
An impossibility theorem for gerrymandering - mathfan
https://dustingmixon.wordpress.com/2017/10/11/an-impossibility-theorem-for-gerrymandering/
======
indubitable
I think this indirectly hints at something much more fundamental: district
based plurality does not make sense as an election system anymore.

I think an eye opening example is to consider a hypothetical single state with
10 representatives. This state has a population, perfectly well distributed,
of 35% democrat, 30% republican, 20% libertarian, and 15% green. And we draw
our districts up with absolutely no bias whatsoever. What 'should' the
election results be? I think any logical person would say about 3.5 democrats,
3 republicans, 2 libertarians, and about 1.5 greens. What we get in reality?
10 democrats as they win a plurality in each and every district. 0
gerrymandering, 35% of the support, 100% of the seats.

To get our above example to have the logical results we want, you end up
having to create really absurd districts that would completely dwarf any sort
of gerrymandering of today. So then the question is quite obvious. Why don't
we do at large proportional elections instead of district based plurality
elections? If a state has 10 representatives then any party is guaranteed at
least 1 seat if they get 10% support, 2 for 20%, and so on.

In the past when politicians and the people were much closer, I think
districts made a lot more sense. You were voting for somebody who you knew or
were at worst split by the most minimal degrees of separation. But today this
isn't true. The average congressional district is now up to 710,767 people [1]
(as of 2010). In 1790, at the time of the first US census, the average
district size was 37,082. Times have changed, but our electoral system has
not.

[1] -
[https://en.wikipedia.org/wiki/List_of_United_States_congress...](https://en.wikipedia.org/wiki/List_of_United_States_congressional_districts)

~~~
andrewla
What I don't like about this analysis is that it enshrines the notion of
political parties. It makes individuals into automata that simply vote for
whatever candidate is thrown at them.

The ideal is that candidates can rest on their own merits, with the party
mechanic existing to provide support to candidates who nominally align with
the party's interests. People vote for candidates, not parties.

Even the reality is that people are not determined by their party affiliation;
New York City, for example, has repeatedly elected Republican mayors despite
being overwhelmingly Democratic in state and national elections. This has even
been used in a derogatory sense, as "Rockefeller Republicans" or "New York
Republicans"; people affiliated with the national party but who skew to the
preferences of voters in their region.

In your hypothetical state, once you created the districts, then the
competition inside that district is between candidates who try to align
themselves with that region; people who, for example, can be a "green-
republican" to draw voters from both blocks, or just a person who speaks to
the needs of voters in that district, regardless of party affiliation.

Using some sort of state-wide rank-preferential or approval system means that
we remove a lot of the strategizing on the part of voters, which is a
positive, but the strategizing on the part of political parties and
individuals seeking office become much more complex, and not in a way that
serves to present candidates that appeal to issues confronted by groups of
people regardless of their party affiliation. Geography is just a heuristic
for this, but not the worst one you could think of.

~~~
indubitable
I think something you're fundamentally getting at is that in today's system
people's views are not necessarily representative of their party's views. Ron
Paul is not really what you'd call a republican, but he certainly ran as one.
The reason is that in our current system, small parties cannot exist - or at
least they can't get any representation in congress, but is there really any
difference? So if you want to actually get a seat you put an R or a D by your
name, or you run in a state like Maine or Vermont - both with total
populations far smaller than many cities.

The reason for this is because of our electoral system. Imagine we have 4
parties each with about a quarter of the support. One of those parties is
going to get 100% of the representation, and the rest are going to get 0. This
incentivizes these parties to begin to merge. Two parties go 'Hey I know we
have nothing in common, but if we work together we can guarantee our voices
are at least heard in congress.' And the other two also see this going on and
does the exact same thing. Next thing you know it you have things like
libertarians and evangelical Christians both being supposedly represented by
the same party, and you get a congressional approval rating in the teens.

Proportional representation changes this. Fractional support being more than
sufficient to get seats in congress means people can break into parties that
actually fundamentally represent their views - instead of being forced to
clump up into super-parties. I think this effect would be particularly
emphasized in our country. We have a phenomenally good system of checks and
balances ensuring that even 'the little guy' in congress can have a meaningful
effect. The problem is our electoral system all but precludes there being any
little guys in congress.

~~~
andrewla
Proportional representation sort-of fixes this. I think rank-preferential
(instant run-off) or approval voting also solves this, even on single
candidate elections, because they remove the spoiler effect. I think switching
to these methods but keeping districting is a good intermediate step that
doesn't "completely change the game" but opens some new doors.

Proportional representation, as I mention above, does remove most incentives
to vote other than your actual preferences, but it does introduce very complex
strategies on the part of candidates and parties. Every election becomes a
state-wide election, so smaller candidates representing regional interests
will find it harder to target their campaign. I'm not sure whether this is a
net positive or negative.

