“Winning Time” and Game Theory
When I was told to write a blog post for a class, my first hope was that I could somehow relate a topic to my favorite sport, so naturally I Googled “game theory basketball.” Low and behold, one of the first hits satisfied my desire. For their final paper in a game theory class, some students at UC Berkley analyzed the tail end of a professional basketball game. It goes through applications of game theory to the decisions of NBA teams late in the game, or as Michael Jordan has so properly deemed it, “Winning Time.”
The paper analyzes two end-of-game situations; one where the team is down by two points and another by three points, both with less than 24 seconds remaining (the length of an NBA shot clock). I will go through the first of these in this post. In these scenarios, a play is usually drawn up to allow the team to run down the clock and attempt either a two or three point shot just before the clock expires. Given rough estimates of league-wide average shooting percentages, this game holds that the approximate success rates for two and three pointers are 50% and 33% respectively.
If it were so simple, attempting a triple would be clearly considered a dominant strategy in the first scenario (down by 2) because it succeeds one third of the time as well as allows the team the chance to 100% win the game at that moment. If you take the two-pointer, the 50% chance of making the shot is multiplied by the 50% chance (assuming teams are fairly equally matched) of pulling out a win in overtime. Put in inequality form:
½ * ½ = ¼ < 1/3
Unfortunately for basketball teams everywhere, this is an oversimplification of the situation because it does not account for the strategy of the defending team. For the sake of calculation, we assume the defending team can either focus their strategy on preventing the three or preventing the two, with no in between or gray area. So, this becomes a simultaneous game where neither team nor coach knows the strategy of their opponent. Taking this into consideration, shot success rates are altered: if the defense looks to prevent a three point shot, the percentage of success of for a three pointer falls to just 15% and the success rate of two point shots increases to 70%; if the D tries to stop the two point shot, two pointer success rates fall to 33% and three pointer success rates rise to 50%.
| Defense | ||||
| Offense | Defend 2PS | Defend 3PS | ||
| 2PS |
16.5%, 83.5% |
35%, 65% |
||
| 3PS |
50%, 50% |
15%, 85% |
||
(all two point percentages, multiplied by 50% for overtime win probability)
This game yields no pure strategy equlibria for either the defense or the offense. One team could effectively counter any of their opposition’s strategies.
What results is a mixed strategy equilibrium. The teams must vary their strategies at random to keep from being predictable. The mixture leaves the opponent indifferent between the two choices. The math used to calculate this mixed strategy equilibrium is shown below:
“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 mixed equilibrium point is made up of percentages: The offense should attempt a three pointer approximately 35% of the time, while the defense should guard against the three 63% of the time. In this equilibrium, the shooting team should pull out a win 28% of the time, and the defending team should be victorious 72% of the time. The most common occurrence in this scenario is that the offensive team will attempt a two to send the game into overtime.
http://faculty.haas.berkeley.edu/rjmorgan/mba211/Chow%20Heavy%20Industries%20Final%20Project.pdf