[EM] Ranked Pairs Feint to Max Gradient Chain Building

[Forest Simmons]

Correction on second example below ...

On Sun, Jan 29, 2023, 10:00 AM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> 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).
>

B approval should be 8/2+6=10, not 14. So the B disapproval is 8.

C's approval should be 6/2+4=7

The corrected pairs are (8,10),(10,8),& (7,11)


> The respective defeat strengths are ...
> 8*4, 14*8, and 10*10
>

Corrected to 8*8, 10*11, and 7*10 ... so the strongest defeat is B>C with
strength 110 ... which does punish A but does not restore the win to C, the
sincere CW.

... 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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