- > 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
This should say while there are two or more undiscarded ... > 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
