Okay in this video we're going to talk a little bit about game theory and its application to sports. So the great mathematician James Von Neumann and the economist, great economist, Oskar Morgenstern They wrote the Theory of Games and Economic Behavior in the 1940s. Now you probably heard about game theory if you saw the movie A Beautiful Mind. And unfortunately John Nash, who the movie is about, and his wife were just killed tragically in a car crash because they didn't wear their seat belts. But he won the Nobel Prize for Economics for something called the Nash Equilibrium. And we'll talk a little bit about an equilibrium and how this applies to football. I guess soccer, on penalty kicks. FIFA's in the news today- not in a good way. And tennis a little bit, decisions in tennis. And you can probably think of other applications. Okay. So if you look at pro-football-reference.com, expected points on a passing play, and expected points on a running play are Brian Burke, formerly of Advanced Football Analytics.com has also analyzed this. You'll find passing plays average about 0.1 more points than running plays. So, why don't teams always pass? Okay? Think I may have spelled Oscar wrong there. Great new book by Oscar, the guy who played Oscar the Grouch of Big Bird by the way, if you're interested. Okay, And the reason you don't always pass is because if the defense knows a pass is coming they're going to stop it. So you have to have some threat of the run. Now we've talked earlier in the video that the running threat is not as important as most people think and we still believe that. But game theory provides a nice explanation of why you don't always pass even if passing seems to be the better play. Okay so let's take a really simple example for what's called the saddle point or a game that has an equilibrium in purer strategies. So let's suppose you have three football plays. Offense play one through three and three defenses, one through three. Here's the yards gained by the offense, given the offense calls this play, the defense calls that play. Now it turns out Offense Two versus Defense Two is in equilibrium or what's called the saddle play. And the equilibrium simply means neither player has incentive to switch. To change strategies. So if you look here, if I play offense two, consider me as the offensive player. Now if I switch from there, I'm going to do worse, I'm going to gain one or zero. If I look at defense two, sorry if I look at this point offense two versus defense two, if the defender switches, he doesn't want to give up yards. He's going to give up more yards. Nobody has incentive to change, so basically this is an equilibrium in pure strategies. There is not always an equilibrium for pure strategies, as we'll see in the following example. Let's suppose the offense can run or pass. And the defense can guess run or guess pass. Now if I run and the defense guesses run, I get slaughtered these five yards. If they guess pass, I gain five yards. If I pass on offense and they guess run, I gain ten yards. I do well. But if they guess pass, I gain zero yards. Now no matter what structure or strategy I pick, there's an incentive for a player to switch. In other words, if I pick If we look at the -0.5, the offensive player would like to switch, 'cause if it was a run defense, I should switch to pass. Okay. If I look here at five here, okay, then basically the defensive player would like to switch. And similarly down here if I play ten the defensive player wants to switch to play zero. If it were here at zero, the offensive player wants to switch. There is no equilibrium in pure strategies. No matter what we pick, somebody has incentive to switch. So what was, that's been known I'm sure for a long time. Or pure strategy equilibriums if they exist, people understood them for a long time, but the brilliant insight of Von Neumann and Morgenstern was- Was to show that basically in mixed strategies we do have an equilibrium always in two person quote unquote, zero sum gains. What one player gains the other player loses. And sports its zero sum games. there is plenty of good stuff on nonzero game theory, you may have heard of the prisoner's but that's not relevant to us in this class. Okay so how do I find a purer strategy? Okay, sort of a mixed strategy, that's sort of an equilibrium. It turns out if the offense runs half the time and the defense passes, plays pass defense three-quarters of the time. Sorry, we will have, neither player will have the incentive to switch and on average the value of the game is the offense gains 2.5 yards. Okay, so let's let q be the probability the offense runs. And let's put that go from, we'll go with zero to 0.05, to 0.1 and it can go to one. And what we would like to know is what's the expected payoff, or yards gained to the room player who has the offense, against run defense and pass defense. Okay, so we'll call this probability. So we'll call this probability for offense. So let's say we put in a- well actually, we don't need that. It's going to be in this column. Okay. So now what's the expected payoff against the run defense? Again, expected value, take probability- value times probability of that value and add them all up. Okay. So with this probability, I would call a run offense. And against the run defense, I would get minus five. And with 1 minus that probability against the run defense, I would get 10. Okay, so if I would never run against the run defense that means I'll always pass I'd get ten and this will drop. If I always run against the run defense that's stupid, I'd get minus five. Now what about the pass defense if this is probably a run. Well, with this probability against the pass defense, I would get five. And with one minus that probability I would pass and I would get zero. Okay. Now, remember this where the idea behind two person zero sum gain theory is, you want to sort of maximize what you're sure to get. Okay? And so basically if I would graph this, Get something like this. All right. So the x-axis is the chance I choose run offense. And then the blue is my expected yards against pass defense. Sorry, run defense. And the orange is expected yards against Pass defense. Okay so you basically assume the other guy knows what you're going to do. Okay so, basically if I would choose the probability of running to be less or equal to 0.5, then the defense is going to choose a pass defense, because I'm not running that much, and I get what's on the orange. Now, if I choose a probability of running that's greater than 0.5, the defense, if they knew that, would put me on the blue. They would pick run defense, because I'm running more of the time. So what I would get is from here to here and there, that's the girth of representing my expected yards gained based on my probability of choosing run. And I want to make that as large as possible, sort of maximize the minimum of the two curves is the idea. And that would be right here where they intersect. And basically that's always going to be true in a two by two game, we're just going to talk about two by two games here to give you the flavor. And so that would mean choose q, the chance of running to be 0.5. And you can see at that point, the two expected yard curves are equal, and then the value of the gain, so to speak, is my expected yardage that I'm sure to gain. Okay, in other words, if I announce I'm going to do 50/50 run or pass, I will gain two and half yards on the average. Flip a coin for run or pass. There's no way the other team can basically hold me below that, so that's the value of the game. Okay, now what about for the defense? Well, they should do pass defense 3 quarters. You can do a similar argument and they would do run defense 1 quarter. Now why is that? Because what's the expected yards they'll give up no matter what I do? Well against the run offense, I would get minus five. I would give up, if I'm the defense, minus five a quarter of the time. And three quarters of the time, I would give up five, okay? And that's 15 quarters minus ten quarters. That's two and a half, okay. Now what about against the pass offense. Well a quarter of the time, okay I'm going to play that run defense in, I would give up ten, and three quarters of the time I would give up zero. So the defense announces I'll do three quarters of the time, pass defense a quarter of the time, run defense. There's nothing you can do to gain on average more than two and a half yards. No matter what you pick on offense, you're going to get two and a half yards. It's no coincidence that when you look at it from the rolling column player standpoint, the value is the same. And the way you solve for the equilibrium mix strategies, if there's not a saddle point and a two-person zero sum gain, that's a two by two, two strategies for each player, you simply set the expected pay-off for each player against each of the other players pure strategy is to be equal. Now we'll continue with this in the next video, showing an interesting thing about football. Let's suppose you get a better running back in the draft. Would you run more, or would you run less? You'd think you'd run more because you've got a better running back. Okay. Well the answer is no, because the defense'll gang up on that better running back and play run defense more, so basically you'll be better off, but you should run less. And we'll see that in the next. The value of the game will go up to you because you have a better running back on offense, but you'll run less. because that threat is going to cause him, the opponent to play run defense often, which is a nice insight. And then we'll talk a little bit about tennis and we'll talk about soccer. So we'll see you in the next video when we continue this interesting discussion.
