In a very interesting article, titled World Cup Game Theory - what economics tells us about penalty kicks, Financial Times columnist Tim Harford gives a simple and intuitive introduction to how to apply game theory to soccer penalty kicks. I really recommend reading his article. (After writing this entry, I detected a flaw in this article. More on that here.)

In this entry, I'll examine the game theory of penalty kicks in some more detail, and we'll arrive at an actual formula for each player of the game. Due to the mathematical nature of this examination, there will be some math that may look dense and deterring, but it really looks more complicated than it actually is. As an attempt to make it easier to follow, I've used some color coding. Maybe, this serves only to make it look messier. Please tell me what you think in the comments.

Now, let's get started

A penalty kick can be reduced into a simple grid game. There are two players: The penalty kicker, who has the choice of which direction to shoot, and the goalie, who has the choice of which way to throw himself.

As Tim Harford points out, there is not enough time for the goalie to see which way the ball is going and to subsequently choose to go in that direction in order to intercept the ball. He must guess, risking going in the complete wrong direction. So the goalie's choice of direction is independent of the shooters choice (1).

Most shooters have a stronger and a weaker side, and should tend to favor their stronger side. But if they always choose to aim at the stronger side, the goalie can exploit this by always going that direction. The shooter can then counter-exploit this by shooting at the other side, where he will most certainly score a goal even though it's his weaker side, since the goalie will be going the other way. The goalie then reacts to this, and we have a never-ending loop of exploitations and counter-exploitations. To solve this problem, we need to find a game-theory optimal solution that offers an equilibrium to the game.

Game-theory optimal play

If both players play game-theory optimally, neither one can better his chances by altering his strategy. If he could, his strategy wouldn't have been optimal. In the same fashion, the optimal strategy is unexploitable, since the opponent can't gain an additional edge on it by switching from his optimal strategy. Remember how the shooter could elect to always shoot to his weaker side to exploit a strategy where the goalie always go to the shooters' stronger side? That means that the goalie's strategy wasn't optimal.

Now, if neither player can gain an edge by altering their strategy, then, by definition, they're indifferent between their choices. If they weren't indifferent, one choice would be better than the other, and the player would gain an edge by opting for that choice. So, in order to find the game-theory optimal strategies, we should look for indifference points.

Strategy values

This game can easily be summarized into a grid as follows:

In this entry, I'll examine the game theory of penalty kicks in some more detail, and we'll arrive at an actual formula for each player of the game. Due to the mathematical nature of this examination, there will be some math that may look dense and deterring, but it really looks more complicated than it actually is. As an attempt to make it easier to follow, I've used some color coding. Maybe, this serves only to make it look messier. Please tell me what you think in the comments.

Now, let's get started

A penalty kick can be reduced into a simple grid game. There are two players: The penalty kicker, who has the choice of which direction to shoot, and the goalie, who has the choice of which way to throw himself.

As Tim Harford points out, there is not enough time for the goalie to see which way the ball is going and to subsequently choose to go in that direction in order to intercept the ball. He must guess, risking going in the complete wrong direction. So the goalie's choice of direction is independent of the shooters choice (1).

Most shooters have a stronger and a weaker side, and should tend to favor their stronger side. But if they always choose to aim at the stronger side, the goalie can exploit this by always going that direction. The shooter can then counter-exploit this by shooting at the other side, where he will most certainly score a goal even though it's his weaker side, since the goalie will be going the other way. The goalie then reacts to this, and we have a never-ending loop of exploitations and counter-exploitations. To solve this problem, we need to find a game-theory optimal solution that offers an equilibrium to the game.

Game-theory optimal play

If both players play game-theory optimally, neither one can better his chances by altering his strategy. If he could, his strategy wouldn't have been optimal. In the same fashion, the optimal strategy is unexploitable, since the opponent can't gain an additional edge on it by switching from his optimal strategy. Remember how the shooter could elect to always shoot to his weaker side to exploit a strategy where the goalie always go to the shooters' stronger side? That means that the goalie's strategy wasn't optimal.

Now, if neither player can gain an edge by altering their strategy, then, by definition, they're indifferent between their choices. If they weren't indifferent, one choice would be better than the other, and the player would gain an edge by opting for that choice. So, in order to find the game-theory optimal strategies, we should look for indifference points.

Strategy values

This game can easily be summarized into a grid as follows:

GS | GW | |

SS | 50% | 95% |

SW | 80% | 30% |

where the rows represent the shooter's strategy choices and the columns represent the goalie's strategy choices. Each cell correspond to a combination of the choices of both players, and the figure is the corresponding chance of a goal. GS means that the goalie throws himself in the shooter's strong direction, and GW that he opts for the shooter's weak side. SS and SW relate to the corresponding shooting strategies.

