Learn how probability, math, and statistics can be used to help baseball, football and basketball teams improve, player and lineup selection as well as in game strategy.

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來自 University of Houston System 的課程

Math behind Moneyball

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Learn how probability, math, and statistics can be used to help baseball, football and basketball teams improve, player and lineup selection as well as in game strategy.

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Module 6

You will learn how two-person zero sum game theory sheds light on football play selection and soccer penalty kick strategies. Our discussion of basketball begins with an analysis of NBA shooting, box score based player metrics, and the Four Factor concept which explains what makes basketball teams win.

- Professor Wayne WinstonVisiting Professor

Bauer College of Business

Okay, let's continue with our discussion of two-person zero-sum game theory as it

applies to sports.

And I just will add in the notes if you're interested,

there's an example on poker bluffing.

We're not going to cover this, but I mean, why do you bluff in poker?

In other words, why do you bet when you have bad cards?

So that your opponent will call when you have the good cards, okay?

And so bluffing plays an important role in poker.

And if you read any of the advanced books on poker,

I think the guy who's considered the guru of poker is David Skolanski.

They'll always talk a lot about bluffing and

game theory will help you understand bluffing a fair amount.

Okay, so let's get back to the football example where we can either run or

pass and the defense can guess run or pass.

So let's suppose this is our current situation right here.

We'll highlight this in red.

Okay, and so run offense versus run defense gains -5 yards, etc.

Now it turns out the mixed strategies here, this is like odds and evens.

It's 50/50 for the offense and defense.

Flip a coin whether you run or pass or play run or pass defense.

And the value is zero, you can see that.

What's your expected payoff if you're the offense against each of

the defense strategies?

0.5 times -5 plus 0.5 times 5 equals 0.

Or 0.5 times 0.5 plus 0.5 times -5 equals 0.

Okay, let's suppose you draft a running back, okay, who's a better running back.

And so the only change we're going to make is we're going to make this number better.

So in other words, if basically the defense doesn't look for

the run, we're going to gain ten yards instead of five yards.

because we have that better running back and

he can break a tackle more often than our current running back.

All right, so what does that mean?

Okay, we've got a better running offense, we should run more, correct?

Wrong!

We should run less.

Okay, it turns out the optimal strategies here are 40% run offense,

60% pass offense, which is less than 50/50.

Why is that the optimal strategy?

because you play this against both of the defensive options,

you average the same amount.

0.4 times -5 is -2.

0.6 times 0.5 is 3, so you'll average 1 yard.

Here you get 0.4 times 10 is 4 yards.

0.6 times -5 is -3 yards.

So the value is 1 yard.

So you're better off You'll average 1 yard you can guarantee then.

0 yards but you run less, now why is that?

Because the defense knows you have good running.

They'll lay for the run more and so you'll run less, but

you'll be better off because you've strengthened your team.

So don't think if you've strengthened a part of your arsenal in sports,

that you'll basically use it more.

I mean, that may not work out.

Okay, so that's really sort of a paradox that game theory sheds some light on.

Now really,

what you'd like to do is validate game theory in a empirical context.

Okay, and we'll get to that in a second when we talk about penalty

kicks in soccer.

I'm taping this right after the FIFA scandal, which is unbelievable.

Okay, I mean, maybe not unsuspected, but still unbelievable.

Okay, but basically, well,

let's talk about the Soccer example.

Well, how could we really use game theory during football?

If we could do something like Madden.

In other words, if you could list the offensive plays

And the defensive plays and have in each one the expected

points that you would gain for each combination,

like if I run my halfback up the middle against the cover 2.

Maybe expected football points, not yards, I might gain 0.2 points per play.

If I throw a long pass against a prevent defense I'll probably

lose a point per play.

And then you could come up with mixed strategies,

the essence of football is mixed strategies, coaches don't want tendencies.

I think if you look, most coaches do close to 50/50 on first down,

even though their passing is much better than their running.

It just seems the teams I've looked at, it's just an engrained belief you need to

do 50/50 run-pass on first down so people can't guess what you're going to do.

But since passing does better in expected points, you should probably pass more.

Okay, until the defense adjusts to that.

Okay, and I don't think coaches really understand that.

I mean profootballreference, we had some examples looking at that.

My colleague Jeff Sagarin and I used to work with Sam Wyche and

the late Dick Cabot helped us, when we'd go to Cincinnati.

And try and have Sam Wyche coaching the Bengals and

he took them to the Super Bowl once.

So he's a pretty good coach.

He would try and fill in the matrix of offensive plays versus defensive plays.

And we'd come up with the mixed strategies.

Okay, we were working on this in 1983, which is 32 years ago, gosh.

Okay, that's unbelievable to me.

But we never got another coach to cooperate with us.

But who knows, that might work out.

But let's talk about soccer and penalty kicks.

Okay, and

then I'll give you a quick tennis example that you can do for homework.

Okay, so in soccer, you can actually have empirical data on basically

whether the goalie and the penalty kicker play game theory.

So at the instance of a penalty kick in soccer, and

penalty kicks are successful at about 80% of the time.

Okay, the kicker decides to kick left or kick right and the goalie moves left or

moves right.

And you can figure out then, if the kicker kicks left and

the goalie moves left, there's like a 58% chance the kick will go in.

But if the kicker kicks left and the goalie moves the other way,

there's 95% chance it'll go in.

So these seem to be,

from watching videotape, the probabilities that the kick will go in.

And of course the kicker wants to maximize this, the goalie wants to minimize this.

So the row player wants to maximize and the column player wants to minimize,

just like we've had in football with offense and defense.

Okay, now if you work out the mixed strategies here, okay, it turns out about

38% that the kicker should kick left and 61% that the kicker should kick right.

And the goalie should move left 42% and the goalie should move right 58%.

And basically, if you look at what the goalies and

kickers do, that's almost exactly the percentages they use.

So they really follow two-person zero-sum game theory.

To my knowledge that's the only data point, only data set that really verifies

that basically Professionals Play Minimax is the title of that paper.

Okay, now another example you could look at,

suppose Venus Williams is at the net and she can hit the ball left to right.

And Serena, the only hope she has is sort of to guess, cheat to the left or

cheat to the right before Venus hits the ball.

And suppose these are the chances that Venus will win the point.

Okay, in other words if Venus hits it to the left, Serena cheats right,

she's almost sure to win the point.

But if Serena cheats in the correct direction

then Venus has only a 50% chance of winning the point.

And you can figure out, we'll do this for a homework problem,

what is the mixed strategy that is optimal for Venus and Serena?

Again, the way you figure out in a 2 by 2 matrix,

the optimal mixed strategies is to set the expected payoff for

one player against each of the other player strategies to be identical.

Now, if you have more than a 2 by 2 matrix, you need to use linear programming

or the Excel solver, and we're just not going to get into that.

If you're interested, you can see my book on, my operations research textbook.

And that's how to solve a general two person zero-sum game.

There's a chapter that discusses that using linear programming Excel solver.

So I mean if you're interested in that, let me know.

I just wanted to introduce you to two-person game theory.

because it's zero-sum, explains a bunch of things in the real world, okay?

And that are surprising, like the better your running game, you may not run more.

And then that basically on penalty kicks in soccer, basically,

two-person zero-sum game theory,

really, the mixed strategy prescribed by that is actually what players use.

So next video, we'll start my favorite sport, which is basketball.

So, start dribbling.