[EM] HBH

[fsimmons at pcc.edu]

In HBH a pecking order is established on the basis of implicit approval or some other monotonic, clone 
consistent order like chiastic approval that has no incentive for order reversals and minimal incentive for 
collapsing (i.e. merging) of ratings.

A monotone, clone consistent measure of distance or proximity of candidtes is also established.

Then, after initializing the variable X as the lowest candidate in the pecking order ...

While there remain two or more uneliminated candidates
  compare X pairwise with the candidate Y with least proximity to X.
  If Y beats X convincingly, then replace X with Y and discard X.
  Else discard Y.
EndWhile

By "beats X convincingly" I mean either Y covers X, or beats X by a strict majority, or is higher than X on 
the pecking order AND beats X pairwise.

The importance of "convincingly" is so that this method will satisfy the plurality criterion.  Like MMPO 
the method has a strong center seeking dynamic, and a danger with all such methods is failure of the 
Plurality criterion, i.e. we don't want a winner to be rated above zero (or ranked above truncation) on 
fewer ballots than some loser has top ratings or rankings.

One possibility for the proximity measure is the sum (over all ballots b) of products of the form

b(X)*b(Y),

where b(Z) means the rating of Z on on ballot b.

Other suitable proximity measures are 

(1) the average rating of X on all ballots that rate Y tops.

(2) the average rating of Y on all ballots that rate X tops.

(3) the smaller of (1) and (2).

(4) a weighted average of (1) and (2), where the weights are the numbers of the respective ballots of each 
kind.

I call the above suggestions "direct" or "explicit" proximity measures, because they reflect a direct 
interpretation of scores on range style ballots.

There are also indirect or implicit measures of proximity.  The simplest of these that is clone consistent 
goes as follows:

First average together all of the ballots that rate X tops.  Then average together all of the ballots that rate 
Y tops.  Call these average ballots ratings f and g, respectively.

The find the sum of all expressions of the form  |f(Z)-g(Z)|*p(Z),

where p(Z) is the proportion of the original ballots that rated Z at the top.

This factor p(Z) is what makes the method clone consistent.  Any other clone free lottery distribution p 
would do just as well.

This sum is an implicit or indirect measure of the distance between X and Y.  If the sum is zero, then the 
functions f and g are identical, which means that the supporters of X and Y have (on average) identical 
ratings for all of the candidates.  The measure is indirect because the candidates' distances from each 
other are judged by how their supporters differ in their ratings of (all of) the candidates.

If this measure of distance were used in HBH, I'm afraid that it would destroy the monotonicty of the 
method, because increasing p(X) makes |f(X)-g(X)| more important in the distance calculation, which in 
turn, could make X and Y either closer together or further apart in this metric, depending on how this 
absolute difference compares with the other absolute differences in the calculation.

However, this indirect measure could be used in an election post mortem to detect insincere voting.  If 
the implicit distances are not roughly consistent with the explicit proximities used in the method, the 
you have evidence of insincere ratings for the purpose of manipulating the election results.