[EM] HBH

[fsimmons at pcc.edu]

> From: Kevin Venzke 
> Hi Forest,
> 
> --- En date de?: Lun 18.7.11, fsimmons at pcc.edu 
> a ?crit?:
> > > The "pecking order" is the Range order. Assume no
> > ties.
> > 
> > I suppose that you could use the range order for the
> > pecking order, but as you mention below that could 
> > lead to some strategic distortions.
> > 
> > The pecking order I had in mind is more like implicit
> > approval; if implicit approval (number of above zero 
> > ratings) has no ties, that's all it is.
> 
> Oops. I understood this when I read it but didn't retain it.
> 
> > > I understand the Plurality argument. Still wrapping my
> > head 
> > > around the
> > > monotonicity and compromise ones. 
> > 
> > The key to montonicity is that from the first time the
> > eventual winner becomes the defending champ, she 
> > has to defeat every challenger, i.e. all of the remaining
> > candidates.? So if a winner W gets barely enough 
> > increased support to change the proximity order by just one
> > pair, say W is now closer to V, than W' 
> > instead of vice-versa, then W' challenges V before W.?
> > Whether or not W' or the current V weathers this 
> > challenge doesn't matter, because W beat the current V
> > before, and then beat W' and all of the other 
> > remaining candidates later.
> > 
> > If the winner gets lots of increased support, just
> > gradually add that support so that it makes this minimal 
> > change in the order, and note that each time this kind of
> > minimal change is made, the winner stays the 
> > same.? If we wanted to get fancy, we would call this a
> > homotopy argument.
> 
> Ok. I'll just trust it for now.

If W is the only one whose rating is raised, then the sequence of eliminations is the same as before right 
up to the point where W entered the fray before (at which point she was the beats all alternative for all of 
the remaining alternatives).  After the change, the order of elimination is different from that point on, but 
W is still the beats all candidate for the remaining alternatives, so the change in order doesn't matter, no 
matter where she enters into the fray she is faced only with a subset of the alternatives that she beat 
before.

> 
> I was thinking a bit while trying to code it. In the 3-candidate case,
> if there is a "Condorcet winner" in the HBH sense then he wins. 
> Otherwisewe have one of two cycles as always.
> 
> I notice that no matter which cycle it is, the winner is going 
> to be
> whichever candidate is deemed further from the approval loser.
> 
> My first thought is that that sounds pretty bad. It sounds like the
> approval loser is set up to spoil the outcome.
> 
> On the other hand, in order to have a cycle, everybody must have 
> majority approval, which makes it very strange to think of there being
> weak noise candidates as such... A candidate lacking majority approval
> (who isn't the AW) has no effect. So it's hard to imagine appalling
> outcomes... I'm pretty sure it can overrule a lone majority contest,
> but it won't even violate minimal defense when it does...
> 
> Kevin Venzke

Kevin,

I've been thinking of two other possible proximity measures, both based on the idea of amalgamation of 
factions, (where each candidate's amalgamateed faction is a weighted average of the factions that rated 
her tops).

One possibility is to say that the most distant alternative from alternative V is the one that rates V the 
lowest in its amalgamated rating.  

The other basic possibility is to say that the most distant alternative  from V is the one that V rates the 
lowest in its amalgamated ratings.

If one were adopted, the other could be used to break ties.

I lean towards the first mentioned possibility because it strongly discourages burial.  The only problem I 
can see with it is that it might encourage top heavy ballots; everybody wants to rate everbody highly to 
delay being drawn into the fray as long as possible.  What do you think?

The proximity measure that we have been using up to now is more of a mutual measure; both have to 
rate each other highly to make the measure judge them to be close.  Also it takes into consideration (to 
some degree) what the other factions think about them.