[EM] HBH

[Toby Pereira]

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: "fsimmons at pcc.edu" <fsimmons at pcc.edu>
To: fsimmons at pcc.edu
Cc: election-methods at lists.electorama.com
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?

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