[EM] Instant Pairwise Elimination (IPE) vote-counting method

[Kristofer Munsterhjelm]

On 14/01/2019 01.55, VoteFair wrote:
> Based on Kristofer's feedback (below), here are the criteria that I
> suspect the Instant Pairwise Elimination (IPE) method fails:
> 
> Majority: fail
> Majority loser: fail

If there's a majority loser X, for every other candidate Y, a majority
of the voters prefer Y to X. So X is thus a Condorcet loser; see below
for that criterion.

Majority failure: Consider

10: A > D > B > C
10: A > C > D > B
11: A > B > C > D
10: B > C > D > A
10: C > D > B > A
10: D > B > C > A

There's a Condorcet cycle between {B, C, D}, and A is the Antiplurality
loser. So IPE will eliminate A first, but A is ranked first by 50.8% of
the voters. Hence IPE definitely fails majority.

> Mutual majority: fail

If a method fails majority, it also fails mutual majority.

A nice thing about methods that eliminate one candidate at a time until
some winner remains, is that if they pass a criterion for a single
candidate, they also pass it for a set. That means that such an
elimination method passes mutual majority iff it passes majority, and
passes Smith iff it passes Condorcet.

> Condorcet: fail

If a method fails majority, it also fails Condorcet.

> Smith/ISDA: fail

... and if a method fails Condorcet, it also fails Smith/ISDA.

> IIA: fail

Every deterministic ranked method fails IIA.

> Consistency: fail

Only positional voting systems (x points for first, y points for second,
etc) pass consistency. IPE is not such a system, so it automatically fails.

What I mean by consistency is here the same-winning-candidate version,
not the same-social-ordering version that Kemeny passes. If you mean
that one, I'd agree with you: it probably fails it, but I can't think of
a proof.

(The same-social-ordering one is sometimes called reinforcement to
distinguish it from consistency; yet in other articles, consistency and
reinforcement are taken to mean the same thing, and which one it is
depends on whether the voting method is defined to output a ranking or a
winner.)

> I'm guessing that IPE passes this criterion:
> 
> Condorcet loser: pass ?

Suppose there's a Condorcet loser. By definition, he loses every
pairwise contest to every other candidate. Thus he'll be eliminated
right away by IPE, and hence IPE passes Condorcet loser.

> And here's my guess about summability:
> 
> Summable: O(N!) ?
> 
> On the surface, these characteristics do not look promising ...

A possible patch: instead of eliminating by Antiplurality, eliminate by
how many candidates beat the candidate in question. That is, eliminate
the candidate who is beaten by more of the others than any other. This
naturally eliminates the Condorcet loser (as in your current version)
without any special case logic. Then break further ties by (e.g.)
Antiplurality.

That should salvage Condorcet (and thus Smith) and Majority (and thus
Mutual Majority). It would probably still fail monotonicity and clone
independence, and it'll stay unsummable for most tiebreaks.

> ... yet I'll say once again that what really matters is HOW OFTEN a
> method complies with the criteria. When we have that data we can more
> meaningfully compare different voting methods -- beyond just looking at
> checklists.
> 
> In fact, this topic, which I would call "Compliance Frequency," is being
> discussed on Reddit, with my suggestions at the following link:
> 
> https://www.reddit.com/r/EndFPTP/comments/afc9mm/quantifying_criteria_failure_rates/edy8gyh

As I've said before, one problem with compliance frequency is that you
don't know the distribution to sample from. The distribution probably
depends on the method in use. E.g. Plurality encourages two-party rule,
so under Plurality, the voters will tend to vote consistent with
two-party rule.

In a way, this is where IRV got it wrong. One way of looking at IRV's
logic is like this: "under Plurality, sometimes fringe candidates make
the wrong major candidate win. So let's patch up that problem by
eliminating fringe candidates before deciding on the winner". That is
all well and fine ... *as long* as voters keep voting for two major
parties. Under the voting distribution induced by Plurality, IRV is very
good!

But if the voters, emboldened by that their votes now matter, change how
they vote in such a way that multiple strong candidates emerge, then IRV
can get confused and you get a Burlington.

You could measure compliance frequency by assuming that every candidate
permutation is equally likely - the so-called impartial culture. But the
same people who would say "but my method X only violates
monotonicity/ISDA/whatever once in a million elections, so that
violation shouldn't count" can then say "but the voters don't vote
according to impartial culture, and in the real world,
monotonicity/ISDA/whatever will only be violated once in a million
elections".

I suppose the only real way to circumvent the problem is to determine
the frequency according to many different distributions. If the method
consistently does badly, then it is bad. If the method consistently does
well, then it is good. If it's a mixed bag, then the advocates can argue
all night about which distributions matter and which don't.

(Another problem is the social rank vs winner distinction. That
distinction very rarely matters when you're only looking for a single
failure, but the frequency of social rank failure may differ from the
frequency of winner-becomes-loser.)

> I had intended to post these compliance-frequency suggestions here on
> this mailing list, but I seldom have time to write
> election-method-reform messages while at my PC, whereas I can type
> Reddit responses on my iPad while eating a meal.
> 
> Again, thank you Kristofer for your very useful feedback! (And no I had
> not already figured out the details that you clarified.)
> 
> Again, thanks to all for considering this suggested method. I'm assuming
> from the lack of criticism that the method is not as seriously flawed as
> IRV, which means it deserves attention as an alternative when a city
> considers adopting something better than plurality voting.

Well, for all its faults, IRV passes majority and is (strictly speaking)
independent of clones, whereas IPE is neither. The lack of harsh words
might be more attributable to that IPE doesn't have "the momentum"
(visible targets get more criticism), and there's no Rob Richie to
attract posters' ire, for IPE.

> My next question is whether anyone wants to put the IPE method on the
> Electrowiki (or Electorama) site? (I tried using my login info from long
> ago, but with no success.) I can supply edited text if that's helpful.
> In advance, thanks!

The Electorama wiki is moving to https://electowiki.org/. You should be
able to register a user there.