[EM] Agenda Based Banks

[Susan Simmons]

Right ... we need strong mono-raise ... when I said Plurality for setting the agenda, I was actually thinking of Equal Top Approval, where the approval cutoff (virtual candidate) is positioned immediately below equal top.

There's a superficial similarity, but a subtle (and crucial) difference, especially in the context of monotonicity.

Mono-raise of X from below to above the approval cutoff is not supposed to knock some other candidate Y from above to below the approval cutoff, for example.

......

You wrote..
> in general, it seems that in a three-cycle, ABB elects the candidate who
beats the winner of the base method pairwise.

True, as long as "winner" means least promising ... as in least approval, for example.

Sent from my MetroPCS 4G LTE Android Device

-------- Mensaje original --------
De: Kristofer Munsterhjelm <km_elmet at t-online.de>
Fecha: 4/8/21 9:16 a. m. (GMT-08:00)
A: Susan Simmons <suzerainsimmons at outlook.com>, election-methods at lists.electorama.com
Asunto: Re: [EM] Agenda Based Banks

On 04.08.2021 00:29, Susan Simmons wrote:
> Actually, in the burial example below, ABB performs strictly better than
> DMC because under DMC, the burial of C by A either rewards A with a win,
> or leaves intact A's sincere 2nd choice as winner ... a zero risk gambit.
>
> By way of contrast, if A's gambit fails under ABB, then A's least
> preferred, B, wins ... a big risk, because A's raising of B enough to
> introduce the necessary cycle might also be enough to move B to the
> necessary agenda level (if it was not already there).
>
> In summary, ABB has both the best burial resistance and the best chicken
> attack resistance of any agenda based Condorcet method that we know of
> (and we know of oodles!).
>
> It seems doubtful that any simpler, monotone, clone free, ISDA, burial
> and chicken resistant method exists ... except perhaps Asset Voting,
> which is too far ahead of its time (and our time ...first promoted only
> 150 years ago by Charles L Dodgson) even for Steven J Brams the great
> academic promoter of Approval, another simple method too far ahead of
> its time.

As the perennial cold shower man, I must ask if this is actually
monotone for all monotone base methods :-)

Suppose that the base method is Plurality. In this election:

5: A>B>C
2: A>C>B
6: B>C>A
1: C>A>B
1: C>B>A

there's an ABCA cycle and the Plurality order is A>B>C. If I understand
correctly, ABB would first deposit A and then find the candidate that
beats A pairwise, i.e. C, and then terminate, giving C as the winner.

Now let 3 of the A>B>C voters raise C to C>A>B:

2: A>B>C
2: A>C>B
6: B>C>A
4: C>A>B
1: C>B>A

Now the Plurality order is B>C>A. B is first deposited, and then the
candidate who beats B (i.e.) A is deposited and wins.

So raising C made C lose. Looks like we need strong mono-raise.

With Plurality as the base method, I think ABB reduces to Eivind
Stensholt's BPW - or in my ABCA jargon, the method where candidate A's
score is "-fpC". If we just had a way of making it fpA-fpC, then all
would be well, at least in the case of Smith set equals three!

(Here's an ugly hack: The inner procedure takes a "tentative first
candidate" A. It inserts A, then it inserts the candidate that beats
everybody in the list, whose [first preference count - the last inserted
candidate's first preference count] is maximized. The inner procedure is
over once nobody beats everybody inside pairwise; the candidate last
inserted is the winner, and his score is the (first pref - last guy's
first pref) count.[1]
The outer procedure runs the inner procedure with every candidate as
tentative first, and elects the winning outcome with the greatest score.
But it's extremely ugly and I have no idea if it's strategy resistant or
monotone beyond a Smith cycle of three.)

In general, it seems that in a three-cycle, ABB elects the candidate who
beats the winner of the base method pairwise.

-km

[1] One could imagine other score functions as well, as long as they
specialize properly. E.g. Q's score is min P: fpQ - fpP, where P is a
candidate from the current chain. But I suspect that properly
generalizing fpA-fpC (so that it preserves DMTBR) will require some kind
of solid coalition logic -- or eliminations.
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