Hi 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, play themselves out in, in real incarnations of these games. So in particular we've mentioned that mixed strategy Nash equilibrium can have some counterintuitive features and they're can be, can be somewhat subtle to solve for. So we might wonder whether people will actually really obey. With the predictions of, of Nash Equilibrium and these settings. So lets have a look at professional soccer penalty kicks. And we'll look at, at some data that was gathered by Ignacio Palacios-Huerta, in, in 2003. where what he did, was he actually looked at a whole series of FIFA games, that he recorded off of television different, different shows, ume e, he looked at 1,417 penalty kicks in, in the Spanish league, England, in Italy and so forth, and so he was 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, and we'll get a look just at this simplified version that correspond to what we analyzed earlier, which is just a left kick. 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, we'll look at what data he actually has from that paper. Okay, so, here's, based on, on what he finds, out of these 1,417 penalty kicks. These are, are sort of the averages. So, in situations where kickers go left and goalies go left, kickers win 58. percent of the time. Goalies win 42% of the time. In situations where the kicker goes left and the goalie goes right, then the, the kicker wins 95% of the time. if the kicker goes right and the goalie goes left, they win. the kicker wins 93% of the time and so forth, so. So these are the actual numbers that, Ignacio finds, based on these, recorded, penalty kicks from the 1417 games. Okay so we, we do see that there's biases here. There's some advantages and dis, disadvantages. so going left against right is slightly better than if, for a kicker than going right versus left. not so different but left left compared to right right, we see a little bit more of a difference. So this is a 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, why don't we do the following? given those numbers, you, we can pause the video. And, solve the game. So you can, take a, a look at this. Try and figure out what the probabilities that the goalie should go left. So, say. The goalie's going to left with probability pg, the kicker's going to go left with probability Piece of k, solve for piece of g and piece of k, with this matrix. So, your going to put pg here, 1-pg here, pk here, 1-pk here, and try an solve for the mix Nash equilibrium of this game. So, take a few minutes. Pause the video, try and solve that, and 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 the, in this game. So what we need, is we need P G to make the kicker indifferent. Right? So if the kicker kicks left, we can figure out what's. payoff they get if the kicker goes right we can figure what payoff they get. So, in particular, the goalie's probability of going left vs right must have the kicker indifferent, so when we look at the kicker's payoff from going left. Compared to their care, payoff to going right, has to be the same. You solve that out and what do you end up with? Pg, is, is, roughly 5/12's in this case, or .42. So, if we, we do the same for the kicker going left, versus the kicker going right, you can go through that and you know, setting the, the Goalies 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 goalies should go left 42% of the time, that leaves them going right 58% of the time. Kickers 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 kickers and goalies score when they go left versus left right, and, and so forth. So if They were doing this facing populations of people going left and right and use of the pay offs, Then this is how they should be behaving. Ok, so what happens in the data, let's take a peek. So the Nash frequencies, goalie going left 42 % of the time, goalie used to go right, Go right 58% of the time. Kicker should go left 38% of the time. Kicker should go right 62% of the time. What are they actually do out of 1,417 games that we recorded? So we have a non-trivial amount of data here. Goalies, 42, 58, right on the money. kickers .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 paths they're getting they're playing almost exactly the Nash equilibrium in terms of the mixed strategy, or in this case, given that this is a 0 sum game, this is the same as the max min strategies. And you know, if we ask a question of exactly how they learned to do this, it's not necessarily true that they are sitting down and looking at a matrix and calculating these things directly. But over time they get a, 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 adjust, they, they, 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 and In keeping them in different over time, so you know, do players randomized well over time? Yeah, pretty well. and you know, Ignacio's paper goes in much more details and this and look at you know, at, at things like, How well they do in terms of mixing you know, is it, do they, 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, it's obviously not randomizing. and so instead there's the 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 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? Did they become predictable? well you know, in, in fact now you see more and more Professional sports team hiring statisticians. Hiring game theorists. Keeping track of exactly what's going on in terms of other, other team's 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, you know? Going through and analyzing these things, is, has become, more of a trend. in other sports, there's, there's similar analyses. There's a very nice paper by Mark Walker and John Waters. the American Economic Review, looking at tennis, an serves. So, you know? Which side, you have to serve into a given area. Do you serve towards the, the left side of it? With the right side of it, the center, which, how does it depend on whether you're right handed, left handed, which directions you're going in, and so forth. so they an, analyze a series of professional tennis, games. And, similarly, they find that minimax play is, is, a, fairly good, predictor of Exactly what's going on, and, you know, there's, there's also questions of how well people really mix over time, but, the, the equilibrium predictions do fairly well. Okay. you know, we see there, there are going to be games that have mixed strategy equilibria. in particular, zero sum and competitive games will tend to have them in, in a lot of situations. players have to be indifferent between what they, the players that they're facing. That gives you some very, interesting comparative statics. you know? We asked the question, do we really see randomization? we found, you know? 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. You know, in nature. If you come up upon a 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 squirrels dart when you're walking by it. it's you know it's following essentially a, a randomized strategy. many bus, business interactions. So if we look at things like audits. tax auditing. that's, that's a game where we are 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 every, they can't audit everybody in the population if there's a cost to auditing. That's going to be a game here they're, they're going to mix and that, random, 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 where Next, Nash Equilibrium, particular Mixed Strategy and Nash Equilibria and will help us understand those things.