[EM] An ABE solution.

[fsimmons at pcc.edu]

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.
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