[EM] CMU's "cake cutting" solution to gerrymandering

[Rob Lanphier]

 Hi folks,

I really appreciate the quick analysis I can count on from this
mailing list, (particularly robert and Kristofer this time). Thanks!

I agree with robert that the gerrymandering issue is ripe for a
solution. Proportional representation is getting some airplay in the
wonkosphere[1], but it still doesn't quite seem like even many of the
voters that would bristle at being called "anti-intellectual" are
willing to listen to us math nerds.

[1]: https://www.vox.com/policy-and-politics/2017/10/11/16453512/gerrymandering-proportional-representation

The cake-cutting solution intuitively seems wonderful. It definitely
implies two sides, but an Approval style variant occurs to me:

1. A list of generated maps (as described by robert) is put in front
of a proportionally-selected commission, which then uses approval
voting for the maps
2. If a single map is approved unanimously, it wins. If multiple maps
are approved unanimously, then ...I don't know....that'd be a fun
exercise. We'll just assume that won't happen for purposes of this
email. If none is approved unanimously, then we keep going:
3. The map with the highest approval rating is selected first, with
the commission divided between those that approved of the map (A), and
those that didn't (B)
4. Group B gets to decide which district to freeze out of the map, and
optionally adjusts the remaining districts
5. Continue using the process described in the original paper.

The interesting thing about this variant is that there's the
additional cake-cutting conundrum of choosing to approve of a
particular map. One complexity comes if the commission has (say) 10
people, and 9 approve of one particular map. That means 9 people are
in group A, and one person is in group B. Does that give too much
power to that one person, where people would try to game their way
into being in group B?  Also, assuming that each group does
majoritarian Robert's Rules of Order style decision making, what would
be the size of an effective super-majority that could ensure that one
party is able to dominate both group A and group B?

It seems to me that the group A/group B divide would cause committee
members to be stingier with their approval then they otherwise might.
The collusion to optimize the map and formation of the A/B teams could
become similar to a game of Diplomacy[2] and could have ugly ways of
gaming the system.  Still, intuitively to me, it seems like the rules
I describe seem to have the same cake-cutting fairness properties
described in the original paper.

[2]: https://en.wikipedia.org/wiki/Diplomacy_(game)

Kristofer: your suggestion of redrawing the districts *after* the
election (thus simulating a multiwinner election) is interesting.
It's a good thought experiment.  What would the ballot look like for
the typical voter, though?

Rob