[EM] Two election method observations

[Kristofer Munsterhjelm]

On 2/21/23 02:24, Forest Simmons wrote:
> Cool ... I think you're onto something good!

Thanks :-)

Observations for three candidates: Let a candidate who is barred from 
being included in an n-1-seat council due to Droop criterion constraints 
be called "unfortunate".

For three candidates, the Droop quota is |V|/3. The possible exclusion 
scenarios are:
	If A has more than |V|/3 first preferences, A is not unfortunate. Ditto 
B and C.
	If there exists some solid coalition {A, B} supported by more than two 
Droop quotas, then C is unfortunate. In other words, if C's last 
preference count is greater than 2|V|/3, then C is unfortunate.
	If a solid coalition of two candidates is supported by more than one 
Droop quota, but not more than two, then we don't need to care about it 
because we can never eliminate both members of the solid coalition. Even 
in a case where we have a Droop quota voting B>A and another Droop quota 
voting C>A, we don't necessarily need to preserve A (although it may 
seem natural to do so).

Hence A is unfortunate if
	fpB > |V|/3 and fpC > |V|/3
or	lpA > 2|V|/3.

Note that an unfortunate candidate may not necessarily exist. E.g.
	34: A>B>C
	33: B>C>A
	32: C>A>B

So if we were to create a method that finds a loser candidate, every 
unfortunate candidate should lead to that unfortunate being kicked out, 
but it should also kick out a candidate that's not unfortunate in the 
cases when such doesn't exist.

Some people have suggested using IRV for this purpose, but the criteria 
for unfortunates show that it doesn't work. For instance:
	1: A>B>C
	3: C>B>A
B has no first preferences and is eliminated first, but A is unfortunate 
because more than two Droop quotas support {B, C}.

Two interesting questions I don't know the answer to:
	1. Is it possible to create a method that in the three-candidate case 
always evicts unfortunates, is decisive, and only uses pairwise information?
	2. For higher number of candidates, do we only have to look at "a 
single Droop quota supports this candidate" and "more than n-1 Droop 
quotas support everybody but this candidate" criteria? Or can size-k 
solid coalitions interact in complex ways to protect certain candidates 
from elimination?

-km