Goodhart’s Law and getting beaten on the near post

I would have loved to do this post with data but I’m not aware of any source from where I could get data for this over a long period of time. Recent data might be available with vendors such as Opta, but to really test my hypothesis we will need data from much farther back – from the times when few football games were telecast, let alone “tagged” by a system like Opta. Hence, in this post I’ll simply stick to building my hypothesis and leave the testing for an enterprising reader who might be able to access the data.

In association football, it is more likely for an attacker to have a goalscoring opportunity from one side rather than from straight ahead. Standing between the attacker and the net is the opposing goalkeeper, and without loss of generality, the attacker can try to score on either side of the goalkeeper. Now, because of the asymmetry in the attacker’s position, these two sides of the goalkeeper can be described as “near side” and “far side”. The near side is the gap between the goalkeeper and the goalpost closest to the attacker. The far side is the gap between the goalkeeper and the goalpost on the farther side.

Red dot is goalkeeper, blue dot is striker.


However, my hypothesis is that this has not been the case recently. For a while now (my football history is poor, so I’m not sure since when) it has been considered shameful for a goalkeeper to be “beaten at the near post”. The argument has been that given the short distance between himself and the near post, the goalie has no business in letting in the ball through that gap. Commentators and team selectors have been more forgiving of the far post, though. The gap there is large enough, they say, that the chances of scoring are high anyway, so it is okay if a goalie lets in a goal on that side.

Introductory microeconomics tells us that people respond to incentives. Goodhart’s Law states that

When a measure becomes a target, it ceases to be a good measure.

So with it becoming part of the general discourse that it is shameful for a goalkeeper to be beaten on the near side, and that selectors and commentators are more forgiving to goals scored on the far side, goalkeepers have responded to the changed incentives. My perception and hypothesis is that with time goalkeepers are positioning themselves closer to their near post, and thus leaving a bigger gap towards the far post. And thus, they are not any more optimizing to minimize the total chance of scoring a goal.

But isn’t it the same thing? Isn’t it possible that the optimal position of the goalkeeper for stopping a shot be the same as that of stopping a shot on the near side? The answer is an emphatic no.

Let us refer to the above figure once again. Let us assume that the chance of scoring when the angle is theta be f(theta). Now, we can argue that this is a super-linear function. That is, if theta increases by 10%, the chances of scoring increase by more than 10%. Again we could use data to prove this but I think it is mathematically intuitive. Given that f(theta) is super-linear, what this means is that 1. The function is strictly increasing, and 2. The derivative f'(theta) is also strictly increasing.

So, going by the above figure, the goalkeeper needs to minimize f(theta_1) + f(theta_2). If the total angle available is theta (= theta_1 + theta_2), then the goalkeeper needs to minimize f(theta_1) + f(theta - theta_1). Taking first derivative and equating it to zero we get,

f'(theta_1) = f'(theta - theta_1)

Because f is a super-linear function, we had argued earlier that its derivative is strictly increasing. Thus, the above equality implies that theta_1 = theta - theta_1 or theta_1 = theta_2 or f(theta_1) = f(theta_2).

Essentially, if the goalkeeper positions himself right, there should be an equal chance of getting beaten on the near and far posts. However, given the stigma attached to being beaten on the near post, he is likely to position himself such that theta_1 < theta_2, and thus increases the overall chance of getting beaten.

It would be interesting to look at data (I’m sure Opta will have this) of different goalkeepers and the number of times they get beaten on the near and far posts. If a goalie is intelligent, these two numbers should be equal. How good the goalkeeper is, however, determined by the total odds of scoring a goal past him.

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