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Hi folks. let's take a look at some data now on

Â Mixed Strategies and begin to see whether or not some of the subtleties that we

Â were talking about earlier actually play, play themselves out in, in real

Â incarnations of these games. So in particular we've mentioned that

Â mixed strategy Nash equilibrium can have some counterintuitive features and

Â they're can be, can be somewhat subtle to solve for.

Â So we might wonder whether people will actually really obey.

Â With the predictions of, of Nash Equilibrium and these settings.

Â So lets have a look at professional soccer penalty kicks.

Â And we'll look at, at some data that was gathered by Ignacio Palacios-Huerta, in,

Â in 2003. where what he did, was he actually looked

Â at a whole series of FIFA games, that he recorded off of television different,

Â different shows, ume e, he looked at 1,417 penalty kicks in, in the Spanish

Â league, England, in Italy and so forth, and so he was looking at high level

Â soccer, and looked at penalty kicks, and what he did is he Kept track of whether

Â people kicked to the left, they kicked to the center, they kicked to the right.

Â And whether they were using their left leg or their right leg.

Â and, and we'll get a look just at this simplified version that correspond to

Â what we analyzed earlier, which is just a left kick.

Â Kick, right kick, and the goalie can either move left or right, which he

Â actually analyzes a subset of the data on page 402, and we'll, we'll look at what

Â data he actually has from that paper. Okay, so, here's, based on, on what he

Â finds, out of these 1,417 penalty kicks. These are, are sort of the averages.

Â So, in situations where kickers go left and goalies go left, kickers win 58.

Â percent of the time. Goalies win 42% of the time.

Â In situations where the kicker goes left and the goalie goes right, then the, the

Â kicker wins 95% of the time. if the kicker goes right and the goalie

Â goes left, they win. the kicker wins 93% of the time and so

Â forth, so. So these are the actual numbers that,

Â Ignacio finds, based on these, recorded, penalty kicks from the 1417 games.

Â Okay so we, we do see that there's biases here.

Â There's some advantages and dis, disadvantages.

Â so going left against right is slightly better than if, for a kicker than going

Â right versus left. not so different but left left compared

Â to right right, we see a little bit more of a difference.

Â So this is a asymmetric game. It's a fairly subtle one.

Â so we have to see whether or not, we're going to end up, with the Nash

Â Equilibrium in this game. Okay.

Â So, why, why don't we do the following? given those numbers, you, we can pause

Â the video. And, solve the game.

Â So you can, take a, a look at this. Try and figure out what the probabilities

Â that the goalie should go left. So, say.

Â The goalie's going to left with probability pg, the kicker's going to go

Â left with probability Piece of k, solve for piece of g and piece of k, with this

Â matrix. So, your going to put pg here, 1-pg here,

Â pk here, 1-pk here, and try an solve for the mix Nash equilibrium of this game.

Â So, take a few minutes. Pause the video, try and solve that, and

Â we'll come back and look at what the solution looks like.

Â Okay, so you've had a chance to look through that.

Â now let's see what actually is happening in the, in this game.

Â So what we need, is we need P G to make the kicker indifferent.

Â Right? So if the kicker kicks left, we can figure out what's.

Â payoff they get if the kicker goes right we can figure what payoff they get.

Â So, in particular, the goalie's probability of going left vs right must

Â have the kicker indifferent, so when we look at the kicker's payoff from going

Â left. Compared to their care, payoff to going

Â right, has to be the same. You solve that out and what do you end up

Â with? Pg, is, is, roughly 5/12's in this case, or .42.

Â So, if we, we do the same for the kicker going left, versus the kicker going

Â right, you can go through that and you know, setting the, the Goalies payoff

Â from going left versus right being indifferent.

Â What do we end up with? We end up with pk, the probability that the kicker goes

Â left is .38 So, in terms of what we found, we found that goalies should go

Â left 42% of the time, that leaves them going right 58% of the time.

Â Kickers should go Left with probability .38 which then puts them going right with

Â probability .62. So we have a simple prediction based on

Â the actual frequencies with which kickers and goalies score when they go left

Â versus left right, and, and so forth. So if They were doing this facing

Â populations of people going left and right and use of the pay offs, Then this

Â is how they should be behaving. Ok, so what happens in the data, let's

Â take a peek. So the Nash frequencies, goalie going

Â left 42 % of the time, goalie used to go right, Go right 58% of the time.

