[EM] Some chance for consensus revisited: Most simple solution

[Jobst Heitzig]

You're absolutely right, Juho -- I modified the condition a number of
times and didn't realize the last version did not imply both factions
prefer C to Random Ballot.

The correct set of situations for which SEC is a solution is
characterized by both factions prefering C to Random Ballot. The latter
is in particular true when alpha=beta and C has the largest total utility.

Sorry for the mistake,
Jobst

Juho Laatu schrieb:
> Makes sense but doesn't this allow also
> 
> 50: A(100) > C(40) > B(0)
> 50: B(100) > C(70) > A(0)
> 
> where 50*40 + 50*70 > max(50,50)*100
> 
> but the A supporters may prefer random ballot from the favourites urn to the possible consensus result (C) and therefore vote (e.g.) for A in their consensus ballot.
> 
> Juho
> 
> 
> 
> --- On Sun, 1/2/09, Jobst Heitzig <heitzig-j at web.de> wrote:
> 
>> From: Jobst Heitzig <heitzig-j at web.de>
>> Subject: [EM] Some chance for consensus revisited: Most simple solution
>> To: election-methods at lists.electorama.com
>> Date: Sunday, 1 February, 2009, 11:02 PM
>> Dear folks,
>>
>> I want to describe the most simple solution to the problem
>> of how to
>> make sure option C is elected in the following situation:
>>
>>    a%  having true utilities  A(100) > C(alpha) >
>> B(0),
>>    b%  having true utilities  B(100) > C(beta)  >
>> A(0).
>>
>> with  a+b=100  and  a*alpha + b*beta > max(a,b)*100.
>> (The latter condition means C has the largest total
>> utility.)
>>
>> The ultimately most simple solution to this problem seems
>> to be this method:
>>
>>
>> Simple Efficient Consensus (SEC):
>> =================================
>>
>> 1. Each voter casts two plurality-style ballots:
>>    A "consensus ballot" which she puts into the
>> "consensus urn",
>>    and a "favourite ballot" put into the
>> "favourites urn".
>>
>> 2. If all ballots in the "consensus urn" have the
>> same option ticked,
>>    that option wins.
>>
>> 3. Otherwise, a ballot drawn at random from the
>> "favourites urn"
>>    decides.
>>
>>
>> Please share your thoughts on this!
>>
>> Yours, Jobst
>>
>>
>>
>> Jobst Heitzig schrieb:
>>> Hello folks,
>>>
>>> I know I have to write another concise exposition to
>> the recent
>>> non-deterministic methods I promote, in particular
>> FAWRB and D2MAC.
>>> Let me do this from another angle than before: from
>> the angly of
>>> reaching consensus. We will see how chance processes
>> can
>>> help overcome the flaws of consensus decision making.
>>>
>>> I will sketch a number of methods, give some pros and
>> cons, starting
>>> with consensus decision making.
>>>
>>> Contents:
>>> 1. Consensus decision making
>>> 2. Consensus or Random Ballot
>>> 3. Approved-by-all or Random Ballot
>>> 4. Favourite or Approval Winner Random Ballot:
>> 2-ballot-FAWRB
>>> 5. Calibrated FAWRB
>>> 6. 4-slot-FAWRB
>>> 7. 5-slot-FAWRB
>>>
>>>
>>>
>>> 1. Consensus decision making
>>> ----------------------------
>>> The group gathers together and tries to find an option
>> which everyone
>>> can agree with. If they fail (within some given
>> timeframe, say), the
>>> status quo option prevails.
>>>
>>> Pros: Ideally, this method takes everybody's
>> preferences into account,
>>> whether the person is in a majority or a minority.
>>>
>>> Cons: (a) In practice, those who favour the status quo
>> have 100% power
>>> since they can simply block any consensus. (b) Also,
>> there are problems
>>> with different degrees of eloquence and with all kinds
>> of group-think.
