[EM] Ranked Pairs Feint to Max Gradient Chain Building

[Forest Simmons]

A curious observation of Kristofer led Kevin and me into a line of inquiry
that has resulted in the following simple but powerful method:

Initialize a chain with the strongest defeat pair as though you were
starting Ranked Pairs, River, or Beatpath CSSD.

Then suddenly switch to Max Gradient Covering by adding to the chain the
candidate with the strongest defeat against the head of the chain among
those that cover it ... it being the current head of the chain.

Repeat this step until the chain has an uncovered head ... to be elected.

Since this method worked so smoothly as an alternate continuation of the
first Ranked Pairs step, Kevin and I thought perhaps we could use the
finish order of any method as a basis for such a chain: initialize a chain
with the highest finish order candidate and then while any candidate covers
the current head of the chain ... add to the chain the highest such
candidate in the finish order.

We were disappointed to find out that almost all seed methods resulted in
non- monotonic combinations of seed plus chain.

Exceptions had to behave like the finish order of Range/Score ... where
strengthening the score of one candidate would not change the relative
finish order among the other candidates.

So Kevin has called our initial success of seeding with one step of RP ...
'beginner's luck".

But it may turn out that no other luck is needed ... if you marry your high
school sweet heart, and everything works out perfectly ... well, if it
ain't broke ...

The one parameter left free in this RP feint covering chain method is the
gauge of defeat strength.

Recently, while musing on swap cost approval, it occurred to me to gauge RP
defeat strength as winning strong approval times losing strong disapproval
... in particular, in the RCV Universal Domain context, strong approval can
be interpreted as Top of ballot strength, while strong disapproval can be
interpreted as Bottom ballot strength.

For example1:

48 C
28 A>B
24 B

Top(C)*Bottom(A)=48*(48+24) , the defeat strength for C>A, is clearly the
strongest by this gauge.

And since C is uncovered, the chain is complete.

Example2:

x AB
y BC
z CA

Assuming max(x,y,z)<(x+y+z)/2, there is a beat cycle of ABC.

The respective strengths fo AB, BC, and CA according tour Top*Bottom gauge
are xz, yx, and zy, respectively. The largest of these will be the one with
the smallest missing factor in the product xyz.

So the defeat strength, in this context is proportional to the reciprocal
of the Top strength of the defeated candidate: A,B, or C wins depending on
which of B, C,or A, respectively has the fewest Top votes.

[Again, we used the fact that all candidates are uncovered ... which makes
the initial chain head the winner. This helps explain Krisyofer's original
observation that got this whole thing started. So you can see why I'm
tempted to call this the KKF method!]

What do you think?

-Forest
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