Quarterback is the highest pressure position in football. The outcome of every play depends on the quarterback’s lightning-fast decisions about how to pass or hand off the ball. On a passing play, a quarterback must look to his receivers, and at each look he must instantly decide whether to throw the ball. If he throws too early, he may have lost the opportunity of a better target that would have emerged later. But if he waits too long, he may end up stuck with a bad option like a sack. So, when should the QB stop looking and throw the ball?
This type of problem faced by a QB is called an “optimal stopping problem.” The best known optimal stopping problem is called The Secretary Problem: a company will interview up to 100 applicants for a secretary position. After each interview, the company must decide whether to hire or reject the applicant. If they are hired, the applicant must accept the job, and all further interviews are cancelled. If they are rejected, the applicant may never be reconsidered. So, when should the company stop interviewing and hire a secretary?
The secretary problem has a well-known decision strategy to maximize the expected quality of the secretary. If the company wants the highest probability of selecting the best secretary, they should interview and reject the first 100/e ≈ 37 applicants, then hire the first applicant they see who is better than all of the first 37. If they want to maximize the expected quality, they should reject the first sqrt(100) = 10 applicants, then hire the first applicant they see who is better than the first 10.
The secretary problem is identical to the passing decision of a quarterback, so we can apply the above logic to help the quarterback make his decision. Let’s say the QB has time to evaluate at most 5 pass options per play. Then, he should reject the first 5/e ≈ sqrt(5) ≈ 2 options and then pass to the next target that is better than either of the first two. This gives a 43% chance of selecting the best pass, and on a scale from 0-10, an expected pass quality of 6.5. In other words, by using this strategy, the QB will throw to the best option more than twice as often as if he throws to his first look every time. Given the power and results of this decision method, I bet that NFL-caliber quarterbacks do actually use some thought processes like this to make in-game pass decisions, even if they are not consciously aware of it.
This is just one of many, many situations where advanced mathematics are relevant to football. Use this advice to draft a team at AdvancedSportistics.com!
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Clayton Kershaw finished the 2014 season with 7.2 Wins Above Replacement.