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

[Kristofer Munsterhjelm]

On 13/01/2019 04.44, VoteFair wrote:
> Here's a suggestion for an easy-to-understand vote-counting method that
> produces very fair results for single-seat elections:
> 
> Voters rank the candidates using up to 7 ranking levels (or 5 ranking
> levels if ovals are marked on a paper ballot and space is limited).
> During counting, each elimination round eliminates the candidate who
> loses every pairwise contest against every other remaining candidate. If
> an elimination round has no pairwise-losing candidate then, for that
> round, each ballot gives one count to the lowest-ranked remaining
> candidate on that ballot, and the candidate with the highest such count
> is eliminated.  The last remaining candidate wins.

This seems very close to Smith//Coombs, but it's not exactly the same
thing. I think it may fail Smith, even.

Suppose that we have a nested inequality situation with an ABCA cycle
and candidates DEF are all beaten by {ABCA} but have a DEFD cycle among
themselves. Then the Smith set is {ABCA}, and the Antiplurality
elimination stage triggers because nobody is beaten by everybody else.
The Antiplurality elimination stage might then eliminate one of ABCA,
and in the worst case, might sequentially eliminate all of them, making
the method fail Smith.

It's probably not monotone either, as very few candidate elimination
methods are, and probably fails independence of clones since both
Antiplurality and Coombs do.

> Unless someone recognizes it as having a different name, I suggest
> calling it Instant Pairwise Elimination (IPE).  The word "instant"
> indicates that this method is similar to instant-runoff voting (IRV) in
> the sense of "instantly" doing multiple elimination rounds.  The word
> "pairwise" makes it clear that the eliminations use pairwise counting,
> rather than the less-fair counting method used in IRV.
> 
> This is a hybrid of Condorcet-loser elimination and the Coombs method.
> For fun, someone on Reddit (u/jpfed) suggested calling it "Coombsdorcet".
> 
> This is not a Condorcet-compliant method!  A test has already found a
> case where the IPE winner is not the Condorcet winner.  Also note that
> the description does not introduce the word Condorcet, even though it
> eliminates Condorcet losers.

Oh, oops. You already figured out what I wrote above :-)

Both Nanson and Baldwin (Borda-elimination methods) are Condorcet
without having an explicit Condorcet step or using a Condorcet matrix.
So are methods of this type:

Arrange the candidates in a line according to some initial ordering.
Start with the last candidate in line, and compare him to the
second-to-last. Eliminate the one of the pair who is beaten one-on-one
by the other. If they tie each other, use whatever tiebreaker you want.
Keep on going up the line until only one candidate is left; he wins.

(Those methods are Smith. They're also summable as long as the methods
used for tiebreaks and initial ordering are.)

> I suggested it on Reddit in the r/EndFPTP subreddit because the
> single-seat voting methods being discussed there most often are
> approval, score, STAR, and IRV, which are easier to understand than
> Condorcet methods, but they have one or both of these disadvantages:
> 
> * Quite vulnerable to tactical/strategic voting
> 
> * Do not work in situations that involve general/runoff elections
> 
> The IPE method seems to be easier to explain to typical
> (non-math-oriented) voters than any of the Condorcet methods (including
> my favorite, the Condorcet-Kemeny method), yet it comes close to
> providing the fairness of Condorcet methods.
> 
> The inspiration for this method is the relative success of the STAR
> method, which is a hybrid of score and runoff.
> 
> Based on the surprisingly favorable response on Reddit, apparently most
> voters are more trusting of a method that eliminates one candidate at a
> time in a way they understand.  This contrasts with Condorcet methods,
> which can identify the winner just by "looking at" a table of pairwise
> counts.
> 
> Also, explicitly identifying "losers," and identifying them one at a
> time, seems to be emotionally appealing to voters.  Perhaps this is part
> of why IRV is seen as appealing.

Do you think the line method above would be received favorably? It is
very simple but has a less explicit loser elimination component.

> Yes, this method is vulnerable to the burial tactic, at least from the
> voter's perspective.  Yet when voting methods finally get measured for
> HOW OFTEN each method fails each fairness criterion, I suspect that this
> burial-criterion failure will not affect the results often enough to be
> significant, especially compared to IRV’s frequent fairness-criteria
> failures.
> 
> In fact, the method may appeal to some voters because it will give them
> the emotional satisfaction of burying "enemy" politicians.

Every voting method permits an angry voter to decide "I'm going to
punish X" and change his vote from A>X>B>C to A>B>C>X. Pervasive burial
susceptibility is more problematic when it creates a feeling that "I
need to put the frontrunner I like the least at the very bottom or he'll
win"; and then everybody does that, and a nobody wins.

I generally think Condorcet methods are (on the whole) robust enough to
disincentivize such massive burial sprees, but I haven't done any
analysis on IPE in particular.

> Please correct me if I'm wrong, but I believe that -- unlike IRV -- use
> of the IPE method would enable polling places to start by sending their
> pairwise counts to the central counting location, and the winner can be
> identified quickly in most(?) cases.  Of course some elections
> (especially if they are highly competitive) will require more ballot
> data to be sent to the central counting location before a winner can be
> calculated.  In elections that involve lots of candidates, the original
> pairwise counts might clarify the elimination sequence for the
> less-popular candidates, which would reduce the amount of ballot data
> that needs to be sent quickly to determine the winner.  By contrast, IRV
> needs almost all the raw ballot data, and the full ranking data with
> lots of candidates does not lend itself to being compressed or summarized.

If there's a clear Condorcet order, then you're right. But as soon as
the Antiplurality elimination step comes into play, the districts will
have to send their ballot data over to the center (or do two-way
communication like in IRV). The reason is that who is next to be
eliminated depends on which candidates have been eliminated so far in a
way that can't be handled by a summed array -- just like IRV.