A team of three William & Mary undergrads hit the back of the net this spring, scoring top honors in an international mathematics competition with their analysis of soccer team strategies.
Team members are Jack Morris ’20, Liz Weech ’20 and Ethan Shelburne ’21. Their submission, “Rock, Paper, Scissors, Shoot!,” was a Moneyball-type analysis of a fictional soccer team called the Huskies. It was part of an international mathematics competition that drew more than 20,000 entries.
The team of Morris, Weech and Shelburne emerged from the competition as not only one of the seven outstanding winners of their problem, but the team also was awarded the Leonhard Euler Award and the $10,000 COMAP Scholarship Award.
“Having followed this team, I had expected them to do well this year,” said the team’s advisor, Anke van Zuylen, associate professor of mathematics at William & Mary. “But this accomplishment completely blew me away. They were one of the few U.S. teams that won an ‘outstanding’ over all problems combined. And only three teams worldwide brought home as many awards as this team.”
COMAP is the Consortium for Mathematics and its Application, a nonprofit organization devoted to the advancement of mathematics education for all ages. Each year, COMAP sponsors a competition, offering a set of open-ended math problems for analysis. The William & Mary team picked problem D, trying to engineer a way to help the coach of the Huskies improve on last year’s 13-15 season. It helped that team members were soccer-literate.
“Jack and I were on the same freshman hall,” Weech said. “We’ve played on the same intramural soccer team for the past four years. He’s pretty good at soccer; I can vouch for that, but I’ve played soccer in some capacity since I was four years old and I really love it.
“I wouldn’t say that I’m necessarily good,” she added. “It’s just something I really enjoy.”
Morris was the team leader and has been competing in COMAP events since high school. He enlisted Weech and another friend their freshman year at William & Mary and submitted a project on Martian colonization.
“We competed. And we did pretty well,” he said. “We got a meritorious rating in our freshman year, which was pretty huge.”
In their sophomore year, the Morris-Weech team took on a problem modeling world language growth. And last year, it was analyzing the most efficient and effective way to evacuate the Louvre in an emergency.
This year, Morris and Weech added Shelburne and the three of them, under the supervision of van Zuylen, organized meetings at the math department to share their previous experience and help other student teams prepare. The students also received tips from William Fox, a lecturer at the math department, who has a long history with the COMAP competition.
Discussion with advisors ended when the team received their question late in the afternoon of Feb. 13. Per contest rules, the teams were on their own, with a problem to ponder and a submission deadline of Feb. 17 at 5 p.m.
“It was the day before Valentine’s Day,” Shelburne said. “I remember it was a Thursday and we started at 5 p.m. That day, we mostly just talked through our overall ideas. And then we really dove in the next day.”
They may have really dove in, as Shelburne said, but the team members were fully cognizant of the significance that Feb. 14 holds.
“I remember during the competition there was talk about, well…what about Valentine’s Day?” Weech said.
The team members took some time on the 14th, “to celebrate Valentine’s Day in our own way,” Morris said. Then, it was back to the Huskies.
The team was presented a request from the Huskies’ coach: explore the complex interactions among the players on the field and come up with a mathematical model to improve play. The William & Mary group was provided by data encompassing all 38 games played last season.
In the nature of soccer, the problem essentially was one of passing strategy. The data provided covered 23,429 passes among 366 players — 30 Huskies and 336 opponents — and was culled from a larger dataset that included nearly 2,000 matches from five European national soccer (football) matches.
Valentine’s Day behind them, the team began digging into the problem. They started working in Swem Library, then moved to Murray House, the headquarters of William & Mary’s elite 1693 Scholars Program. (Morris and Shelburne are both 1693 scholars.)
“Being a computer science major, I started to read in data and putting into networks,” Weech said. “Jack did a lot of the basic visualization to put in our paper. Ethan really excelled in finding all these network metrics and what the relevant equations were. He put them on a whiteboard and figured out how it related to soccer.”
The William & Mary Huskies analysis blended math with soccer savvy. Shelburne took the lead in creating a new set of metrics (Offensive Duel Centroid, Defensive Duel Centroid and Press Time). The idea was to analyze ball-possession changes to characterize the aggressiveness of each team’s playing style.
