Re: [Election-Methods] method design challenge +new method AMP
On May 9, 2008, at 20:27 , Jobst Heitzig wrote: Dear Juho, you wrote: Yes, but as I see it the reasons are different. In a typical non- deterministic method like random ballot I think it is the intention to give all candidates with some support also some probability of becoming elected. Not at all! At least in those non-deterministic methods which I design the goal is to make it probable that the voters implement a strategic equilibrium in which a compromise option (instead of the favourite of a mere majority) will be elected with (near) certainty. Ok, there are also such methods (more complex than basic random ballot). I interpreted the "stronger than majoritarianism" search of a compromise candidate to be an additional requirement that determines one subclass of (deterministic and nondeterministic) election methods. Juho But for such an equilibrium to exist in the first place, the method cannot be majoritarian, since then the majority would have no incentive at all to cooperate. Instead, all voters must have some power, not only those belonging to the majority, and therefore each voter is given control over an equal amount of winning probability. Still, the goal is not that they assign this amount to their favourite option but that they "trade" it in some controlled way, in order to elect a compromise which makes all the cooperating voters better off than without the trading! Since at the same time, voting shall be secret, the trading cannot be expected to be performed by open negotiations between the voters, but it must be facilitated by some mechanism which trades winning probabilities automatically depending on the preference information on the voters' ballots. If then in certain situations it happens that not much trading actually takes place, so that the winning probabilities remain with the voters' favourites, then this is only an indication that no sufficiently attractive compromise options existed in that situation. But whenever such an option does exist, the goal of non-deterministic methods like DFC, D2MAC, and AMP is that voters recognize that they are better off with the compromise than with the benchmark Random Ballot solution, and that they can bring about the election of the compromise by safely indicating their willingness to trade their share of the winning probability, without running the risk of being cheated by the other faction(s). D2MAC is quite good at this if only the compromise option is sufficiently attractive, but not in a situation which is as narrow as the one I gave at the beginning of this thread. AMP is better there, but it is not monotonic unfortunately. Yours, Jobst In the deterministic methods electing some non- popular extremist is typically an unwanted feature and a result of the method somehow failing to elect the best winner. *No* election or decision method should be applied without first checking the feasibility of options with respect to certain basic requirements. This sorting out the "constitutional" options cannot be subject to a group decision process itself since often the "unconstitutional" options have broad support (Hitler is only the most extreme example for this). In other words, without such a feasibility check *before* deciding, also majoritarian methods can produce a very bad outcome (think of Rwanda...). Ok, this looks like an intermediate method where one first has one method (phase 1) that selects a set of acceptable candidates and then uses some other method (phase 2) (maybe non-deterministic) to elect the winner from that set. There is need for pure non-deterministic methods like random ballot, and pure deterministic methods, and also combinations of different methods may be useful. Also in the case where the no-good candidates are first eliminated I see the same two different philosophies on how the remaining candidates are handled. Either all remaining candidates (with some support) are given some probability or alternatively one always tries to elect the best winner. The intention was thus not to say non- deterministic methods would not work properly but that there are two philosophies that are quite different and that may be used in different elections depending on the nature of the election. Due to this difference I'm interested in finding both deterministic and non-deterministic solutions for the challenge. Juho Yours, Jobst ___ ___ _ EINE FÜR ALLE: die kostenlose WEB.DE-Plattform für Freunde und Deine Homepage mit eigenem Namen. Jetzt starten! http://unddu.de/? [EMAIL PROTECTED] ___ Inbox full of spam? Get leading spam protection and 1GB storage with All New Yahoo! Mail. http://uk.docs.yahoo.com/nowyoucan.html Election-Methods mailing list - see http://electorama.com/em for list info __
Re: [Election-Methods] method design challenge +new method AMP
