Monthly Archives: September 2018

Jonathan Mattingly’s work on Gerrymandering

Probability

My last two posts were about a hurricane and a colonscopy, so I thought it was time to write about some math again.

For the last five years Mattingly has worked on a problem with important political ramifications: what would a typical set of congressional districts (say the 13 districts in North Carolina) look like if they were chosen at “random” subject to the restrictions that they contain a roughly equal number of voters, are connected, and minimize the splitting of counties. The motivation for this question can be explained by looking at the current congressional districts in North Carolina. The tiny purple snake is district 12. It begins in Charlotte goes up I40 to Greensboro and then wiggles around to contain other nearby cities producing a district with a large percentage of Democrats.

To explain the key idea of gerrymandering, suppose, to keep the arithmetic simple, that a state has 2000 Democrats and 2000 Republicans. If there are four districts and we divide voters

District           Republicans       Democrats

1                           600                            400

2                           600                            400

3                           600                            400

4                           200                            800

then Republicans will win in 3 districts out of 4. The last solution extends easily to create 12 districts where the Republicans win 9. With a little more imagination and the help of a computer one can produce the outcome of the 2016 election in North Carolina election in which 10 Republicans and 3 Democrats were elected, despite the fact that the split between the parties is roughly 50-50.

The districts in the North Carolina map look odd, and the 7th district in Pennsylvania (named Goofy kicks Donald Duck) look ridiculous, but this is not proof of malice.

Mattingly with a group of postdocs, graduate students, and undergraduates has developed a statistical approach to this subject. To explain this we will consider a simple problem that can be analyzed using material taught in a basic probability or statistics class. A company has a machine that produces cans of tomatoes. On the average the can contains a pound of tomatoes (16 ounces), but the machine is not very precise, so the weight has a standard deviation (A statistical measure of the “typical deviation” from the mean) of 0.2 ounces. If we assume the weight of tomatoes follows the normal distribution then 68% of the time the weight will be between 15.8 and 16.2 ounces. To see if the machine is working properly an employee samples 16 cans and finds an average weight of 15.7 pounds.

To see if something is wrong we ask the question: if the machine was working properly then what is the probability that the average weight would be 15.7 pounds or less. The standard deviation of one observation is 0.2 but the standard deviation of the average of 16 observations is 0.2/(16)1/2  = 0.005. The observed average is 0.3 below the mean or 6 standard deviations. Consulting a table of the normal distribution or using a calculator we see that if the machine was working properly then the probability of an average of 15.7 or less would occur with probability less than 1/10,000.

To approach the gerrymandering, we ask a similar question: if the districts were drawn without looking at party affiliation what is the probability that we would have 3 or fewer Democrats elected? This is a more complicated problem since one must generate a random sample from the collection of districts with the desired properties. To do this Mattingly’s team has developed methods to explore the space of possibilities and then making successive small changes in the maps. Using this approach one has make a large number of changes before you have a map that is `independent.” In a typical analysis they generate 24,000 maps. They found that using the randomly generated maps and retallying the votes, ≤3 Democrats were elected in fewer than 1% of the scenarios. The next graphic shows results for the 2012, 2016 maps and one drawn by judges.

Mattingly has also done analyses of congressional districts in Wisconsin and Pennsylvania, and has helped lawyers prepare briefs for cases challenging voting. His research has been cited in many decisions including the three judge panel who ruled in August 2018 that the NC congressional district were unconstitutional. For more details see the Quantifying Gerrymandering blog

https://sites.duke.edu/quantifyinggerrymandering/author/jonmduke-edu/

Articles about Mattingly’s work have appeared in

(June 26, 2018) Proceedings of the National Academy of Science 115 (2018), 6515–6517

(January 17, 2018)  Nature 553 (2018), 250

(October 6, 2017) New York Times  https://www.nytimes.com/2017/10/06/opinion/sunday/computers-gerrymandering-wisconsin.html

The last article is a good (or perhaps I should bad) example of what can happen when your work is written about in popular press. The article, written by Jorden Ellenberg is, to stay within the confines of polite conversation, simply awful. Here I will confine my attention to its two major sins.

