 Okay, folks, let's take a look at some data now on mixed strategies and begin to see whether or not some of the subtleties that we were talking about earlier actually play themselves out in real incarnations of these games. So in particular, we mentioned that mixed strategy Nash equilibria can have some counter intuitive features and there can be somewhat subtle to solve for. So we might wonder whether people actually really obey the predictions of Nash equilibria in these settings. So let's have a look at professional soccer penalty kicks and we'll look at some data that was gathered by Ignacio Palacio Suerta in 2003 where what he did was he actually looked at a whole series of FIFA games that he recorded off of television different shows. He looked at 1417 penalty kicks in the Spanish League, England, Italy and so forth. And so he's looking at high level soccer and looked at penalty kicks and what he did is he kept track of whether people kicked to the left, they kicked to the center, they kicked to the right and whether they were using their left leg or their right leg. And we're going to look at this simplified version that corresponds to what we analyzed earlier which is just a left kick, right kick and the goalie can either move left or right which he actually analyzes a subset of the data on page 402 and we'll look at what data he actually has from that paper. Okay, so here's based on what he finds out of these 1417 penalty kicks. These are sort of the averages. So in situations where kickers go left and goalies go left, kickers win 58.% of the time, goalies win 42% of the time. In situations where the kicker goes left and the goalie goes right then the kicker wins 95% of the time. If the kicker goes right and the goalie goes left, the kicker wins 93% of the time and so forth. So these are the actual numbers that Ignacio finds based on these recorded penalty kicks from the 1417 games. Okay, so we do see that there's biases here. There's some advantages and disadvantages. So going left against right is slightly better for a kicker than going right versus left. Not so different but left to left compared to right to right. We see a little bit more of a difference. So this is an asymmetric game. It's a fairly subtle one. So we have to see whether or not we're going to end up with the Nash equilibrium in this game. Okay, so why don't we do the following. Given those numbers, we can pause the video and solve the game. So you can take a look at this, try and figure out what the probabilities that the goalie should go left. So let's say the goalie is going to go left with probability pg, the kicker is going to go left with probability p sub k, solve for p sub g and p sub k with this matrix. So you're going to put pg here, 1 minus pg here, pk here, 1 minus pk here and try and solve for the mixed strategy Nash equilibrium of this game. So take a few minutes, pause the video, try and solve that and then we'll come back and look at what the solution looks like. Okay, so you've had a chance to look through that. Now let's see what actually is happening in this game. So what we need is we need pg to make the kicker indifferent, right? So if the kicker kicks left, we can figure out what payoff they get. If the kicker goes right, we can figure out what payoff they get. So in particular, the goalie's probability of going left versus right must have the kicker indifferent. So when we look at the kicker's payoff from going left compared to their payoff to going right, it has to be the same. You solve that out and what do you end up with? pg is roughly 512 in this case or .42. So if we do the same for the kicker going left versus the kicker going right, you can go through that and setting the goalie's payoff from going left versus right being indifferent. What do we end up with? We end up with pk, the probability that the kicker goes left is .38. So in terms of what we found, we found that goalie should go left 42% of the time. That leaves them going right 58% of the time. The kicker should go left with probability .38, which then puts them going right with probability .62. So we have a simple prediction based on the actual frequencies with which kicker's and goalie's score when they go left versus left, right, and so forth. So if they were doing this, facing populations of people going left and right, and these are the payoffs, then this is how they should be behaving. So what happens in the data? Let's take a peek. So the Nash frequencies, goalie going left 42% of the time, goalie should go right 58% of the time, kicker should go left 38% of the time, kicker should go right 62% of the time. What do they actually do out of 1,417 games that were recorded? So we have a non-trivial amount of data here. Goalie's 42.58, right on the money. Kicker's 40.60, so very, very close to the 38.62. So in fact, when we see professional soccer players playing, and we look at the payoffs, they're playing almost exactly the Nash equilibrium in terms of the mixed strategy, or in this case, given this is a zero-sum game, this is the same as the Max Min strategies. And if we ask a question of exactly how they learn to do this, it's not necessarily true that they're sitting down and looking at a matrix and calculating these things directly. But over time, they should be indifferent between going left and right. So if the other players are going in one direction or the other too often and they start adjusting, they can get a better payoff from going one direction or the other, they'll take advantage of that. And so things have to adjust in keeping them indifferent over time. So do players randomize well over time? Yeah, pretty well. And Ignacio's paper goes in much more detail on this and looks at things like how well they do in terms of mixing. If you wanted to mix 50-50, one way to do it would be to go left one time, then right the next time, then right left the next time, and so forth, and just alternate. That's obviously not randomizing. And so instead, there's a question of whether people randomize so that they're really unpredictable over time. And Ignacio finds that they do fairly well even in terms of the strings of kicks that they have. There's other questions you could ask. How well do they perform under pressure? If it's a big game and it's a very important kick, do they tend to go towards their stronger foot? Do they become predictable? Well, in fact, now you see more and more professional sports teams hiring statisticians, hiring game theorists, keeping track of exactly what's going on in terms of other teams' tendencies. What do they tend to do in this situation? What do they tend to do in that situation? What's our best strategy in response to that? So going through and analyzing these things has become more of a trend. In other sports, there's similar analyses. There's a very nice paper by Mark Walker and John Wooders, American Economic Review, looking at tennis and serves. So which side you have to serve into a given area? Do you serve towards the left side of it, the right side of it, the center? How does it depend on whether you're right-handed, left-handed? Which directions you're going in and so forth? So they analyze a series of professional tennis games. And similarly, they find that Minimax Play is a fairly good predictor of exactly what's going on. And there's also questions of how well people really mix over time. But the equilibrium predictions do fairly well. OK, we see there are going to be games that have mixed strategy equilibria. In particular, zero-sum and competitive games will tend to have them in a lot of situations. Players have to be indifferent between what the players that they're facing. That gives you some very interesting comparative statics. We ask the question, do we really see randomization? We find, yes, in professional sports, we do see randomization. There's lots of other things in the world where you see randomization. So predator-prey games, in nature, if you come up with a kind of squirrel, a squirrel thinks you're trying to catch it, what does it do? It randomizes a bit. So it's very unpredictable to figure out which way the squirrel is going to dart when you're walking by it. It's following essentially a randomized strategy. Many business interactions. So if we look at things like audits, tax auditing, that's a game where you're going to see a situation where it's competitive. And tax authorities don't necessarily want you to know exactly whether you're going to be audited or not. They might want you to have some uncertainty. So they can't audit everybody in the population if there's a cost to auditing. That's going to be a game where they're going to mix. And that randomization might help tax authorities. So there's a lot of settings where random checks, random audits are essentially optimal strategies as part of some game. And we're next, Nash Equilibria, in particular mixed strategy in Nash Equilibria, will help us understand those things.