Dear Juho, you wrote: > Yes, but as I see it the reasons are different. In a typical non- > deterministic method like random ballot I think it is the intention > to give all candidates with some support also some probability of > becoming elected. Not at all! At least in those non-deterministic methods which I design the goal is to make it probable that the voters implement a strategic equilibrium in which a compromise option (instead of the favourite of a mere majority) will be elected with (near) certainty. But for such an equilibrium to exist in the first place, the method cannot be majoritarian, since then the majority would have no incentive at all to cooperate. Instead, all voters must have some power, not only those belonging to the majority, and therefore each voter is given control over an equal amount of winning probability. Still, the goal is not that they assign this amount to their favourite option but that they "trade" it in some controlled way, in order to elect a compromise which makes all the cooperating voters better off than without the trading! Since at the same time, voting shall be secret, the trading cannot be expected to be performed by open negotiations between the voters, but it must be facilitated by some mechanism which trades winning probabilities automatically depending on the preference information on the voters' ballots. If then in certain situations it happens that not much trading actually takes place, so that the winning probabilities remain with the voters' favourites, then this is only an indication that no sufficiently attractive compromise options existed in that situation. But whenever such an option does exist, the goal of non-deterministic methods like DFC, D2MAC, and AMP is that voters recognize that they are better off with the compromise than with the benchmark Random Ballot solution, and that they can bring about the election of the compromise by safely indicating their willingness to trade their share of the winning probability, without running the risk of being cheated by the other faction(s). D2MAC is quite good at this if only the compromise option is sufficiently attractive, but not in a situation which is as narrow as the one I gave at the beginning of this thread. AMP is better there, but it is not monotonic unfortunately. Yours, Jobst > In the deterministic methods electing some non- > popular extremist is typically an unwanted feature and a result of > the method somehow failing to elect the best winner. > > > *No* election or decision method should be applied without first > > checking the feasibility of options with respect to certain basic > > requirements. This sorting out the "constitutional" options cannot > > be subject to a group decision process itself since often the > > "unconstitutional" options have broad support (Hitler is only the > > most extreme example for this). > > > > In other words, without such a feasibility check *before* deciding, > > also majoritarian methods can produce a very bad outcome (think of > > Rwanda...). > > Ok, this looks like an intermediate method where one first has one > method (phase 1) that selects a set of acceptable candidates and then > uses some other method (phase 2) (maybe non-deterministic) to elect > the winner from that set. > > There is need for pure non-deterministic methods like random ballot, > and pure deterministic methods, and also combinations of different > methods may be useful. > > Also in the case where the no-good candidates are first eliminated I > see the same two different philosophies on how the remaining > candidates are handled. Either all remaining candidates (with some > support) are given some probability or alternatively one always tries > to elect the best winner. The intention was thus not to say non- > deterministic methods would not work properly but that there are two > philosophies that are quite different and that may be used in > different elections depending on the nature of the election. > > Due to this difference I'm interested in finding both deterministic > and non-deterministic solutions for the challenge. > > Juho > > > Yours, Jobst > > ___ > >___ _ > > EINE FÜR ALLE: die kostenlose WEB.DE-Plattform für Freunde und > > Deine Homepage mit eigenem Namen. Jetzt starten! http://unddu.de/? > > [EMAIL PROTECTED] > > ___ > Inbox full of spam? Get leading spam protection and 1GB storage with > All New Yahoo! Mail. http://uk.docs.yahoo.com/nowyoucan.html > > > Election-Methods mailing list - see http://electorama.com/em for list > info pgpUbFPDuaKZJ.pgp Description: PGP signature Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
On May 9, 2008, at 10:46 , Jobst Heitzig wrote: Dear Raphfrk you wrote: One issue with random processes is that they don't work well for a legislature. A majority would just keep asking that the vote be repeated until they win it. Saying that a re-vote cannot occur unless the situation changes would require that a definition of a change in the situation be decided. Alternatively, laws could be considered social contracts which have a duration and certain terms of termination which would have to be met by any later decisions to change the law. Also, people have a certain level of distrust for random processes. I don't think people would accept a President who was elected even though he only had a 1% chance of winning. I am not sure what the threshold is before it would be acceptable (some people would object to a 49% candidate winning instead of a 51% candidate). This is probably true. I would not recommend such a method for elections of Presidents or the like but for bodies who frequently make individual decisions on issues. Probabilistic methods are actually proportional methods (at least if they aim at giving n% probability to a candidate with "n% support", or some other probabilities that the voters like more). I don't know what the other (non-proportional) methods should be called here since "dictatorship of majority" is not valid in this particular case. Maybe "always elect the best" (according to some criterion) is more accurate. Juho Yours, Jobst __ _ EINE FÜR ALLE: die kostenlose WEB.DE-Plattform für Freunde und Deine Homepage mit eigenem Namen. Jetzt starten! http://unddu.de/? [EMAIL PROTECTED] ___ The all-new Yahoo! Mail goes wherever you go - free your email address from your Internet provider. http://uk.docs.yahoo.com/nowyoucan.html Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
