Game Theory Applied to Crunch Time in Basketball
This paper was written by a few students at the University of California – Berkley as a final paper for their game theory class. It is particularly interesting because it applies game theory to late game decisions of NBA teams, marrying my love of basketball to key game theory concepts discussed in lecture.
One of the scenarios it analyzed consisted of a situation where a team is down by two points with less than 24 seconds left in the game. In this situation, the coach usually calls a timeout and designs a play that results in their team shooting a two or three point shot just before the expiration of the shot clock. The simple sequential game analysis of the situation shows that this situation can be set up as a game where the shooting team can choose to shoot either a two or three; Applied with rough NBA league wide shooting percentages, this game shows that the success rate a two pointer is 50% and 33% for a three pointer. Therefore, shooting a three is considered the dominant strategy for the shooting team in this game because it not only succeeds one third of the time, but it also allows the team to outright win the game. This value is greater than the 50% chance of the team making a two point shot multiplied by the 50% chance of winning in overtime.
However, this oversimplifies the event because it does not take into consideration the strategy of the defending team. A better analysis would be a simultaneous game because each coach must implement his strategy during the timeout before knowing the strategy of the opposing team. Taking this into account changes the percentages of shot success: if the defense is looking to stop a three pointer, the percentage of success of the shooting team drops to a meager 15% whereas the success of making a two point shot rises to 70%; if the defense looks to defend the two point shot, it lowers two point success rate to 33% while raising three point success rate to 50%. This situation yields no dominant strategy for either team. The other team could effectively counter any predictable strategy by their opposition.
As a result, the best solution consists of a mixed strategy equilibrium. Each team must mix up their plays in an unpredictable manner to remain asymmetric and keep their opponent off guard. This is integral for victory. At the equilibrium point, each team is randomizing its strategies enough so that the other team does not have a dominant strategy and thus, must also mix its strategies in an optimum percentage. The mixture leaves the opponent indifferent between the two choices. The math is then shown in the paper:
Shooter’s Best Response:
Let q = % of time defender defends 3
The expected payoff to the shooter is:
q x .15 + (1-q) x .5 if shooting a 3
q x .35 + (1-q) x .165 if shooting a 2
Therefore, the shooter should shoot the 3 if:
q x .15 + (1 – q) x .50 > q x .35 + (1 – q) x .165
q < .626, meaning the shooter should always shoot the three if the defender defends against the three pointer less than 62.6% of the time.
Defender’s Best Response:
Let p = % of time shooter shoots 3
The expected payoff to the defender is:
p x .85 + (1 – p) x .65 if defending 3
p x .50 + (1 – p) x .835 if defending 2
Therefore, the defender should defend the three if:
p x .85 + (1 – p) x .65 > p x .50 + (1 – p) x .835
p > .346, meaning the defender should always defend the 3 if the shooter shoots the three more than 34.6% of the time
The equilibrium point consists of the shooting team attempting a three pointer 35% of the time, with the defending team defending against it 63% of the time. This predicts that the shooting team should win the game 28% of the time, while the defending team 72% of the time. For that reason, the most common outcome for this type of end game situation is that the shooting team attempts a much higher percentage two point shot and send the game to overtime.