How mathematicians are trying to make NFL schedules fairer

ESPN had a nice article about applied mathematicians at the University of Buffalo who are working with the NFL to create fairer schedules. A few quotes:

“This is a field I’ve worked in for 46 years, including 43 as a professor,” Karwan said by phone last week. “I’ve worked on very difficult problems that take more than 12 hours on the supercomputer to solve. And this is by far the hardest any of us have ever seen.”


In developing the schedule, NFL assigns “penalty points” to outcomes such as three-game road trips, games between teams with disparate rest, and road trips following a Monday night road game. In their final proof of concept in 2017 before receiving the grant, Karwan and Steever took the 2016 schedule and lowered the penalty total by 20 percent…

The first step is based in both math and reality. Before creating the schedule, the NFL identifies a small number of games — usually between 40 and 50 — to lock in. The league refers to this as “seeding.” It helps accommodate expectations from television partners for key games in certain time slots, as well as about 200 annual requests from owners who prefer their stadiums not be used in a given week because of concerts, baseball games, marathons and other potential complications…

At that point, the NFL asks its computers to run schedule simulations until it finds one that has an acceptable penalty total. Usually that means juggling the 40 to 50 pre-seeded games. Karwan and Steever believe the key to improving the schedule is to better choose those pre-seeded games, allowing the computer to see stronger schedules that would otherwise be blocked by the initial choices through a process known as integer programming.

Not surprisingly, this research was publicized by the MIT Sloan Sports Analytics Conference, an annual conference dedicated to the integration (insert rim shot) of mathematics and sports.

Stay Focused

From Kirk Cousins, quarterback of the Washington Redskins:

Sometimes our guests ask why I have this hanging above my desk. It’s an old high school math quiz when I didn’t study at all and got a C+… just a subtle reminder to me of the importance of preparation. If I don’t prepare I get C’s!


The Running Nerd: The US Marathoner Who Is Also a Statistics Professor

I loved these articles about Jared Ward, an adjunct professor of statistics at BYU who also happens to be a genuine and certifiable jock… he finished the 2016 Olympic marathon in 6th place with a time of 2:11:30.

Ward started teaching at his alma mater after graduating from BYU with a master’s degree in statistics in April 2015…

Ward wrote his master’s thesis on the optimal pace strategy for the marathon. He analyzed data from the St. George Marathon, and compared the pace of runners who met the Boston Marathon qualifying time to those who did not.

The data showed that the successful runners had started the race conservatively, relative to their pace, and therefore had enough energy to take advantage of the downhill portions of the race.

Ward employs a similar pacing strategy, refusing to let his adrenaline trick him into running a faster pace than he can maintain.

And, in his own words,

[A]t BYU, on our cross-country field, on the guys side, there were maybe 20 guys on the team; half of them were statistics or econ majors. There was one year when we thought if we pooled together all of the runners from our statistics department, we could have a stab with just that group of guys at being a top-10 cross-country team in the nation…

To be a runner, it’s a very internally motivated sport. You’re out there running on the road, trying to run faster than you’ve ever run before, or longer than you’ve ever gone before. That leads to a lot of thinking and analyzing.We’re out there running, thinking about what we’re eating, what we need to eat, energy, weightlifting, how our body feels today, how it’s going to feel tomorrow with how much we run today. We’re gauging all of these efforts based on how we feel and trying to analyze how we feel and how we can best get ourselves ready for a race. As opposed to all the time on a soccer field, you’re listening to do a drill that your coach tells you to do, and then you go home.

I think we have a lot of time to think about what we are doing and how it impacts our performance. And statistics is the same way. It’s thinking about how numbers and data lead to answers to questions.

Yes, I think there’s probably some sort of connection there to nerds and runners.