Dear Raphfrk you wrote: > One issue with random processes is that they don't work well for a > legislature. A majority would just keep asking that the vote be > repeated until they win it. > Saying that a re-vote cannot occur unless the situation changes would > require that a definition of a change in the situation be decided. Alternatively, laws could be considered social contracts which have a duration and certain terms of termination which would have to be met by any later decisions to change the law. > Also, people have a certain level of distrust for random processes. > I don't think people would accept a President who was elected even > though he only had a 1% chance of winning. I am not sure what the > threshold is before it would be acceptable (some people would object > to a 49% candidate winning instead of a 51% candidate). This is probably true. I would not recommend such a method for elections of Presidents or the like but for bodies who frequently make individual decisions on issues. Yours, Jobst ___ EINE FÜR ALLE: die kostenlose WEB.DE-Plattform für Freunde und Deine Homepage mit eigenem Namen. Jetzt starten! http://unddu.de/[EMAIL PROTECTED] Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
On May 9, 2008, at 1:09 , Jobst Heitzig wrote: Usually I consider Random Ballot a "benchmark" method for this very reason: the "default" winning probability of a candidate should equal the proportion of the voter who favour her. Any deviances from this default distribution should be justified somehow, for example by an increase in some measure of "social utility". I commented this point also in my reply to raphfrk. Random ballot is a perfect "benchmark" for many elections. But there are also elections that should be "benchmarked" against different methods / criteria. Sometimes the intention is to elect a candidate that is e.g. considered to be a good compromise, and one could e.g. intentionally try to avoid electing extremists. It would be good to always make it clear what kind of election method one is looking for. Both probability based and "deterministic" methods are needed. Juho ___ Now you can scan emails quickly with a reading pane. Get the new Yahoo! Mail. http://uk.docs.yahoo.com/nowyoucan.html Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
On May 9, 2008, at 0:56 , Jobst Heitzig wrote: For A1,A2 to be considered clones, the ratings would have to be something like 51: A1 100 > A2 99 > C 55 > B 0 49: B 100 > C 55 > A1 1 > A2 0 Could be also e.g. A > C 99 > B 0 and after inserting the clones A1 100 > A2 99 > C 98 > B 0 There are thus many cases where separating clones from non-clones is not easy. In this example also the number of rating levels impacts the outcome. You also seem to think so, since you wrote: One approach to try to avoid this problem would be to use a more limited clone concept: candidates that are ranked/rated equal with each others. But that would never really occur in practice. I think one should define the notion "clone" like this: A1,A2 are clones if and only if on each ballot, the difference in ratings between any pair of options is smallest for the pair A1,A2. Yes, this is one possible definition (that can be used to formulate the clone criterion). Juho ___ All new Yahoo! Mail "The new Interface is stunning in its simplicity and ease of use." - PC Magazine http://uk.docs.yahoo.com/nowyoucan.html Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
One approach to clones is to allow the voters to indicate them. One approach is to use the preference strengths. Instead of voting A1>A2>C>B the voter could vote A1>A2>>C>>B, which indicates that the voter in some sense considers A1 and A2 to be clones from his/her point of view. (Btw, all the various clone criteria that require all voters to vote in a certain way (to group the clones together) are quite theoretical since of course it seldom happens that all voters vote that way.) I might contribute one more method as a response to the challenge in line with these thoughts, but not today. (Just to show that rankings and clones may also live happily together :-).) Juho ___ All new Yahoo! Mail "The new Interface is stunning in its simplicity and ease of use." - PC Magazine http://uk.docs.yahoo.com/nowyoucan.html Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
> you wrote > > If a majority has a 100% chance of getting their candidate elected, > > then there is no incentive for them to trade.? If the voters are 100% > strategic, they will know this. > > Yes, although some Range Voting supporters try hard to convince us of > the opposite, it seems. Well, your example is under the assumption of perfect information. I think in the near perfect balance of your example and without perfect information, range or approval would probably elect the compromise as a reasonably large number of voters would approve the compromise as well as their favourite instead of risking their hated candidate wins. As the majority for A gets larger (say >55%), A's supporters would be advised to switch to bullet voting just for A. I think that approval is a good and practical voting method. However, the ideal method is one that gives people an incentive for honest ratings and elects the candidate that is socially optimal, i.e a candidate who would be a 'Kaldor-Hicks optimal' candidate. If there is more than one optimal, then use random ballot to decide which one, but a tie isn't that likely. One issue with random processes is that they don't work well