Â Kicker should go left 38% of the time. Kicker should go right 62% of the time.

Â What are they actually do out of 1,417 games that we recorded? So we have a

Â non-trivial amount of data here. Goalies, 42, 58, right on the money.

Â kickers .40 .60 so very very close to the .38 .62, so in fact when we see

Â professional soccer players playing and we look at the paths they're getting

Â they're playing almost exactly the Nash equilibrium in terms of the mixed

Â strategy, or in this case, given that this is a 0 sum game, this is the same as

Â the max min strategies. And you know, if we ask a question of

Â exactly how they learned to do this, it's not necessarily true that they are

Â sitting down and looking at a matrix and calculating these things directly.

Â But over time they get a, they should be indifferent between going left and right.

Â So if the other players Are going in one direction or the other too often, and

Â they start adjust, they, they, they can get a better payoff from going one

Â direction or the other, they'll take advantage of that.

Â And so things have to adjust and In keeping them in different over time, so

Â you know, do players randomized well over time? Yeah, pretty well.

Â and you know, Ignacio's paper goes in much more details and this and look at

Â you know, at, at things like, How well they do in terms of mixing you

Â know, is it, do they, if you wanted to mix 50, 50 one way to do it would be to

Â go left one time then right the next time, then right left the next time and

Â so forth and just alternate, it's obviously not randomizing.

Â and so instead there's the question of whether people randomize so that they're

Â really unpredictable over time. And Ignacio finds that they do fairly

Â well even in terms of the strings of Of kicks that they have.

Â there's other questions you could ask. How well do they perform under pressure,

Â if it's a big game and it's a very important kick? Do they tend to go

Â towards their stronger, foot? Did they become predictable? well you know, in, in

Â fact now you see more and more Professional sports team hiring

Â statisticians. Hiring game theorists.

Â Keeping track of exactly what's going on in terms of other, other team's

Â tendencies. What do they tend to do in this

Â situation? What do they tend to do in that situation? What's our best strategy

Â in response to that? So, you know? Going through and analyzing these things, is,

Â has become, more of a trend. in other sports, there's, there's similar

Â analyses. There's a very nice paper by Mark Walker

Â and John Waters. the American Economic Review, looking at

Â tennis, an serves. So, you know? Which side, you have to

Â serve into a given area. Do you serve towards the, the left side

Â of it? With the right side of it, the center, which, how does it depend on

Â whether you're right handed, left handed, which directions you're going in, and so

Â forth. so they an, analyze a series of

Â professional tennis, games. And, similarly, they find that minimax

Â play is, is, a, fairly good, predictor of Exactly what's going on, and, you know,

Â there's, there's also questions of how well people really mix over time, but,

Â the, the equilibrium predictions do fairly well.

Â Okay. you know, we see there, there are

Â going to be games that have mixed strategy equilibria.

Â in particular, zero sum and competitive games will tend to have them in, in a lot

Â of situations. players have to be indifferent between

Â what they, the players that they're facing.

Â That gives you some very, interesting comparative statics.

Â you know? We asked the question, do we really see randomization? we found, you

Â know? Yes, in professional sports, we do see, randomization.

Â there's lots of other things in the world where you see randomization.

Â So, predator-prey games. You know, in nature.

Â If you come up upon a squirrel, a squirrel thinks you're trying to catch

Â it. What does it do? it randomizes a bit, so

Â it's very unpredictable to figure out which way the squirrels dart when you're

Â walking by it. it's you know it's following essentially

Â a, a randomized strategy. many bus, business interactions.

Â So if we look at things like audits. tax auditing.

Â that's, that's a game where we are going to see a situation where it's

Â competitive. and tax authorities don't necessarily

Â want you to know exactly whether you're going to be audited or not.

Â They might want you to have some uncertainty so they can't audit every,

Â they can't audit everybody in the population if there's a cost to auditing.

Â That's going to be a game here they're, they're going to mix and that, random,

Â randomization might help. Tax authorities.

Â So there's a lot of settings where random checks, random audits, are essentially

Â optimal strategies as part of some game. And where Next, Nash Equilibrium,

Â