Goalkeepers and rationality

At Stumbling and Mumbling the author is discussing why goalkeepers don’t maximise their chance of making a save from a penalty kick. According to this paper they only stood still during a penalty kick 6.3% of the time, even though 28.7% of kicks were down the middle.

Mr Mumbling puts forward three reasons why the goalkeeper may stand still less than is optimal:

  1. It puts pressure on the striker in some way,
  2. It is a social norm – way of minimising regret (as a dive looks cooler than standing still),
  3. Goalkeepers also value not getting yelled at, it is less likely people will make fun of you if you miss a penalty kick when diving than when you miss the kick when not moving.

These are all good reasons which probably explain the phenomenon, however I have a couple of other ideas:

Firstly, if each penalty kick is a separate event, would the goalkeeper not choose the strategy that maximises his chance of stopping the ball.  Down the middle is occurs less than a third of the time.  If there are only two other choices, left or right, then either one or both of them is more likely to be kicked towards.  As a result, if the keeper thought like this it could make sense to do some combination of diving.

However, the incredibly obvious issue with this is that there are more than two other choices – there is top left, top right, bottom lef, and bottom right (at least).  I would be interested to find out what they defined as middle and other, and how high the probability is of stopping it in these place even if the keeper picks it right!

The second reason I have involves assuming that penalty kicks are not independent.  This makes sense as goalkeepers and strikers in the modern era study each other, and so will make their decisions on where to dive and shoot based on prior information.

If this is the case then the goalkeeper needs to jump around a touch more randomly in order to maximise their probability of stopping the striker.  Think of it this way, if the keeper always stayed in the middle, the striker would always shoot to the corners, which cannot be stopped (making this sort of like the original Phillips curve relationship).  Now this begs the question, why doesn’t the distribution of diving match the distribution of shots (the question we are trying to answer from the beginning).

In this strategic case, the probabilities may differ because the probability of stopping the ball may be different in different areas.  It’s harder to stop a ball that goes in the corner, and as a result the excess jumping to the corners may be a strategy to minimise how much they aim at the corner, or conversely a strategy to make it harder to score from a corner (by forcing the striker to shoot past the net).  Until we know more about the keepers and strikers ability to save and score from different shots we can’t make a conclusion – however this does not imply that the current probabilities are sub-optimal.

Also, the goalkeeper may be making a dive subject, not only to wanting to save the ball now, but also to shape the belief structure of future penalty takers.  If this is the case there are another two reasons why the keeper may dive more than he needs to:

  1. It may change the penalty takers beliefs in such a way that the keepers payoff from his strategy is greater,
  2. If goals have different payoffs in different games, then keepers may act sub-optimally in the prior games in order to  (attempt to) influence penalty takers beliefs in the main game.

All in all, we have plenty of reasons why keepers head to the corner more often than the ball.