  1. Ellenberg refers several times to the Duke team but never mentions them by name. I guess our not-so-humble narrator does not want to share the spotlight with the people who did the hard work. The three people who wrote the paper are Jonathan Mattingly, professor and chair of the department, Greg Herschlag, a postdoc, and Robert Ravier, one of our better grad students. The paper went from nothing to fully written in two weeks in order to get ready for the court case. However, thanks to a number of late nights they were able to present clear evidence of gerrymandering. It seems to me that they deserve to be mentioned in the article, and it should have mentioned that the paper was available on the arXiv, so people could see for themselves.
  2. The last sentence of the article says “There will be many cases, maybe most of them, where it’s impossible, no matter how much math you do, to tell the difference between innocuous decision making and a scheme – like Wisconsin’s – designed to protect one party from voters who might prefer the other.” OMG. With many anti-gerrymandering lawsuits being pursued across the country, why would a “prominent” mathematician write that in most cases math cannot be used to detect gerrymandering?

Two faces of Hurricane Florence

Reflections

The title is supposed to evoke images of a woman who is a soccer mom by day and serial killing hooker by night. The coastal part of the state of North Carolina saw the second side of Florence. In Durham, which is 150 miles from Wilmington, we mostly  her softer side.

In the days leading up to Sunday September 9, the storm was hyped by the weather channel and the local news stations. Florence started as category 1 in the eastern half of the Atlantic Ocean but it was predicted that as she reached the warmer weather in the Atlantic, she would grow to category 4 and smash into coast somewhere in the Carolinas (and she did). The message of a coming disaster was reinforced by our neighborhood listserv. People who lived through Hurricane Fran in 1996, saw many trees go down, and lived a week or more without power, and they were not anxious for a repeat performance. Emails were filled with long lists of things we should do in order to prepare.

Monday September 10. At my house we had our semi-annual inspection of the heating/cooling system. It is hard to get work done while the technician is in the house, so I googled “How do you prepare for a hurricane?” The answers were similar to what I had seen on the listserv, with one new funny one: remove all of the coconuts from your trees, since they can become cannonballs in a storm. When the inspection was finished about 11:30 (with no repairs needed!), I went to the grocery store with my little list: water and nonperishable foods. The store was a zoo and water was flying off the shelves, but I came home with 48 quart bottles of Dasanti, canned fruit, canned soup, bread, peanut butter, energy bars, etc.

Tuesday September 11  was the 17th anniversary of a day that changed the world. Donald and Melania visited the new memorial to the plane that went down in Pennsylvania. I came down with a cold and stayed home from school. By this time, there was more weather than news on news. I recall hearing one weatherman pontificate “at this point we are as good with predictions 72 hours out as we used to be at 24 hours.” Then he showed a track that had the eye of a category 1 storm over Durham about five days later. Tuesday afternoon Duke announced that all classes were cancelled as of 5PM Wednesday. , with no classes on Thursday and Friday. UNC and NCState also closed and in a sign of looming disaster cancelled their football games. For UNC this was a blessing. Having lost to Cal and to the East Carolina in the two previous weeks, they were happy to escape from another loss at the hands of U of Central Florida.

Wednesday September 12. The morning weather forecast announced a dramatic change in the track of the storm. It was now predicted to turn left after land fall, hang out at the beach for a couple of days and then head off to Atlanta. The new storm track was great news for Durham but not for Myrtle Beach South Carolina which was given a short deadline to evacuate. Wednesday was more or less a normal day. I met with a graduate student at 11, had lunch, taught my class, and announced the shifted schedule for the homework and exams due to the cancellation of class on Friday.

