[Election-Methods] Cumulative Approval

[Juho]

On Apr 5, 2008, at 19:55 , Chris Benham wrote:

> 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.

First, thanks for the thorough analysis.

C could be said to be the strongest candidate above. (I also referred  
to the idea that the sum of first and second positions could be  
relevant, but I'm not sure what this method will do in all cases and  
for what purposes that type of utility function would be good.)

One thing that talks in favour of B is that for the AB voters it is  
easier to compromise than for the BC voters since A seems to be a  
clear loser.

When comparing B and C (or any potential two winners) the sequential  
nature of the algorithm may be a problem since the route that the  
algorithms takes may (in some "semi-random" way) decide the winner.  
In this case we may have two possible approval states that are final  
(equilibriums) in the sense that no further compromises would take  
place.

31: AB>C
32: B>CA
37: C>A=B
=> B leads, no further compromises

31: AB>C
32: BC>A
37: C>A=B
=> C leads, no further compromises

An alternative to the serial process is to use some "static" rule  
that determines e.g. which one of the "leader + stable approval  
state" alternatives is the best. This is typically (not really sure  
about this case) computationally complex so I started with a  
sequential algorithm.

So, which one of the two cases above is better / more righteous?  
Should the B supporters compromise or not? I don't have a clear  
answer to offer now. There may be multiple working utility functions.

> 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.

Yes. The method is closer to Condorcet here.

> 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.

I think here it should elect B.
- A leads, CB compromises
- B leads, CA compromises
- B leads and wins

>
> 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.

Yes, B voters can truncate and win. In this case the new method seems  
to perform worse than IRV (but equal to Condorcet wv).

>
> 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.

Yes.

>
>
> Chris Benham

There seems to be a mixture of strategic voting cases. The new method  
resembles in some aspects the Condorcet methods. The main idea was  
anyway to patch some of the flaws in IRV. Do you have an overall  
estimate on the problems? Better or worse than IRV?

Did you consider some of the strategic cases that you listed to be so  
serious that they would clearly make the new method too vulnerable to  
strategic attacks?

I note that high number of sincere ties (especially on one voter  
group only, and among the strong candidates) as in some of the  
examples is probably not common in real life elections. I wonder how  
much this has impact. One could as well also evaluate the impact of  
strategic ties. I leave ffs e.g. the performance of the new method  
when all the ties would be broken to fractional votes (e.g. X=Y to  
X>Y and Y>X).

Juho



>
>
>
>
>
>
>
> 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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