[EM] Some thoughts on general possibly strategy-resistant Condorcet methods

[Kristofer Munsterhjelm]

In my post about strategy resistatn Condorcet methods, I wrote that my 
program found the following for a three candidate ABCA cycle:

A's score: A>B * min(A>B, B>C)/fpC
(And similar but rotated for B and C)

The problems with this are that it's not cloneproof and that it's not 
general (doesn't say anything about how to proceed with four candidates, 
five etc).

But then I noticed that the min(A>B, B>C) term does look a lot like a 
beatpath calculation. So how about this, for a general score:

Let A's score against C be infinity if C doesn't beat A pairwise, otherwise

[first leg of beatpath from A to C] * [value of beatpath from A to C]/fpC,

where fpC is the number of first preferences for C.

Then let A's score be the minimum of A's score against anyone else. Ties 
are broken by looking at the next-to-minimum scores, and then the next 
to next and so on up.

This still isn't entirely cloneproof, though, but the following should be:

A's score against C is (assuming C beats A)

[first leg of beatpath from A to C] * [value of beatpath from A to 
C]/[number of approvals for C]

Is that an interesting approach? I guess it's rather opaque, and I'm not 
sure how much of the strategy resistance is lost by going from first 
preferences to approvals. It also requires voters to mark an approval 
threshold, and I don't entirely like methods that require that.

Similarly, the fpA - fpC method could be generalized to either:

A's score against C is (assuming C beats A pairwise)

number of approvals for A - number of approvals for C (approach 1), or
number of voters who approved only A (approach 2)

Of perhaps tangential interest is that the three-candidate first 
preference variant (where A's score is fpA - fpC,) passes the CD criterion.

Another possible generalization is to turn these into chaining methods. 
Start with A being the winner according to some base method (e.g. the 
Approval winner), then switch to the new A that beats/covers the old A 
and for which one of the scores

a) [first leg of beatpath from A to old A] * [value of beatpath from A 
to old A]/[number of approvals for old A],
b) number of approvals for new A - number of approvals for old A
c) number of voters approving of the new A but not the old A

is maximized. Interestingly, c) reduces to UncAAO even though I didn't 
set out to make it so.