# Game Theory 101: Soccer Penalty Kicks

In honor of the 2010 Planet Cup, William Spaniel displays how to optimizing kicking and defending penalty kicks. Should the kicker kick to his stronger facet much more often? The solution could surprise you. Observe: This is a total info sport. If the goalie does not truly know which aspect is the kicker’s weak one, then the equilibrium will be diverse.

@niallsp I think it is like that in the video…no?

@JimBobJenkins oh, so in the video it should have been UL = Sigma KL and UR = Sigma KL?

@CogitoErgoCogitoSum The MSNE is not an arbitrary distribution as long as both of the negative numbers are the same, and it wouldn’t be sensible to model it with different numbers. (That would be like I am more okay with giving up a goal to the right than I am to the left, which is…strange.)

@niallsp The utilities are for the goalie and the sigmas (not thetas) are for the kicker. Both sigmas are for kicking left, so that should resolve the issue.

how come it was UL = theta KL and UR = theta KR but when you set them equal they both became KL?

@CogitoErgoCogitoSum The -1 is for how much the soccer players Team lost for the opposition scoring the point. since it was one point the goalie loses one point.

Hold on. You arbitrarily stick a -1 in there… and understandably so. But you could just as easily have stuck in a -2, -3, so forth. Or different values for each. The point of putting a negative point of any value in there is to force the goalie to make a reasonable decision. But if their values are arbitrary then your MSNE has an arbitrary distribution.

So because the kicker knows that the goalie knows his stronger side and will defend that side more frequently, it does not make sense to play his stronger side more frequently, thus the kicker will assign a higher probability to his weaker side (knowing that the goalie will cover that side less frequently). So this is actually realistic (given that both have complete information). I think now I got it, thanks a lot!!!

@billaudesvarennes This is saying that if both players know the kicker’s stronger side, this is not a sensible strategy.

By the way, these lectures series are very helpful and intuitive. Great job!!!!!

@billaudesvarennes However, this does not have any implication for the actual way this game is going to be played, since the kicker will definitely prefer his right side, right?

If I am understanding this correctly, the probability distribution of the kicker’s strategy is supposed to make the goalie indifferent between his pure strategies (DL or DR), thus the kicker must player KR with a lower probability to ensure the goalie’s indifference between sides is still given. Otherwise, if the kicker would play his stronger side (KR) with a higher probability, the goalie could not be indifferent between DL and DR anymore, he would cleary prefer DR, is this correct so far?