Game Theory and Defending Against a 2-on-1
https://puckplusplus.com/2017/12/16/game-theory-and-defending-against-a-2-on-1/
One of the oldest questions in hockey is how to properly defend against a 2-on-1. Should the defender play the pass or the shot? From a young age defenders are taught to play the pass and let the goalie defend the shot. A shot attempt is more likely to be successful when assisted by a pass; therefore the defender is covering the higher percentage play by blocking the pass. The problem with always playing the pass is that the opponents will start strengthening their shot instead of passing, resulting in a 1-on-0. This split second decision has turned into a Game Theory Matrix. The defender must decide whether to cover the pass or the shot and the attacker must decide whether to pass or shoot.
After finding many stats and developing accurate estimates the following matrix is created. Each number represents the probability the shooter makes the shot.
| Attacker Shoot | Attacker Pass | |
| Defender Play Shot | 0.124 | 0.214 |
| Defender Play Pass | 0.142 | 0.134 |
The matrix above shows that the defender is in the best position when correctly playing the attacker’s shot. If the defender plays the shot and the attacker elects to pass, the attacker is in his best position. Based on what we learned in class it is clear that there is no Pure Nash Equilibrium. However, we can use the equation: P * xG(Pass | Defend Pass) + (1 – P) * xG(Shoot | Defend Pass) = P * xG(Pass | Defend Shot) + (1-P) * xG(Shoot | Defend Shot), to estimate that an attacker should shoot 82% of the time and pass only 18% percent of the time. We can then use the equation; Q * xG(Defend Pass | Pass) + (1 – P) * xG(Defend Shot | Pass) = P * xG(Defend Pass | Shoot) + (1-P) * xG(Defend Shot | Shoot), to estimate that a defender should play the shot 8% of the time and play the pass 92% percent of the time.
These findings strongly support a defender playing the pass. They also show that an attacker should shoot the puck instead of passing on a 2-on-1 break.