[EM] Re: Quota-Limited Weighted Approval

Fri May 28 00:47:02 PDT 2004

  Participants,
On  Thurs.Feb. 26  this year, I  posted  a strange idea for a 
single-winner ranked-ballot voting method I called
"Quota-Limited Weighted Approval".

> Voters rank the candidates. Equal preferences and truncation ok.
> (1) Symetrically complete the ballots.
> (2) Based on these now symetrically completed ballots, give each 
> candidate a weight of  1 for each ballot on which it is
> ranked in first place. (The total weight of  the candidates will now 
> be equal to the total number of  original before-step-1
> ballots.  Any candidate with a weight  equal to or greater than half 
>  the total weight of all the candidates wins).
> (3) Each ballot fully approves the highest-ranked  candidates whose 
> combined weight is less than half  the total weight of
> all the candidates. Each ballot also fractionally approves the next 
> highest-ranked candidate, so that  the combined weight
> of  the candidates approved by each ballot  is equal to half  the 
> total weight of all the candidates.
> (4) The candidate with the highest  approval score wins.

Some explanation of  what I meant by  "Symetrically complete the 
ballots."  The idea is that ballots that don't strictly rank
all  the candidates are "incomplete". To quote Woodall, "The symetric 
completion of a ballot is obtained by replacing  all
possible completions of it with equal weight, chosen so that the total 
weight is 1.  For example, if there are five candidates
ABCDE, then the symetric completion of a ballot marked  A>B, consists of 
six ballots, each with a weight 1/6, marked
A>B>C>D>E,  A>B>C>E>D, A>B>D>C>E,  A>B>D>E>C, A>B>E>C>D, A>B>E>D>C."
He doesn't consider methods which allow equal non-last ranking, so I add 
this example. The "symetric completion" of  a
ballot marked A=B>C>D>E, consists of  two ballots, each with a weight of 
1/2, marked A>B>C>D>E  and B>A>C>D>E.

I went on to make some claims about  QLWA's properties:

> This method meets (mutual) Majority, Independence of  Clones, 
>  Participation,  Reverse Symetry, Symetric Completion,
> Woodall's  Plurality criterion, and Independence of  Pareto-Dominated 
>  Alternatives. It is independent of  any losers with
> no first preferences.
> It fails Condorcet, Later-no-harm, Later-no-help, and  Steve Eppley's 
>  "resistance to truncation" criterion.
> It might be ok regarding his other  two "defensive strategy" criteria: 
> "minimal defense" and "non-drastic defense".

Some of these are wrong. It fails Participation, Reverse Symetry,  the 
Plurality Criterion, and  "Minimal Defense".
BTW, "Symetric Completion" is s criterion of  Woodall's which says "A 
truncated ballot should be treated in the same way
as its symetric completion". Adapting that to methods that allow equal 
non-last ranking, I mean "A ballot that doesn't strictly
rank all the candidates should be treated in the same way as its 
symetric completion". Perhaps that should be distinguished
as "Generalised" SC.
The method has a  tendency to elect the CW, and to resist the Burying 
strategy. It has the property in 3-candidate elections
that if  the first-preference winner gets all the second preferences of 
the first-preference runner-up, then the first-prefernce
winner (FPW) is elected. Here are some examples from James G-A in which 
QLWA  resists Burying.

His most recent  (44 of  the Bush voters are Burying  the sincere CW, 
Kerry) :
44: Bush>Nader>Kerry
05:Bush>Kerry>Nader
38:Kerry>Nader>Bush
05:Kerry>Bush>Nader
08:Nader>Kerry>Bush

Weights:    Bush 49,   Kerry 43,   Nader 8.
QLWA approvals  (100 ballots, "Quota" = 50)
44:Bush>.125 Nader
05:Bush>.0232558 Kerry
38:Kerry>.875 Nader
05:Kerry>.1428571 Bush
08:Nader>.9767441 Kerry

QLWA final scores-      Bush: 49.714285       Kerry:50.930231     
Nader:46.75.   Kerry wins.