Sources: and

Exponents and the decathlon

During the Olympics, I stumbled across an application of exponents that I had not known before: scoring points in the decathlon or the heptathlon. From

Decathlon, which at the Olympics is a men’s event, is composed of 10 events: the 100 meters, long jump, shot put, high jump, 400 meters, 110-meter hurdles, discus throw, pole vault, javelin throw and 1,500 meters. Heptathlon, a women’s event at the Olympics, has seven events: the 100-meter hurdles, high jump, shot put, 200 meters, long jump, javelin throw and 800 meters…

As it stands, each event’s equation has three unique constants — $latex A$, $latex B$ and $latex C$— to go along with individual performance, $latex P$. For running events, in which competitors are aiming for lower times, this equation is: $latex A(BP)^C$, where $latex P$ is measured in seconds…

B is effectively a baseline threshold at which an athlete begins scoring positive points. For performances worse than that threshold, an athlete receives zero points.

Specifically from the official rules and regulations (see pages 24 and 25), for the decathlon (where P is measured in seconds):

  • 100-meter run: 25.4347(18-P)^{1.81}.
  • 400-meter run: 1.53775(82-P)^{1.81}.
  • 1,500-meter run: 0.03768(480-P)^{1.85}.
  • 110-meter hurdles: 5.74352(28.5-P)^{1.92}.

For the heptathlon:


  • 200-meter run: 4.99087(42.5-P)^{1.81}.
  • 400-meter run: 1.53775(82-P)^{1.88}.
  • 1,500-meter run: 0.03768(480-P)^{1.835}.

Continuing from FiveThirtyEight:


For field events, in which competitors are aiming for greater distances or heights, the formula is flipped in the middle: $latex A(PB)^C$, where $latex P$ is measured in meters for throwing events and centimeters for jumping and pole vault.

Specifically, for the decathlon jumping events (P is measured in centimeters):

  • High jump: 0.8465(P-75)^{1.42}
  • Pole vault: 0.2797(P-100)^{1.35}
  • Long jump: 0.14354(P-220)^{1.4}

For the decathlon throwing events (P is measured in meters):

  • Shot put: 51.39(P-1.5)^{1.05}.
  • Discus: 12.91(P-4)^{1.1}.
  • Javelin: 10.14(P-7)^{1.08}.

Specifically, for the heptathlon jumping events (P is measured in centimeters):

  • High jump: 1.84523(P-75)^{1.348}
  • Long jump: 0.188807(P-210)^{1.41}

For the heptathlon throwing events (P is measured in meters):

  • Shot put: 56.0211(P-1.5)^{1.05}.
  • Javelin: 15.9803(P-3.8)^{1.04}.

I’m sure there are good historical reasons for why these particular constants were chosen, but suffice it to say that there’s nothing “magical” about any of these numbers except for human convention. From FiveThirtyEight:

The [decathlon/heptathlon] tables [devised in 1984] used the principle that the world record performances of each event at the time should have roughly equal scores but haven’t been updated since. Because world records for different events progress at different rates, today these targets for WR performances significantly differ between events. For example, Jürgen Schult’s 1986 discus throw of 74.08 meters would today score the most decathlon points, at 1,384, while Usain Bolt’s 100-meter world record of 9.58 seconds would notch “just” 1,203 points. For women, Natalya Lisovskaya’s 22.63 shot put world record in 1987 would tally the most heptathlon points, at 1,379, while Jarmila Kratochvílová’s 1983 WR in the 800 meters still anchors the lowest WR points, at 1,224.

FiveThirtyEight concludes that, since the exponents in the running events are higher than those for the throwing/jumping events, it behooves the elite decathlete/heptathlete to focus more on the running events because the rewards for exceeding the baseline are greater in these events.

Finally, since all of the exponents are not integers, a negative base (when the athlete’s performance wasn’t very good) would actually yield a complex-valued number with a nontrivial imaginary component. Sadly, the rules of track and field don’t permit an athlete’s score to be a non-real number. However, if they did, scores might look something like this…



This Is Why There Are So Many Ties In Swimming

From the excellent article “This Is Why There Are So Many Ties In Swimming“, ties in swimming are allowed by the sport’s governing body because of the inevitability of roundoff error.