Thursday September 13 was the calm before the storm.  My wife and I took a walk around the neighborhood in the morning, went out to a Mexcian Lunch at La Hacienda at the northern edge of Chapel Hill, and grilled some chicken for dinner. (In NC barbecuing refers to cooking a large piece of meat over a slow fire until you can pull it apart with your hands. Grilling gets the job done in 15-20 minutes.) Feeling more confident about the future we scaled back from cooking four chicken breasts to have leftovers that could be eaten over the next few days, to making only two.

Friday September 14. Florence made land fall at Wilmington as a category 2 hurricane, which is definitely more than half as strong as a category 4. It was amusing to see several weathermen competing to see who could be filmed in the eye of the hurricane, where the winds suddenly drop to 0. A woman on the neighborhood listserv described it as an eerie experience. Most of the other things that happened along the coast were not at all funny. New Bern was about 30 miles from the coast, but it was on the banks of the Neuse River. Rains plus hurricane winds resulted in flooding that left hundreds of people (who stupidly chose not to evacuate) in need of rescue. Up in Durham things were much more sedate. There was very little wind or rain. However since the weatherman had told us to shelter in place, we stayed in for most of the day, as did most of our neighbors.

Saturday September 15 was more or less a repeat of Friday. Twisting a line from Big Bang in which Penny is talking about here relationship with Leonard. “This is a new boring kind of hurricane.”   I don’t know what normal people do during a hurricane, but it is a great chance to get some work done. On Friday I read a couple of papers from the arXiv and thought up a new problem for one of my grad students to work on. Saturday I decided I would use the lull to finally finish up the 5th edition of Probability; Theory & Examples, so I sifted through emails I had saved from readers and corrected some typos. Not everything I do is math. I watched the third round of the Evian Championship, the fifth major of the LPGA season. In more keeping with more manly pursuits I watched parts of Duke’s 40-27 victory over Baylor in Waco, Texas. The win was remarkable because Duke’s starting quarterback, a junior who might make it to the pros, was out. In addition, I watched Texas beat the USC Trojans, 37-14. It’s not just that my son is a CS professor in Austin. Having been at UCLA for nine years, when Peter Carroll was there, I love to see USC lose.

Sunday September 16. Duke’s severe weather policy (which covers not only the university but also the hospitals) ended at 7AM, so we figured that we had reached the end of the hurricane. We got up and went to the Hope Valley Diner for breakfast at about 7:30. When we first started going there it was called Rick’s. However, the owner got tired of having a restaurant named after her ex-husband. In keeping with the intellectual climate of Durham, our usual waitress is a second year medical student at UNC Greensboro. Weather was light rain as it had been for the last couple of days, so we went to the Southpoint Mall to get out of the house. Being in a Christian region almost all the stores don’t open until noon. At the mall Susan bought some Crocs, and I bought a couple of books that I read before going to sleep. However, mostly we walked around like many of the families who came there with their small kids.

Monday September 17. Our hopes of nicer weather were quickly dashed. It was raining very hard when we woke up. Curiously the direction of the flow was now from SW to NE in contrast with the last few days of SE to NW. Almost immediately there was a tornado alert on TV accompanied with its loud obnoxious noise, and robophone call to tell us of the event, which came a few minutes after channel 14 told us that the warning had expired.  The tornado warning came from an area well north of us, so we weren’t really worried when soon there was a second one even further north. This was too much excitement for Duke, so they cancelled classes until noon, irritating people who traveled through awful weather to get to their 8:30AM classes. Soon after the despair of facing another day of rain set it, the sun came out and Susan and I took a walk.

Tuesday and Beyond. Unfortunately the end of the rain does not bring the end of the misery for people near the coast, as we learned with Hurricane Matthew. Many of the rivers there have large basins (of attraction). Many will only crest Wednesday or Thursday. The Cape Fear River will reach 60+ feet compared to its usual 20. But don’t worry. Trump will be coming soon to inspect the damage. When he came he clumsily read from a prepared statement, that soon we will be getting lots and lots of money. I guess his advisers didn’t tell him that the state is so heavily gerrymandered that he will probably see 10 Republican congressmen elected from 13 districts.