for a legislature. A majority would just keep asking that the vote be repeated until they win it. Saying that a re-vote cannot occur unless the situation changes would require that a definition of a change in the situation be decided. Also, people have a certain level of distrust for random processes. I don't think people would accept a President who was elected even though he only had a 1% chance of winning. I am not sure what the threshold is before it would be acceptable (some people would object to a 49% candidate winning instead of a 51% candidate). > > Optimal utility via trade requires that voters have something to > > trade, and fractions of a win probability seems to be quite a > > reasonable solution. > > I cannot really imagine any other thing unless we consider money > transfers... There is a method called CTT voting where money is used and in theory, it provides an incentive for everyone to be honest. However, it requires each voter to act completely independently. Also, it weakens the secret ballot. Also, it only requires payment in a small number of cases, where the result is close. In a landslide, nobody has to pay. http://rangevoting.org/CTT.html It would probably not work in practice, especially for large elections. Most of the issues are listed on the linked page. For a legislature, you could trade future votes instead of money. One issue is that it needs to be a secret ballot for CTT to work, (otherwise parties can control the votes by disciplining party members who vote the 'wrong' way after the vote is completed and made public). However, it needs to be a public ballot so that the voters know if their legislator is voting in their best interests. In Germany, motions of no confidence in the Chancellor (head of government) are handled by a secret ballot that requires an absolute majority of their legislature to pass. This is used to change the head of government, so I guess not all legislatures have public votes for everything. Raphfrk Interesting site "what if anyone could modify the laws" www.wikocracy.com AOL's new homepage has launched. Take a tour at http://info.aol.co.uk/homepage/ now. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
Dear Raphfrk, you wrote > There needs to be some system for providing an incentive for people > to give their honest ratings.? A random system with trading seems > like a reasonable solution. I am glad that I am no longer alone with this opinion... > If a majority has a 100% chance of getting their candidate elected, > then there is no incentive for them to trade.? If the voters are 100% > strategic, they will know this. Yes, although some Range Voting supporters try hard to convince us of the opposite, it seems. > OTOH, a support of a majority should be better than support of a > minority. Absolutely! Usually I consider Random Ballot a "benchmark" method for this very reason: the "default" winning probability of a candidate should equal the proportion of the voter who favour her. Any deviances from this default distribution should be justified somehow, for example by an increase in some measure of "social utility". (The underlying rationale for methods like D2MAC or AMP is even stronger: every voter should have full control over "her" share of the winning probability, so that in particular when she bullet votes, this share must goes to her favourite. Only such methods are truly "democratic".) > Optimal utility via trade requires that voters have something to > trade, and fractions of a win probability seems to be quite a > reasonable solution. I cannot really imagine any other thing unless we consider money transfers... Yours, Jobst pgpQdRGiMeeWy.pgp Description: PGP signature Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
Dear Juho, you wrote: > One observation on clone independence and electing a centrist > candidate using rankings only and when one of the "extremists" has > majority. ... > It is thus impossible for the algorithm in this case and > with this information (rankings only) to satisfy both requirements > and to be fully clone independent. D'accord. This is a good reason to consider rankings insufficient, since from rankings only one cannot determine whether to apparent clones are truly clones in the sense that they are (nearly) equivalent in all relevant aspects. From ratings information, however, one can see this. Therefore I would not at all consider A1,A2 clones in your ratings example: > A=100 C=55 B=0 => A1=100 A2=56 C=54 B=0 > B=100 C=55 A=0 => B=100 C=56 A1=54 A2=0 For A1,A2 to be considered clones, the ratings would have to be something like 51: A1 100 > A2 99 > C 55 > B 0 49: B 100 > C 55 > A1 1 > A2 0 You also seem to think so, since you wrote: > One approach to try to avoid this problem would be to use a more > limited clone concept: candidates that are ranked/rated equal with > each others. But that would never really occur in practice. I think one should define the notion "clone" like this: A1,A2 are clones if and only if on each ballot, the difference in ratings between any pair of options is smallest for the pair A1,A2. (Analogously, a set S of options should be called a clone set if and only if on each ballot, all rating differences between two options in S are smaller than all rating differences between other pairs of options. Even more generally, a system Y of disjoint sets S1,...,Sk of options could be called a clone partition if and only if on each ballot, all rating differences between two options which are contained in the same member of Y are smaller than all rating differences between other pairs of options.) With this definition, the problem you described cannot really occur: Assume the rankings are > 51: X1>X2>X3>X4 > 49: X4>X3>X2>X1 If X1,X2 are clones, X2 cannot be considered a good compromise since 49 voters don't like her. Similarly, if X3,X4 are clones, X3 cannot be considered a good compromise since 51 voters don't like her. Yours, Jobst pgp9sEhTbdSjT.pgp Description: PGP signature Election-Methods mailing list - see http://electorama.com/em for list info
