Game Theory and Soccer

I like game theory. Unfortunately, it doesn't have much real-world practicality to it. However, there is one important exception: soccer. This is especially true of penalty kicks.

Since it is World Cup time, I'm gonna bring out a portion of the lecture "Best responses in soccer and business partnerships" in Yale Professor Ben Polak's Game Theory course. For those not interested in a lecture, A short interview of some noone from last year's tourney is "The Game Theory of Penalty Kicks".

Those who not wanting read the entire lecture, the cliff note version is here:

So there's a lesson here, and it's pretty much (just to make the lesson resonate again): imagine there you are in the World Cup, you're playing for England, you have to justify your actions not only to your teammates and your manager, and your boss, but to about 60 million rather angry fans. What's the lesson here? I'm hoping it was going to be obvious, what's the lesson here? The lesson is, do not shoot to the middle. Let me qualify that lesson slightly, unless you're German. Germans can do whatever they like.

For those who don't wanna watch the vid, here is a printout of what the lecturer true on the whiteboard.

So this is a game that occurs in soccer and just to give an idea of how important it is for those people who are unfortunate enough not to be soccer fans here, the last World Cup was decided on penalty kicks. In England's case, England goes out of every single World Cup and every single European competition because it loses on penalty kicks, usually to Germany, it has to be said...

So what we're going to do is we're going to look at some numbers that are approximately the probabilities of scoring when you kick the penalty kick in different directions. But just make sure everyone ─ do I need to explain what's going on here? Is everyone familiar with this situation? There's one guy who's going to run up and kick the ball. The goal keeper is standing at the goal. And their aim is to kick it into the goal. That's probably enough. You've all seen this right? If you haven't seen this, go see it. I mean come on! So things you should do in life: read Shakespeare and see a soccer game.

So the rough numbers for this are as follows ─ and actually later on in the class I'll give you some more accurate numbers, but these will do for now. There are three ways, the goal─ the attacker could kick the ball. He could kick the ball to the left, the middle, or the right. And I shouldn't just say he here of course, I mean this is he or she but if I get that wrong going on, please forgive me for it. The goalie can dive to the left or the right. In principle the goalie could stay in the middle. We'll come back and talk about that later. So this is the guy who is shooting, he's called the shooter and this is the goalie.

These are roughly ─ well, let me put up the payoffs for this game and then I'll explain them. So you'll notice that I'm just going to put in numbers here and then the negative of the number and the numbers are roughly like this: (4,-4). So the numbers are (4, -4), (9, -9), (6, -6), (6, -6), (9, -9) and (4, -4). And the idea here is that the number 4 represents 40% chance of scoring if you shoot the ball to the left of the goal and the goal keeper dives to the left. So the payoff here is something like u1(left) if the goal keeper dives to the left is equal to 4, by which I mean there's a 40% chance of scoring.

So the number for the--The payoff for the shooter is his probability of scoring and the payoff for the goal keeper is just the negative of that. Let's keep things simple. As I said before, for now we'll ignore the possibility that the goal keeper could stay put. So how should we start analyzing this important game? Well we start with the ideas we learned already several weeks ago now, or more than a week ago, which is the idea of dominant strategies. Does either player here have a dominated strategy? Does either player have a dominated strategy? No, it's kind of clear that they don't.

Let's just look at the shooter, for example. So you might think that maybe middle dominates left, but notice that middle has a higher payoff against left than shooting to the left. It has as lower payoff if the goalie dives to the right. So, not surprisingly in this game, it turns out, that if the goalie dived to the left you're best off shooting to the right, second best off shooting to the middle, and worst off shooting to the left. That's if the goalie dives to the left. And if the goalie dives to the right, you're best off shooting to the left, second best off shooting to the middle, and worst off by shooting to the right; and that's kind of common sense.

Okay, so if we had stopped the class after the first week where all we learned to do was to delete dominated strategies, we'd be stuck. We'd have nothing to say about this game and as I said before, this is the most important game, so that would be bad news for Game Theory. But luckily, we can do a little bit better than that. Before I do that, let's just take a poll of the class. How many of you, if you were playing for, I guess it's going to be America, which is a sad thing to start with, never mind. You guys are playing for America and you're taking this penalty kick and it's the last kick in the World Cup, how many of you, show of hands, how many of you would shoot to the left? How many of you would shoot to the middle? How many of you would shoot to the right?

