[EM] Idea for generalizing Resistant set into hopefully always strategyfree system

Tue May 12 12:33:11 PDT 2026

So I did some more work trying to work out specific criteria compliance,
and it still looks hopefull so far.

Assuming I got things right, the proposed signle winner system satisfies
the following criteria, with a brief proof sketch provided as well.

Monotonicity

Increasing the rank of a candidate in the T set cannot make them fail the
qualification in any sub-elections they previously qualified,
so increaasing a candidate cannot remove that candidate from the T set.
It also cannot lower the probability of being elected in the tiebreakr during
the Random Ballot part, nor afterward as well.

Decreasing the rank of a candidate outside the T set cannot make them
succed the qualification in any sub-election they previously failed,
so decreasing a candidate cannot add that candidate to the T set.
It also cannot increase the probability of being elected in the tiebreakr during
the Random Ballot part, nor afterward as well.

LN-Help+Harm

Strictly speaking LN-Help and LN-Harm are dependent on the inference rules
for infering complete rankorder from incomplete ballots,
which I left unspecified, so they are technically not applicable.
I also assumed a default rankorder of all candidates ranked equal
in case a ballot is considered spoiled or not provided,
which means criteria compliance also depends on the assumptions
on what make a ballot spoiled or not.
In practice however, they are satisfied for at least some inference rules.

For a given candidate, whether they statisfies the qualification in any sub-election
does not depend in the relative order of strictly lower ranked candidates,
thus the rankings of strictly lower ranked candidates can be changed
freely on any ballot as long as they all remain strictly lower ranked to
the candidate in question.
This is sufficient for LN-Help+Harm compliance for at lest some inference rules,
assuming LN-Help and LN-Harm generalize to assume the provided ballots
are not spoiled.
Example of such a inference rule is all unlisted/unranked are considered equally
strictly lower ranked the all listed/ranked candidates.

Mutual Majority

So from the above we would obviously think this one autofails becuase
of the impossibility theorem the proves the impossibility of a signle
system satisfying Mutual Majority + Monotonicity + LN-Help + LN-Harm,
but it turns out the proof at
http://www.votingmatters.org.uk/ISSUE6/P4.HTM
(Reference [3] on https://electionwiki.org/wiki/Monotonicity)
does not apply to us (at least as far as I can tell).
My attempt at creating counterexamples to diesprove satisfaction
of this critera have also failed so far,
though that propably does not count for much since I am still not that good at it.

Clone Independence (for perfect clone)

In the case the set T contains only a single member and that meber is replaced
by a set of perfect clones, thoose clone will all be members of T since
each of them will qualify the corresponding sub-elections as the original candidate.

In the case of T containing multiple members and one of thoose members are
replaced by a set of perfect clones, thoose clones will all be members of T in
addition to the other candidates since each clone will qualifiy the corresponding
sub-elections as the original candidates.
When we get to Random Ballot tiebreaking, the probability of electing from the
clone set will then be the same as the probability of the previously replaced candiadate.
We will then with certainty get a tiebreaker between all clones in the clone set,
or we would assuming the original candidate did not already have perfect clones
participating in the election, but that one exception does not affect the satisfaction
of the criteria.
Similarly if all perfect clones are replaced by a singular candidate,
that candidate will have the same probability of being elected as that of electing
from the previous clone set by the same argument as above.

AFB
I am still working on this one, no result either way so far.

Participation

We autosucced on just about every form of participation there is since
you can always get an at least as good outcome by participating
since the outcome of not participating is the same as one of the
possible rankorders to provide with the ballots.

But in case the spirit of the Participation criteria is supposed to be
when the total rankorder increases and/or decreases,
then only the goalposts have been moved to whether a persones
becomes an eligible voter, or stops being an eligible voter.
Most versions of the criteria (at least of thoose I have seen) don't
deal with the concept of a default rankorder,
so we need to generalize the criteria to apply to the case here.

One of the more common versions (as far as I have seen) is satisfied for
what I think is a reasonable generalization.
Becomming both an eligible voter and providing a ballot, no spoiling,
where ones strict first/top preference candidate is ranked strictly
first/topped cannot make that candidate loose nor decrease their
probability of winning since this cannot kick them out of the T set,
nor reduce their proability of winning during Random Ballot tiebreaking.
The multiple topped ranked candidates case is somewhat messy,
but I think the same strategy can be used to prove that as well.

I am also quite sure that the mentioned generalization to multiwinner
proportional representation is flawed at this point,
it was a nice and neatly fitting generalizatoin but I have lost hopes for it,
though no proof either way.

