[EM] Re: Chain Climbing --> Chain Filling

[Jobst Heitzig]

Dear Forest!

You wrote:
> 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.

That's right.

You continued:
> 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 is a nice idea, but upon closer inspection it turns out not to be
monotonic, unfortunately. Look at this example: Assume the approval list
from top to bottom is A,B,C,D,E, and the defeats are
A>B>C>D>E>A>D>B<E>C>A. Then the chain filling begins A>B, C is skipped,
D is added to give A>D>B, and E is skipped. Hence A wins. Now reverse
the defeat C>A into A>C without changing approvals. Then the chain
filling begins A>B>C, D is skipped, and E is added to give E>A>B>C.
Hence E wins although the original winner A was raised.

> 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.

That's a good point! I think that for this reason, we should proceed
trying to find a variation which works top down.

Yours, Jobst