>>> (c) Finally, the method is time-consuming, and hardly
>> applicable in
>>> large groups or when secrecy is desired.
>>>
>>>
>>> Let us address problem (a) first by replacing the
>> status quo with a
>>> Random Ballot lottery:
>>>
>>>
>>> 2. Consensus or Random Ballot
>>> -----------------------------
>>> Everybody writes her favourite option on a ballot and
>> gives it into an
>>> urn. The ballots are counted and put back into the
>> urn. The number of
>>> ballots for each option is written onto a board. The
>> group then tries to
>>> find an option which everyone can agree with. If they
>> fail within some
>>> given timeframe, one ballot is drawn at random from
>> the urn and the
>>> option on that ballot wins.
>>>
>>> Pros: Since the status quo has no longer a special
>> meanining in the
>>> process, its supporters cannot get it by simply
>> blocking any consensus -
>>> they would only get the Random Ballot result then. If
>> there is exactly
>>> one compromise which everybody likes better than the
>> Random Ballot
>>> lottery, they will all agree to that option and thus
>> reach a good
>>> consensus.
>>>
>>> Cons: Problems (b) and (c) from above remain. (d)
>> Moreover, it is not
>>> clear whether the group will reach a consensus when
>> there are more than
>>> one compromise options which everybody likes better
>> than the Random
>>> Ballot lottery. (e) A single voter can still block the
>> consensus, so the
>>> method is not very stable yet.
>>>
>>>
>>> Next, we will address issues (b), (c) and (d) by
>> introducing an approval
>>> component:
>>>
>>>
>>> 3. Approved-by-all or Random Ballot
>>> -----------------------------------
>>> Each voter marks one option as "favourite"
>> and any number of options as
>>> "also approved" on her ballot. If some
>> option is marked either favourite
>>> or also approved on all ballots, that option is
>> considered the
>>> "consensus" and wins. Otherwise, one ballot
>> is drawn at random and the
>>> option marked "favourite" on that ballot
>> wins.
>>> Pros: This is quick, secret, scales well, and reduces
>> problems related
>>> to group-think. A voter has still full control over an
>> equal share of
>>> the winning probability by bullet-voting (=not mark
>> any options as "also
>>> approved").
>>>
>>> Cons: (b') Because of group-think, some voters
>> might abstain from using
>>> their bullet-vote power and "also approve"
>> of options they consider
>>> well-supported even when they personally don't
>> like them better than the
>>> Random Ballot lottery. Also, (e) from above remains a
>> problem, in
>>> particular it is not very likely in larger groups that
>> some options is
>>> really approved by everyone.
>>>
>>>
>>> Now comes the hardest part: Solving problems (b')
>> and (e) by no longer
>>> requiring full approval in order to make it possible
>> to reach "almost
>>> unanimous consensus" when full consensus is not
>> possible. In doing so,
>>> we must make sure not to give a subgroup of the
>> electorate full power,
>>> so that they can simply overrule the rest. Instead, we
>> must make the
>>> modification so that still every voter has full
>> control over an equal
>>> share of the winning probability. This is why we
>> cannot just lower the
>>> threshold for consensus from 100% to, say, 90%. What
>> we do instead is this:
>>>
>>> 4. Favourite or Approval Winner Random Ballot (FAWRB),
>>>    simplest version, using two ballots
>> (2-ballot-FAWRB)
>> -------------------------------------------------------
>>> Still, each voter marks one option as
>> "favourite" and any number of
>>> options as "also approved" on her ballot.
>> The option getting the largest
>>> number of "favourite" or "also
>> approved" marks is nominated as
>>> "compromise". Two ballots are drawn at
>> random. If the nominated
>>> compromise is marked on both as "favourite"
>> or "also approved", it wins.
>>> Otherwise, the option marked as "favourite"
>> on the first of the two
>>> ballots wins.
