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Should a football team run or pass?A game theory approach
Laura A. McLay(c) 2012
Based on Mathletics by Wayne Winston
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?
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.
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
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 }
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).
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
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}
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)
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)
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
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
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