

When to go for itYou're the head coach of a football team, and early in the game^{1}, your team faces a 4th and 1 from your opponent's 45. Do you go for it or do you punt? Because they are riskaverse, coaches often play it safe and punt the ball away. But is it the right decision? 1. We'll focus on early in the game since late game score/time strategy is highly situation dependent. Mathematical ModelingImplicit in a coach's decision to punt is his assessment that having the ball 1st and 10 from his own 45 (as his opponent would after a failed fourth down play) is a very desirable situation for his opponent. What he discounts is that a successful conversion gives his team a 1st and 10 inside his opponent's 45, which is an even better situation for his team than the one he risks giving his opponent.
Carter and Machol^{2} studied the value of field position (based on expected points scored) and determined that the relationship is linear. Specifically, the value (V) of a firstandten X yards from your opponent's goal line is V = 5.91  0.077X, which implies that the breakeven point is your own 23 yardline.
DiscussionBased on these calculations, it makes sense to go for it if your probability of success is at least 55% (in which case, the values of going for it and punting are equal). We'll use 3rd down data because there's much more of it. In 2004, there were 2075 plays in which the offense faced a 3rd and 1. The offense converted a first down 1390 times, or 67%. Therefore, in this situation, punting is a losing proposition. We should note that not all coaches are as riskaverse as we initially stated. University of Toledo head coach Tom Amstutz gets it. His teams routinely go for it on fourth down. In 2004, Toledo attempted 38 plays on fourth down and converted 23 (61%). Economist David Romer^{3} has studied fourth down strategy extensively and concludes that coaches are much more conservative on fourth down than they should be. 2. Carter, Virgil and Machol, Robert (1971) Operations Research on Football, Operations Research, 19 (2) 541544. 3. Romer, David (2003) It's Fourth Down and What Does the Bellman Equation Say? A DynamicProgramming Analysis of Football Strategy. Back to Top 
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