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

[C.Benham]

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

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