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Game theory at the Super Bowl

Mike Olinick sent a link to the following:

Game theory says Pete Carroll’s call at goal line Is defensible
by Justin Wolfers, "The Upshot" blog, New York Times, 2 February 2015

This year's Super Bowl game ended in dramatic fashion. Trailing by 4 points with time running out, the Seattle Seahawks had the ball at the New England Patriots one-yard line. Instead of handing off to their star running back, Seattle attempted a pass, which was intercepted by New England. Social media lit up, and, with customary understatement, many sports fans had soon labeled this the worst play call in history. Looking at the situation more calmly in his blog post, Justin Wolfers notes that the decision may have been rational. The run sounds logical, but your opponent would also know this, and could defend accordingly. He frames the run vs. pass problem in the language of game theory, which would recommend a mixed strategy involving a random choice between pass and run.

Jim Greenwood noted that the comments section also included some interesting discussion. Indeed, there is even a link to a simulation analysis described at Slate:

Tough call: Why Pete Carroll’s decision to pass was not as stupid as it looked
by Brian Burke, Slate, 2 February 2015

This article acknowledges that the play was the most consequential in Super Bowl history, as measured by the difference between the probability of winning before the play and the probability after. In this case, Seattle's chance of winning was reduced from 88% to almost zero). But this by itself not evaluate the decision to pass. As a number of other commenters pointed out it was not a simple run vs. pass decision. Burke writes:

But an interception wasn’t the only added risk of a passing play. There was also the possibility of a sack and higher probabilities of a penalty or turnover. There are any number of possible combinations of outcomes to consider on Seattle’s three remaining downs—too many to directly evaluate. So I ran the situation through a game simulation. The simulator plays out the remainder of the game thousands of times from a chosen point—in this case from the second down on. I ran the simulation twice, once forcing the Seahawks to run on second down and once forcing them to pass. I anticipated that the results would support my logic (and Carroll’s explanation) that running would be a bad idea. It turns out I was wrong. The simulation—which is different than Win Probability—gave Seattle an 85 percent chance of winning by running and a 77 percent chance by passing. It turns out the added risk of a sack, penalty, or turnover was not worth the other considerations of time and down.

Discussion
But where does the 88% come from?

More on Gini

The analysis received news coverage elsewhere, for example:

How airline seating reflects income inequality
by Michael Hiltzik, Los Angeles Times, 2 December 2014

Some math doodles

<math>P \left({A_1 \cup A_2}\right) = P\left({A_1}\right) + P\left({A_2}\right) -P \left({A_1 \cap A_2}\right)</math>


Accidental insights

My collective understanding of Power Laws would fit beneath the shallow end of the long tail. Curiosity, however, easily fills the fat end. I long have been intrigued by the concept and the surprisingly common appearance of power laws in varied natural, social and organizational dynamics. But, am I just seeing a statistical novelty or is there meaning and utility in Power Law relationships? Here’s a case in point.

While carrying a pair of 10 lb. hand weights one, by chance, slipped from my grasp and fell onto a piece of ceramic tile I had left on the carpeted floor. The fractured tile was inconsequential, meant for the trash.

BrokenTile.jpg

As I stared, slightly annoyed, at the mess, a favorite maxim of the Greek philosopher, Epictetus, came to mind: “On the occasion of every accident that befalls you, turn to yourself and ask what power you have to put it to use.” Could this array of large and small polygons form a Power Law? With curiosity piqued, I collected all the fragments and measured the area of each piece.

Piece Sq. Inches % of Total
1 43.25 31.9%
2 35.25 26.0%
3 23.25 17.2%
4 14.10 10.4%
5 7.10 5.2%
6 4.70 3.5%
7 3.60 2.7%
8 3.03 2.2%
9 0.66 0.5%
10 0.61 0.5%
Montante plot1.png

The data and plot look like a Power Law distribution. The first plot is an exponential fit of percent total area. The second plot is same data on a log normal format. Clue: Ok, data fits a straight line. I found myself again in the shallow end of the knowledge curve. Does the data reflect a Power Law or something else, and if it does what does it reflect? What insights can I gain from this accident? Favorite maxims of Epictetus and Pasteur echoed in my head: “On the occasion of every accident that befalls you, remember to turn to yourself and inquire what power you have to turn it to use” and “Chance favors only the prepared mind.”

Montante plot2.png

My “prepared” mind searched for answers, leading me down varied learning paths. Tapping the power of networks, I dropped a note to Chance News editor Bill Peterson. His quick web search surfaced a story from Nature News on research by Hans Herrmann, et. al. Shattered eggs reveal secrets of explosions. As described there, researchers have found power-law relationships for the fragments produced by shattering a pane of glass or breaking a solid object, such as a stone. Seems there is a science underpinning how things break and explode; potentially useful in Forensic reconstructions. Bill also provided a link to a vignette from CRAN describing a maximum likelihood procedure for fitting a Power Law relationship. I am now learning my way through that.

Submitted by William Montante