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Field Goals, the Super Bowl, and Mathematical Models

There has been a tremendous increase in the use of mathematical analysis to make policy, inform organizational decisions, and explain news and events.  I generally think this is a positive thing:  I understand that math gives us a powerful set of tools for understanding and processing the world.  But I also understand the limitations inherent in mathematical modeling.

All mathematical models rely on assumptions that limit their impact.  Mathematicians are typically aware of the assumptions about objects, relationships, and contexts that their models make.  Politicians, journalists, and others who invoke mathematics to make a point seem less aware.  This often leads to bold, unjustified claims based on what “the math” has told them.

An inconsequential but illustrative example of this occurred during Super Bowl 49.

At the start of the second half with the game tied at 14, Seattle drove into scoring position, and faced a 4th-and-1 at New England’s 8 yard line.  Seattle basically had two options:  kick the field goal (a high-percentage play for 3 points), or try to make a 1st-down, and ultimately a touchdown (a lower-percentage play for 7 points).  Seattle opted for the field goal and went ahead 17-14.

Lots of people on Twitter second-guessed the decision, including The Upshot’s David Leohnardt.

Here, David Leonhardt is applying a simple expected value argument.  “Going for it” on 4th-and-1 at the opponent’s 8-yard-line likely produces more total points in the long run than kicking field goals, which suggests that Seattle should have gone for it.  It’s not a bad argument; in fact, I used a similar analysis on the NFL’s new overtime rule.

But in order to apply this argument, it’s important to understand what assumptions this model makes.  For example, this model assumes that the amount of time remaining in the game is irrelevant.  Of course, it’s not:  it’s easy to construct a situation in which “time remaining” is the determining factor in the kicking a field goal (say, the game is tied, and only seconds remain).

This model also assumes that all points are of equal worth.  But they aren’t.  Depending on the game situation, the extra four points a touchdown gives you may be irrelevant, or of significantly less value than the three points the field goal gives you (imagine a team up by six points late in the game).

There are lots of factors this analysis does not consider.  This doesn’t mean that the expected value argument is invalid.  It just means that, like all mathematical models, what it says depends on the assumptions it makes.  And the more we use mathematical models to drive our decisions, the more important it is to be clear about the assumptions that are made and the consequences they entail.

A Saturday Morning Optimization Problem

I recently faced an interesting optimization problem.

Through my local grocery store’s rewards program, I earned a one-time 20% discount, to be applied to a single future shopping trip.  Naturally I wanted to maximize the value of my discount, and the more I spent, the more I would save.  But like all optimization problems, there were a number of constraints involved.

shopping bags


First, I wanted to buy only things I would actually use.  This prevented me from buying things like saffron (expensive things that would drive up the value of my 20% discount) because I wouldn’t use them.  It also limited the quantity of high-priced proteins I would buy, as such things need to be consumed quickly to be enjoyed.

Second, I could only buy what I could carry, since I walk to and from the grocery store.  This put global constraints on the volume and weight of my purchases, which made me think about maximizing cost per-unit-weight/volume at a local level.

All in all, I’d say I did pretty well!  With some planning and foresight, the total value of my 20% discount ended up being around $46.  And I don’t think I’ll need to buy dried basil any time soon.

shopping receipt


Pinscreen Approximations

I’ve always enjoyed playing around with pinscreens, but only recently did I realize what cool mathematical concepts they display!

Pinscreen Approximation

For example, the image above shows an approximation of the volume of a hemisphere using right cylinders.  A pinscreen Riemann sum!

And the images below suggest how we might approximate the areas of a circle and a square using pinscreens.

pinscreen circle

pinscreen square

Compute the ratio of raised pins to total pins, and multiply by the total area of the pinscreen.  A pinscreen Monte Carlo method!

Any other cool math hiding in there?



02/15/2015 — Happy Permutation Day!

Today we celebrate the second Permutation Day of the year!  I call days like today permutation days because the digits of the day and the month can be rearranged to form the year.


Additionally, today is a transposition day, as a single swap of two digits is all that’s need to turn the day-month into the year.

Celebrate Permutation Day by mixing things up!  Try doing things in a different order today.  Just remember, for some operations, order definitely matters!

Exploring Compound Interest

Go to a <a href="">related post</a> about a topic one blogger calls “incredibly important to share with your kids.” »My latest piece for the New York Times Learning Network is a math lesson exploring personal savings and the power of compound interest.  The piece was inspired by a new program in Illinois that creates an automatic payroll-deduction savings program for all state residents.

In addition to exploring the basic ideas of savings and compounding, students are invited to analyze the merits of this state-run program.

The automatic retirement savings program mentioned in the article is described as a zero-fiscal-cost program because it does not require any government funding to run. This is because the savers themselves pay the costs, in the form of fees to financial institutions, amounting to 0.75 percent of their total savings each year.

Have students compute the costs associated with maintaining the account for each of the typical savers they profiled in the previous activity. One way to do this is to compute 0.75 percent of the total value of the savings account each year, before interest is computed. This is an estimate of the amount that would be paid in fees that year, and thus should be subtracted from the amount in savings.

The entire piece is freely available here.  Hopefully students will get a sense of the power and value of long-term savings, and maybe ask a few good questions about the the true price of zero-fiscal-cost programs.