[EM] Test implementation of FAWRB

[Jobst Heitzig]

Dear folks,

feel free to test my simple implementation of FAWRB at 
http://www.impro-irgendwo.de/groucho/fawrb.html

It's really easy and I hope it is some fun, too.

And please send comments or corrections if some of my English is too bad :-)

Yours, Jobst



Jobst Heitzig schrieb:
> Dear Forest,
> 
> I'm sorry again for answering so late - I always find time to read your 
> messages and even think some time about them but not for answering...
> 
> What occupies me most at this moment with your idea of using some 
> "degree of cooperation" to determine the weights in a mix of a random 
> ballot lottery and some compromise option are the following questions:
> 
> 1. What option should qualify as the "compromise" to which we consider 
> transferring the winning probability from the favourite of the first 
> drawn ballot? Originally, you suggested taking the most approved option 
> of those approved on the first drawn ballot. Later I got the impression 
> that it would be strategically equivalent and more transparent if only 
> the approval winner was used as a possible compromise. This also enables 
> us to split the method into two simple phases, an idea I will describe 
> in the first method definition below. But the "most approved" variant 
> can also be improved in some way I will describe in the second method 
> definition below.
> 
> 2. How should "degree of cooperation" be measured? We started with the 
> average approval rate of all options, then saw that the resulting method 
> was non-monotonic and switched to the approval rate of the approval 
> winner. Today I had a new idea for this which I will describe in the 
> second method definition below.
> 
> 3. How should this measure of cooperation (designated by x) be 
> transformed (via a function f) into a winning probability f(x) of the 
> compromise? We started with f(x)=x^4 which made sure that when the 
> compromise is "60% good" it would be an equilibrium if everyone 
> cooperated, so that the compromise would then come to be the sure 
> winner. I then suggested to use to use f(x)=1/(5-4x) instead, since that 
> function was the largest possible in which this equilibrium result was 
> true.
>   Meanwhile I realized that with the latter function, this equilibrium 
> often is not very stable, while the other equilibrium ("no cooperation 
> at all") is not only stable but globally "attractive" (by "attractive" I 
> mean attractive in the dynamic system whose state is the pair (x,y) of 
> cooperation rates in the two faction and whose dynamics is that both 
> factions replace their cooperation rate with the optimal one given the 
> other faction's rate. By "globally" I mean that the system converges to 
> the equilibrium from almost all possible initial states). With the 
> original function f(x)=x^4 the "full cooperation" equilibrium is 
> globally attractive when the compromise is at least "60% good", but I 
> still feel that this choice of f vanishes too fast as x falls below 1. A 
> good intermediate choice for f could be this:
>   f(x) = 1/(5-4x) for x>=5/6, and
>   f(x) = 108/125 * x^2 for x<=5/6.
> This is the maximal function which makes sure the "full cooperation" 
> equilibrium exists when the compromise is at least "60% good" and that 
> it is also globally attractive when the compromise is at least "66.67% 
> good". More precisely, in the case of two homogeneous factions, the 
> "full cooperation" equilibrium will be globally attractive whenever the 
> sum of the two factions' ratings of the compromise is at least 4/3 (no 
> time for a proof here, may follow later).
>   Since this choice of f is a bit complicated, the choice for f I favour 
> at the moment is this:
>   f(x) = (5x^2 + x^14)/6.
> This is very near to the above piecewise function but much easier to 
> implement (see the method definitions below).
> 
> 
> Now let me suggest two more method variations:
> 
> 
> **
> ** Definition of method Two-Phase-FAWRB
> ** -------------------------------------
> ** Phase I: Perform a standard approval election to find the
> ** "compromise option" for Phase II. The question on the ballot reads
> ** "Which options do you consider good compromise options?".
> ** Designate the approval winner of this phase by X.
> ** ---------
> ** Phase II:
> ** 1. On a new ballot, everyone
> **    (i) specifies her favourite option and
> **    (ii) answers the question "Would you cooperate to elect X?"
> ** 2. A die is tossed. If it shows six then 15 of these ballots are
> **    drawn at random, otherwise only 3 are drawn.
> ** 3. If all drawn ballots answered "yes", X is elected.
> **    Otherwise the favourite on the first drawn ballot is elected.
> **
> 
> 
> **
> ** Definition of method FMAC-RB
> ** (Favourite or Most Approved Compromise Random Ballot)
> ** ----------------------------------------------------------------
> ** Phase I: Perform a standard approval election to find the
> ** "compromise ranking" for Phase II. The question on the ballot reads
> ** "Which options do you consider good compromise options?".
> ** ---------
> ** Phase II:
> ** 1. On a new ballot, everyone
> **    (i) specifies her favourite option and
> **    (ii) marks any number of additional options as "also approved"
> ** 2. A die is tossed. If it shows six then 15 of these ballots are
> **    drawn at random, otherwise only 3 are drawn.
> ** 3. On each drawn ballot, find amoung those options approved
> **    (=favourite or "also approved") on the ballot the one which
> **    had the highest approval score in phase I.
> **    If this "most approved compromise" option is the same for all
> **    drawn ballots, that option is elected.
> **    Otherwise the favourite on the first drawn ballot is elected.
> **
> 
> 
> Optionally, the ballot for phase II in both methods only contains those 
> options which received at least 5% in phase I.
> 
> 
> Both methods implement the function f(x) = (5x^2 + x^14) / 6 by using 
> the die.
> 
> The second method reintroduces the "most approved" idea but makes the 
> definition of "degree of cooperation" depend only on the approval rate 
> of the most approved option of the first drawn ballot. In this way the 
> method remains monotonic and will still enable more than one set of 
> factions to cooperate. Consider, for example this situation:
> 26%: A1>A>>A2 >>> B1=B2=B
> 25%: A2>A>>A1 >>> B1=B2=B
> 25%: B1>B>>B2 >>> A1=A2=A
> 24%: B2>B>>B1 >>> A1=A2=A
> Here it would be desirable if not only the A-voters had an incentive to 
> cooperate to give A a good winning probability, but if also the B voters 
>  had some possibility and incentive to give B a similarly good winning 
> probability. When only the approval winner (A) is considered a possible 
> compromise, this would not be possible. If the "most approved 
> compromise" is used but the degree of cooperation would depend only on 
> the approval winner, such a cooperation would be possible but there 
> would be strategic incentives for the B-voters to cheat. With the second 
> of the above methods, however, all voters have incentive to approve 
> "their" compromise (A or B). However, both A and B will only get about 
> 5/6*(1/2)^3=5/48 winning probability anyway, so perhaps this is not much 
> of an improvement. I posted the second method more as an indication in 
> which direction we could proceed to reintroduce the "most approved" part 
> in a strategically relevant way.
> 
> 
> I hope this was not totally incomprehensible...
> 
> Jobst
> 
> 
> fsimmons at pcc.edu schrieb:
>> 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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