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
[EMAIL PROTECTED] 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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