A mixed Nash Equilibrium strategy for soccer penalty kicks is mentioned at some Vanderbilt lecture.

Quick recollection of Nash Equilibrium,

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A Nash equilibrium, named after John Nash, is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy.

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So onto the soccer thing,

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Economist Ignacio Palacios-Huerta analyzed 1,417 penalty kicks from five years of professional soccer matches among European clubs. The success rates of penalty kickers given the decision by both the goalie and the kicker to kick or dive to the left or the right are as follows:

Goalie | |||

Left | Right | ||

Kicker | Left | 58% | 95% |

Right | 93% | 70% |

In all cases, left and right is from the kicker's perspective. If the goalie guesses the kicker's direction correctly, he will block about 3 or 4 kicks out of 10. If the goalie guesses wrong, the kicker's chance of success is very high. Next, we calculate the mixed strategy equilibrium. Let *p* be the probability that the goalie jumps to the left and *1-p* be the probability he jumps right. To make the kicker indifferent, we must solve:

payoff from kicking left | = | payoff from kicking right |

58p + 95(1-p) | = | 93p + 70(1-p) |

The result is *p*=42%. The goalie must jump left 42 out of 100 times to make the kicker indifferent between kicking left and right.

Next, we turn to the kicker's strategy that makes the goalie indifferent. First, note that the table above has only the kicker's payoffs represented by the probability of success. The goalie's payoffs are the opposite: the probability of a miss. We can rewrite the above table to represent the chance that the kicker misses by subtracting the numbers in the table from 100.

Goalie | |||

Left | Right | ||

Kicker | Left | 42% | 5% |

Right | 7% | 30% |

Let *q* be the probability that the kicker kicks to the left and *1-q* be the probability he kicks right. To make the goalie indifferent, we must solve:

payoff from jumping left | = | payoff from jumping right |

42q + 7(1-q) | = | 5q + 30(1-q) |

The result is *q*=39%. The kicker must kick left 39 out of 100 times to make the goalie indifferent between jumping left and right.

Surprisingly, the game is not very symmetric between kicking left and right which, in turn, implies that the relative frequencies of left and right for the goalie and the kicker should not be 50-50. How well does game theory predict actual behavior? Here is the actual behavior of kickers and goalies in the 1,417 observed penalty kicks:

- Kickers:
- Predicted proportion of kicks to the left: 39%
- Observed proportion of kicks to the left: 40%

- Goalies:
- Predicted proportion of jumps to the left: 42%
- Observed proportion of jumps to the left: 42%

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Yep, now let's explain this to the kids...

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