[EM] An ABE solution

[fsimmons at pcc.edu]

Mike, thanks for your comments. I'll respond in line below.

> From: MIKE OSSIPOFF 
> Hi Forest--
> 
> Thanks for answering my question about MTA vs MCA. Your argument 
> on that question is convincing, and
> answers my question about the strategy difference between those 
> two methods.
> 
> Certainly, electing C in the ABE avoids the ABE problem. I'd 
> been hoping that the election of C can be attained
> without diverging from Plurality's results enough to upset some 
> people, as MMPO and MDDTR seem to
> do.
> 
> So the method that you describe might avoid the public relations 
> (non)problems of methods that elect
> A.
> 
> I have a few questions about the method that you describe:
> 
> 1. What name do you give to it? In this post I'll call it "Range 
> till cover-winner" or RCW

Good Idea.

> 
> 2. The covering relation doesn't look at pairwise ties?

Different variants handle ties differently, but I favor using dominance in the win/loss/tie matrix, in which 
the entry in row x column y is respectively one, minus one, or zero depending on whether candidate x 
beats, is beaten by, or ties with candidate y, respectively.  We consider a candidate to be tied with 
itself, so for each x, the diagonal (x, x) entry is zero.

Note that this matrix can be gotten by subtracting the transpose of the pairwise matrix from itself, and 
then replacing each entry by its sign, where sign(t) is 1, -1, or 0, depending on whether t is positive, 
negativve, or zero.  In other words, replace each entry in the pairwise margins matrix with its sign.

Row x dominates row y iff the two rows are not identical and every entry in row x is at least as great as 
the corresponding entry in row y.

Candidate x covers candidate y iff row x dominates row y in the above sense.

> 
> 3. Does the ballot ask the voter for cardinal ratings of the 
> candidates, or is the range score
> calculated a la Borda? 

Cardinal ratings are assumed, and for public proposal I contemplate three slots.

If the ballots are ordinal, you can transform them (clone free and monotonically) to cardinal ballots via my 
algorithm under the heading "Borda Done Right."

> 
> 4. How does it do by FBC? And by the criteria that bother some 
> people here about MMPO (Kevin's
> MMPO bad-example) and MDDTR (Mono-Add-Plump)? 

I think it satisfies the FBC.  In fact, it reduces to Approval in the two slot version or if all voters rate only 
at the extremes., Approval strategy is probably near near optimal.

It satisfies Mono-Add-Plump.  Proof:  First note that addition of a ballot that truncates all of the 
candidates doesn't change the winner since it doesn't change the covering relations nor does it change 
the range score (cardinal rating) order. Now, on such a ballot raise the winner from trunctated status to 
non-truncated status, and leave the rest of the candidates at the bottom.  By the monotonicity of the 
method, the winner is preserved.

I believe it satisfies the Plurality criterion.  At least in Kevin's MMPO bad example it ties the two outside 
candidates.

> 
> There's much hope that, by electing C instead of A, RCW can 
> avoid those criticisms.
> 
> I'm also interested in how it does by 1CM and 3P, but I'll look 
> at that, instead
> of asking you to do everything for me, especially since I'm the 
> one promoting those
> two criteria.

I don't hve those two criteria on the tip of my tongue, but I'll look them up and see if I can help you on 
that.

> 
> Mike

Forest



> 
> 
> 
> 
> Here?s my current favorite deterministic proposal: Ballots are 
> Range Style, say three slot for simplicity.
> 
> When the ballots are collected, the pairwise win/loss/tie 
> relations are
> determined among the candidates.
> 
> The covering relations are also determined. Candidate X covers 
> candidate Y if X
> beats Y as well as every candidate that Y beats. In other words 
> row X of the
> win/loss/tie matrix dominates row Y.
> 
> Then starting with the candidates with the lowest Range scores, 
> they are
> disqualified one by one until one of the remaining candidates X 
> covers any other
> candidates that might remain. Elect X.
> 
> For practical purposes this method is the same as Smith//Range. 
> Where they
> differ, the member of Smith with the highest range score is 
> covered by some
> other Smith member with a range score not far behind.
> 
> This method resolves the ABE (approval bad example) in the 
> following way:
> Suppose that the ballots are
> 
> 49 C
> 27 A(top)>B(middle)
> 24 B
> 
> No candidate covers any other candidate. The range order is 
> C>B>A. Both A and
> B are removed before reaching candidate C, which is not covered 
> by any
> remaining candidate. So the Smith//Range candidate C wins.
> 
> If the ballots are sincere, then nobody can say that the Range 
> winner was a
> horrible choice. But more to the point, if the ballots are 
> sincere, the A
> supporters have a way of rescuing B: just rate hir equal top 
> with A.
> 
> Suppose, on the other hand that the B supporters like A better 
> than C and the A supporters know this. Then the threat of C 
> being elected will deter B faction defection, and they will 
> rationally vote A in the middle:
> 
> 49 C
> 27 A(top)>B(middle)
> 24 B(top)>A(middle)
> 
> Now A covers both other candidates, so no matter the Range score 
> order A wins.
> 
> This completely resolves the ABE to my satisfaction.
> 
> The method also allows for easy defense against burial of the CW.
> 
> In the case
> 
> 40 A>B (sincere A>C>B)
> 30 B>C
> 30 C>A
> 
> where C is the sincere CW, the C supporters can defend C's win 
> by truncating A. Then the Nash equilibrium is
> 
> 40 A
> 30 B>C
> 30 C
> 
> in which C is the ballot CW, and so is elected.
> 
> 
> Now for another topic...
> 
> 
> MTA vs. MCA
> 
> I like MTA better than MCA because in the case where they differ 
> (two or more
> candidates with majorities of top preferences) the MCA decision 
> is made only by
> the voters whose ballots already had the effect of getting the 
> ?finalists? into
> the final round, while the MTA decision reaches for broader support.
> Because of this, in MTA there is less incentive to top rate a 
> lesser evil. If
> you don?t believe the fake polls about how hot the lesser evil 
> is, you can take
> a wait and see attitude by voting her in the middle slot. If it 
> turns out that
> she did end up as a finalist (against the greater evil) then 
> your ballot will
> give her full support in the final round.
>