What is a football team's best mix of running and passing plays?

The problem The problem• An offense can run or pass the ball An offense can run or pass the ball• The defense anticipates the offense’s choice and chooses a run or pass offense. and chooses a run or pass offense• Given this strategic interaction, – what is the best mix of pass and run plays for the offense? – what is the best mix of pass and run defenses?

Idealized payoffs (yards) Idealized payoffs (yards) Run defense (x) Pass defense (1‐x) Offense runs (q) ‐5 5 Offense passes (1 q) Offense passes (1‐q) 10 0We consider a zero sum game. The offense wants the most yards. The defense wants the offense to have the fewest yards.A pureA pure strategy is deterministic: the offense or defense makes the same is deterministic: the offense or defense makes the same decision all the timeA mixed strategy is a random strategy that assigns probabilities to the available A mixed strategy is a random strategy that assigns probabilities to the availablechoices.

Case 1: Defense chooses a pure strategy• The offense chooses a mixed strategy The offense chooses a mixed strategy – Run with probability q – Pass with probability 1‐q Pass with probability 1 q• If a run defense is chosen the expected gain is: If a run defense is chosen, the expected gain is: q(‐5) + (1‐q)10 = 10‐15q• If a pass defense is chosen the e pected gain is If a pass defense is chosen, the expected gain is: q(5) + (1‐q) 0 = 5q

Case 1: Defense chooses a pure strategy• For any value of q chosen by the offense the For any value of q chosen by the offense, the defense wants to minimize the yards: min{ 10‐15q, 5q } min{ 10 15q 5q }• The offense should choose q (0 < q < 1) that maximizes the min{ 10‐15q, 5q }

Case 1: Defense chooses a pure strategyExpected payoff q The offense should run half the time, gaining 2.5 yards per attempt (on average). yards per attempt (on average)

Case 2: Offense chooses a pure strategy• The defense chooses a mixed strategy The defense chooses a mixed strategy – Run defense with probability x – Pass defense with probability 1 x Pass defense with probability 1‐x• If th ff If the offense runs, the expected gain is: th t d i i x(‐5) + (1‐x)(5) = 5 – 10x• If the offense passes, the expected gain is: x(10) + (1 x)(0) 10x x(10) + (1‐x)(0) = 10x

Case 2: Offense chooses a pure strategy• For any value of x chosen by the defense the For any value of x chosen by the defense, the offense wants to maximize the yards: max{ 5 – 10x, 10x } max{ 5 10x 10x }• The defense should choose x (0 < x < 1) that minimizes the max{ 5 – 10x, 10x}

Case 2: Offense chooses a pure strategyExpected payoff x The defense should choose a run defense 1/4 of the time, allowing 2.5 yards per attempt (on average). (The offense gain and defensive loss are always identical)

Idealized payoffs (yards) Idealized payoffs (yards) Run defense (x) Pass defense (1‐x) Offense runs (q) r‐k r+k Offense passes (1 q) Offense passes (1‐q) p+mk p mk p‐mkSuppose the defense chooses run and pass defenses with equal likelihoods.The offense would gain r yards per run, on average.The offense would gain p yards per pass, on average.The correct choice on defense has m times more effect on passing as it does on The correct choice on defense has m times more effect on passing as it does onrunning (range of 2mk vs. 2k)

Idealized payoffs, cont d. Idealized payoffs, cont’d. Run defense (x) Pass defense (1‐x) Offense runs (q) r‐k r+k Offense passes (1 q) Offense passes (1‐q) p+mk p mk p‐mkSuppose the defense chooses a pure strategy.If a run defense is chosen, the expected gain is: q(r‐k) + (1‐q)(p+mk) = (p+mk) + (r‐k‐p‐mk)q q(r k) + (1 q)(p+mk) (p+mk) + (r k p mk)qIf a pass defense is chosen, the expected gain is: q(r+k) + (1‐q) (p‐mk) = (p‐mk)+(r+k‐p+mk)q

Case 3: Idealized inputs Case 3: Idealized inputsExpected payoff q • q = m/(m+1) [Does not depend on r or p!] • Lik i Likewise, x = 1/2 + (r‐p)/(2km+m) for the defense 1/2 + ( )/(2k + ) f th d f

Case 3: intuition Case 3: intuition• For m=1 For m=1 – Offense runs pass and run plays equally• For m>1 For m>1 – Offense runs more since the defensive call has more of an effect on passing plays more of an effect on passing plays• For m<1 – Offense passes more since the defensive call has less of an effect on passing plays

http://punkrockor.wordpress.com	4187
http://www.newsblur.com	13
http://newsblur.com	12
http://dev.newsblur.com	3
http://webcache.googleusercontent.com	2

What is a football team's best mix of running and passing plays?

by Laura McLay

Accessibility

Categories

Upload Details

Usage Rights

5 Embeds 4,217

Statistics

What is a football team’s best mix of running and passing plays? Webinar Transcript