We've got kind of an even split there, pretty much an even split. We're going to assume these are the correct numbers and we're going to see if that even split is really a good idea or not. So how should we go about thinking about this? What I suggest we do is we do what we did last time and we start to draw a picture to figure out what my expected payoff is, depending on what I believe the goalie is going to do. So this is the same kind of picture we drew last time.

So on the horizontal axis is my belief, and my belief is essentially the probability that the goalie dives to the right. Now as I did last time, let me put in two axes to make the picture a little easier to draw. So this is 0 and this is 1. And you probably have lines in your notes but I don't, so let me just help myself a bit. So this is 2, 4, 6, 8, 10, so this is going to be 2, 4, 6, 8, and 10 and over here 2, 4, 6, 8, and 10, 2, 4, 6, 8, and 10. This would be the basis of my picture.

So it starts with a possibility of shooting to the left. Let's do this in red. So I shoot to the left and the goalie dives to the left, my payoff is what? It's 4. If I shoot to the left and there's no probability of the goalie diving to the right, which means that they dive to the left, then my payoff is 4, meaning I score 40% of the time. If I shoot to the left and the goalie dived to the right, then I score 90% of the time, so my payoff is .9. By the way why is it 90% of the time and not 100% of the time? I could miss; okay, I could miss. That happens rather often it turns out, well 10% of the time.

So we know this is going to be a straight line in between, so let's put this line in. So what's this? It's the expected payoff to Player I of shooting to the left as it depends on the probability that the goal keeper dives to the right. And conversely, we can put in … well let's do them in order.

So middle: so if I shoot to the middle and the goal keeper dives to the left, then my payoff is .6, is 6, or I score .6 of the time, and if I shoot to the middle and the goalie dives to the right I still score 60% of the time, so once again it's a straight line in between. So this line represents the expected payoff of shooting to the middle as a function of the probability that the goal keeper dives to the right.

Finally ─ let's do it in green ─ let's look at the payoffs, expected payoffs, if I shoot to the right. So if I shoot to the right and the goalie dives to the left, then I score with probability .9, or my payoff is 9. Conversely, if I shoot to the right and he or she dives to the right, then I score 40% of the time, so here's my payoff .4. And here's my green line representing my expected payoff as the shooter, from shooting to the right, as a function of the probability that the goalie dives to the right.

Did everyone understand how I constructed this picture? Easier picture than the one we constructed last time. So what does everyone notice from this picture? What's the first thing that jumps out at you from this picture? Assuming these numbers are true, what jumps out at you from this picture? Can we get some mikes up here? So Ale, can we get this guy? Stand up first, the guy in red. What's your name? Don't hold the mike; just shout.

Student: There's no point at which the 6, at which it shooting in the middle gets a higher payoff.

Professor Ben Polak: Exactly, exactly. So the thing that I hope jumps out at you from this picture is (no great guesses about figuring out this is a ½), so if the probability that the goalie's going to jump to the right is less than a ½, then the best you can do is represented by this green line, which is shoot to the right. So the goalie is going to shoot to the right with the probability less than a ½, sorry he's going to dive to the right with the probability less than a ½, you should shoot to the right.

Conversely, if you think the goalie's going to shoot to the right with probability more than a ½, then the best you can do is represented by the pink line, and that's shooting to the left, or if you think the goalie's going to dive to the right with the probability more than a ½, the best you can do, your best response is to shoot to the left. And there is no belief you could possibly hold given these numbers in this game that could ever rationalize shooting the ball to the middle. Is that right? So no: to say it another way, middle is not a best response to any belief I can hold about the goal keeper, to any belief.

So there's a lesson here, and it's pretty much (just to make the lesson resonate again): imagine there you are in the World Cup, you're playing for England, you have to justify your actions not only to your teammates and your manager, and your boss, but to about 60 million rather angry fans. What's the lesson here? I'm hoping it was going to be obvious, what's the lesson here? The lesson is, do not shoot to the middle. Let me qualify that lesson slightly, unless you're German. Germans can do whatever they like.