[EM] Landau vs Banks confusion

[Forest Simmons]

Chris,

Not so quick; don't throw the baby out with the bath water!

Max Covering based on an approval list or any other monotonically generated
list of the candidates elects a candidate monotonically from the set of
uncovered candidates, i.e. the Landau Set, but not necessarily from the
Banks set, because (in general) some Landau candidates may not be Banks
candidates.

All uncovered candidates are members of Landau.

But only candidates that stand at the head of (one or more) maximal totally
ordered chains (in the defeat relation) qualify as members of the Banks
set.

We have several methods (including Copeland) that elect monotonically from
Landau, but so far only one from Banks (chain climbing).

Copeland elects monotonically from Landau, but is no more clone independent
than Borda.  As far as Landau methods go, the Max Covering schema is the
only one I know of (besides chain climbing itself) which is both monotone
and clone free

If chain climbing satisfied Independence from Pareto Dominated Alternatives
(IPDA), I would be content to stop looking for a better Banks method.

Max Covering does satisfy the IPDA criterion, so perhaps some tweak on it
might get us where we want. We would start with the covering relation which
is a partial order on the set of candidates. Then we would single out some
maximal chain in the covering relation. as in the Max Cover procedure.
Finally beef up this chain with additional candidates while relaxing the
order relation from covering to mere defeat until it become a maximal
totally ordered chain with respect to the defeat relation.

In the resulting chain every candidate would defeat all of the candidates
below it and would be defeated by all of the candidates above it, and no
other candidate could be included into the chain without destroying this
total order property.  The candidate at the head of the chain would beat
all of the other candidates in the chain.  And since that chain was built
up to be maximal with respect to the defeat relation that candidate would
be a member of the Banks set.

Can this be done monotonically?  The devil is in the details.

Our recent attempt got pretty close, but did not (to our disappointment)
actually satisfy monotonicity.as it first appeared.

Back to the drawiing board!
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