>>>
>>> Pros: Full consensus can be reached if some option is
>> approved by
>>> everyone. Such an option will win with certainty. If
>> no such option
>>> exists, also partial consensus can be reached: if,
>> say, 90% agree to the
>>> best compromise option, that option will win with at
>> least 81%
>>> probability (=90%*90%). On the other hand,
>> bullet-voting still assures
>>> that my favourite gets my share of the winning
>> probability: if 5%
>>> bullet-vote, their favourite gets at least 5% of the
>> winning
>>> probability. Problem (b') shall no longer exist
>> since by not approving I
>>> do not destroy the consensus complete but only lower
>> the compromise's
>>> probability a bit.
>>>
>>> Cons: (f) The incentive to approve a good compromise
>> is only there when
>>> I prefer the compromise quite a lot to the Random
>> Ballot lottery, not
>>> when I prefer it only slighty. (g) If the process of
>> nominating options
>>> does not prevent this, there is the possibility that a
>> really harmful
>>> option is elected with some small probability.
>>>
>>>
>>> (Another method which achieves almost the same as
>> 2-ballot-FAWRB is the
>>> older D2MAC which is very similar.)
>>>
>>>
>>> The game-theoretic reason for problem (f) is this:
>>>
>>> Consider a situation in which C is the compromise and
>> all N voters
>>> approve it (N being large for simplicity). Now
>> consider that I ask
>>> myself whether it would server my better not to
>> approve C but to
>>> bullet-voter for my favourite, A. If I remove my
>> approval for C, the
>>> winning probabilities change in the following way: C
>> no longer wins with
>>> probability 1 but only with approximately probability 
>> 1 - 2/N  (more
>>> precisely  1 - 2/N + 1/N²). My favourite A's
>> winning probability grows
>>> from  0  to  1/N,  since A now wins whenever my ballot
>> is the first of
>>> the two drawn ballots. But at the same time, also the
>> other voters'
>>> favourites' winning probabilities grow, since
>> another voter's favourite
>>> now wins when my ballot is the second drawn ballot. In
>> other words, the
>>> probability of ending up with a Random Ballot lottery
>> result grows from
>>>  0  to approximately 1/N,  too. Therefore,
>> bullet-voting only makes
>>> sense when the utility I assign to the compromise C is
>> smaller than the
>>> mean of (i) the utility I assign to my favourite A and
>> (ii) the utility
>>> I assign to a Random Ballot lottery. In other words,
>> it is better to
>>> cooperate in the election of C only when I rate C
>> higher than half the
>>> way up from my rating of the Random Ballot lottery to
>> my favourite's
>>> rating.
>>>
>>>
>>> Important: Although the FAWRB process always uses a
>> chance process,
>>> namely drawing ballots, it will still usually lead to
>> a deterministic or
>>> almost deterministic result! This is because with the
>> incentives in
>>> place, people are usually very good at finding
>> compromises which they
>>> then will (almost) all approve of, giving them 100%
>> (or almost 100%)
>>> winning probability! Just as in "Consensus or
>> Random Ballot", the very
>>> fact that the voters don't like the Random Ballot
>> lottery when a
>>> compromise exists will lead to the compromise being
>> elected and the
>>> Random Ballot being avoided.
>>>
>>>
>>> The next step towards my recommended version of FAWRB
>> reduces this
>>> problem (f) by replacing the fixed number of three
>> ballots by a more
>>> sophisticated drawing process:
>>>
>>>
>>> 5. Favourite or Approval Winner Random Ballot,
>>>    calibrated version, using 3 or 15 ballots
>> (calibrated FAWRB)
>> ---------------------------------------------------------------
>>> Still, each voter marks one option as
>> "favourite" and any number of
>>> options as "also approved" on her ballot.
