[EM] An ABE solution

[Chris Benham]

Forest,
 
I think you have wandered into the realm of sophistry. Yes, if the voters are constrained to two
rating or preference levels then Approval meets "2-slot Condorcet", barring exact pairwise ties
"2-slot Condorcet" is decisive, and since Approval meets the FBC then compliance with FBC
and "2-slot Condorcet" are compatible.
 
But when we talk about the Condorcet criterion it is generally understood that there are more
than 2 candidates and that the voters are free to give full strict rankings (which isn't possible on
2-slot ballots).
 
I'm happy with the concept "n-slot Condorcet" (where n is 3 or greater) meaning that if we assume
that no voter wanted to express more than n number of preference-levels and they all expressed
their sincere rankings then a sincere CW must win.
 
(So, I think the 3-slot version of  your suggested "RCW" method is a promising "3-slot Condorcet"
method.)
 
"Indeed, the three slot case does appear to satisfy the FBC as well."

Isn't there a "not" missing from that sentence?
 
"But I did say that "IF" it is strategically equivalent to Approval (as Range is, for 
example) then for "practical purposes" it satisfies the FBC."
 
The main "practical purpose" of the FBC as I see it is to assist in marketing the method by giving
voters an absolute guarantee.  Also, as you suggest by capitalizing, that is a big "if". 

The strength of a voter's incentive to submit an approval-like ballot will depend on their sincere
ratings (or utilities) and how well informed they are (about say which candidates are the front-runners).
So do voters who don't know who the front-runners are and whose sincere ratings are evenly 
spaced have an incentive to only use the Top and Bottom ratings slots?  If so, it would be much
weaker than it is with Range.
 
Chris Benham


________________________________
From: "fsimmons at pcc.edu" <fsimmons at pcc.edu>
To: Chris Benham <cbenhamau at yahoo.com.au> 
Cc: em <election-methods at electorama.com>; MIKE OSSIPOFF <nkklrp at hotmail.com> 
Sent: Wednesday, 23 November 2011 9:01 AM
Subject: Re: An ABE solution

You are right that although the method is defined for any number of slots, I suggested three slots as 
most practical.

So my example of two slots was only to disprove the statement the assertion that the method cannot be 
FBC compliant, since it is obviously compliant in that case.  

Furthermore something must be wrong with the quoted proof (of the incompatibility of the FBC and the 
CC) because the winner of the two slot case can be found entirely on the basis of the pairwise matrix.  
The other escape hatch is to say that two slots are not enough to satisfy anything but the voted ballots 
version of the Condorcet Criterion.  But this applies equally well to the three slot case.

Either way the cited "therorem" is not good enough to rule out compliance with the FBC by this new 
method.

Indeed, the three slot case does appear to satisfy the FBC as well.  It is an open question.  I did not 
assert that it does.  But I did say that "IF" it is strategically equivalent to Approval (as Range is, for 
example) then for "practical purposes" it satisfies the FBC.  Perhaps not the letter of the law, but the 
spirit of the law.  Indeed, in a non-stratetgical environment nobody worries about the FBC, i.e. only 
strategic voters will betray their favorite. If optimal strategy is approval strategy, and approval strategy 
requires you to top rate your favorite, then why would you do otherwise?