~~~
indubitable
Smaller candidates today already stand stand practically 0 chance of getting
elected. Literally every single member in the house of representatives is
either a democrat or a republican. That is a consequence of a simple question:
'If a person receives 10% of the vote in a state with 10 representatives, how
many seats should they receive?' The current answer is 0.

In our current system the one and only power of smaller candidates is a
spoiler effect. This enables them to have some influence on the super-party
that's closest to their own ideology, even if implicitly. Rank preferential /
Australian voting / etc not only fail to change the 10% support = 0%
representation issue, but they even remove the spoiler effect which means that
smaller parties can safely be completely ignored unless they look to gain a
plurality themselves - which is rather contrary to the notion of smaller.

On this note, I'm not seeing why you think that proportional would make
campaigning more difficult for 'smaller candidates.' It would mean they might
actually stand a chance of finally getting a seat in congress. A basic example
from Europe would be something like 'The Pirate Party.' They're never likely
to gain substantial support, but they have been able to receive representation
at the national level in a variety of nations, proportional to their support,
exclusively due to proportional systems.

~~~
KGIII
Sanders and King are both Independents but caucused with the Democrats.

This doesn't really negate your point, I just figured you may be interested.
They may not literally all be a D or an R, but it's trivial and I'm mostly
being pedantic.

~~~
Armisael16
If we're being pedantic, both Sanders and King are senators.

~~~
KGIII
Right, my bad. I read that as congress. I'm nit sure why I missed the
specificity, considering I read it twice and then looked up the information on
Wikipedia.

My bad, indeed.

------
catpolice
At this point I feel like it's worthwhile to remember to why we even have
districts.

We could, in principle, get rid of the districts and just have some kind of
statewide scheme for electing representatives proportional to the outcome of a
popular vote. The idea behind retaining districts, as far as I understand it,
is that people in similar locales will often have very correlated interests
(e.g. with regard to decisions that affect that area) and that choosing
representatives from particular locales is supposed to guarantee that people
in that area are represented (and make the representatives to some degree
beholden to them).

Ideally, we'd split the states into districts in a way that doesn't deviate
from overall proportional representation very much, hence the idea of the
efficiency gap. So gerrymandering can break that efficiency constraint.

But the notion of geographic correlation of interests makes it such that the
population is unlikely to be distributed in the kind of uniformly random way
you need to get weirdly shaped, efficient districts, and conversely having
weirdly shaped districts at all effectively assumes that geographic
correlation of interests isn't an important factor.

In other words: if we have to make weirdly shaped districts to maintain
efficiency, then the assumptions that caused us to require districts in the
first place probably fail and we should just ditch them.

~~~
sova
Ideally: the Representative who is chosen for a district (the weird shapes we
are talking about) _represents_ everyone of the populous of that district.
That way in Congress, when someone asks "hey do your constituents want a
factory next to that highway plus a bunch of taxbreaks?" the representative
knows which way to vote, as s/he _represents_ the local people. The weird
shapes, even if they increase "efficiency" don't actually account for locality
much. It'd be better to use something like commuting maps to draw regions to
ensure locality.

~~~
cr0sh
Not only that, usually the "representative" doesn't even live in or near the
area they are representing (sure, they may have an address in the area, but
that isn't difficult to do).

------
weeksie
Totally. This is why in the more recent gerrymandering thread I was making
such a big deal about the efficiency gap rather than geometry for detecting
this stuff. Geometry is just a flawed heuristic for a multitude of reasons.

That's why the efficiency gap is so exciting as a measure.

~~~
sova
Looking at the paper, counting the number of "wasted votes" is actually the
most solid metric it seems!

------
tveita
It's strange that there's so much focus on district shapes, when any voting
system with proportional representation would make gerrymandering pretty much
ineffective.

Every voting system has its flaws, but the US system seems to have
particularly few advantages.

~~~
waegawegawe
It's not really "strange." The chances of us ever going to a proportional
system en masse are near zero. The chances of making extreme gerrymandering
illegal are significantly better (the Supreme Court recently heard a case
about it).

Many people prefer to spend time thinking within the context of what is likely
to happen. That's not to say we don't need folks who dream big, but we need
more than _just_ those folks.