Now, those figures are just made up for the purpose of illustration. I don't claim that they're realistic in any way. Notice though, that I've taken into account the chance that the shooter misses the goal even if the goalie goes the wrong way, and that there is a bigger risk for this when he shoots to his weaker side. Notice, also, that the chance of a goal is greater if he opts for the stronger side and the goalie goes the right way, than if he opts for the weaker side and the goalie goes that way.

So, we have 2 strategies for each player: SS and SW for the shooter, and GS and GW for the goalie. We have 4 strategy pairs, SSGS, SSGW, SWGS and SWGW, with corresponding outcome values. (2)

Calculating the goalie's strategy

If the shooter chooses the strategy SS S% of the time, he will chose the strategy SW 1-S% of the time (3). Similarily, if the goalie chooses the strategy GS G% of the time, he will chose strategy GW 1-G% of the time. We should now solve for S and G, which are the strategy variables for the two players.

The expected value for the shooter of strategy SS is:

E(SS) = G * SSGS + (1 - G) * SSGW

In plain English, this means that the shooter will obtain an outcome value of SSGS (50% in our example) the G% of times when the goalie chooses the strategy GS, and he'll obtain a outcome value of SSGW (95% in our example) the 1-G% of times when the goalie chooses the strategy GS.

Similarily,

E(SW) = G * SWGS + (1 - G) * SWGW

Now, here comes the magic. The shooter is indifferent when E(SS) = E(SW), as explained above. To find this point, we'll insert the two equations above into that equation:

E(SS) = E(SW)

G * SSGS + (1 - G) * SSGW = G * SWGS + (1 - G) * SWGW

G * SSGS + SSGW - G * SSGW = G * SWGS + SWGW - G * SWGW

G * SSGS + G * SWGW - G * SSGW - G * SWGS = SWGW - SSGW

G * (SSGS + SWGW - SSGW - SWGS) = SWGW - SSGW

G = SWGW - SSGW / (SSGS + SWGW - SSGW - SWGS)

And that's the formula for the indifference point for the shooter. When the goalie chooses his actions according to this formula, the shooter will be indifferent to his strategy choices. Plugging our example figures into the formula:

G = SWGW - SSGW / (SSGS + SWGW - SSGW - SWGS)

G = 0.3 - 0.95 / (0.5 + 0.3 - 0.95 - 0.8)

G ~ 0.684

The goalie should go for the shooter's stronger side about 68.4% of the time.

Calculating the shooter's strategy

Similarly, we calculate the expected values of the goalie's strategy choices:

E(GS) = S * SSGS + (1 - S) * SWGS = S * SSGS - S * SWGS + SWGS

E(GW) = S * SSGW + (1 - S) * SWGW = S * SSGW - S * SWGW + SWGW

Notice that high outcome values are bad for the goalie, as that means a large probability of a goal. But that doesn't matter to us. We don't care about the exact values of the strategies, as long as they're equal. Now for the indifference equation:

E(GS) = E(GW)

S * SSGS - S * SWGS + SWGS = S * SSGW - S * SWGW + SWGW

S * (SSGS + SWGW - SSGW - SWGS) = SWGW - SWGS

S = SWGW - SWGS / (SSGS + SWGW - SSGW - SWGS)

Again, plugging in the example figures:

S = SWGW - SWGS / (SSGS + SWGW - SWGS - SSGW)

S = 0.3 - 0.8 / (0.5 + 0.3 - 0.95 - 0.8)

S ~ 0.526

Conclusion

The shooter should opt for his stronger side only 52.6% of the time, while the goalie goes that way 68.4% of the time. Both players are then indifferent to their choices. This means that either one of them could choose any action, and still obtain exactly the same result. But biasing towards one decision opens up for the opponent to exploit that bias, and that's why they should stay with the frequencies prescribed by the solution (4). They cant do better by changing, but they can do worse, if their opponent catches on.

Next entry on this topic: Soccer Penalty Article Flawed?

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Notes:

(1) Actually, some goalies have developed an ability to, based on the movements of the shooter prior to the impact with the ball, anticipate in what direction the shooter will aim. This gives him a greater time frame in order to decide which way to go. On the other hand, some shooters have developed a counter-technique of bluffing which way he's going to aim, so this boils down to a game of it's own. For simplicity, we'll assume that there are no prior indications as to which way the shooter will aim.

(2) Here, we consider the probability of a goal, for each strategy pair, an outcome. Even if the eventual, actual outcome of the kick is still uncertain, the probability of a goal, given a specific strategy pair, serves as a value of that strategy pair for each player.

(3) As explained in the entry Probability Theory for Dummies.

(4) If the opponent isn't playing the optimal strategy, however, one might consider making exploitative adjustments to the optimal strategy. Bear in mind, though, that this opens up for counter-exploitation, and should thus only be done if one don't expect the opponent to catch on.

External links in this post:

World Cup Game Theory - What economics tells us about penalty kicks

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## 2 comments:

Yeah I'm a 12th grader and I don't understand this article one bit, and I would like you to tickle my butthole with your love stick and call me Samantha

Too bad this wont work in fifa.

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