1. We'll focus on early in the game since late game score/time strategy is highly situation dependent.

Implicit 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.

Background: Before continuing, we'll have to decide on the potential outcomes.  We'll make the following assumptions:

1. If you punt, your opponent's average starting field position will be his own 15.  (Based on 413 punts in the 2004 season for which the line of scrimmage was between mid-field and the punting team's opponent's 40 yard-line.)
2. There are only two possible outcomes: a one-yard gain (resulting in a successful attempt) and no-gain (resulting in a turnover on downs).  This assumption is likely to favor punting, so if going for it still looks attractive, we can be confident of that decision.
3. The probability of a successful 4th down attempt is p.

Carter and Machol² studied the value of field position (based on expected points scored) and determined that the relationship is linear.  Specifically, the value (V) of a first-and-ten X yards from your opponent's goal line is V = 5.91 - 0.077X, which implies that the break-even point is your own 23 yard-line.

Calculations: If you go for it on 4th down, there are two possible outcomes: success (which results in a 1st and 10 from the opponent's 44) or failure (which results in a turnover on downs and your opponent having 1st and 10 from his own 45.  So the value of going for it (V_go) is a weighted average of these two results.  Using Carter and Machol's result:

V_go = p[5.91 - 0.077(44)] - (1-p)[5.91 - 0.077(55)] = 4.197p - 1.675

Conversely, if you punt, your opponent takes the ball at his 15, a situation which is worth:

V_punt = - [5.91 - 0.077(85)] = 0.635

Discussion

Based 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 risk-averse 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 Romer³ 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) 541-544.

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