Mixing it up

Recently I watched a soccer game in which a goal was scored similarly to the one in the video below:

When an attacker has only the goalkeeper in front of him but is approaching the goal form an angle, he has two choices. He can try and place the ball either between the keeper and the "near post" or between the keeper and the "far post." Shooting towards the near post is the less risky option; the striker is less likely to miss the target altogether than when he aims for the far post. However, a near post shot is easier for the keeper to save (provided that he dives in the right direction). In the game I was watching the striker shot towards the near post and scored, just like in the video. The announcer then said: "In situations like this one, when the striker scores on a near post shot it's always the keeper's fault." I hear that line quite often actually. It's wrong, and a little game theory can show why.

A one-on-one when the striker is charging the goal at an angle is a 2x2 simultaneous move game. The striker has two options: "shoot near" or "shoot far," and the keeper can either "dive near" or "dive far". The important thing here is that the keeper as well as the striker make their decisions simultaneously, i.e. each picks his strategy not knowing what the other will do. (If the keeper waits to see which way the striker shoots before deciding where to dive, it will be too late to stop the shot. If the striker takes too long to shoot, the keeper will simply scoop the ball from under his feet.) Now saying that a near post goal is always the keeper's fault is equivalent to saying that keepers should always dive towards the near post; since near post shots are easy to save if you dive in the same direction, this would ensure that no goals (or almost no goals) would be scored on those kind of shots. But it wouldn't be a reasonable thing for a goalkeeper to do. If the keeper always dives near, then the striker could always shoot far towards the unprotected corner of the goal, and score with a very high rate. What the keeper needs to do instead is to play a mixed strategy: randomize between diving near and diving far. This will force the striker to randomize between shooting near and shooting far, and reduce the rate at which goals are scored in these situation to a minimum. Now since near shots are easier, the keeper should protect the near post more than the far post; i.e., he should dive near with probability higher than diving far. But, in equilibrium, he can't dive near every single time, and thus every once in a while a striker will score shooting towards the near post. And it won't be the keepers fault at all.