[EM] Ranked Pairs Feint to Max Gradient Chain Building

[Forest Simmons]

So far our best defeat strength gauge is the product of winning approval
and losing disapproval.

This still leaves open how to define approval and disapproval in this
context.
Up until now I've been using Equal Top for approval and equal Bottom for
disapproval.

Another possibility is implicit approval and disapproval.

Let's try something more symmetrical between Top and Bottom... a compromise
between those two previous ideas:

For use in defeat strength calculations where approval and disapproval
scores are needed, (for now) we will try ...

For each candidate X let the approval factor a(X) be the number of ballots
on which X is ranked Top (or equalTop) plus half the number of ballots
where X is ranked strictly be tween Top and Bottom ... that is half the
number of ballots on which X is ranked ahead of at least one candidate,
while at least one other candidate is ranked ahead of X.

Also let the disapproval d(X) be given by the difference between the number
of ballots and a(X).

Example:

48 C
28 A>B
24 B

The respective approval scores for A, B and C are 28, 38, and 48.

The corresponding disapprovals are
72, 62, and 52.

The defeat strengths for A>B, B>C, and C>A are ...
28*62, 38*52, and 48*72, respectively.

The strongest defeat is C>A.  And since C is uncovered, no other defeat can
depose C from the head of the chain.

Example2

8 A>B (sincere was A>C)
6 B>C
4 C (sincere was C>A)

The sincere ballots yield C as the only uncovered candidate, hence the
winner.

The truncation in the C faction was a preemptive defense against the burial
of C in the A faction ... let's see if it is protective:

The respective approval/ disapproval pairs are (8, 10), (14, 4), and (10,
8).

The respective defeat strengths are ...
8*4, 14*8, and 10*10 ... not quite enough to retain the win for C, but
enough to punish A by electing A's last choice.

I hope you all will experiment with this streamlined method to get a feel
for it.

So far it seems pretty robust to me.

Any bad results?

It's almost simple enough to replace Gross Loser Elimination as the most
viable RCV election method.

What do you think?

Do I need to go over again the simplicity and relevance of "Covering" ...
both in concept and calculation?

-Forest


On Thu, Jan 26, 2023, 12:53 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> 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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