[EM] HBH

[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
   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?