Laura Albert McLay's slides from Oberlin College (2015)

1. Should a football team run or pass?
A game theory approach
Laura Albert McLay
Badger Bracketology
@lauramclay
@badgerbrackets
http://bracketology.engr.wisc.edu/
© 2015

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

3. The solution:
Linear programming!

4. Definitions
• Players
• Actions
• Information
• Strategies
• Payoffs
• Equilibria
Eventually we’ll relate this to linear programming!

5. Definitions
• Players: we have 2
• Actions: discrete actions of actions available to each
player and when they are available (order of play)
• Information: what each player knows about variables at
each point in time
• Strategies: a rule that tells each player which action to
choose at each decision point
• Payoffs: the expected utility/reward each player receives
as a function of every players’ decisions
• Equilibria: strategy profiles consisting of best strategies
for each of the players in the game

6. Payoff matrix
• A two person game is between a row player R and a
column player C
• A zero-sum game is defined by a × payoff matrix
where is the payoff to C if C chooses action
and R chooses action
o R chooses from the rows ∈ {1, … , }
o C chooses from the columns ∈ {1, … , }
o Note: deterministic strategies can be bad!
• Zero-sum: my gain is your loss. Examples?

7. Rock-Paper-Scissors
• Payoff matrix?
=

8. Rock-Paper-Scissors
Payoff matrix?
=
0 1 −1
−1 0 1
1 −1 0
Why is this zero sum?
Strategies:
• Deterministic: pure strategies
• Random/stochastic: mixed strategies

9. Strategies
Payoff matrix
• A strategy for a player is a probability vector
representing the portion of time each action is used
o R chooses with probability
= , , … ,
o C chooses with probability
= , , … ,
o We have: ≥ 0, = 1, … ,
∑ = 1

10. Payoffs
Expected payoff from R to C:
, = ! ! =
Note:
• and are our variables
Problem:
• We want to solve this as a linear program but , is a
quadratic function with two players with opposing goals.

11. Solution
Game theory to the rescue!

12. Theorem
Expected payoff from R to C:
, = ! ! =
Theorem:
There exist optimal strategies ∗ and ∗
such that for all strategies and :
, ∗ ≤ ∗, ∗ ≤ [ ∗, ]
Note we call ∗, ∗ the value of the
game.
Hipster mathematician

13. Reflect on the inequality
, ∗
≤ ∗
, ∗
≤ [ ∗
, ]
• ∗, ∗ ≤ ∗, : C guarantees a lower bound
(worst−case) on his/her payoff
• , ∗ ≤ ∗, ∗ : R guarantees an upper bound
(worst-case) on how much he/she loses
• Fundamental problem: finding ∗ and ∗
Both R and C play
optimal strategies
C plays optimal,
R plays suboptimal
R plays optimal,
C plays suboptimal

14. Objective function analysis
• Suppose C adopts strategy
• Then, R’s best strategy is to find the that minimizes
:
min
*
• And therefore, C should choose the that maximizes
these possibilities:
max
-
min
*
This will give us ∗ and ∗. This is hard!

15. Useful result
• Let’s focus on the inner optimization problem:
min
*
o This is easy since it treats as “fixed” so we have a linear
problem.
Lemma: min
*
= min /
where / is the pure vector of only selecting action (e.g.,
/ = [1 0 0 … 0])
Idea: a weighted average of things is no bigger than the
largest of them.

16. Put it together
We now have:
max
-
min /
subject to ∑ = 1
≥ 0, = 1,2, … ,
This is a linear program!!

17. Reduction to a linear program
• Now introduce a scalar 1 representing the value of
the inner minimization (min / ):
max
2,3
1
subject to 1 ≤ / , = 1,2, … ,
∑ = 1
≥ 0, = 1,2, … ,
1 free

18. Reduction to a linear program
Matrix-vector notation
max 1
1/ − ≤ 0
/ = 1
≥ 0
/ is the vector of all 1’s
Block matrix form
max
0
1 1
− /
/ 0 1
≤
=
0
1
≥ 0
1 free

19. Now do the same from R’s perspective
Everything is analogous to what we did before!
• R solves this problem:
min
*
max
-
• Lemma: max
2
= max /
• That gives us the following linear program:
min
*
max /
subject to ∑ = 1
≥ 0, = 1,2, … ,
• Introduce a scalar 4 representing the value of the inner
maximization (max / ):

