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 <[email protected]> wrote: > >> From: Jobst Heitzig <[email protected]> >> Subject: [EM] Some chance for consensus revisited: Most simple solution >> To: [email protected] >> 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 >>>> <[email protected]> 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 > > > > ---- Election-Methods mailing list - see http://electorama.com/em for list info
