| 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) |
Let p be the probability for the shooting team to shoot a 3-pointer. The optimal value of p is such that the defending team get no incentive to defend a two over three.
The defensive team’s incentive to defend two = 0.82 (1-p) + 0.5 p
The defensive team’s incentive to defend three = 0.69 (1-p) + 0.77 p
Equating,
0.82 (1-p) + 0.5 p = 0.69 (1-p) + 0.77 p
0.82 p – 0.5 p – 0.69 p + 0.77 p = 0.82 – 0.69
p = 0.325
Let q be the probability for the defending team to defend a 3-pointer. At the optimal value of q, the shooting team get no incentive to shoot a two over three.
The shooting team’s incentive to shoot two = 0.18 (1-q) + 0.31 q
The shooting team’s incentive to shoot three = 0.5 (1-q) + 0.23 q
Equating,
0.18 (1-q) + 0.31 q = 0.5 (1-q) + 0.23 q
0.18 q – 0.31 q – 0.5 q + 0.23 p = 0.18 – 0.5
q = 0.8
Reference
Game theory applied to basketball by Shawn Ruminski: Mind Your Decisions.
