[Election-Methods] Cumulative Approval

[Chris Benham]

Juho wrote:
"I presented only some positive examples. Also various bad failure  
cases would be appreciated if you can find good examples."

Juho,

31: A>B
32: B>C
37: C

C is clearly the strongest candidate, having both more first preferences
and more second preferences than either of the other candidates.

Your suggested "Cumulative Approval" method elects B.

Unlike  IRV or  Bucklin, your suggested method fails Later-no-Help.

31: A>B
32: B>C
37: C>A

The C voters have added a second preference for A, which causes
the winner to change from B to C.

Like  IRV it fails Mono-raise and so is vulnerable to the Pushover
strategy.

49: A
27: B>A
24: C>B

Your suggested method elects B.


45: A
04: C>A  (was A)
27: B>A
24: C>B

Now your method elects A.

Of course it fails Later-no-Harm.

49: A
24: B
27: C>B

B wins, but if  the B voters change to B>C then C wins.

The method fails Kevin Venke's "Sincere Favourite" (and therfore
fails FBC).
http://nodesiege.tripod.com/elections/#critsf

31: A=B
32: B>C
37: C>A

The method elects C.

31: B>A  (was B=A)
32: B>C
37: C>A

Now  it elects B. Some voters have changed the winner from not 
one of their most preferred to one of their most preferred by dropping
one of their most preferred from the top-most ranking (or rating) on
their ballots.  This is a failure of  Sincere Favourite.


Chris Benham







Juho juho4880 at yahoo.co.uk 
Thu Apr 3 13:57:22 PDT 2008 

Here's one new method (as far as I know, tell if you have seen this  
before) for your consideration.

One viewpoint to this method is that it tries to make the sequential  
process of IRV better than what it is in the basic IRV. On the other  
hand this can be seen also as an Approval method where the approval  
cutoff can move.

The voters will rank the candidates (equal rankings are ok). The vote  
counting algorithm is based on collecting cumulatively approvals for  
the candidates.

The algorithm follows roughly the following philosophy.

(a) tentatively elect a winner
(b) voters are given the chance to compromise and approve more  
candidates to find a better winner (approvals are final and can not  
be canceled)
(c) those voters whose so far approved candidates are weakest at one  
moment shall compromise first

I explained this rough philosophy before the algorithms since this  
hopefully helps when going through the detailed descriptions below.

First one rather complex procedural description of the method.

(1) all voters approve their favourite candidate(s)
(2) find those candidates that have most approvals (=>leaders) (also  
partial tie breaking possible here)
(3) find those voters that have not yet approved all the leaders
(4) take from that set only those voters who still have not approved  
all the candidates that they prefer to the least preferred leader(s)
(5) take from that set only those voters who approve the lowest  
number of the leaders
(6) take from that set only those voters whose best approval result  
among the approved candidates is lowest
(7) these voters will change their vote to approve also the next  
candidate(s) (in their order of preference)
(8) if there were still such voters jump back to point (2)
(9) elect the candidate with highest number of approvals (use tie  
breaker if needed)

This description was quite complex because of the all tie related  
concerns. The following version of the algorithm is a bit simpler. It  
breaks all ties in the results as soon as they are encountered  
(unlike the algorithm above that allowed multiple leaders to exist).  
In large public elections both approaches typically yield the same  
results since ties are very rare in large elections. Rows from (2) to  
(5) have been modified.

(1) all voters approve their favourite candidate(s)
(2) find the candidate that has most approvals (leader) (use tie  
breaker if needed)
(3) find those voters that have not yet approved the leader
(4) take from that set only those voters who still have not approved  
all the candidates that they prefer to the leader
(5)
(6) take from that set only those voters whose best approval result  
among the approved candidates is lowest
(7) these voters will change their vote to approve also the next  
candidate(s) (in their order of preference)
(8) if there were still such voters jump back to point (2)
(9) elect the candidate with highest number of approvals (use tie  
breaker if needed)

In all tie breaking cases above the simplest tie breaker is basic  
lottery, but also other additional criteria could be used.

Here's one example calculation (a typical simplified left-centre- 
right example).

Votes:
49: A>B>C
12: B>A>C
12: B>C>A
27: C>B>A

- first all approve their favourites

49: A>BC
12: B>AC
12: B>CA
27: C>BA

- A is the leader
- BCA and CBA voters could compromise
- BCA voters will compromise since B has only 24 approvals (C has 27)

49: A>BC
12: B>AC
12: BC>A
27: C>BA

- A is still the leader
- now the CBA voters must compromise

49: A>BC
12: B>AC
12: BC>A
27: CB>A

- B is the leader
- all voters have either already approved B or do not have any more  
compromise candidates before B
- B is the winner

Another example (typical theoretical example of a loop of three).

Votes:
35: A>B>C
33: B>C>A
32: C>A>B

- all will approve their favourites first

35: A>BC
33: B>CA
32: C>AB

- A is the leader
- only BCA voters can compromise, and they will

35: A>BC
33: BC>A
32: C>AB

- C is the new leader
- ABC voters can compromise, and they will

35: AB>C
33: BC>A
32: C>AB

- B is now the leader
- CAB voters can compromise, and they will

35: AB>C
33: BC>A
32: CA>B

- B is the leader
- no voters will compromise
- B wins

Note that typically Condorcet methods elect A. This method is in a  
way more compromise seeking (also second preferences count as  
approval votes) and B will be elected. B has the lowest number of  
last positions in the votes.

I hope these examples were sufficient to demonstrate how the method  
is expected to work.

The reason why I thought this method is worth writing a mail is that  
it has some interesting strategy related properties. Surely it has  
also some weak spots like all this type of serial selection processes  
tend to have. But maybe they are marginal enough to make this method  
useful in some environments.

In the first example IRV would have eliminated B first, and then  
elected A. The CBA voters could have compromised and made B the  
winner. In this method CBA voters however got B without the need to  
resort to strategic/modified votes.

In the first example and with a Condorcet method the ABC voters could  
have attempted to bury B under C and thereby make A the winner (quite  
difficult task though). Also this method would elect A if sufficient  
number of the ABC voters are strategic. This method however seems to  
have slightly higher tolerance against this type of strategic voting.  
Let's assume that 25 out of the 49 ABC voters are strategic. This is  
typically enough to make A win in Condorcet methods.

Votes:
24: A>B>C
25: A>C>B (strategic)
12: B>A>C
12: B>C>A
27: C>B>A

- the method proceeds as in the first example until...

24: A>BC
25: A>CB
12: B>AC
12: BC>A
27: CB>A

- B is the leader
- now we have the strategic ACB voters that can and must compromise

24: A>BC
25: AC>B
12: B>AC
12: BC>A
27: CB>A

- C is the leader
- but luckily not a winner since BAC and ABC voters can compromise
- the ABC voters will compromise since A is less approved than B

24: AB>C
25: AC>B
12: B>AC
12: BC>A
27: CB>A

- B is again the leader
- no compromises left
- B wins

The 24 sincere ABC voters were enough to break the strategy (19 would  
have been enough I think).

I presented only some positive examples. Also various bad failure  
cases would be appreciated if you can find good examples. I hope I  
didn't make too many mistakes above and the descriptions were clear.

The algorithms where not very simple (the (uncommon) ties were the  
most complex part), but I guess the basic idea of allowing voters to  
compromise and give support also to their second preferences if their  
more preferred candidates are not about to win is clear enough for  
regular voters to understand.

Juho


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