[EM] Mono-switch-Plump criterion compliance claims corrected

C.Benham cbenham at adam.com.au
Tue Oct 25 21:08:56 PDT 2016 

Forest has pointed out that my supposed example of  Approval Sorted 
Margins failing mono-switch-plump is nonsense.

So (at least for the time being) I am not able to show that Approval 
Sorted Margins fails the mono-switch-plump criterion.

(BTW I am still very happy with the Margins-Sorted Losing Votes (erw) 
Elimination method.)

Chris Benham


On 10/24/2016 10:28 PM, C.Benham wrote:
>
> The Mono-switch-plump criterion is much stronger than I previously 
> thought, and is probably simply incompatible with the
> Condorcet criterion.
>
> I used to think that its met by two of my favourite Condorcet 
> methods,  Margins-Sorted Losing Votes (erw) Elimination (equivalent in 
> the 3 candidate case
> to the "MMLV(erw)M" I discuss in the May 2014 post) and Approval 
> Sorted Margins.  Consider this election under MSLVerwE :
>
> 40: A
> 29: C>A
> 03: B
> 28: B>C
>
> A>B 69-31,    B>C 31-29,    C>A 57-40.   LV(erw) scores:  A40 > B31 > 
> C29.  No adjacent pair is out-of-order pairwise, so MSLV(erw)E elects A.
>
> But if we switch the 3 B plumping ballots to A then C becomes the 
> Condorcet winner (C>B 29-28,  C>A 57-43).
>
> 43: A
> 29: C>A
> 28: B>C
>
> And now this election under Approval Sorted Margins:
>
> 30: C
> 04: C>A
> 33: A>B
> 32: B
>
> A>B 37-32,   B>C 64-34,   C>A 34-33.    (Implicit) Approval scores: 
> B64 > A37 > C34.  The adjacent pair with the smallest (absolute 
> margin) difference
> in their scores (A > C) is pairwise out of order so we flip that to 
> give B > C > A.  Now neither adjacent pair is pairwise out-of-order, 
> so the order is
> final and so Margins Sorted Approval elects B.
>
> But if we switch two of the 32 B plumping ballots to A then A becomes 
> the Condorcet winner (A>B 39-34,  A>C 35-34).
>
> 30: C
> 04: C>A
> 33: A>B
> 02: A
> 30: B
>
> I doubt that IBIFA meets the criterion.
>
> But I remain sure that it's met by Bucklin (and similar methods like 
> MTA and MCA and QLTD).
>
> Chris Benham
>
> On 11 May 2014  Chris Benham  posted  to EM:
>
>>
>>>  Mono-switch-plump:
>>>
>>> *The probability of candidate X winning must not be reduced if one 
>>> or more ballots that
>>> plump for any not-X  are replaced by an equal number of ballots that 
>>> plump for X.*
>>
>> Previously I showed that this is failed by the following methods:
>>
>> Schulze (aka Beatpath), Ranked Pairs, River, MinMax (all equivalent 
>> with 3 candidates) if they use Winning Votes to weigh pairwise defeats.
>>
>> IRV and the Condorcet methods based on IRV  (such as Benham and Woodall)
>>
>> Total Approval Chain Climbing.
>>
>> I claim that it is met by  Margins,  any positional method, IBIFA, 
>> Bucklin and Bucklin-like methods like Median Ratings and MCA and MTA.
>>
>> And also it is met by MMLV(erw)M.     To support that claim I'll just 
>> talk about the  Margins Sort version with 3 candidates.
>>
>> Plumping ballots for any X always contribute to X's   score and 
>> switching plumping ballots to X might get rid of one of X's pairwise 
>> defeats.
>>
>> If X has no pairwise defeats then that will always be still the case 
>> after switching some plumping ballots to X and so X will still win. X 
>> can't
>> be a winner with all pairwise defeats so we are only concerned about 
>> the case when X has just one (and so will the other 2 candidates).
>>
>> Say we designate the candidate with the highest score 1, the 
>> second-highest 2 and and the lowest 3.   The algorithm in this 
>> 3-candidate cycle
>> situation  elects 1 unless 2 both pairwise beats 1 and has a score 
>> that is closer to 1's than to 3's.
>>
>> If winning candidate X is in position 2 then the effect of plumping 
>> ballots being switched from 1 to 2  will be to just make 2 still 
>> closer to 1,
>> and the effect of plumping ballots being switched from 3 to 2 will 
>> have the same effect (and make 3 further away).
>>
>> If winning candidate X is  1  and pairwise beats 2 and loses to 3, 
>> then the only hope of making 1 lose is to switch some plumping 
>> ballots from
>> 2 to 1 sufficient for 2 and 3 to change places but that won't work 
>> because then 2 and 3 will be adjacent candidates that are out of 
>> pairwise
>> order and will be much closer together score-wise than the other such 
>> pair and they'll be switched back to give the final order 1>2>3.
>>
>> And if X is 1 and losing to 2  then it means that 1's distance 
>> (scorewise) from 2 is such that 2 and 3 are switched in the order, 
>> and switching
>> any plumping ballots to 1 will only increase that distance.
>>
>> I hope that (almost confused) waffle is not too confusing or opaque.
>>
>> Chris Benham
>>
>>
>>
>>
>>
>>  Mono-switch-plump:
>>
>> *The probability of candidate X winning must not be reduced if one or 
>> more ballots that
>> plump for any not-X  are replaced by an equal number of ballots that 
>> plump for X.*
>>
>> Mono-raise is the traditional monotonicity criterion, but I don't see 
>> why anyone would
>> see failure of  Mono-switch-plump as less embarrassing than failing 
>> Mono-raise.
>>
>>
>> 25 A>B
>> 26 B>C
>> 23 C>A
>> 22 C
>> 04 A
>>
>> B>C  51-45       C>A 71-29       A>B 52-26
>>
>> Top Preferences:  C45 > A29 > B26
>>
>> When there are three candidates the MinMax , Beatpath (aka Schulze), 
>> Ranked Pairs and River algorithms
>> are all equivalent. When they use Winning Votes as the measure of 
>> defeat strength they all elect C.
>>
>> IRV  (aka the Alternative Vote) and  Benham (and Woodall) also elect 
>> C.  But if we replace the 4A ballots
>> with 4C ballots the winner with all these methods changes from C to B.
>>
>> 25 A>B
>> 26 B>C
>> 23 C>A
>> 26 C
>>
>> B>C  51-49       C>A 71-29       A>B 48-26
>>
>> Top Preferences:  C45 > B26 > A25
>>
>> Total Approval Chain Climbing  also fails.
>>
>> 25 A>B
>> 06 A>C
>> 32 B>C
>> 27 C>A
>> 08 C
>> 02 B
>>
>> C>A>B>C,   Approvals C73 > B59 > A58
>>
>> TACC  elects C, but if the 2B  ballots are changed to 2C, then the 
>> winner changes to A.
>>
>> 25 A>B
>> 06 A>C
>> 32 B>C
>> 27 C>A
>> 10 C
>>
>> C>A>B>C,     Approvals C75 > A58 > B57
>
>
>
> ----
> Election-Methods mailing list - see http://electorama.com/em for list info
>
>
> No virus found in this message.
> Checked by AVG - www.avg.com <http://www.avg.com>
> Version: 2016.0.7859 / Virus Database: 4664/13260 - Release Date: 10/23/16
>

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20161026/70945ac3/attachment.htm>

More information about the Election-Methods mailing list