They applied those new metrics — and found that they weren’t a good predictor of success. So, the team turned their attention to a new model that incorporated the opponent’s strategy using four well-known soccer playing styles — Tiki-Taka, Long Ball, High Pressing and Parking the Bus. And crunched those numbers.
The report includes citations from the surprisingly voluminous literature of mathematical analysis of the world’s most popular sport. The William & Mary team didn’t shy away from expressing disagreement with some conclusions rendered by published soccer scholars and offering their own alternatives.
Among the eigenvalues and regression analyses and copious formulae was a lot of soccer sense. For instance, because the goal of their analysis is to improve team interactions, the paper downplays the relevance of distinguishing between successful and unsuccessful shots on goal: “Whether a shot goes in generally depends on the individual ability of both the shooter and the goalkeeper, rather than the playing style and the team network as a whole.”
Their 20-page report ended with a letter to the team coach. Shelburne described it as something that might be told to the coach in the Huskies’ clubhouse on the eve of a big match. There was no math, but the letter ended with a simple table that proposes a set of Huskies responses to various styles of opponent play.
“We essentially showed the coach what strategies respond best to different playing styles by opposing teams,” Shelburne said. “We had three different possible counter-strategies — one to maximize the chance of victory, one to maximize the chance of a tie and one to minimize the chance of losing. Each one of those could be appropriate in a different scenario.”
For example, if the opposing team is showing a Long Ball playing style, the Huskies should counter with High Pressing to maximize chances of a win. It’s unknown if the COMAP contest judges included soccer fans, but Fox said the well-written narrative was certainly instrumental in the paper’s high scoring. The results were announced April 27.
Fox ran the numbers that frame the William & Mary team’s achievement. There were more than 25,000 entries in the entire competition, with some 24,000 coming from Chinese teams, he said. Problem D alone had more than 7,000 entries.
“The United States had probably 600 entries, and all the other countries in the world had maybe 300 combined,” Fox explained. “For this William & Mary team to be one of the top two solutions, out of 25,000? That’s huge. There ought to be a ticker-tape parade for these students.”
He explained that the COMAP contest has two divisions. The William & Mary team entered problem D in the Interdisciplinary Contest in Modeling (ICM), which requires a math-based, interdisciplinary approach to real-world problems.
The International COMAP Scholarship Award goes to the four top teams in the competition. The award amounts to $10,000, with $9,000 divided among Morris, Weech and Shelburne and $1,000 going to William & Mary, in the name of van Zuylen, the team advisor.
William & Mary also won the Leonhard Euler Award, presented to a team selected by the head judge of the ICM's Problem D. The COMAP web site states that the head judge looks for a paper with Meritorious/Finalist/Outstanding rating that contains especially creative and innovative modeling and shows good understanding of interdisciplinary science.
Fox issues his praise for the team from the seat of experience. He has been involved in the COMAP competition for more than 30 years, beginning as a team advisor and then as a contest judge — he even served a term as contest director.
“I still do judging because I enjoy it,” Fox said, noting that he is no longer a team advisor and did not read the William & Mary submission until after the contest. “So, I’ve been doing this a long time and I know what it takes in order to have a winning paper.”
The judging, Fox noted, places a premium on cogent, understandable explanation of the solution. The judges begin with what he called a triage process, winnowing out the weaker papers.
“We bring in the top 100 papers in each category to what’s called the final reads,” he said. “And the judges get together — this year we did it on Zoom conference.”
The judges read and reread the papers over three days and Fox said each of those top papers were reviewed nine or ten times. He noted that the submissions are anonymous and he himself didn’t see the William & Mary paper until after the fact.
“There are guides out there for what it takes to win,” Fox said. “I’ve actually written books about it. But not that many teams follow the guidance. I don’t know if Jack’s team followed the guides, but they had to, to be one of the two teams that bubbled up to the top to win the $10,000 scholarship.”
Graduation has broken up the Morris-Weech-Shelburne team. Morris will enter a graduate program in research and technology at MIT. Weech will go to New York City to work at Google as a software engineer, although she says she’s not sure when she will be able to start.
Shelburne has another year at William & Mary and will organize a new team to take on next year’s COMAP competition.