>> The option getting the largest
>>> number of "favourite" or "also
>> approved" marks is still nominated as
>>> "compromise". A die is tossed. If it shows a
>> six then 15 ballots are
>>> drawn at random, otherwise only 3 ballots. If the
>> nominated compromise
>>> is marked on all these ballots as
>> "favourite" or "also approved", it
>>> wins. Otherwise, the option marked as
>> "favourite" on the first of the
>>> drawn ballots wins.
>>>
>>> Pros: As in 2-ballot-FAWRB, but now voters will also
>> approve compromises
>>> they only find slightly better than the Random Ballot
>> lottery (more
>>> precisely: which they rate higher than 1/5 of the way
>> up from their
>>> rating of the Random Ballot lottery to their
>> favourite's rating).
>>> Cons: Problem (g) from above remains. (h) When there
>> are more than one
>>> possible compromise options, say C1 and C2, some
>> voters may apply
>>> "approval strategy" and refuse to approve of
>> C1 in order to get C2
>>> nominated instead of C1. When C1 is nominated anyway,
>> they thereby
>>> reduce C1's winning probability unnecessarily.
>>>
>>>
>>> Mathematical note: The reason why the mentioned
>> "approval limit" moves
>>> from 1/2 down to 1/5 of the way from Random Ballot to
>> favourite is that
>>> the expected number of ballots drawn moved from 2 to
>> 5.
>>>
>>> Next, we tackle problem (h) by decoupling the
>> nomination of the
>>> compromise from the later agreement to the nominated
>> compromise. This
>>> can be achieved by simply splitting the "also
>> approved" slot into two
>>> slots named "good compromise" (used for both
>> nomination and agreement)
>>> and "agreeable" (used only for agreement):
>>>
>>>
>>> 6. Favourite or Approval Winner Random Ballot,
>>>    version with four slots (4-slot-FAWRB)
>>> ----------------------------------------------
>>> Each voter marks one option as "favourite",
>> any number of options as
>>> "good compromise" and any number of options
>> as "agreeable" on her
>>> ballot, the unmarked options being implicitly regarded
>> as "bad". The
>>> option getting the largest number of
>> "favourite" or "good compromise"
>>> marks (but not counting "agreeable" marks!)
>> is nominated as
>>> "compromise". A die is tossed. If it shows a
>> six then 15 ballots are
>>> drawn at random, otherwise only 3 ballots. If the
>> nominated compromise
>>> is marked on all these ballots as
>> "favourite", "good compromise", or
>>> "agreeable", it wins. Otherwise, the option
>> marked as "favourite" on the
>>> first of the drawn ballots wins.
>>>
>>> Pros: Voters can now use approval strategy for the
>> nomination step
>>> without reducing the final winning probability of the
>> nominated
>>> compromise: The can just give only some one of the
>> potential compromise
>>> options the "good compromise" mark and
>> giving the other acceptavle
>>> compromise options the "agreeable" mark.
>>>
>>> Cons: Only problem (g) might remain.
>>>
>>>
>>> The final step is only needed when there is the
>> possibility that some
>>> really bad option can actually make it onto the
>> ballot. It is not needed
>>> when options are first checked by some independent
>> authority for their
>>> feasibility, as is often implicitly done in political
>> systems by supreme
>>> courts or the like.
>>>
>>> So, if (g) is really a problem, we can try to reduce
>> it by introducing
>>>    some mechanism by which a really large majority
>> (say, 90%) can
>>> prevent an option from being accepted on the ballot.
>> This leads me to
>>> the final version of FAWRB:
>>>
>>>
>>> 7. Favourite or Approval Winner Random Ballot,
>>>    version with supermajority-veto (5-slot-FAWRB)
>>> -------------------------------------------------
>>> Each voter marks one option as "favourite",
>> any number of options as
>>> "good compromise", any number of options as
>> "agreeable", and maybe some
>>> options as "harmful" on her ballot, the
>> unmarked options being
>>> implicitly regarded as "bad". Every option
>> receiving more than 90%
>>> "harmful" marks is removed before we
>> continue as in 4-slot-FAWRB: Of the
>>> remaining options, the one getting the largest number
>> of "favourite" or
>>> "good compromise" marks (but not counting
>> "agreeable" marks!) is
>>> nominated as "compromise". A die is tossed.
