Penalty Shoot Out

P(l)ayoff table
What's the best strategy for taking or defending penalty kicks?  Essentially there are three options for each player: go left, right or centre.  But what proportion of the time should each player take each of these options?

We can try to answer this with a toy model and see what happens.  In this model the goal is always saved if the choices match and is always conceded if they don't.  What makes this model interesting is the "utility" of each outcome to each player.

We could choose 1:0 to the striker for a goal and 1:0 to the goalie for a save, but we can make things more realistic by applying a bit of psychology.  Let's face it: if the goalie doesn't move and the ball goes left or right s/he will look pretty stupid.  For that reason I have given the goalie a minus one in this case.  Likewise if the striker shoots towards the middle and the goal is saved then the striker will look silly, so in that case he or she gets a minus one instead of zero.

Given this payoff table, what are the rational strategies?  Well it's clear that there's no pure strategy Nash Equilibrium where the striker chooses one of left/right/centre and so does the goalie, but neither can improve given the choice of the other.  So Nash's theorem tells us there must be a mixed strategy equilibrium where each player chooses randomly using fixed probabilities, and cannot do any better given the probabilities chosen by the other.

Let the probabilities that the striker goes left, centre, and right be $x, y, z$ respectively.  For the goalie call these probabilities $u, v, w$.  We know that each player chooses values that make the other indifferent to his or her options.  (Why? Because otherwise the other player could improve their chances by choosing a pure strategy!)

The expected outcomes for the goalie if the goalie dives left, centre, or right are therefore equal.  So
$$
\begin{align}
x &= -x + y -z \\
 &= z \\
1 &= x + y + z
\end{align}
$$
Which is solved by
$$
x = 0.2\\
y = 0.6\\
z = 0.2\\
$$

Likewise the expected outcomes for the striker are the same whichever way the striker kicks.  So
$$
\begin{align}
v + w &= u - v + w \\
&= u + v \\
1 &= u + v + w
\end{align}
$$
Which is solved by$$
u = 0.4\\
v = 0.2\\
w = 0.4\\
$$

The upshot is that the goalie rarely stays centre even though that's where most the shots are sent.

Even more counter intuitively, the behaviour of each player is determined entirely by the utility (in the head of) the other player, and not by their own utilities.

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