Feature Column

Risk Analysis and Romance

Happily ever after for Courtney Milan's math-major heroine Maria Camilla Lopez involves a master's degree focused on risk analysis. Let's explore real-world research in risk and management, from food bank strategies to the moons of Jupiter.

Ursula Whitcher
AMS | Mathematical Reviews, Ann Arbor, Michigan
Email Ursula Whitcher

February brings Valentine's Day, and with it an opportunity to play one of my favorite games: what research would this fictional character be working on? This time, our protagonist is Maria Camilla Lopez, the heroine of Courtney Milan's novel Hold Me.

Courtney Milan is a genuine polymath. She has bachelor's degrees in mathematics and chemistry, and a master's degree in physical chemistry. She then switched gears to earn a law degree, clerked for Supreme Court Justices Sandra Day O'Connor and Anthony Kennedy, and worked as a law professor before leaving academia for a full-time writing career.

Courtney Milan

Courtney Milan (Photo by Jovanka Novakovic)
 

The fictional Maria Lopez is just finishing her own bachelor's degree in math. Maria is a nontraditional student. She took time off between high school and college to work, saving money for hormones and gender affirmation surgery. To keep herself intellectually engaged, Maria started an anonymous blog about hypothetical disasters. She funnels her real anxiety and wide-ranging curiosity into mathematical models of subjects such as international cyberattacks and zombie plagues. As the book begins, she's running a Monte Carlo simulation of grocery supply chain failures during an apocalyptic pandemic. (Hold Me was published in 2016, but Maria's puzzles have all-too-enduring relevance!)

Maria is becoming closer and closer friends with one of the regular commenters on her blog, a man who goes by the handle ActualPhysicist. They share science jokes and pictures of their day. Maria even shares a photo of the gorgeous, bright red, hand-decorated high heels she's wearing as a sort of armor, to meet with an acquaintance who has been dismissive and rude. The only problem is, her acquaintance, Jay Thalang, is ActualPhysicist.

Hold Me is a romantic comedy, so eventually Maria and Jay work things out. This entails Jay admitting what a jerk he has been. His rudeness stemmed from a combination of stress, sexism, youthful trauma, and a form of loneliness many mathematicians will relate to—the loneliness of having your closest friends scattered all over the world.

Hold Me cover

The cover of Hold Me
 

I want to focus on the resolution of another of Maria's problems, the question of what to do after graduation. She applies to entry-level positions in actuarial science, but she wants to do something weirder and riskier: use her expertise in imaginary disasters to advise companies on preventing real ones. To do so, she needs credentials. She seeks them in a very specific place: Stanford's Management Science and Engineering (MS&E) department. Novels are full of fictional departments at fictional universities, but Management Science and Engineering is a real program. It focuses on mathematically informed approaches to solving business and policy problems, drawing on disciplines such as operations research, statistics, and computer science.

What kinds of projects would Maria find intriguing? Let's explore some of the real research at MS&E that could engage someone with a strong mathematical background and experience modeling a wide range of scenarios.

 

Elisabeth Paté-Cornell and the Europa Clipper

 

The Engineering Risk Research Group headed by Professor Elisabeth Paté-Cornell, the founding chair of MS&E, provides an obvious source of projects for Maria. Paté-Cornell, whose father was an officer in the French Marine Corps, was born in Dakar, Senegal in 1948. Growing up, she was interested in both mathematics and literature, but decided that a more technical career would offer her more job opportunities while still allowing her to indulge her literary interests. Thus, she majored in mathematics and physics at Aix-Marseille University, where she earned bachelor's degrees in mathematics and physics in 1968. Though her undergraduate program was highly theoretical, Paté-Cornell knew she wanted to attack more applied problems. She did a master's degree in the exciting new field of computer science at the Institute Polytechnique in Grenoble. Based on advice from one of her professors there, she came to Stanford for a second master's degree in operations research. She combined all of this experience for her PhD from Stanford's Engineering-Economic Systems department, where she worked on risk analysis and models of earthquakes. Over the course of her career, Paté-Cornell has pursued research on a huge variety of topics, including space shuttle heat shielding and the risk of nuclear war. She has analyzed lessons from disasters such as the failure of the Fukushima Daichi nuclear plant and the Deep Water Horizon oil spill, studied hospital trauma centers, and considered terrorism risks.

Elisabeth Paté-Cornell

Elisabeth Paté-Cornell (Professional photo used under CC-BY-SA 4.0)
 

Recently, Paté-Cornell mentored Stanford mechanical engineering PhD student Yiqing Ding and four MS&E master's students, Sean Duggan, Matthew Ferranti (now an economics PhD student at Harvard), Michael Jagadpramana, and Rushal Rege, in a study of radiation risk in outer space. Their subject was NASA's Europa Clipper spacecraft, which is due to launch toward Jupiter's moon Europa in 2024. Europa is covered in smooth water ice, streaked with lines or cracks. Scientists hypothesize that a moon-wide liquid ocean layer lies between the ice and Europa's rocky core. Learning about Europa's structure is made more difficult because the moon orbits within a belt of radiation trapped by Jupiter's magnetic field. Radiation is a danger to both spacecraft and the scientific instruments they carry. To manage the radiation risk, instead of orbiting Europa itself, the Clipper will enter an elliptical orbit around Jupiter. Each time the Clipper flies by Europa it will pass by at a different angle, slowly building a detailed picture of the moon's surface.

