What is a football team's best mix of running and passing plays?
-
Upload
laura-mclay -
Category
Sports
-
view
5.533 -
download
1
Transcript of What is a football team's best mix of running and passing plays?
![Page 1: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/1.jpg)
Should a football team run or pass?A game theory approach
Laura A. McLay(c) 2012
Based on Mathletics by Wayne Winston
![Page 2: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/2.jpg)
The problemThe problem
• An offense can run or pass the ballAn offense can run or pass the ball
• The defense anticipates the offense’s choice and chooses a run or pass offenseand 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?
![Page 3: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/3.jpg)
Idealized payoffs (yards)Idealized payoffs (yards)
Run defense (x) Pass defense (1‐x)
Offense runs (q) ‐5 5
Offense passes (1‐q) 10 0Offense passes (1 q) 10 0
We consider a zero sum game. The offense wants the most yards. The defense wants the offense to have the fewest yards.
A pure strategy is deterministic: the offense or defense makes the sameA pure strategy is deterministic: the offense or defense makes the same decision all the time
Amixed strategy is a random strategy that assigns probabilities to the availableA mixed strategy is a random strategy that assigns probabilities to the available choices.
![Page 4: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/4.jpg)
Case 1: Defense chooses a pure strategy
• The offense chooses a mixed strategyThe offense chooses a mixed strategy– Run with probability q
– Pass with probability 1‐qPass 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
![Page 5: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/5.jpg)
Case 1: Defense chooses a pure strategy
• For any value of q chosen by the offense theFor 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 }
![Page 6: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/6.jpg)
Case 1: Defense chooses a pure strategy
Expected payoff
q
The offense should run half the time, gaining 2.5 yards per attempt (on average)yards per attempt (on average).
![Page 7: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/7.jpg)
Case 2: Offense chooses a pure strategy
• The defense chooses a mixed strategyThe 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 th t d i i• If the offense runs, the expected gain is:
x(‐5) + (1‐x)(5) = 5 – 10x
• If the offense passes, the expected gain is:
x(10) + (1‐x)(0) = 10xx(10) + (1 x)(0) 10x
![Page 8: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/8.jpg)
Case 2: Offense chooses a pure strategy
• For any value of x chosen by the defense theFor 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}
![Page 9: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/9.jpg)
Case 2: Offense chooses a pure strategy
Expected 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)
![Page 10: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/10.jpg)
Idealized payoffs (yards)Idealized payoffs (yards)
Run defense (x) Pass defense (1‐x)
Offense runs (q) r‐k r+k
Offense passes (1‐q) p+mk p‐mkOffense passes (1 q) p+mk p mk
Suppose 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 onThe correct choice on defense has m times more effect on passing as it does on running (range of 2mk vs. 2k)
![Page 11: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/11.jpg)
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) p+mk p‐mkOffense passes (1 q) p+mk p mk
Suppose 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)qq(r‐k) + (1‐q)(p+mk) = (p+mk) + (r‐k‐p‐mk)q
If a pass defense is chosen, the expected gain is:q(r+k) + (1‐q) (p‐mk) = (p‐mk)+(r+k‐p+mk)q
![Page 12: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/12.jpg)
Case 3: Idealized inputsCase 3: Idealized inputsExpected payoff
q
• q = m/(m+1) [Does not depend on r or p!]
Lik i 1/2 + ( )/(2k + ) f th d f• Likewise, x = 1/2 + (r‐p)/(2km+m) for the defense
![Page 13: What is a football team's best mix of running and passing plays?](https://reader033.fdocuments.net/reader033/viewer/2022052900/5559d426d8b42a98208b4d2d/html5/thumbnails/13.jpg)
Case 3: intuitionCase 3: intuition
• For m=1For 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 playsmore of an effect on passing plays
• For m<1– Offense passes more since the defensive call has less of an effect on passing plays