[EM] SODA

[fsimmons at pcc.edu]

Yes, you are right!

Now I would like to suggest a way to make this method clone proof:

The key is to use the solid coalition structure of the factions to determine the sequential order of play 
(i.e. "delegation"), from largest coalition to smallest.  I believe that completely solves the problem.

Here's an example where A got split into A1 and A2. 

16 A1>A2>B
12 A2>A1>B
24 B>A1=A2
48 C

Even though the C faction is the biggest faction, and the A1 faction is the second smallest faction, 
candidate A1 is the first to delegate in this new order.  Here's why:

The largest coaltion (besides the entire set of factions) is the coalition made up of the set of factions 
{A1, A2, B} with 52 percent of the electorate (versus 48 percent for the coalition {C faction}).  Within the 
large coalition, the largest subcoalition is {A1, A2} with 26 percent of the entire electorate (versus 24 
percent for the coalition {B faction}).

Within this subcoalition the larger of the two subcoalitions is the A1 faction.  Since there are no further 
subcoalitions, candidate A1 plays first.

Then A2 goes next, because we finish the {A1, A2} subcoalition (which was larger than the B 
subcoalition) before letting B play. C goes last because at the root of the coalition tree C was the branch 
on the smaller side.

In sum the order of play is A1, A2, B, C.

The process of deciding the order of play can be summarized more succinctly with a recursive 
description:

Start at the root of the coalition tree, and recursively order the leaves (i.e. the individual factions) of the 
respective branches in descending order of the branch sizes.

I think that in selling the method, we can make the precise sequential order a technical detail easily 
glossed over by simply referring to it as the natural "clone independent sequential order," or something 
like that.


----- Original Message -----
From: Jameson Quinn 
Date: Tuesday, July 5, 2011 7:25 pm
Subject: Re: [EM] SODA
To: fsimmons at pcc.edu
Cc: election-methods at lists.electorama.com

> 2011/7/5, fsimmons at pcc.edu :
> >
> > I thought that A was required to make her approvals consistent 
> with her
> > ordering, i.e. to approve
> > everybody ranked above her cutoff. Doesn't that mean she is 
> required to
> > approve herself?
> >
> > Maybe I'm thinking of an older version of SODA.
> >
> > I hope you are right that there is nothing to fix.
> 
> Let's do this slowly. Here's the scenario:
> 
> 34 A>B>C
> 35 B>C>A
> 31 C>A>B,
> 
> B delegates first. B delegates to B,C. Totals are now C 66, B35, A34.
> A's turn. If A does not delegate, C will be winning when it 
> comes to
> C's turn, and so C will not delegate. So A delegates to A,B. Totals
> are now B69, C66, A34. C's turn. C is unhappy with B and so delegates
> to C,A - but it's not enough. Final totals are B69, C66, A65.
> 
> I believe that the correct strategy for any combination of delegable
> and undelegable votes (including minor, non-Smith candidates) in a
> 3-candidate Smith set is always for everyone to approve two 
> members of
> the Smith set if they care between the bottom two. This gives 
> the same
> result as minimax and most Condorcet methods. I haven't proven this,
> and I don't have a general understanding of strategy for larger Smith
> sets.
> 
> It is possible, when there are 3 or more near-clones A1, A2, A3...
> running against a different candidate B with almost 50% - that 
> is, B
> can beat any combination of fewer than all the A's, and B has no
> preference among the A's - that the true Condorcet winner among the
> A's is subject to center squeeze, and the A's are forced to throw
> their support to whichever of them has the most delegable votes, in
> order to prevent B from winning. The upshot is that SODA, even
> assuming candidates are honest in their pre-vote rankings and
> strategic in their delegation, does not pass the Condorcet criterion,
> but does pass the majority Condorcet criterion (that is, a pairwise
> winner always wins if each of the pairwise wins constitutes a
> majority). But I can't find any nonmonotonic scenario pairs, so this
> "Plurality within the faction" is the worst result I can find. I think
> that it's both unlikely and, really, not so bad.
> 
> JQ
> 
> >
> >
> > ----- Original Message -----
> > From: Jameson Quinn
> > Date: Tuesday, July 5, 2011 1:07 pm
> > Subject: Re: [EM] SODA
> > To: fsimmons at pcc.edu
> > Cc: election-methods at lists.electorama.com
> >
> >> 2011/7/5
> >>
> >> > Jameson suggested that the SODA candidates make their approval
> >> decisions> sequentially instead of
> >> > simultaneously.
> >> >
> >> > The problem with this is that if a winning candidate moves to
> >> first place
> >> > in the sequence by an increase
> >> > in support, she may become a losing candidate:
> >> >
> >> > Assume sincere preferences are
> >> >
> >> > 35 A>B>C
> >> > 34 B>C>A
> >> > 31 C>A>B
> >> >
> >> > If approval decisions are made in descending order of faction
> >> size A, B, C,
> >> > then B wins.
> >> >
> >> > If B gains more support so that the totals become
> >> >
> >> > 34 A>B>C
> >> > 35 B>C>A
> >> > 31 C>A>B,
> >> >
> >> > the sequential order becomes B, A, C, and the winner will 
> be C.
> >> >
> >>
> >> No. B still wins. If A feels that C is winning, then A can
> >> delegate to B,
> >> and then B cannot lose. So C cannot be the winner. And therefore
> >> B will
> >> delegate to C, to force A's hand. Whether or not C delegates
> >> then is
> >> irrelevant.
> >>
> >> Of course, if A actually prefers C to B, and has managed to keep
> >> B ignorant
> >> of this fact, then C will win. But then, in such a case, A could
> >> have gotten
> >> the same result by being honest from the start.
> >>
> >> >
> >> >
> >> > How can we fix this?
> >> >
> >> >
> >> I don't think there's anything that needs fixing, though you 
> may find
> >> another example to show I'm wrong.
> >>
> >>
> >> > How about allowing the largest faction (in this example 49 C)
> >> to go second,
> >> > and making the second
> >> > largest faction (in this example 27 A>B) go first?
> >> >
> >> > That would also work in the example above. How bad would it
> >> be in a worst
> >> > case example?
> >> > ----
> >> > Election-Methods mailing list - see http://electorama.com/em
> >> for list info
> >> >
> >>
> >
>