[EM] Mono-switch-Plump criterion compliance claims corrected (ASM example fixed)

C.Benham cbenham at adam.com.au
Wed Oct 26 11:13:01 PDT 2016 

The Approval Sorted Margins example I gave earlier didn't work, so below 
I've substituted one that does.

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:
>>
>> 43: A
>> 04: A>C
>> 19: B>C
>> 07: B
>> 27: C>B
>>
>> B>A 53-47,    A>C 47-46,   C>B 31-26.    (Implicit) Approval scores: 
>> B53 > C50 > A47.
>>
>> Both adjacent pairs are out-of-order pairwise and the approval score 
>> differences are the same (3) in both
>> cases so we flip the order of the lower-ordered pair to give B>A>C.  
>> Now no adjacent pair is pairwise out-of-order
>> so that order is final and B wins.
>>
>> Now say we change two of the A-plumping ballots into B-plumping 
>> ballots. Then C will be the Condorcet winner.
>>
>>
>> 41: A
>> 04: A>C
>> 19: B>C
>> 09: B
>> 27: C>B
>>
>> C>A 46-45,   C>B 31-28,   B>A 55-45
>>
>> 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
>>
>>
>>
>>
>>
>>
>
>

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20161027/666f9ae9/attachment-0001.htm>

More information about the Election-Methods mailing list