[EM] Resistant Set Questions

[Gustav Thorzen]

So I got qurious about the Resistant set and tried to read up on it,
but the electowiki page and its references left some thing unclear,
and other things I was wondering about left out.
I am assuming complete rankorders are provided by and/or infered from the ballots,
using some not-insane inference rule (like every unmentioned candidates ranked equally last).
I also assume that outside of question (0),
we have a total of m candidates, n voters, and k candidates in each sub-election (k<=m).

0) In each sub-election, does the definition of the Resistant set have
k as the total number of candidates (no?)
or as number of candidates in that specific sub-election (yes?)?

1) The set appears to be defined only for strict rankorder ballots,
does the following generalization make sense?
If we assign each candidate a rank equal to
the number of candidate ranked the same or higher (self included),
and and apply the definition,
do we still get its desireable properties.

And as a clarifying example,
if a=b>c>d=e=f>g then
a and b are considered 2:nd ranked,
c 3:d ranked,
d and e and f 6:th ranked,
g 7:th ranked,
and if a>b=c then
a is 1:st ranked,
b and c 3:rd ranked,

2) Regardless if we stick to the original or the generalization,
if a candidate a satisfy the following in all their sub-elections,
v(a>{rest of the k candidates}) > n/k,
then is a the only member of the Resistant set?
And if yes, is it known whether that is a sufficient and/or a neccecary condition?

3) This one is probably wrong if (2) is wrong, but in any case,
If I create the smalest non-empty set of all candidates a satisfying
v(a>{rest of the k candidates}) > n/k for each subelection
where a is the only set member,
then is this new set the same as the Resistant set when
strict rankorder ballots are used?

4) If we create a new set as the intersection (set theory)
of the Resistant set and the MB-Smith set (that is candidates member of both),
does NewSet//M satisfy Majority criterion and fail Monotonicity,
since Resistant//M fails Monotonicity if M satisfies Majority.
Here MB-Smith means the smalest non-empty set where each member
Majority-Beats each non-member, unlike the regular Smith set,
where each member simply Plurality-Beats non-member.
MB-Smith is therefore a superset of the regular Smith, or PB-Smith, set.

Gustav