[EM] coherent, house monotonic, droop proportional

[Ross Hyman]

Thanks Gustav,

In the paragraph before Claim 4 I wrote:

"For the proofs of claims 4-9, which constitute a proof of Droop
proportionality, consideration is restricted to fully ranked ballots,
and for simplicity, to ballot sets for which there are no ties for
elections or exclusions."

I should have moved the part about "no ties' to apply to all the
claims.Thanks for pointing that out.

I am asking if my proof of Droop proportionality is correct for ballot
sets for which there are no ties.

Best,
Ross

On Fri, Apr 24, 2026 at 3:49 PM Gustav Thorzen via Election-Methods
<election-methods at lists.electorama.com> wrote:
>
> On Tue, 14 Apr 2026 08:14:43 -0500
> Ross Hyman via Election-Methods <election-methods at lists.electorama.com> wrote:
>
> > Dear all,
> > Back in June I posted a preprint on arxiv describing an STV-like
> > method that satisfied Droop proportionality like conventional STV,
> > and, unlike conventional STV, was also coherent and house monotonic.
> > I never submitted it to publication because my gut was telling me that
> > there was an error. Well, it took a while, but I found the error and I
> > have now revised the paper: https://arxiv.org/abs/2506.12318
> > Unfortunately, the new method is cumbersome, but I believe the new
> > procedure corresponds to the proof. I would greatly appreciate it if
> > anyone can look it over to see if it is correct.
> > Best,
> > Ross
> > ----
>
> So I took a look at this (arxiv version 3 to be exact) and tried to
> work my way through it.
> I think I finally have a good enough grasp to check some of the
> claims and proofs in it.
>
> But first some definitions I want to make sure I understood correctly:
> Coherence - Same as https://en.wikipedia.org/wiki/Coherence_(fairness)?
> (also using the Oklahoma example.)
> Droop proportionality - Generalization of Mutual Majority criterion for more
> then 1 winner systems?
> Quota-Based Phragmen - Refereing to the systems in the
> (Salmi, Olli 2002) and (Woodall, Douglas R. 2003) references?
>
> and since you never provide any sort of (explicit) name to any of the methods,
> the one you propose is the one describes halfway through section 5 Top-down Phragmen?
>
> Assuming all the above are answered yes, my results of checking your claims are as follows:
> (1) Trivially true from your definition, though it feels like the definition could be simplified.
> (2) I found no errors here, but I am not confident I got all possible edge cases covered.
> I leave this to someone else to check.
> (3) No errors in the proof, but I will go into detail about what is proven later on.
> (4) Trivially false unless you use "exceeds" as a shorthand for
> "greater then or equal to" rather then "strictly greater then",
> which would be a first for me.
> An example would an exactly equal number of rankorders ranking Alice > Bob
> as thoose ranking Bob > Alice.
> We get a perfect tie where the winner is determined by random tiebreaking and whoever
> of the two gets lucky will be a winner with votes exactly equal to Q_1, not exceeding it.
> You have not stated this is supposed to break the norm and be a fully deterministic system
> without any form of randomness no matter what, so random tiebreaking is implicitly assumed.
> (5) False for pretty much the same reason as (4) is false.
> (6) I found no errors here, but I am not confident I got all possible edge cases covered.
> I leave this to someone else to check.
> (7) False for pretty much the same reason as (4) is false.
> (8) False for pretty much the same reason as (4) is false.
> (9) I found no errors here, but I am not confident I got all possible edge cases covered.
> I leave this to someone else to check.
>
> When it comes to claim (3) about LN-Help+Harm,
> we sort of run into some problems about have ballots are counted.
> You mention in passing the proofs for the claims all assumes complete rankorder,
> implicitly also that they are to be strict,
> as well as reducing to regular IRV for the single winner case,
> which also typically assumes strict rankorders.
> But voter symmetry + candidate symmetry and complete strict rankorders
> forces the inference rule for incomplete ballots to cause spoilage
> whenever two or more candidates are left out on the provided ballot.
> This causes LN-Help+Harm failures in edge cases not covered in your proof,
> for example (signle winner 4 candidates here for simplicity):
> 10 A>B>C>D
> 09 B>A>C>D
> 01 B>A (spoiled forcefully by voter+candidate symmetry+complete rankorder, or otherwise)
> results in A winning, but if the incomplete ballot is extended to add later preferences
> of C and D (B>A>C>D or B>A>D>C, does not matter), the ballot is not spoiled
> leaving a tiebreaker between A and B each with 50% chance by voter+candidate symmetry.
> This is a LN-Help failure for B, and a LN-Harm failure for A,
> even if we use inference for only 1 missing candidate cases (which is still possible).
>
> What I normally see is for IRV to simply allow incomplete preference orders,
> which it looks like you could do to solve the above problem,
> but then things get extremely sensitive to what part of the ballots are keept.
> For example, how would A>{everyone else}>B be counted?
> Doing it in any way other then simply turning it into A top and nothing else
> leads to LN-Help/Harm failures, but it has other possibly unwanted
> strategic side effects.
>
> There are basiclly two versions of LN-Help/Harm each,
> one about what happens when we add our preferences to incomplete ballots,
> and one about when we reorder strictly lower preferences.
> Your proof looks like it covers the later version for each,
> but depending on how ballots are counted,
> the first version is failed for complete preference orders
> (I will not assume you are willing ditch voter of candidate symmetry
> without explicitly stating such.)
>
> Normally I deal with this by proving both cases,
> or refere to whatever version others use,
> or avoid any claims that depends on the difference.
>
>
> Hopefully this is still worth something,
> even if a bit late.
> Gustav
>
> ----
> Election-Methods mailing list - see https://electorama.com/em for list info