~~~
waqf
I agree with the substance of what you say but I don't think it's wrong to
invite everyone to have a general sense of the big picture.

And by big-picture standards, because of the way the US is constituted, the
questions that make their way to SCOTUS are usually pretty strange.

------
awjr
I suspect Gerrymandering isn't uniquely a US problem. I do a lot of UK Census
spatial analysis and the district centroids that are used create problems. In
the article the 'ear muffs' shape would create a centroid for the voting
district outside of the district, given that the shape of the district was
designed to give a certain ethnic group a voice, whereas the process for
creating districts in the UK are done usually on the idea of 'belonging' to a
town/village/area and are decided and audited at a national 'neutral' level.

The data scientist in me wants centroids around circles with fuzzy edges. If
we could all agree that fuzzy circles is the way forward we could solve
gerrymandering over night.

The reality is though, that I create these amazing pieces of analysis then
have to explain away why that district is odd, y'know, cos the shape is just
weird.

Locally they are redoing the district boundaries which is giving me an
opportunity to submit my own district shapes that work better with Census data
:D

~~~
sova
Fuzzy circles! Great idea. That, or repeated boundary drawing that brute
forced every possible district combination and gave a probabilistic result of
elections based on where districts were drawn.

------
sova
For reference, there is a NY Times article about the plague of redistricting
and also a simple excel tool that someone made to make this very easy:
[https://www.nytimes.com/2017/08/29/magazine/the-new-front-
in...](https://www.nytimes.com/2017/08/29/magazine/the-new-front-in-the-
gerrymandering-wars-democracy-vs-math.html)

As a human who takes pride and joy in the Constitution and its tenets I find
the crowbar-able gap space of redistricting an important valley of electoral
attention that needs ample light of attention.

------
rndmize
> Now suppose you are tasked with drawing 5 districts for this region.

Or, we could go with proportional representation, make it a single district,
and now we can reasonably expect to get 2-3 blue reps and 2-3 red ones,
handling both the districting problem and efficiency/fairness problem in one
go.

~~~
shadofx
How would that deal with popular independents? Would they get two seats if
they pull 40% of the electorate?

~~~
erentz
Under an STV you're still voting for people. The independent with 40% of the
vote would be elected and then a proportion of their unneeded votes would be
transferred to the subsequent candidates on those voting papers.

[https://en.wikipedia.org/wiki/Single_transferable_vote#Findi...](https://en.wikipedia.org/wiki/Single_transferable_vote#Finding_the_winners)

------
importantbrian
I think the real answer to all of this is to restore a reasonable ratio of
population to representatives. It is much more difficult to Gerrymander 133
districts than with 9. It will still happen but the possible advantage one
party can get is much smaller.

The constituents also get much better representation, because they are much
closer to the person representing them. The representative is intimately
familiar with their constituents concerns because they live in and around
them.

It also strips a lot of the money out of politics. Representatives don't have
to spend all their time fundraising to raise war chests of millions of
dollars. You can win an election in a 30,000 person district by knocking on
doors and sending out inexpensive mailers.

[http://www.thirty-thousand.org/](http://www.thirty-thousand.org/)

------
oh_sigh
> Our paper proves that in _some cases_ , it’s impossible to get a small
> efficiency gap without drawing bizarrely shaped districts.

Italics mine. Is there any reason to believe that any real life situation
would fall under their impossibility? The key is to get a system which is good
enough to prevent gerrymandering for the world that exists, not for any
possible world.

------
dragontamer
So the question is if "Minimizing Efficiency Gap" is what we should be aiming
for.

Its a solid proposal, but I guess its up to the courts to decide if
"Efficiency Gap" is a good metric. I guess I'll be interested in hearing the
arguments as they come up.

~~~
cousin_it
It seems to me that a system with zero efficiency gap would be equivalent to
one big election with proportional representation. If that's what people
really want, why talk about voting districts at all?

~~~
djrogers
You're either thinking about this too broadly, or as a system filled with
static entities. Within voting districts, politics is local - individuals can
sway voters, voters can decide to show up/skip voting, etc.

Also, the concept of a 'blue/red' district can be very blurry - there are many
examples of 'blue' districts (districts with democrat congress reps, and a
history of voting blue) that voted red in the last presidential election, and
vice-versa for previous elections.

Historically, Americans prefer to vote for people, not parties.

~~~
rjsw
Do people in the US have the same feeling for what area is 'local' as they did
when districts were drawn up ?

------
js2
I'm confused about the example.

