[EM] Deep Proportional Ordering

[Filip Ejlak]

I'm quite a fan of proportional ranking methods. Sure, they can't be fully
Droop proportional, but house monotonicity has its advantages, like giving
meaningful information on which candidate achieved which place.
Proportional Ordering method, proposed by James Green-Armytage in 2004, has
a nice, logical definition. However, I think there is some risk of the
ordering uncontrollably degenerating into more of a Condorcet ranking (for
example if the 1st place winner gets squeezed during the 2nd place contest,
etc.).
Therefore I want to propose a refinement of this method. It is something
that not only produces a list of winners, but also a list of sets of voters
that are represented by these winners - so another advantage is that when a
candidate has to be replaced, theoretically no by-election or recount is
needed if you have a Smith ranking for given voters set's preferences;
instead you take the next candidate from this ranking as the replacement.

Deep Proportional Ordering - definition:

At each kth step:
1. Perform a CPO-STV tally (k+1 candidates, k seats) considering only the
outcomes that include the previously chosen k-1 candidates.
2. Perform a Meek STV tally with the CPO-STV-winning set (k candidates, k
seats) where each previously chosen candidate must reach Droop quota
(exact, not rounded) and the new CPO-STV winner must reach the remaining
double Droop quota.
3. Take the set of voters that were apportioned to the double Droop quota
and apply a single-winner Smith method to it; add the winner to the ranking
at the kth place.

Repeat these steps until all candidates have been ranked.
If any Meek STV tally in Step 2 ended with some wasted votes, 2/(k+1) of
these votes should also be apportioned to the voters set of the new winner.

And now a *bonus feature*...
…you can make the ordering less proportional and more centrist-focused - to
the degree of your choice - by decreasing the quota for the candidates that
have already been chosen (i.e. the smaller quota in step 2 - so the big
quota gets even bigger). Setting the quota to 0% of the default value
changes the result into a Condorcet ranking (because the previous winners
don't affect new contests). 100% is of course a standard DPO, while
anything in between will be a mix of these two approaches, so with rather
low non-zero values I think it should be indeed suitable for returning a
set of "different shades of centrists". This bonus feature can be used with
the original Proportional Ordering method as well.
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