[EM] Method X

[Filip Ejlak]

Method X is cloneproof as well! That makes it even more superior to Kemeny
(I think that cloneproofness is a must for a good electoral method). I also
think that IIB *is* met, because if a given ballot is not exhausted, it is
not irrelevant.
I'm building something similar to the quadelect program so I can check
similar things; indeed the results show that Smith//Method_X is monotonic
(if ties are resolved properly), cloneproof and DMTBR-compliant (and of
course ISDA-compliant). Finding a method that combines it all is an amazing
achievement! A polytime version, if such exists, would certainly be the
Holy Grail Method. If it doesn't exist, then the non-polytime version would
probably still be the best option - as it was already mentioned, real-life
elections wouldn't be at all as computation-costly as it is theoretically
possible.
Perhaps basing the elimination process on first. pref. Copeland score or
"friendly" score instead of simple plurality score should be considered - I
guess it might at least make Method X naturally ISDA-compliant, but the
criteria compliance and strategy resistance would need to be checked again.

Below I present the closest thing to a non-monotonicity example that I
found; by the way, you can check if the X score results are correct (and
the implementation is OK):

1: A>B>D>C
1: B>C>A>D
1: B>C>D>A
1: C>A>B>D
1: C>B>A>D
1: C>D>A>B
1: D>A>B>C
2: D>C>A>B
1: D>C>B>A

The scores are A:0, B:5, C:5, D:5, so there's a B-C-D tie.
But then we raise B by changing CBAD to BCAD and DCBA to DBCA.
The new scores are A:4, B:5, C:0, D:6, so D wins - B is no longer in the
winning tie.
So if we want to 100% guarantee the monotonicity in all edge cases, we
should make sure that such a tie is not resolved by making B the winner.


śr., 2 sie 2023 o 01:21 Kristofer Munsterhjelm <km_elmet at t-online.de>
napisał(a):

> This is method X, the monotone burial-resistant method from the previous
> post: (It doesn't really have a name yet.)
>
> - Each candidate A obtains his score by eliminating candidates in
> rounds, one candidate per round, until one other candidate (B) remains
> or A himself is forced to be eliminated. In the former case, A's score
> is A>B. In the latter case, A's score is zero.
>
> - When figuring out A's score, the method chooses the sequence of
> candidates to eliminate so as to maximize that score.
>
> - In a round, the method can never eliminate a candidate who has more
> than 1/n of the total number of (non-exhausted) first preferences, where
> n is the number of remaining candidates in the round in question.
>
> - Unlike IFPP, it never eliminates more than one candidate per round.
>
> - Highest score wins.
>
> That's it!
>
> Now for the bad news:
>         - it's not summable (a summary takes O(n2^n) space),
>         - it's not polytime (ditto),
>         - its use of quotas means it would probably fail IIB,
>         - and I have no idea why it works.
>
> I can't say I wasn't tempted to name it after myself since it's kind of
> a big deal (if I can verify its monotonicity), but given its drawbacks,
> maybe it's not a good idea? It's like the Kemeny of monotone
> burial-resistant methods: slow and impractical, but it shows what's
> *possible*.
>
> I would very much like some independent verification or replication,
> though.
>
> -km
> ----
> Election-Methods mailing list - see https://electorama.com/em for list
> info
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20230803/a89047f8/attachment-0001.htm>