[EM] Sequential reweighting schemes

[Jameson Quinn]

One way to get proportional representation is to sequentially elect
candidates, then reweight ballots to "use them up" so that the majority
faction does not dominate every election. There are essentially two common
ways to do this reweighting. The two ways are equivalent in party list
systems, where they are called Largest Remainder and Highest Averages, but
they are different with systems that accept, directly or asset/delegated,
free rankings or ratings. I'm sure I'm not the first person to realize this,
so I wouldn't be surprised if there were already existing terms for these
two ways, but since I don't know them, I'll invent my own. Here they are:

   - Quota-based reweighting. This method uses the Droop or Hare quota Q,
   and reduces the votes of an elected candidate with V(C) votes by a factor of
   (V(C) - Q)/V(C). Essentially, it "uses up" exactly one quota worth of votes.
   This corresponds to the Largest Remainder method. It's implicitly used by
   STV, and explicitly by AT-TV.
   - Proportion-based reweighting. Under this method, a ballot's weight is
   always a simple formula based on the number of already-elected candidates
   from that ballot E(B). The possibilities are D'Hondt, which corresponds to
   the Droop quota, and Saint-Lague, which corresponds to the Hare quota.
   Proportion based reweighting is called the Highest Averages method when used
   in a party list context. This is used by RRV and PAV.

Markus Schulze, in his
papers<http://www.votingmatters.org.uk/ISSUE18/I18P2.PDF>on free
riding and Schulze STV<http://home.versanet.de/~chris1-schulze/schulze2.pdf>,
argues convincingly that free riding is the most important strategic concern
in a PR system, easily overwhelming single-winner-style strategies such as
burial. (Particularly, Hylland free riding is unavoidably a factor in any PR
system.) So how do quota- and proportion-based reweighting affect free
riding?

Consider two parties or factions A and B and the most-popular candidate of
each, Alex and Beth. Party A has several quotas worth of votes, party B has
scarcely more than one (for whichever quota or equivalent is being used).
Under proportion-based rewieghting, Alex's voters and Beth's voters alike
will lose half (D'Hondt) or two-thirds (Saint-Lague) of their power when
their candidate is elected. Under quota-based reweighting, voters for Alex
will lose only a fraction of their voting power for electing further
candidates, while voters for Beth will lose almost all of it. That is to
say, with quota-based, voters who are closer to being decisive in electing a
given candidate "pay more" for that privilege, while voters for an obvious
winner "pay" relatviely less. Again, in a party list system, this
distinction does not matter, because by the time the last candidate from a
given party list is elected, the different reweightings will have balanced
out. But in non-list systems, this matters, because people might prefer
candidates from several different parties.

Which is better? Proportion-based reweighting has a clear advantage in that
it gives an ordered list, and any size of representative body can be found
by simply taking the first N from that list. Quota-based, on the other hand,
will start by electing several candidates from the largest party; in a body
of 150 seats with a 100-seat party against a 50-seat one, the first 50 seats
will be from the first party and the final 100 will alternate between the
two.

But I would argue that quota-based has a less-exploitable free-rider
incentive. There is less of an incentive to avoid obvious winners, while the
incentives for voting for scant winners are mixed - a positive incentive
because you may be decisive, but a negative one because your vote will then
lose force.

Therefore, when an ordered list is not required, I would advise using a
quota-based reweighting. That's why I designed AT-TV with such a rule.

JQ
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