Re: [Election-Methods] method design challenge + new method AMP
I wanted to consider this afresh. At 01:58 PM 4/28/2008, Jobst Heitzig wrote: Hello folks, over the last months I have again and again tried to find a solution to a seemingly simple problem: The Goal - Find a group decision method which will elect C with near certainty in the following situation: - There are three options A,B,C - There are 51 voters who prefer A to B, and 49 who prefer B to A. - All voters prefer C to a lottery in which their favourite has 51% probability and the other faction's favourite has 49% probability. - Both factions are strategic and may coordinate their voting behaviour. First of all, a method which elects C unambiguously from these conditions is problematic. This is a very close election, but the majority prefer A. Voters, however, may not like taking risks; for the A voters to purely shoot for an A victory is very dangerous, slip up and they get B. As stated, there is no way to know who is the best winner from the SU measure, which, however, Jobst fixes: Those of you who like cardinal utilities may assume the following: 51: A 100 > C 52 > B 0 49: B 100 > C 52 > A 0 That's down in the noise, however. Note that Range Voting would meet the goal if the voters would be assumed to vote honestly instead of strategically. With strategic voters, however, Range Voting will elect A. That depends on their knowledge. If this is a zero-knowledge situation, and they vote according to personal expected outcome maximization, and it is Range 100, all voters will vote C 52. Which shows up to problems: human beings don't have preferences that fine, unless special abstractions allow them to assign precise utilities, such as known monetary return from outcomes. However, all voters would vote, in Range 2 and Range 10, C as midrange. In Range 2, this would give us A 5100, B: 5000, C:4900. A wins. But with Range 100, and assuming, instead, that 52% is the *average* sincere rating, which makes more sense than assuming they are uniform, they would, if it's zero-knowledge and they vote intelligently to maximize personal utility, elect B. The certainty of this increases with the number of voters. Contrary to Jobst's first intuition, Range *does* satisfy this, if the resolution is adequate. Imagine a continuum of voters who are described, in sum, as above. The numbers Jobst gives for the C utility are averages. Given that all voters vote "strategically," all voters vote the extremes. So the only question is how they will rate C. The claim is often made that "strategic" voters in Range will bullet vote. However, a bullet vote risks the election of the least-desired candidate. The issue is often complicated by Range critics by aasuming some weak utility, then assuming that the voter will, knee-jerk, bullet vote for the favorite, as if the voter is strongly motivated. What has been done is to assume two contradictory voter utilities: a weak one and a strong one. A voter who votes, say, a straight party ticket, without regard for the individual candidates, has a strong utility for the party's candidate, not a weak one. Very strong. Only if you grab this voter by the scruff and shake him and say, "No, you are not going to elect Abe Lincoln. Now that we have settled that, and stricken Lincoln's name from the ballot, whom do you prefer, Adolf Hitler or Josef Stalin?" Never mind that the voter's response may well be a serious abstention, caused by rapidly leaving the jurisdiction in question. If the voter has any sense and is able to do so. If we assume that the voters exist in a curve such that 52 is the median vote (as well as the average one), and they vote strategically, using the best game theory to do so, Approval probably satisfies the problem. At 50% utility, bullet voting and voting for two are exactly matched, take your pick, if there are enough voters. (When I studied this in detail, I found that bullet voting was slightly better in expected outcome, but it was only with very small numbers of voters. With many voters, they are equal. But if the expected utility is higher than 50%, in Approval, the zero-knowledge strategic vote is to vote for both. And a lot of words have been written to the contrary without actually studying the game matrix and election probabilities.) If the median vote is 52, what is the average vote, if voters can only vote 0 or 100? (Approval). Under the strategic voting assumption, it would surely be above 50. And probably more than 1% above, with high certainty, but I haven't done the exact math, and the shape of the curve is not precisely stated. If they can vote Range 100, and if they know their utilities and the math and vote strategically, they vote their average vote, which, if the bell curve is symmetrical, would probably be 52. C wins. Why is this contrary to what so many have written? Well, zero knowledge. So let's look at the opposite extreme? Perfect knowledge. Range 100. But the
Re: [Election-Methods] method design challenge + new method AMP
How about using STV or some other proportional method to select the
n-1 worst candidates and then elect the remaining one?
