Good idea. Let's play with it. ----- Original Message ----- From: Toby Pereira Date: Wednesday, July 20, 2011 4:44 pm Subject: Re: [EM] HBH To: [email protected] Cc: [email protected]
> I was thinking - Schulze STV compares every result against every > other result > that differs by just one candidate, which could be a lot of work > for a computer! > So could your HBH system be used for STV elections? Determine > the order of > comparison and compare two results that differ by one candidate > and the "losing > candidate" is eliminated. So each pairwise comparison eliminates > a candidate and > it's all done much more quickly. > > > > > ________________________________ > From: "[email protected]" > To: [email protected] > Cc: [email protected] > Sent: Mon, 18 July, 2011 19:25:01 > Subject: [EM] HBH > > HBH stands for Hog Belly Honey, the name of an inerrant > "nullifier" invented by > a couple of R.A. Lafferty > > characters. The HBH is the only known nullifier that can "posit > moral and > ethical judgments, set up and > > enforce categories, discern and make full philosophical > pronouncements," in > other words eliminate the > > garbage and keep what's valuable. The main character, the "flat > footed genius," > Joe Spade, picks the > > name "Hog Belly Honey," for it "on account it's so sweet." > > The whole idea of HBH is just starting at the bottom of a > pecking order and > pitting (for elimination) the > > current champ against the most distant challenger. I hope you > will keep that in > mind as we introduce > > the necessary technical details. > > HBH is based on range style ballots that allow the voters to > rate each > alternative on a range of zero to > > some maximum value M. [Keep this M in mind; we will make > explicit use of it > presently.] > > Once the ballots are voted and submitted, the first order of > business is to set > up a "pecking order" for > > the purpose of resolving ties, etc. Alternative X is higher in > the pecking > order than alternative Y if > > alternative X is rated above zero on more ballots than Y is > rated above zero. > If both have the same > > number of positive ratings, then the alternative with the most > ratings greater > than one is higher in the > > pecking order. If that doesn't resolve the tie, then the > alternative with the > greatest number of ratings > > above two is higher, etc. > > In the practically impossible case that two alternatives have > exactly the same > number of ratings at each > > level, ties should be broken randomly. > > The next order of business is to establish a proximity relation > between > alternatives. For our purposes > > closeness or proximity between two alternatives X and Y is given > by the number > > Sum over all ballots b, min( M*(M-1), b(X)*b(Y) ). > > [The minimization with M*(M-1) clinches the method's resistance > to compromise, > as explained below.] > > This proximity value is a useful measure of a certain kind of > closeness of the > two alternatives: the larger > > the proximity number the closer the alternatives in this limited > sense, while > the smaller the number the > > more distant the alternatives from each other (again, in this > limited sense). > > For the purposes of this method, if two alternatives Y and Z > have equal > proximity to X, then the one that > > is higher in the pecking order is considered to be closer than > the other. In > other words, the pecking > > order is used to break proximity ties. > > Next we compute the majority pairwise victories among the > alternatives. > Alternative X beats alternative > > Y majority-pairwise if X is rated above Y on more than half of > the ballots. > > For the purposes of this method, the "victor" of a pair of > alternatives is the > one that beats the other > > majority pairwise, or in the case where neither beats the other > majority-pairwise it is the one that is > > higher in the pecking order. Of the two, the non-victor > alternative is called > the "loser." In other words, > > the pecking order decides pairwise victors and losers when there > is no majority > defeat. [This convention > > on victor and loser is what makes the method plurality > compliant, as explained > below.] > > Next we initialize an alphanumeric variable V with the name of > the lowest > alternative in the pecking > > order, and execute the following loop: > > While there remain two or more discarded alternatives > discard the loser between V and the alternative most distant > from V, > and replace V with the name of the victor of the two. > EndWhile > > Finally, elect the alternative represented by the final value of V. > > This HBH method is clone free, monotone, Plurality compliant, > compromise > resistant, and burial > > resistant. > > Furthermore, it is obviously the case that if some alternative > beats each of the > other alternatives majority > > pairwise, then that alternative will be elected. > > Let's see why the method is plurality compliant: > > If there is even one majority defeat in the sequence of > eliminations, every > value of V after that will be the > > name of an alternative that is rated positively on more than > half of the > ballots. If none of the victories are > > by majority defeat, then the winner is the alternative highest > on the pecking > order, i.e. the one with the > > greatest number of positive ratings. > > Let's see why the method is monotone: > > Suppose that the winner is moved up in the ratings. Then its > defeat strengths > will only be increased, and > > any proximity change can only delay its introduction into the > fray, so it will > only face alternatives that > > lost to it before. > > Let's see why it is compromise resistant: > > Since Favorite and Compromise are apt to be in relatively close > proximity, and > pairwise contests are > > always between distant alternatives, if Compromise gets > eliminated, it will > almost certainly be by > > someone besides Favorite, so there can hardly be any incentive > for rating > Favorite below Compromise. > > Furthermore, there is no likely advantage of rating Compromise > equal to > Favorite, because rating > > compromise just below Favorite already makes the maximum > possible contribution > M*(M-1) to their > > proximity sum, i.e. the best you can do to make sure they are > pitted against > each other only after all of > > the other alternaties have been eliminated (if at all). > > How about burial? > > I don't have such an easy argument for burial resistance, but > the experiments I > have conducted show > > that more likely than not it won't pay off. I hope that Kevin > will run his > simulations on the method for > > (hopefully) more support on that account. > > I realize that the method sounds complicated from the > description above, but all > of the complication is > > from the details of tie breaking, including what to do when > defeats are not > majority-pairwise. > > Other than that, as mentioned at the beginning, it is just > starting at the > bottom of the pecking order and > > pitting (for elimination) the current champ against the most > distant challenger. > > Aint that sweet? > > ---- > Election-Methods mailing list - see http://electorama.com/em for > list info > ---- Election-Methods mailing list - see http://electorama.com/em for list info
