Should a football team run or pass? A linear programming approach to game theory

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

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?

The solution:
Linear programming!

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

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

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?

Rock-Paper-Scissors
• Payoff matrix?
=

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

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

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.

Solution
Game theory to the rescue!

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

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

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!

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.

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

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

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

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 / ):

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

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

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

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

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)

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

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.

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

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

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

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)

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

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

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

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

Should a football team run or pass? A linear programming approach to game theory

More Related Content

What's hot (20)

Viewers also liked (13)

Similar to Should a football team run or pass? A linear programming approach to game theory (20)

More from Laura Albert (12)

Recently uploaded (20)

Should a football team run or pass? A linear programming approach to game theory