# Dynamic programming, Nash equilibria, and going for it on 4th down

Last weekend marked the first big college football Saturday of the year.  The only game I really cared about was the Northern Iowa – Iowa game: I went to Iowa, and I grew up in Cedar Falls (home of the UNI campus).  Iowa came from behind and won 17-16 after blocking two field goal attempts in the final seconds.  My brother and I were talking on the phone during the 4th quarter.  My brother, a UNI fan, was bothered by the conservative coaching that he feels let Iowa back into the game, and that spun into a more general conversation about risk-averse coaching.  I don’t know anything about sabermetrics, but I did read Moneyball, and I love sports.

Our conversation reminded me of a paper by economist David Romer that I had always intended on reading: “It’s Fourth Down and What Does the Bellman Equation Say?”  (I actually recommend this updated 2005 version.)  So I read it.  It received some attention from the sports world when it came out because Romer’s claim is that the conventional wisdom is wrong: it’s often a much better idea to try to convert on 4th down rather than kick the ball away to the other team.  It depends on field position and yards-to-go, of course.  He sets up a not-too-complicated dynamic programming model where he is able to place values on particular game situations, and then compares the difference between kicking and “going for it”.  It’s interesting but I have some problems with it – in particular the use of 3rd down outcomes rather than 4th down outcomes.  The justification is that because teams don’t go for it on 4th down very often there is not enough data, so 3rd down data is a reasonable substitute.  I have issues with this because for one thing, playcalling is very different on 3rd down, especially when one is approaching field goal range.  Players are also taught to handle 3rd down differently – throw the ball away and avoid taking a sack.  Romer does address these sorts of issues, but it still bothers me.

To overcome the lack of 4th down “going for it” data, Christopher Adams from the FTC (!) uses Madden NFL 07 (!!) for simulation purposes and constructs a game theoretic model for 4th down attempts in this paper.  He comes to a different conclusion: the conventional wisdom may not be so bad after all.  I am looking forward to reading the Adams paper in detail – it’s in my backpack.  I hope to do some experimentation in this area once I get a grip on the concepts.  I would like to write a paper about the prevent defense that Gregg Easterbrook and Bill Simmons despise so much!  But right now, I’ve got to get back to work 😉