Here is a game theory analysis of an end-of-game basketball scenario. This paper by Shawn Ruminski appeared in Presh Talwalkar’s blog, Mind Your Decisions. Here is the problem.
The fourth quarter is approaching its end. Team A is leading the game with two points and with the shot clock turned off. The ball is now with Team B. They have two choices: aim for a win with a three-point attempt or a tie (and Overtime) with a two-pointer attempt. What is the right strategy?
Here are some assumptions:
- Open 2-point Field Goal %: 62.5
- Open 3-point Field Goal %: 50.0
- Contested 2-point Field Goal %: 35.7
- Contested 3-point Field Goal %: 22.8
- The chance of winning the O/T is 50:50
Team B has two choices: attempt 2 or 3 points, and Team B has two: defend 2 or 3. Following are the payoffs.
If Team B goes for a 2-pointer and Team A defends against a 2-pointer. The probability of team B winning is 0.357 x 0.5 = 0.179 (FG% for a contested 2 pt followed by winning in O/T).
If Team B goes for a 2-pointer and Team A defends against a 3-pointer. The probability of team B winning is 0.625 x 0.5 = 0.313 (FG% for an open 2 pt followed by winning in O/T).
If Team B goes for a 3-pointer and Team A defends against a 2-pointer. The probability of team B winning is 0.5 x 1 = 0.5.
If Team B goes for a 3-pointer and Team A defends against a 3-pointer. The probability of team B winning is 0.228 x 1 = 0.228
| Team A (Defend) | |||
| Defend 2 | Defend 3 | ||
| Team B (Shoot) | Shoot 2 | (0.18, 0.82) | (0.31, 0.69) |
| Shoot 3 | (0.50, 0.50) | (0.23, 0.77) |
As you can see from the payoff matrix, there is no dominant strategy for either team; therefore, there must be a mixed strategy.
Reference
Game theory applied to basketball by Shawn Ruminski: Mind Your Decisions.
