[EM] Thinned List Chain Climbing

[fsimmons at pcc.edu]

Dear EM list folks,

I grew up on a sugar beet farm.  Evey spring after the beets sprouted we thinned
them to allow the remaining ones to reach their potential.  It turns out that
"chain climbing" can enjoy a similar benefit from the "thinning" of an over
crowded "seed list."

The seed list for TACC is a list of the alternatives in order of approval.  Lack
of thinning makes ordinary TACC fail the IPDA (Independence from Pareto
Dominated Alternatives).

In what follows let the letter L represent any list of alternatives such that
(a) it is monotonically generated, so that if only alternative A improves
relative to other alternatives on one or more ballots, then it is the only one
that moves up the list, and (b) if alternative B is ranked ahead of alternative
C on at least one ballot, as well as on all ballots where they are not ranked
equal (or both truncated), then B appears no lower in the list L than C.

For example ...

(1)  L could be the list of candidates in random ballot order (where additional
ballots are drawn at random if necessary to refine the order until it is a
complete ranking of the alternatives).

OR

(2) L could be the list of candidates in the order of fewest truncations on the
ballot.

The first of these orders can be used to break ties in the second order without
disturbing the desired properties.

Now for the thinning:  

To thin L just remove every alternative that is covered by an alternative higher
(i.e. earlier) in the original list.  In the first example, remove X if it is
convered by some Y with fewer truncations.  In the second example, remove X if
it is covered by some Y that is preferred over X in the random ballot order. 
Remember that Y covers X iff it pairwise beats X along with every alternative
that X beats.

After thinning L, apply regular chain climbing to the thinned list.

This method (applied to the kinds of lists we are talking about) satisfies
Independence from Pareto Dominate Alternatives.

In fact, we can say more.  If we agree that "Y strongly covers X with respect to
L" when X is covered by Y from higher up the list L, then we can say that the
method is Independent from Strongly L-Covered Alternatives.

It preserves, monotonicity, and to the degree L respects clone sets, the method
satisfies clone independence.

Furthermore, if the order of L restricted to the Smith Set is the same as it
would be were L recreated from the same ballots with the non-Smith alternatives
removed, then the method is Independent from non-Smith Alternatives.  The above
examples (properly interpreted) satisfy this condition.

When we take into account that the method (no matter which L) always picks from
the Banks set, we have a truly respectable list of properties.

Furthermore, for the above two example orders for L, the method has unparalleled
resistance to  strategic cycle creation by burial and truncation, which is the
Achilles Heel of most otherwise respectable Condorcet methods.

Among ourselves, let's call the general technique "Thinned List Chain Climbing."

For the stochastic method based on Thinned Random Ballot Order (example 1
above), let's call it ThRnBltOCC.

For the determinsitic method based on Thinned Fewest Truncations Order, let's
call it ThFwTrOCC.

Then when we make a public proposal, we can call it the True Majority Choice
method, or whatever title will sell the best without fibbing.

What do you think?

Forest