Jump-Shot Or Drive? (Using Mixed Strategy Nash Equilibria to Predict Player Behavior) By Patrick Long
Game Theory A branch of economics Deals with Strategic interactions between entities Frames theses interactions as “Games” Has applications from Evolutionary Biology to simple games (Rock – Paper – Scissors)
Mixed Strategy Equilibria Games Two by two payoff Matrix Because there is no Nash Equilibrium the players place “weights” on the options that their opponent might take. Both Players seek to make their opponents indifferent between their two options. We can solve for the mixed strategy equilibrium:
Mixed Strategy Games (continued….) 60p + 20(1-p) = 30p + 90(1+p) 40p + 20 = 90 – 60p P = .7 1-P = .3 40q + 70(1-q) = 80q + 10(1-q) 70 – 30q = 10 + 70q Q = .6 1-Q = .4
Mixed Strategy Games continued… When Offensive Players improves either his jump-shot or drive, the Defender will counter this by placing more weight on that option. Thus the player ends up using their worse action, more as a result. The new weights would be: Q = .5 1-Q = .5
Assumptions of the Model Rationality Perfect (near perfect information) No Pure Strategy Nash Equilibrium Simultaneous, or Near Simultaneous Game
Hypothesis In a one on one basketball game, players defend the jump-shot or drive, based upon their knowledge of their opponents abilities. If one player improves their raw shooting ability, and their opponent observes this, then they will switch to defending their jump- shot more, which will enable their opponent to drive more. Cases where this does not happen includes players who have a “Pure Nash Equilibrium” to always drive or always shoot a jump-shot. Evidence for this mixing should be observable via a positive correlation between shooting & Driving Percentages Athletic and Anthropomorphic variables play an unknown role in player’s behavior.
Is This Actually True?
Project Procedure Select Basketball Teams to supply Players for the Study Control For Factors that could influence the result (Scheduling, Injuries, Location) Have each player shoot a sample of Jump Shots to determine a raw shooting percentage. Have each player play one on one against each other to determine how often they drive vs. shot a Jump-Shot Collect Data on player’s Athletic Ability and Anthropomorphic Attributes to determine changes
The Participants
Player Profile Data (Athletic Variables)
Player Profile (Shooting/Driving Data)
Raw Shooting Percentage vs. Driving Percentage
Outliers
Raw Shooting Percentage vs. Driving Percentage (Outliers Omitted)
Other Statistically Significant Correlations
Driving Percentage to ¾ Sprint
Possible Explanations For Outliers Athletic Ability (well Below Relative Strength Composite Averages) Absolute Strength vs. Relative Strength Basketball IQ (General IQ) Basketball Player’s attitudes Coaching
Pure Nash Equilibrium A Basketball Player who is an exceptionally Good Jump-Shooter would have a Pure Nash Equilibrium. Exceptional Drivers, Shooters, or extremely Poor Defenders are the Most Likely Examples.
Multiple Regression Model
The Impact?
Applications and Uses Scouting & Drafting Player Development Better Understanding of the Game of Basketball Application of Game Theory to a new Area (To the real world)
Reasons for Further Research Sample size, composition, distribution etc. Other Outliers More complex models and Experiments
Sources Gibbons, Robert. Game theory for applied economists. Princeton University Press, 2015. Spaniel, William. Game theory 101: the complete textbook. CreateSpace, 2011. Shea, Stephen M., and Christopher E. Baker. Basketball Analytics: Objective and Efficient Strategies for Understanding How Teams Win. CreateSpace Independent Pub. Platform, 2013. https://www.theatlantic.com/entertainment/archive/2015/06/nba-data- analytics/396776/ https://www.basketball-reference.com/draft/NBA_2009.html
Questions?