20. Reduction to a linear program
Matrix-vector notation
min 4
4/ − ≥ 0
/ = 1
≥ 0
/ is the vector of all 1’s
Block matrix form
min
0
1 4
− /
/ 0 4
≥
=
0
1
≥ 0
4 free

21. OK, so now we have two ways to solve
the same problem
Let’s examine how these solutions are related.

22. Minimax Theorem
• Let ∗ denote C’s solution to the max-min problem
• Let ∗
denote R’s solution to the min-max problem
• Then:
max
2
∗
= min
*
∗
Proof:
From strong duality, we have 4∗ = 1∗. Also
1∗ = min / ∗ = min
*
∗ from C’s problem
4∗ = max
2
∗ / = max
2
∗ from R’s problem

23. We did it!

24. Let’s work on an example
Example from Mathletics by Wayne Winston (2009), Princeton University Press, Princeton, NJ.

25. Football example: offense vs. defense
5(7, 8) Offense runs (7:) Offense passes (;< = 1 − :)
Run defense ( ) -5 10
Pass defense ( = 1 − ) 5 0
The offense wants the most yards. The defense wants the offense to
have the fewest yards. This is a zero sum game.
Using this information, answer the following two questions:
(1) What fraction of time should the offense run the ball?
(2) If they adopt this strategy, how many yards will they achieve per
play on average?
Idealized payoffs (yards)

26. Case 1: Look at the offense
• The offense chooses a mixed strategy
o Run with probability
o Pass with probability = 1 −
• Solve the linear program:
max 1
subject to
1 ≤ −5 + 10
1 ≤ 5
+ = 1
, ≥ 0

27. Case 1: Look at the offense
We know that = 1 − , which simplifies the
formulation to:
max 1
subject to
1 ≤ −5 + 10 (1 − ) = 10 − 15
1 ≤ 5
, ≥ 0
Let’s solve the problem visually.

28. Case 1: Look at the offense
We want the largest value of 1 that is “under” both lines.
This happens when = 1/2 (and = 1/2): run half the time,
pass half the time.
1∗ = min / ∗ = 2.5 yards per play, on average.
Expected payoff
, proportion of time offense runs the ball
Run defense
10 − 15
Pass defense
5

29. Case 2: Look at the defense
• We still do not know the optimal defensive strategy.
• The defense chooses a mixed strategy
o Run defense with probability
o Pass defense with probability = 1 −
• Solve the linear program:
min 4
subject to
4 ≥ −5 + 5 = 5 − 10
4 ≥ 10
+ = 1
, ≥ 0

30. Case 2: Look at the defense
We want the smallest value of 4 that is “over” both lines.
This happens when = 1/4 (and =3/4): prepare for run a quarter of the
time, prepare for a pass three quarters of the time.
This yields 4∗ = 2.5 yards per attempt (on average). The offense gain and
defensive loss are always identical!
Expected payoff
Run offense
5 − 10
Pass offense
10
, proportion of time defense prepares for run

31. Football example #2:
offense vs. defense
5(7, 8) Offense runs (7:) Offense passes (;< = 1 − :)
Run defense ( ) I − J K + J
Pass defense ( = 1 − ) I + J K − J
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 on running (range of 2 J vs. 2J)
Idealized payoffs (yards)

32. Football example #2:
offense vs. defense
Offense problem Defense problem
min 4
subject to
4 ≥ (I − J) + (I + J)
4 ≥ (K + J) + (K − J)
+ = 1
, ≥ 0
max 1
subject to
1 ≤ (I − J) + (K + J)
1 ≤ (I + J) + (K − J)
+ = 1
, ≥ 0

33. Football example #2:
offense problem
After a lot of algebra…
= /( + 1) [Does not depend on I or K!]
Likewise, = 1/2 + (I − K)/(2J + ) for the defense
Expected
payoff
, proportion of time offense runs the ball
Run defense
K + J + (I − K − ( + 1)J)
Pass defense
K − J + (I − K + ( + 1)J)
K + J
K + J

34. Intuition
The correct choice on defense has times more effect
on passing as it does on running
• For = 1
o Offense runs pass and run plays equally
• For > 1
o Offense runs more since the defensive call has more of an
effect on passing plays
• For < 1
o Offense passes more since the defensive call has less of an
effect on passing plays

35. Related blog posts
• Happiness is assuming the world is linear
• Why the Patriots’ decision to let the
Giants score a touchdown makes sense
• Introducing Badger Bracketology 1.0
• Some thoughts on the College Football
Playoff