>> If it shows a six then 15
>>> ballots are drawn at random, otherwise only 3 ballots.
>> If the nominated
>>> compromise is marked on all these ballots as
>> "favourite", "good
>>> compromise", or "agreeable", it wins.
>> Otherwise, the option marked as
>>> "favourite" on the first of the drawn
>> ballots wins.
>>> Pros: The "harmful" slot allows a 90%
>> majority to keep harmful extremist
>>> options from having a chance.
>>>
>>> Cons: This supermajority-veto can be used to oppress
>> minorities which
>>> are smaller than 10%, because they have no longer full
>> control over
>>> their share of the winning probability.
>>>
>>>
>>> Hopefully that explains some things.
>>> I will also put the definitions into the Electowiki
>> within a few days.
>>> Yours, Jobst
>>>
>>>
>>> Raph Frank schrieb:
>>>> On Sat, Oct 25, 2008 at 8:02 PM, Greg Nisbet
>>>> <gregory.nisbet at gmail.com> wrote:
>>>>> Ok now the actual criticism. I know that FAWRB
>> is nondeterministic.
>>>>> Here is why that is bad.
>>>>>
>>>>> Factions (both unwilling to compromise):
>>>>>
>>>>> A 55%
>>>>> B 45%
>>>>>
>>>>> you view A as gaining a "55% chance of
>> victory".
>>>>> This reasoning is flawed. Instead of viewing A
>> as getting .55 victory
>>>>> units, think of it as a random choice between
>> two possible worlds:
>>>>> A-world and B-world
>>>>>
>>>>> A-world is 10% more likely to occur, however
>> they share remarkable
>>>>> similarities.
>>>>>
>>>>> In both worlds >=45% of the people had no
>> say whatsoever.
>>>> The trick with his method is that neither A-world
>> or B-world is likely
>>>> to actually occur.  It creates an incentive to
>> find a compromise,
>>>> called say, AB-world.
>>>>
>>>> If all voters vote reasonably, then the result is
>> a high probability
>>>> that the AB option will be picked.
>>>>
>>>> The utlities might be
>>>> ..... A-AB-B
>>>> 55: 100-70-0
>>>> 45: 0-70-100
>>>>
>>>> In effect, each A supporter agrees to switch his
>> probability to AB in
>>>> exchange for a B supporter switching to AB.
>>>>
>>>> So, the initial probabilities would be
>>>>
>>>> A: 55%
>>>> AB: 0%
>>>> B: 45%
>>>>
>>>> Expected utility
>>>> 55: 55
>>>> 45: 45
>>>> Total: 100
>>>>
>>>> However, after the negotiation stage, the results
>> might be
>>>> A: 10%
>>>> AB: 90%
>>>> B: 0%
>>>>
>>>> Expected utility
>>>>
>>>> 55: 10% of 100 and 90% of 70 = 73
>>>> 45: 90% of 70 = 63
>>>> Total: 136
>>>>
>>>> I don't 100% remember the method (and it could
>> do with a web
>>>> description :p ), but that is what it is
>> attempting to do.
>>>> The idea is not that it is random.  The idea is
>> that it says "OK, if
>>>> you can't all agree on a compromise, then we
>> will pick a winner at
>>>> random".
>>>>
>>>> The threat that a random winner will be picked is
>> what allows the
>>>> negotiation.  If a majority can just impose its
>> will, then there is no
>>>> point in compromising.
>>> ----
>>> Election-Methods mailing list - see
>> http://electorama.com/em for list info
>> ----
>> Election-Methods mailing list - see
>> http://electorama.com/em for list info
> 
> 
>       
>