[EM] Some chance for consensus (was: Buying Votes)

[fsimmons at pcc.edu]

Jobst,

That was a great exposition of how to find the consensus candidate when there is one.

Not quite as important, but still valuable, is achieving partial cooperation when that is the best that can be 
done:

25  A1>A>>A2
25  A2>A>>A1
25  B
25  C

Here there isn't much hope for consensus, but it would be nice if the first two factions could still cooperate 
on gettiing A elected, say 25% of the time. (50% seems too much to hope for)

It seems to me that if we require our method to accomplish the potential cooperation in this scenario while 
achieving consensus where possible, the ballots would have to have more levels, and there would have to be 
an intermediate fall back between the consensus test and the random ballot default.

What do you think?

My Best,

Forest

----- Original Message -----
From: Jobst Heitzig 
Date: Sunday, October 26, 2008 6:33 am
Subject: Some chance for consensus (was: [EM] Buying Votes)
To: Raph Frank 
Cc: Greg Nisbet , election-methods at lists.electorama.com, Jacob Taylor , Forest W Simmons 

> 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 
> 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.
> 
>