optimal decision making in football mse 339 project

1. Optimal Decision Making in FootballMS&E 339 Project

3. Use dynamic programming techniques to answer two primary questions about decision-making in football. What is the optimal policy to follow for deciding whether to run an offensive play, punt or kick a field goal at each possible situation that could arise in the course of a football game? If you implemented such a policy, how much of a performance improvement would you realize when competing against an opponent playing a standard strategy? Project Objectives

4. Key rules 2 Teams, 60 minute game (2 halves), highest score wins Basic scoring plays: Touchdown (7 points), Field Goal (3 points) Field is 100 yards long Advancing the ball: 4 plays (downs) to gain 10 yards if successful, down will reset to 1st down if unsuccessful, other team will gain possession of the ball Teams have the option of punting the ball to the other team (typically reserved for 4th down) which gives the other team possession but in a worse position on the field Teams can attempt to kick a field goal at any point Common Strategies Coaches typically rely on common rules of thumb to make these decisions Motivating Situation 4th down and 2 yards to go from the opponent�s 35 yard line Chance of successfully kicking field goal is ~40% Chance of gaining 2 yards is 60% Expected punt distance would be ~20 yards Which is the right decision? And when? Football Primer

5. Sackrowitz (2000) �Refining the Point(s)-After-Touchdown Decision� Backwards induction (based on the number of possessions remaining) to find optimal policy No quantitative assessment of the difference between optimal strategy and the decisions actually implemented by NFL coaches Romer (2003) �It�s Fourth Down and what does Bellman�s Equation Say?� Uses play-by-play data for 3 years of NFL play to solve a simplified version of the problem to determine what to do on fourth down Key assumption is that the decision is made in the first quarter Results are that NFL coaches should generally go for the first down more frequently Others Carter and Machor (1978) Bertsekas and Tsitiklis (1996) Carroll, Palmer and Thorn (1998) Brief Literature Review

6. Model setup Model one half of a game Approximately 500,000 states. One for each combination of: Score differential Team in possession of ball Ball position on field Down Distance to go for first down Time remaining The half was modeled as consisting of 60 time periods (equivalent to 60 plays) Reward value created for each state represents the probability that team 1 will win the game Transition probabilities We estimated all probabilities required for the model Solution approach Backwards induction to find optimal decision at each state Problem Formulation

7. Solution Technique

8. Optimal vs. Heuristic

9. Optimal vs. Heuristic

10. Comparison of Play Selection

11. Results

12. Near Goal Results

13. Model Limitations

14. Estimating reward values State sampling For each time period, sample 1,000 states according to a series of distributions that should represent the most commonly reached states at certain points in an actual game Outcome sampling For each feasible action in each state, sample one possible outcome for each action and set the Q value corresponding to that action equal to the sample�s Q value The state�s Q value is set to the maximum Q value returned Approximate DP Approach

15. Estimating reward values (continued) Fitting basis functions Given our sample of 1,000 states with Q values, we fit linear coefficients to our basis functions to solve the least squares problem The basis functions that we employed were: Team in Possession of Ball Position of ball Point differential Score indicators Winning by more than 7 Winning by less than 7 Score tied Losing by less than 7 Down indicators 3rd down for us 3rd down for them 4th down for us 4th down for them Approximate DP Approach

16. Basis Functions

17. Determining approximate policy Using the basis functions, can calculate Q values for all states Iterate through all states and determine the optimal action at each state based on the relevant Q values for the potential states that we could transition to. Comparison to heuristic policy Employ backwards induction to solve for the exact reward values for all states given that team 1 is playing the approximate policy and team 2 is playing the heuristic policy ADP vs. Exact Solution

18. ADP v. Exact Results

19. Comparison of Play Selection

20. Comparison of Performance

21. Optimal Policy The implementation of the optimal policy resulted in an average increased winning percentage of 6.5% in the initial states which we considered representative The algorithm was able to run on a PC in 32 minutes (incorporating some restrictions on the state space to achieve this performance) Approximate Policy The implementation of the approximate policy resulted in an average increased winning percentage of 3.5% in initial representative states The algorithm ran in 2.3 minutes Next Steps Get transition probabilities from real data Incorporate more decisions Improve the heuristic and basis functions Conclusions