Forest

----- Original Message -----
From: Chris Benham 

> Forest,
>  
> "When the range ballots have only two slots, the method is 
> simply Approval, which does satisfy the 
> FBC."
>  
> When you introduced the method you suggested that 3-slot ballots 
> be used "for simplicity".
> I thought you might be open to say 4-6 slots, but a complicated 
> algorithm on 2-slot ballots
> that is equivalent to Approval ??
>  
> "Now consider the case of range ballots with three slots: and 
> suppose that optimal strategy requires the 
> voters to avoid the middle slot.  Then the method reduces to 
> Approval, which does satisfy the FBC."
>  
> The FBC doesn't stipulate that all the voters use "optimal 
> strategy", so that isn't relavent.
> 
> http://wiki.electorama.com/wiki/FBC
>  
> http://nodesiege.tripod.com/elections/#critfbc
> 
> Chris  Benham
>  
>  
>  
>  
> 
> 
> ________________________________
> From: "fsimmons at pcc.edu" 
> To: C.Benham 
> Cc: em ; MIKE OSSIPOFF 
> 
> Sent: Tuesday, 22 November 2011 11:11 AM
> Subject: Re: An ABE solution
> 
> 
> 
> From: "C.Benham" 
> 
> > 
> > Forest Simmons, responding to questions from Mike Ossipff, 
> wrote 
> > (19 Nov 
> > 2011):
> > 
> > > > 4. How does it do by FBC? And by the criteria that bother some
> > > > people here about MMPO (Kevin's MMPO bad-example) and 
> MDDTR 
> > > (Mono-Add-Plump)?
> > >
> > > I think it satisfies the FBC.
> > 
> > Forest's definition of the method being asked about:
> > 
> > > Here’s my current favorite deterministic proposal: Ballots 
> are 
> > Range 
> > > Style, say three slot for simplicity.
> > >
> > > When the ballots are collected, the pairwise win/loss/tie 
> > relations are
> > > determined among the candidates.
> > >
> > > The covering relations are also determined. Candidate X 
> covers 
> > > candidate Y if X
> > > beats Y as well as every candidate that Y beats. In other 
> > words row X 
> > > of the
> > > win/loss/tie matrix dominates row Y.
> > >
> > > Then starting with the candidates with the lowest Range 
> > scores, they are
> > > disqualified one by one until one of the remaining 
> candidates 
> > X covers 
> > > any other
> > > candidates that might remain. Elect X.
> > 
> > 
> > Forest,
> > 
> > Doesn't this method meet the Condorcet criterion? Compliance 
> > with 
> > Condorcet is incompatible with FBC, so
> > why do you think it satisfies FBC?
> 
> When the range ballots have only two slots, the method is simply 
> Approval, which does satisfy the 
> FBC.  Does Approval satisfy the Condorcet Criterion?  I would 
> say no, but it does satisfy the "votes only 
> Condorcet Criterion." which means that the Approval winner X 
> pairwise beats every other candidate Y 
> according to the ballots, i.e. X is rated above Y on more 
> ballots than Y is rated above X.
> 
> Now consider the case of range ballots with three slots: and 
> suppose that optimal strategy requires the 
> voters to avoid the middle slot.  Then the method reduces to 
> Approval, which does satisfy the FBC.
> 
> 
> > 
> > 
> > http://lists.electorama.com/pipermail/election-methods-
> > electorama.com/2005-June/016410.html
> > 
> > > Hello,
> > >
> > > This is an attempt to demonstrate that Condorcet and FBC are 
> > incompatible.> I modified Woodall's proof that Condorcet and 
> > LNHarm are incompatible.
> > > (Douglas R. Woodall, "Monotonicity of single-seat 
> preferential 
> > > election rules",
> > > Discrete Applied Mathematics 77 (1997), pages 86 and 87.)
> > >
> > > I've suggested before that in order to satisfy FBC, it must 
> be 
> > the case
> > > that increasing the votes for A over B in the pairwise 
> matrix 
> > can never
> > > increase the probability that the winner comes from {a,b}; 
> > that is, it 
> > > must
> > > not move the win from some other candidate C to A. This is 
> > necessary 
> > > because
> > > if sometimes it were possible to move the win from C to A by 
> > increasing> v[a,b], the voter with the preference order B>A>C 
> > would have incentive to
> > > reverse B and A in his ranking (and equal ranking would be 
> > inadequate).>
> > > I won't presently try to argue that this requirement can't 
> be 
> > avoided 
> > > somehow.
> > > I'm sure it can't be avoided when the method's result is 
> > determined solely
> > > from the pairwise matrix.
> 
> Note that in our method the Cardinal Ratings order (i.e. Range 
> Order) is needed in addition to the 
> pairwise matrix; the covering information comes from the 
> pairwise matrix, but candidates are dropped 
> from the bottom of the range order.
> 
> In the two slot case can the approval order be determined from 
> the pairwise matrix?  If so, then this is a 
> counterexample to the last quoted sentence above in the 
> attempted proof of the incompatibility of the CC 
> and the FBC.
> 
> Forest
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