Europa Clipper flybys

Schematic illustration shows Europa Clipper flybys
 

Ding, Paté-Cornell, and their group write that past quantitative analyses of radiation risk in space exploration have focused on possible radiation exposure to individual astronauts. The radiation risk to the Europa Clipper is different because of its cumulative exposure over multiple flybys, and because the difficulties of exploration near Jupiter limit our information about how intense those exposures might be. Furthermore, different parts of the spacecraft and its payload may have different radiation tolerances.

Instead of assuming a constant radiation dose on each flyby, the MS&E group built a probabilistic model that allowed for radiation to be higher or lower, according to a log-normal distribution. In other words, the logarithm of the radiation dose was normally distributed; in one example they considered, the most likely radiation dose in a single twelve-hour flyby was just under 2000 rad, but the potential dose was far higher. (For comparison, doctors treating cancer might target a tumor with 2000 rad over the course of five days.) After constructing their model, the group ran simulations, modeling approximately 1000 missions with about 70 flybys in each mission. The extra flybys allowed them to see how long it might take for multiple instruments to fail. Their model showed that multiple instruments were likely to fail in quick succession as radiation accumulated.

log normal distributions

Log normal distribution curves for different parameters
 

Of course, radiation is only one form of risk to the Europa Clipper mission. Paté-Cornell has written elsewhere about the importance of incorporating multiple types of error, including human error, in complete risk analyses. Systematic attempts to measure risk encourage us to contemplate dangers we might otherwise ignore. In an essay entitled "Improving Risk Management: From Lame Excuses to Principled Practice," Paté-Cornell and Louis Anthony Cox Jr. (University of Colorado) write:

Deliberate exercises in applying "prospective hindsight" (i.e., assuming that a failure will occur in the future, and envisioning scenarios of how it could happen) and probabilistic analysis using systems analysis, event trees, fault trees, and simulation can be used to overcome common psychological biases that profoundly limit our foresight. These include anchoring and availability biases, confirmation bias, group-think, status quo bias, or endowment effects.

 

Volunteers and Food Systems

 

Maria hopes that graduate school will help her connect with businesses and community groups that are trying to make better choices. She would find new opportunities to do so by collaborating with Professor Irene Lo, who researches ways to use operations research for social good. Lo majored in math at Princeton University and received a PhD from Columbia's Industrial Engineering and Operations Research department in 2018. She has studied school choice algorithms and problems in graph theory.

Irene Lo

Irene Lo (Professional photo used by permission)
 

Recently, Lo put her expertise in matching to the test in a collaboration with Food Rescue U.S. (FRUS), a nonprofit that connects businesses that have extra food with food banks that need it. Coordinating food pickup is a hard problem. Food Rescue U.S. uses an app to connect volunteers who want to help with donor businesses that have food ready to share. Lo, Yale School of Management professor Vahideh Manshadi, and the PhD students Scott Rodilitz (Yale) and Ali Shameli (Stanford) teamed up with Food Rescue U.S. to look for ways to maximize volunteer engagement. Volunteers are more likely to keep contributing to an organization when it's easy for them to find ways to participate. One strategy Food Rescue U.S. uses to keep volunteers involved is "adoption": a volunteer can promise to visit a particular site at the same time every week. Adoption makes food delivery more predictable for both volunteers and businesses. But if too many sites are adopted, volunteers logging into the app for the first time won't have anything to do. This conundrum illustrates an economic concept called market thickness: buyers and sellers (or, here, volunteers and donors) can only accomplish their goals when sufficient numbers of people participate in the process.

Lo, Manshadi, Rodilitz, and Shameli built a mathematical model to study matching between volunteers and donor sites. Choose a scaling parameter $n$ that controls the overall size of the market, and suppose there are $na$ donor sites and $nb$ volunteers. Suppose the probability that a volunteer likes an available donor site is $c/n$, where $c$ is another fixed parameter (so matching is easier when $c$ is large, and tougher when $c$ is small). When the first volunteer arrives, the probability that none of the sites works for them is $(1-\frac{c}{n})^{na}$. Thus, the probability that the volunteer can find a good match is $1-(1-\frac{c}{n})^{na}$. If they are successful, the number of available sites drops by 1. Write $M$ for the total number of matches after all volunteers have arrived. Lo, Manshadi, Rodilitz, and Shameli showed that as the scaling factor $n$ grows large, $M/n$ converges (almost surely) to $a + b - \frac{1}{c} \log(e^{ca}+e^{cb}-1)$.

Using this mathematical model, Lo and her collaborators then considered multiple rounds of volunteer and donor matching, and explored how removing some donor sites due to adoption would change the overall matching process. They identified two simple and appealing optimal strategies, depending on market characteristics: either all of the donor sites should be adopted, or none of them should be removed from the pool. More complicated efforts at optimization did not increase the number of overall matches. They point out that this theoretical prediction matches real-world observations about the differences between volunteer pools in different places:

Our interviews with site directors reveal that there are inherent differences between the volunteer pools in different locations. For example, some FRUS sites are in college towns, and thus, the volunteer base consists of many engaged students who are more likely to be attentive to last-minute needs. In other sites, however, a majority of volunteers are professionals who may not be as flexible in their level of engagement.

Nonprofits could use this insight to find new, subtle ways to encourage their volunteers to keep coming back. For example, instead of showing the same "adopt" button to everyone logging into the app, Food Rescue U.S. could encourage adoption in big cities and discourage it in college towns. Our heroine Maria Lopez, who knows a lot about building online communities, might have other ideas to test!

 

Further reading

 

Ursula Whitcher
AMS | Mathematical Reviews, Ann Arbor, Michigan
Email Ursula Whitcher


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