[EM] Re: Approval/Condorcet Hybrids

[Ted Stern]

On 10 Dec 2004 at 13:29 PST, Forest Simmons wrote:
> Jobst's recent postings about complaints and their rebuttals, and "short 
> ranked pairs" has led me to the following Approval Condorcet hybrid:
>
> Ballots are ordinal with approval cutoff, equal rankings allowed.
>
> Let U(A) be the set of uncovered candidates that cover the approval winner 
> A.  The member of U(A) with the highest approval is the method winner W.
>
> In particular, if the approval winner A is covered only by itself, then 
> A=W is the method winner.
>
> It seems that W has a strong position:  W covers A (i.e. beats every 
> candidate that is beaten by the approval winner A), is maximal in this 
> respect (i.e is not covered by any other candidate), and has the maximum 
> approval among such candidates.
>
> What does it mean if W is different from A?  It means that W beat A head 
> to head, and that W beat every candidate that A beats head to head, so 
> ultimately it means that given more accurate polling information, the 
> voters (in all likelihood) would have adjusted their approval cutoffs to 
> give more approval to W than to A;  i.e. approval winner A getting more 
> approval than W was probably a mistake due to disinformation about the 
> relative strength or weakness of the other candidates.

<snip -- remainder of discussion available at
         http://thread.gmane.org/gmane.politics.election-methods/6030>

Hi Forest,

I've been thinking some more about your sprucing-up ideas and playing with
some examples.  One thing I find is that in elections with lots of candidates,
any strong method (Ranked Pairs, Schulze, River or Short Ranked Pairs) quickly
becomes mind-numbingly complicated for the average voter, who would lose
confidence in the outcome, regardless of any Immunity from Majority Complaint
(or similar) property.

I like the first step of your Sprucing Up process, eliminating covered
candidates.  It quickly de-noises the candidate pool, and uncovered candidates
have short beatpaths to counter complaints.  But it isn't hard to find
examples where the set of uncovered candidates is clone-free and large enough
(5 or more) to still be too large for the average voter to follow the
remaining steps.

In your Approval/Condorcet hybrid example above, I like choosing the winner
from U(A).  The winner will almost certainly be in even smaller set of
candidates, with high voter preference, and with all the strong properties of
being in the uncovered set.  I can't imagine any non-pathological real
election example with more than 3 or 4 candidates in U(A).  And even 3 would
be an extreme case.

But choosing the candidate with the highest approval, even from only those
members of the uncovered set who cover A, still seems tenuous and prone to all
the problems of standard Approval/Condorcet hybrids.  One of AV's biggest
weaknesses is its loss of preference information -- there could still be a
member of U(A) who defeats W pairwise.  Also, your argument that, if W differs
from A, this implies "that W beat every candidate that A beats head to head"
does not follow.  It only implies that W has highest approval in U(A).

I think a better winner could be found by determining a strong clone-proof
method winner from U(A) [with 3 candidates it won't really make any difference
which method, will it?  But I prefer Ranked Pairs].  And if approval weighting
(a la James Green-Armytage's approval-weighted pairwise) is used to rank
defeats, approval is still part of the process.  Call this winner V.

All arguments you make for W above apply equally well to V, and V should be
immune to any complaint from W, A, or covered candidates (via short
beatpaths).

To summarize,

   1) Determine Approval winner A.

   2) Find U(A), the set of uncovered candidates that cover A.

   3) Use any strong clone-proof method on U(A), with approval-weighted
      ranking (approval-weighted ties broken by wv, margin, random ballot).

Has this been proposed already?  Each step is from another suggestion, but I
haven't seen any combination of this sort.  But I haven't dug so far into the
archives.

Here's a simple Graded Ballot well suited to this method.  For clarity, the
approved grades A,B,C could be colored differently from the disapproved grades
D,E,F.

,----
|    Grade at least one candidate.
|
|    Equal grades are allowed.
|
|    The highest possible grade is A+ (strongly approve).
|
|    The default grade for ungraded candidates is F- (strongly disapprove).
|
|    The default minor grade for graded candidates is 0.
|
|    Any grade higher than F-, even D+ and lower, is a vote for that candidate
|    over all other candidates with lower grades.
|
|               Approve     Disapprove
|                                        minor grade
|              A   B   C    D   E   F     +   0   -
|    X1       ( ) ( ) ( )  ( ) ( ) ( )   ( ) ( ) ( )
|                                        
|    X2       ( ) ( ) ( )  ( ) ( ) ( )   ( ) ( ) ( )
|                                        
|    X3       ( ) ( ) ( )  ( ) ( ) ( )   ( ) ( ) ( )
|                                        
|    X4       ( ) ( ) ( )  ( ) ( ) ( )   ( ) ( ) ( )
|                                        
|    X5       ( ) ( ) ( )  ( ) ( ) ( )   ( ) ( ) ( )
| 
`----

With a machine-aided paper ballot, this could be filled out with only one or
two entries per candidate, and ungraded candidates would be filled in
automatically as F- to prevent tampering.

Ted
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