[EM] HBH

[fsimmons at pcc.edu]

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: fsimmons at pcc.edu
Cc: election-methods at lists.electorama.com

> 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" 
> 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?
> 
> ----
> Election-Methods mailing list - see http://electorama.com/em for 
> list info
>