[EM] HBH (typo correction)

[fsimmons at pcc.edu]

-
> 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?
>
>