40:A>C>B  (sincere is A>B>C)
46:B>A>C
07:C>A>B
07:C>B>A
All the A voters are Burying B.

Weights:   A40,   B46,   C14.
QLWA approvals (100 ballots, Quota = 50)
40:A>.7142857C
46:B>.1A
07:C>.9A
07:C>.7826096B

QLWA final scores-   A:50.9     B:51.49826    C:42.571428.     B wins.

28:A>B>C
27:B>C>A   (sincere is B>A>C)
23:C>A>B
22:C>B>A

Weights:   A28,   B27,  C45
QLWA approvals
28:A>.8148148B
27:B>.5111111C
23:C>.1785714A
22:C>.1851851B

QLWA final scores-   A:32.107142     B:53.888886    C:58.8    The 
Burying backfires, and  C wins.

Take this mad example:
101:A
001:B>A>C
101:C>B>A
Ranked Pairs(winning votes) says B>A>C. I think anyone who is wearing 
their  Independence of  Irrelevant Alternatives (IIA)
spectacles would agree that A should win. Certainly a big  "this isn't 
IRV" wake-up call for the A supporters.

"Symetrically completing" the ballots, this becomes:
50.5:A>B>C
50.5:A>C>B
1: B>A>C
101: C>B>A

Weights:      A101,    B1,    C101.
QLWA approvals  (203 ballots, Quota = 101.5)
50.5: A>.5B
50.5:A>.0049504C
1:    B>.9950495A
101:C>.5B

QLWA final scores-    A:101.9950495     B:76.75    C:101.2499952.     A 
wins.

Here is an example of QLWA failing Woodall's  "Plurality" criterion, 
which says that  if  that if  there is a candidate x which has
more first-place votes than candidate y has non-last- place  votes, then 
candidate y cannot win.

79:A>B>C
41:B
80:C
Here C   has a "plurality" over A, so to meet the criterion, A is not 
allowed to win.  QLWA  elects A.

Here is the example which has caused me to dump QLWA as my pet 
 single-winner method. It is one of  Woodall's example
elections.

11:A>B>C>D>E
10:B>C>A>D>E
10:C>A>B>D>E
10:E>A>B>D>C
10:E>B>C>D>A
10:E>C>A>B>D
02:D>E>A>B>C
63 ballots. All candidates in the Smith set. RP and BP pick A. Borda 
picks B.

Weights-  A:11     B:10     C:10    D:2    E:30
QLWA  approvals  (Q = 31.5)
31:ABC>.25D
10:E>.1363636A
10:E>.15B
10:E>.15C
02:D>.983333E

Final scores-   A:32.363636     B:32.5     C:32.5     D:9.75    E:31.966666

QLWA gives this as a tie between B and  C.  

C should definitely have a lower probabilty of winning than A or  E.  A 
has a higher Borda-score, and simply more support,
than C. E has some claim, based on viewing D as a near-irrelevant 
alternative. I  think this shows that QLWA fails Mono-Raise
and Mono-add-Top.
In terms of  the monotonicity criteria listed by Woodall, I now have 
QLWA failing all of them except Mono-add-Plump and
Mono-append.  It fails  Mono-raise-random, Mono-sub-top, 
Mono-raise-delete, Mono-sub-plump, Mono-add-Top,
Mono-raise, Mono-remove-bottom, Participation, Later-no-harm, 
Later-no-help; and his Plurality Criterion.
Against this list, IRV fares much better.  QLWA meets  Steve Eppley's 
 "Non-Drastic Defense" Criterion (at least one of the
versions), but not is other  "defensive" criteria. It meets another one 
he mentions, Independence of Pareto-Dominated Alternatives
(IPDA).

Chris Benham

http://groups.yahoo.com/group/election-methods-list/files/wood1996.pdf

http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-December/011480.html