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

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