First of all, it seems highly unlikely such a distribution of red and blue
votes would occur. But more importantly, blue is a majority of every 3x3
square - why shouldn't it win every district?

~~~
dragontamer
> why shouldn't it win every district?

Because we are supposed to have a representative government. If there's 55%
Blue voters and 45% Red Voters, you're supposed to ideally have 55% Blue
representatives and 45% Red representatives.

The example is a counterexample of this concept. Instead of 55% Blue / 45% Red
split, the map as a whole becomes 100% blue. This is almost the very
definition of "Tyranny of the Majority".

~~~
Retric
If votes are 80-20 then it's not necessarily possible or reasonable to draw 80
vs 20 winning districts.

~~~
dragontamer
Which states are 80-20 in America?

Maryland is an example of a highly gerrymandered state (towards Democrats,
although there are many Republican examples too). Maryland voted 60.5% Clinton
/ 35.3% Trump, which suggests that they have (roughly) 2 Democrats for every 1
Republican in the state.

However, due to the significant gerrymandering in Maryland, there is ONE
Republican and SEVEN Democrats in the House. That's certainly not fair towards
Republicans who live in Maryland.

EDIT: Apparently North Carolina is the Republican example to talk about, in
case you want some "balance".

~~~
Retric
There are actually more extreme examples. Consider, Hillary Clinton got 90.9%
of the DC vote Trump got 4.1%, I am not sure how you could slice up DC to get
10% Republican representation.

In Maryland's case it might be able to make 2 districts Republican, but that's
assuming a very concentrated geographic minority which may or may not actually
exist.

IMO, If we really want proportional representation then we should use a
proportional system.

~~~
dragontamer
DC is a single city and doesn't even have one representative in Congress
(Okay, they nominally have one. But she's non-voting so.... that's not really
useful). They're technically a US Territory and have no more power than say
Puerto Rico.

So once again, if you're talking about US Representatives and Voting
Districts, it only makes sense to talk about well... Voting Districts. Name me
one US State with 80/20 split and more than 1 US Representative. Ultimately,
your hypothetical doesn't exist!

\---------

That's the cool thing about politics: its real. There's no need to make up
hypotheticals when we can just draw from the real world.

FYI: the discussion at hand is about US Representatives, and not about the
Electoral College. Washington DC is therefore irrelevant, as it has no
representatives. Washington DC does get 3 Electoral college votes, but that
has nothing to do with Gerrymandering.

There's a variety of states who have "Representatives At Large", such as
Wyoming, which are basically immune to Gerrymandering. The population is so
low that Wyoming only gets 1-Representative, so the entire State is the whole
district. There's really no "fair" way to cut up a single Representative (its
all or nothing), but that's mostly due to the very low population of these
states.

Basically, Gerrymandering can only be an issue in a state with more than 1
Representative.

~~~
Retric
These rules are not limited to National elections. State elections are also
based on redistricting. DC is part of the US and has District Elections so
these rules will apply to it.

[https://www.dcboe.org/election_info/election_results/v3/2016...](https://www.dcboe.org/election_info/election_results/v3/2016/November-8-General-
Election)

So, no you can't dodge the question. And I ask you to draw a DC map with some
Republican representation.

PS: AKA draw a map such that WARD EIGHT MEMBER OF THE COUNCIL is likely
Republican.

~~~
dragontamer
I'm not sure why you're so obsessed with Washington DC in particular. If you
live there, you probably know much more about it than I do.

But my general interest is how this whole event will play out in the greater
scope of the country. I don't believe any US State with more than 1
Representative has an 80/20 split.

Perhaps Washington DC really is an edge case that needs to be thought out
more. But I don't think its representative of the problems of Gerrymandering
that exist in multiple states right now. (In particular, what this Gill v
Whitford Supreme Court case is bringing up in Wisconsin)

~~~
Retric
You can find plenty of examples on both sides South Carolina sends 7
republicans an 1 Democrat to the house, but it's own biased house is 75
Republican to 45 Democrat. Which is a very safe majority of power, while still
better representing the actual voting. It also contains wacky maps like this
one:
[https://www.govtrack.us/congress/members/SC/2](https://www.govtrack.us/congress/members/SC/2)

