[EM] Chain Climbing --> Chain Filling

[Forest Simmons]

Ted,

Thanks for your thoughtful critique.  I have been thinking along similar 
lines for different reasons, mainly a desire to achieve IDPA.

Unfortunately, reverse TACC is not monotonic with respect to approval.  If 
the winner moves up to the top approval slot without also becoming the CW, 
she will turn into a loser.

However, the following "chain filling" method is monotonic:

Working from top to bottom of the approval list, fill in a chain by 
incorporating each candidate that can be included transitively. The 
candidate at the top of the resulting maximal chain is the winner.

This technique transforms (mutatis mutandi) each chain climbing method 
into a chain filling method.

The chain filling slows the descent enough that even the Approval Winner 
can win the method without being the CW.

Suppose for example that pairwise A beats B beats C beats A, and that the 
approval order from greatest to least is A>B>C.

The maximal chain is A>B, without C, which cannot fit into this chain 
transitively.

There is no CW, yet the approval winner wins, which could never happen in 
reverse TACC.

As mentioned above, I wanted to work top down so that I would come to the 
Pareto dominators before getting to the Pareto dominated candidates. 
Then it doesn't matter if the Pareto dominated candidates are eliminated 
at the beginning; the rest of the chain will be the same, including the 
top candidate.

[If approval ties are broken by random ballot, then Pareto dominators will 
be above Pareto dominated candidates.]

Filling the chain does indeed give us full monotonicity as well:

If the winner moves up relative to any of the other candidates, either in 
approval or pairwise, the chain remains the same, since the approval order 
of the rest of the candidates is the same, as well as their pairwise 
comparisons with each other, and it doesn't matter at what stage the top 
member of the chain is added in, as long as it is not later than before.

My Best,

Forest