With the World Cup starting today, now is a great time to discuss the game theory behind soccer penalty kicks. This blog post will do three things: (1) show that penalty kicks is a very common type of game and one that game theory can solve very easily, (2) players behave more or less as game theory would predict, and (3) a striker becoming more accurate to one side makes him *less* likely to kick to that side. Why? Read on.

**The Basics: Matching Pennies**

Penalty kicks are straightforward. A striker lines up with the ball in front of him. He runs forwards and kicks the ball toward the net. The goalie tries to stop it.

Despite the ordering I just listed, the players essentially move simultaneously. Although the goalie dives after the striker has kicked the ball, he cannot actually wait to the ball comes off the foot to decide which way to dive—because the ball moves so fast, it will already be behind him by the time he finishes his dive. So the goalie must pick his strategy before observing any relevant information from the striker.

This type of game is actually very common. Both players pick a side. One player wants to match sides (the goalie), while the other wants to mismatch (the striker). That is, from the striker’s perspective, the goalie wants to dive left when the striker kicks left and dive right when the striker kicks right; the striker wants to kick left when the goalie dives right and kick right when the goalie dives left. This is like a baseball batter trying to guess what pitch the pitcher will throw while the pitcher tries to confuse the batter. Similarly, a basketball shooter wants a defender to break the wrong way to give him an open lane to the basket, while the defender wants to stay lined up with the ball handler.

Because the game is so common, it should not be surprised that game theorists have studied this type of game at length. (Game theory, after all, is the mathematical study of strategy.) The common name for the game is *matching pennies*. When the sides are equally powerful, the solution is very simple:

If you skipped the video, the solution is for both players to pick each side with equal probability. For penalty kicks, that means the striker kicks left half the time and right half the time; the goalie dives left half the time and dives right half the time.

Why are these optimal strategies? The answer is simple: neither party can be exploited under these circumstances. This might be easier to see by looking at why all other strategies are *not* optimal. If the striker kicked left 100% of the time, it would be very easy for the goalie to stop the shot—he would simply dive left 100% of the time. In essence, the striker’s predictability allows the goalie to exploit him. This is also true if the striker is aiming left 99% of the time, or 98% of the time, and so forth—the goalie would still want to always dive left, and the striker would not perform as well as he could by randomizing in a less predictable manner.

In contrast, if the striker is kicking left half the time and kicking right half the time, it does not matter which direction the goalie dives—he is equally likely to stop the ball at that point. Likewise, if the goalie is diving left half the time and diving right half the time, it does not matter which direction he striker kicks—he is equally likely to score at that point.

The key takeaways here are twofold: (1) you have to randomize to not be exploited and (2) you need to think of your opponent’s strategic constraints when choosing your move.

**Real Life Penalty Kicks**

So that’s the basic theory of penalty kicks. How does it play out in reality?

Fortunately, we have a decent idea. A group of economists (including Freakonomics’ Steve Levitt) once studied the strategies and results of penalty kicks from the French and Italian leagues. They found that players strategize roughly how they ought to.

How did they figure this out? To begin, they used a more sophisticated model than the one I introduced above. Real life penalty kicks differ in two key ways. First, kicking to the left is not the same thing as kicking to the right. A right-footed striker naturally hits the ball harder and more accurately to the left than the right. This means that a ball aimed to the right is more likely to miss the goal completely and more likely to be stopped if the goalie also dives that way. And second, a third strategy for both players is also reasonable: aim to the middle/defend the middle.

Regardless of the additional complications, there are a couple of key generalizations that hold from the logic of the first section. First, a striker’s probability of scoring should be equal regardless of whether he kicks left, straight, or right. Why? Suppose this were not true. Then someone is being unnecessarily exploited in this situation. For example, imagine that strikers are kicking very frequently to the left. Realizing this, goalies are also diving very frequently to the left. This leaves the striker with a small scoring percentage to the left and a much higher scoring percentage when he aims to the undefended right. Thus, the striker should be correcting his strategy by aiming right more frequently. So if everyone is playing optimally, his scoring percentage needs to be equal across all his strategies, otherwise some sort of exploitation is available.

Second, a goalie’s probability of not being scored against must be equal across all of his defending strategies. This follows from the same reason as above: if diving toward one side is less likely to result in a goal, then someone is being exploited who should not be.

All told, this means that we should observe equal probabilities among all strategies. And, sure enough, this is more or less what goes on. Here’s Figure 4 from the article, which gives the percentage of shots that go in for any combination of strategies:

The key places to look are the “total” column and row. The total column for the goalie on the right shows that he is very close to giving up a goal 75% of the time regardless of his strategy. The total row for the striker at the bottom shows more variance—in the data, he scores 81% of the time aiming toward the middle but only 70.1% of the time aiming to the right—but those differences are not statistically significant. In other words, we would expect that sort of variation to occur purely due to chance.

Thus, as far as we can tell, the players are playing optimal strategies as we would suspect. (Take that, you damn dirty apes!)

**Relying on Your Weakness**

One thing I glossed over in the second part is specifically how a striker’s strategy should change due to the weakness of the right side versus the left. Let’s take care of that now.

Imagine you are a striker with an amazingly accurate left side but a very inaccurate right side. More concretely, you will always hit the target if you shoot left, but you will miss some percentage of the time on the right side. Realizing your weakness, you spend months practicing your right shot and double its accuracy. Now that you have a stronger right side, how will this affect your penalty kick strategy?

The intuitive answer is that it should make you shoot more frequently toward the right—after all, your shot has improved on that side. However, this intuition is not always correct—you may end up shooting *less* often to the right. Equivalently, this means the more inaccurate you are to one side, the more you end up aiming in that direction.

Why is this the case? If you want the full explanation, watch the following two videos:

The shorter explanation is as follows. As mentioned at the end of the first section of this blog post, players must consider their opponent’s capabilities as they develop their strategies. When you improve your accuracy to the right side, your opponent reacts by defending the right side more—he can no longer so strongly rely on your inaccuracy as a phantom defense. So if you start aiming more frequently to the right side, you end up with an over-correction—you are kicking too frequently toward a better defended side. Thus, you end up kicking more frequently to the left to account for the goalie wanting to dive right more frequently.

And that’s the game theory of penalty kicks.

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When do I get more than one Nash equilibrium in game theory and how do I tell if I only have one Nash equilibrium?

I suggest a complete different game theory: penalties should be made at the START of the game. This would result for both parties to play better. At the end of the game the outcome of the penalties should be added. We would definitely enjoy the game more!