Still, my concern is not about the results. My concern is how easy it is to
game this proposal.

~~~
dragontamer
> Still, my concern is not about the results. My concern is how easy it is to
> game this proposal.

Approximately the same difficulty to game the US Court system.

At the end of the day, Judges will adapt the rules to future cases. Law is an
innately human process, not the cold machine logic that most of us Computer
Science folks deal with.

If a major mistake is made in the current rubric, future lawyers and judges
will bring it up on a case-by-case basis. Unlike code, Law is pliable and
changes on local conditions. The issue isn't the "edge cases" which are dealt
with as they come up.

If the proposal isn't working for say, Washington DC, then you can expect a
case to bubble up through the Washington DC Courts, to maybe the US Court of
Appeals, and eventually back to the Supreme Court if its major enough. If its
purely a Washington DC issue (and not applicable anywhere else), there might
not be any need for the US Court of appeals or US Supreme Court to hear about
the case!

------
dqpb
Gerrymandering is so ridiculous. Clearly geographic coordinates is the wrong
data structure for systematically biasing votes for/against certain
demographics.

A voting system should be explicit about exactly which demographic criteria
should carry weighted bias, what value that weight should be, and what it's
purpose is.

It should also be clear to each voter which demographic categories they fall
under, what the final strength of their vote will be, and what the
justification is for that outcome.

------
crb002
There is also gerrymandering against independents and third parties.

A fair metric minimizes average distance from a citizen to the centroid of
their district. The lowest energy Voronoi diagram wins. Nobody gets screwed.

------
sova
The Supreme Court (id est The Highest Court of the Land) is currently
evaluating if they can rely on an algorithm to detect Constitutionality of
district shapes. You're kidding, right? As a computational scientist this is
concerning to put it mildly.

Trying to ensure proportionality in representation... is that the intended
goal? Is this even a meaningful pursuit in a bicameral system? The Supreme
Court makes rulings that span time infinitely forward, meaning that all those
different parties that have existed since the inception of the Union have
dissolved and new ones have been formed since, but the rulings of the Supreme
Court have not this luxury of death and recycling. Nay, a ruling by the court
is very potent and quite a footprint on the legal plain.

So there is no easy way to ensure that a district is Constitutionally
Compliant, but we can definitely point out "unconstitutional districting" \--
right? Do you agree?

~~~
cmac2992
From my understanding, the Baker v Carr in 1962 decision enabled federal
courts to intervene in and to decide redistricting.

Back in the day redistricting was even crazier. You could just draw an
arbitrary number of districts with arbitrary population sizes. That seems bad
on the surface. With this system, election outcomes could potentially be
_entirely_ determined by how districts were gerrymandered.

The efficiency standard seems like trick to get closer to proportional
representation. All of this comes down to the 14th amendment being "one person
one vote" rather than something like "all votes must have equal weight". A
small language that has made things really complicated.

IMO just switching to proportional representation is a much cleaner solution.
But the efficiency standard seems like a decent solution to keeping races
competitive and from being severely gerrymandered.

~~~
sova
Thanks for your words and insight. Yes, proportional voting would be great!
Your state gets 5 house seats? Let's have priority-voting where a voter picks
choices 1-5 for representation. Jane is first choice, then Miyoko, Sofia,
Leben, and Pierre. Jane may not get all the "first place" votes but she could
get enough to end up in one of the five house seats our state gets.

Reading up on the 14th amendment, it does ensure that every male over 21 is
able to vote (assuming they did not lose the right through crime, also a big
issue today as a third of our populous is disenfranchised). It does not,
however, say anything about how voting for reps is done at the state level,
and I wonder if this becomes a States' matter as soon as we part from the 14th
amendment. Pardon my gap-riddled knowledge of such important affairs

~~~
KGIII
Most States don't take away your right to vote if you are convicted of a
felony. Felons get to vote and can even, more often than not, vote while
incarcerated. To vote while incarcerated, they use absentee ballots from the
district where they resided prior to detention.

~~~
waqf
Only two states allow felons to vote while incarcerated:
[http://www.ncsl.org/research/elections-and-
campaigns/felon-v...](http://www.ncsl.org/research/elections-and-
campaigns/felon-voting-rights.aspx)

~~~
KGIII
D'oh. I had looked up, previously, to find out that felons can usually vote
and extrapolated Maine to be the same as the rest, with regards to voting
while incarcerated. My bad, the rest is correct however.

I know Maine allows it, and even encourages it. I actually started a campaign
for State Senate at one point. As I got closer to election, I learned more
about the system and decided I couldn't participate in that. So, I never
turned in all the signatures and didn't run.

But, thanks for the correction. For some reason, I'd assumed the rest of the
States allowed it. I understand that, in Maine, some candidates have even
campaigned, in person, in the prisons. Notably, we don't actually have a lot
of prisoners or very dangerous prisons.