Hopefully of some interest
Gustav

On Sun, 10 May 2026 16:56:18 +0200
Gustav Thorzen via Election-Methods <election-methods at lists.electorama.com> wrote:

> So I made some progress on understanding the Resistant set better,
> and realized it is possible to create a system neatly interpolating
> between the signle member strategyfree scenario
> and the Random Ballot method,
> while also looking like it satisfies Mutual Majority and Clone Independence,
> both perfect clones and clone coalitions.
> 
> 
> But first some definitions:
> 
> N = Total number of voters.
> M = Total number of candidates.
> m = Number of candidates in the sub-election, when talking about one.
> T = The set of candidates to tiebreak with Random Ballots.
> t = Number of members of T participating in the sub-election, when talking about one.
> 
> Voters have true preferences of complete rankorders, ties allowed.
> Every voter is a priori assumed to have the complete rankorder of every candidate tied,
> and the ballot format and counting and inference rules allows every voter to change
> that to any other complete rankorder of their choice,
> and should the ballot be spoiled, e.g. not participating or an invalid ballot is provided,
> then their initially assumed rankorder remains unchanged by the system.
> This means the number of rankorders are always fixed equal to N.
> I will use "lor" as a shorthand for the logical-or, and "xor" for exclusive-or.
> 
> For a given rankorder, a candidates rank is defined as
> the number of candidates of the same rank lor higher,
> including the candidate itself.
> 
> For some examples with M=7 and the rankorder
> a=b=c>d>f=g>h
> a, b, and c, are each ranked both 3:rd
> d is ranked 4:th
> f, and g, are each ranked 6:th
> h is ranked 7:th
> 
> In a subelection between b, c, and d,
> m = 3
> b, and c, are each ranked both 2:nd
> d is ranked 3:rd
> 
> In a subelection between d, and h,
> m=2
> d is ranked both 1:st
> h is ranked 2:nd
> 
> 
> To the method itself.
> 
> For a positive integer starting at 1 and up to M, in that order,
> check if it possible to create a set T with that many members,
> where each member candidates individually satisfies the following condidtion:
> In every sub-election where that member candidate participates,
> they are ranked t:th lor higher on some number of rankorders
> strictly greater then the number N/(m+1-t) (no rounding here).
> 
> The first time we can create such a set T,
> we send the members of for Random Ballot tiebreaking,
> with all candidates not in T marked as disqualified.
> Should the Random Ballot tiebreaking fail to disqualify
> all but 1 of the members of T (as if, but technically possible),
> we tiebreak the remaining members with by picking
> one winner at random with each remaining candidate
> assigned an equal probability of being picked (I think this is called Sortition).
> 
> 
> So about its properties.
> 
> First lets go over its 1 big problem (well 2 if it is not strategyfree),
> the amounts of computaions scales at an absurdly fast pace
> as M increases unless someone can identify a more efficient way.
> The naive method seems to take something around O(N*M^M),
> which is way to large for practical purposes.
> Even with computers this is beyond silly,
> only ever feasable for relatively small M.
> 
> I also came up with a similar version which I only conjecture
> to have full/complete/strict honesty as a strong nash equilibrium
> regardless of true preferences (assuming this one is strategyfree)
> but even that one is beyond impractical (N*2^M at least, likely much much worse).
> 
> Generalizing both to multiwinner scenarios also came naturally,
> for the system here, you basically replace the N/(m+1-t) with
> N/(m+k-t) where k is the number of winners and use a the
> k winners PR Random Ballot version, but I have nothing idicating
> it would keep any honesty incentives,
> and is so much worse to count I could not get myself to bother trying to
> figure out the time and effort it takes to calculate.
> 
> Then there is the question if this proposed system actually is strategyfree.
> It seems like it could be, and it also looks to satisfy
> Mutual Majority and Clone Independences,
> but I can only say for certain with M=1 or M=2,
> and "looks like it is true" for M=3 is nowhere near "always true".
> A number of theorems claim only Random Ballots is strategyfree,
> but the ones I found include a bunch of other critera
> beyond voter and candidate/outcome symmetry,
> so it looks like they would be subverted here.
> 
> 
> All in all, I figured it might be worthwhile if anyone finds it interesting.
> I am almost finished with my attempt at a good+feasable fully deterministic system
> (since no randomness limits to how good they can be),
> and this proposed system here was strikingly similar to how it
> would generalize if made to include randomness.
> 
> Gustav
> 
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