Juho
On Apr 28, 2008, at 20:58 , Jobst Heitzig wrote:
> Hello folks,
>
> over the last months I have again and again tried to find a
> solution to
> a seemingly simple problem:
>
> The Goal
> -
> Find a group decision method which will elect C with near certainty in
> the following situation:
> - There are three options A,B,C
> - There are 51 voters who prefer A to B, and 49 who prefer B to A.
> - All voters prefer C to a lottery in which their favourite has 51%
> probability and the other faction's favourite has 49% probability.
> - Both factions are strategic and may coordinate their voting
> behaviour.
>
>
> Those of you who like cardinal utilities may assume the following:
> 51: A 100 > C 52 > B 0
> 49: B 100 > C 52 > A 0
>
> Note that Range Voting would meet the goal if the voters would be
> assumed to vote honestly instead of strategically. With strategic
> voters, however, Range Voting will elect A.
>
> As of now, I know of only one method that will solve the problem (and
> unfortunately that method is not monotonic): it is called AMP and is
> defined below.
>
>
> *** So, I ask everyone to design some ***
> *** method that meets the above goal! ***
>
>
> Have fun,
> Jobst
>
>
> Method AMP (approval-seeded maximal pairings)
> -
>
> Ballot:
>
> a) Each voter marks one option as her "favourite" option and may name
> any number of "offers". An "offer" is an (ordered) pair of options
> (y,z). by "offering" (y,z) the voter expresses that she is willing to
> transfer "her" share of the winning probability from her favourite
> x to
> the compromise z if a second voter transfers his share of the winning
> probability from his favourite y to this compromise z.
> (Usually, a voter would agree to this if she prefers z to
> tossing a
> coin between her favourite and y).
>
> b) Alternatively, a voter may specify cardinal ratings for all
> options.
> Then the highest-rated option x is considered the voter's "favourite",
> and each option-pair (y,z) for with z is higher rated that the mean
> rating of x and y is considered an "offer" by this voter.
>
> c) As another, simpler alternative, a voter may name only a
> "favourite"
> option x and any number of "also approved" options. Then each
> option-pair (y,z) for which z but not y is "also approved" is
> considered
> an "offer" by this voter.
>
>
> Tally:
>
> 1. For each option z, the "approval score" of z is the number of
> voters
> who offered (y,z) with any y.
>
> 2. Start with an empty urn and by considering all voters "free for
> cooperation".
>
> 3. For each option z, in order of descending approval score, do the
> following:
>
> 3.1. Find the largest set of voters that can be divvied up into
> disjoint
> voter-pairs {v,w} such that v and w are still free for cooperation, v
> offered (y,z), and w offered (x,z), where x is v's favourite and y is
> w's favourite.
>
> 3.2. For each voter v in this largest set, put a ball labelled with
> the
> compromise option z in the urn and consider v no longer free for
> cooperation.
>
> 4. For each voter who still remains free for cooperation after this
> was
> done for all options, put a ball labelled with the favourite option of
> that voter in the urn.
>
> 5. Finally, the winning option is determined by drawing a ball from
> the
> urn.
>
> (In rare cases, some tiebreaker may be needed in step 3 or 3.1.)
>
>
> Why this meets the goal: In the described situation, the only
> strategic
> equilibrium is when all B-voters offer (A,C) and at least 49 of the
> A-voters "offer" (B,C). As a result, AMP will elect C with 98%
> probability, and A with 2% probability.
>
>
>
>
> Election-Methods mailing list - see http://electorama.com/em for
> list info
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