly smaller than 1/(5-4x) and has the same value of 1 
and slope of 4 for x=1. Therefore, it still encourages unanimous 
cooperation in our benchmark situation
   50: A(1) > C(gamma) > B(0)
   50: B(1) > C(gamma) > A(0)
whenever gamma > (1+1/(1+(slope at x=1)))/2 = 0.6, just as the other 
methods did.

The advantage of using (1 + 3x + 3x^7 + x^8) / 8 is that then there is a 
  procedure in which you don't need any calculator or random number 
generator, only three coins:


**
** Method 3-coin-FAWRB
** ---------------------
** Ballots: Approval with one option marked "favourite".
** Tally:
**  1. Determine the approval winner, X.
**  2. Draw a ballot at random;
**     if it does not approve of X, its favourite, Y, wins.
**  3. Otherwise, toss three coins;
**     if they show no heads, X wins.
**  4. If it's one head, draw one further ballot;
**     if it's two heads, draw another seven ballots;
**     if all three show heads, draw another eight ballots.
**  5. If all drawn ballots approve of X, she wins;
**     otherwise, Y wins.
**


Isn't this guaranteed fun?

Jobst



I wrote:
> 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
>>
> 
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