In 1972, Sweden’s Gunnar Larsson beat American Tim McKee in the 400m individual medley by 0.002 seconds. That finish led the governing body to eliminate timing by a significant digit. But why?

In a 50 meter Olympic pool, at the current men’s world record 50m pace, a thousandth-of-a-second constitutes 2.39 millimeters of travel. FINA pool dimension regulations allow a tolerance of 3 centimeters in each lane, more than ten times that amount. Could you time swimmers to a thousandth-of-a-second? Sure, but you couldn’t guarantee the winning swimmer didn’t have a thousandth-of-a-second-shorter course to swim. (Attempting to construct a concrete pool to any tighter a tolerance is nearly impossible; the effective length of a pool can change depending on the ambient temperature, the water temperature, and even whether or not there are people in the pool itself.)

Tennis and best 2-out-of-3 vs. best 3-out-of-5

I recently read a very interesting article on regarding men’s and women’s tennis that reminded me of the following standard problem in probability.

Player X and Player Y play a series of at most n games, and a winner is declared when either Player X or Player Y wins at least n/2 games. Suppose that the chance that Player X wins is p, and suppose that the outcomes of the games are independent. Find the probability that Player Y wins if (a) n = 3, (b) n = 5.

The easiest way to solve this is to assume that all n games are played, even if that doesn’t actually happen in real life. Then, for part (a), we can use the binomial distribution to find

  • P(X = 0) = P(Y = 3) = (1-p)^3
  • P(X = 1) = P(Y = 2) = 3p(1-p)^2
  • P(X = 2) = P(Y = 1) = 3p^2(1-p)
  • P(X = 3) = P(Y = 0) = p^3

Adding the first two probabilities, the chance that Player Y wins is (1-p)^3 + 3p(1-p)^2 = (1-p)^2 (1+2p).

Similarly, for part (b),

  • P(X = 0) = P(Y = 5) = (1-p)^5
  • P(X = 1) = P(Y = 4) = 5 p (1-p)^4
  • P(X = 2) = P(Y = 3) = 10p^2 (1-p)^3
  • P(X = 3) = P(Y = 2) = 10 p^3 (1-p)^2
  • P(X = 4) = P(Y = 1) = 5 p^4 (1-p)
  • P(X = 5) = P(Y = 0) = p^5

Adding the first three probabilities, the chance that Player Y wins is (1-p)^5 + 5p(1-p)^4 + 10p^2(1-p)^3 = (1-p)^3 (1+3p+6p^2).

The graphs of (1-p)^2 (1+2p) and (1-p)^3 (1+3p+6p^2) on the interval 0.7 \le p \le 0.9 are shown below in blue and orange, respectively. The lesson is clear: if p > 0.5, then clearly the chance that Player Y wins is less than 50%. However, Player Y’s chances of upsetting Player X are greater if they play a best 2-out-of-3 series instead of a best 3-out-of-5 series.

best2outof3Remarkably, this above curve has been observed in real-life sports: namely, women’s tennis (which plays best 2 sets out of 3 — marked WTA below) and men’s tennis (which plays best 3 sets out of 5 in Grand Slams — marked ATP below). The chart indicates that when two men’s players ranked 20 places apart play each other in Grand Slams, an upset occurs about 13% of the time. However, the upset percentage is only 5% in women’s tennis. (This approximately matches the above curve near p = 0.8.)

However, in tennis tournaments that are not Grand Slams, men’s tennis players also play a matches with a maximum of 3 sets. In those tournaments, the chances of an upset are approximately equal in both men’s tennis and women’s tennis.

However, since the casual tennis fan (like me) only tunes into the Grand Slams but not other tennis matches, this fact is not widely known — which gives the misleading impression that top women’s tennis players are not as tough, skilled, etc. as men’s tennis players.

The FiveThirtyEight article argues that top women’s tennis players don’t make it to the latter stages of Grand Slam tournaments than top men’s players because of the two tournaments are held under these different rules, and that women’s tennis would be better served if their matches were also played in a best-3-out-of-5 format.