Game Development Reference
In-Depth Information
In soccer, penalty kicks launch the ball at up to 125 mph. At this speed,
the ball travels from foot to goal in about a fifth of a second. This is not
enough time for the goalie to jump and try to block the ball after it has
been kicked. His only choice is to jump before the kick. At the same time,
the kicker must choose a side without knowing which way the goalkeeper
is going to go.
This is a matching pennies game. The goalie wants to match kick
sides with the kicker, while the kicker wants the opposite.
In this game, the kicker's payoff is the likelihood that he will score.
The unequal payoffs come from the fact that every player kicks better to
one side than the other. His chances of scoring when blocked correctly are
better on his good side than on his bad side, and his chances of scoring
unblocked are better on his good than bad side as well. (Of course, his
payoff is still better on his bad side if he is not blocked than if he is blocked
on his good side; otherwise, kicking on his good side would be a pure
equilibrium strategy and he would thoughtlessly do that every time.)
The kicker's best strategy is to randomly choose his good and bad
sides in different proportions, kicking on the good side most of the time,
but mixing in an occasional off-side kick to keep the goalie honest. At the
same time, the goalkeeper must mirror him, blocking his good side most
of the time and occasionally going the other way.
We can calculate the exact proportions from the chances that the kicker
will score with each of the four possible kick side/block side combinations.
One research study gathered data from hundreds of kicks in European
league games and came up with the following table of goal chance percent-
ages (which is effectively a payoff matrix):
Goalie jumps to
good side
Goalie jumps to
bad side
Kicker uses
good side
63.6% chance of goal
94.4% chance of goal
Kicker uses
bad side
89.3% chance of goal
43.7% chance of goal
Using these numbers, a bit of math reveals that the best strategy for the
kicker is to use his good side 59.7% of the time and his bad side 40.3%
of the time. This mixed strategy gives an overall success rate of 74.0%
regardless of what the keeper does. Any deviation from these percentages
allows the goalie to improve his results by switching to a pure strategy of
