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In the last lecture, I explained Nash equilibrium in random strategies and

it's called a mixed strategy equilibrium.

Sometimes it's better for you to be unpredictable.

So make yourself unpredictable is very important in

certain strategy consideration like a simple game of rock, paper and scissors.

And a similar situation frequently arises in sports games, okay?

So, in this lecture, I'm going to explain the concept of

mixed strategy equilibrium can be fruitfully applied to sports games.

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Okay, probably a good example is penalty kick in football or soccer game.

Okay, so, penalty kick is a part of soccer game where one player, kicker, is playing

this game of penalty kick against the other player, the goalkeeper, or goalie.

So, it's a two players game.

And the strategies could be very complicated but

usually either the kicker kicks to the right side, or the left side.

Kicking to the middle is very rare.

So we can safely assume that kicker, one of the players,

have, has two strategies, kick to the left and kick to the right.

Okay, what's a strategy of goalkeeper goalie?

Since the ball is coming very fast, okay.

So, ideally, the goalkeeper can see which

direction the ball is coming and he can jump to the right side, okay?

But in practice,

the ball is coming very fast, so the goalkeeper has to guess in advance, okay?

So simultaneously with the kicking,

goalkeeper, goalie can easily jump to right or jump to the left.

So, goalkeeper has two strategies, jumping to right or jumping to the left.

So it's a simultaneous move game with two players, goalkeeper, goalie and

the kicker simultaneously choose one of the two strategies.

Okay, so that's the nature of penalty kick.

And here obviously it's very important to be unpredictable.

If kicker always kicks to the same side to, to the left for

example, it's predicted by goalie, and his goal is lost.

So like just like a rock, paper and scissors game.

It's very important to be unpredictable.

So let's analyze the Nash equilibrium in penalty kick.

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So here is the payoff table of penalty kick game.

Two play as kicker and goalie.

And the kicker has two strategies, kicking to the left and kicking to the right.

And goalie has two strategies, jumping to the left or jumping to right, okay?

And then you can analyze or, or you can examine the probability that kicker wins.

What is the probability that kicker wins when kicker kick,

kicks the ball to the left side, and goalie jumps to the same side?

You have certain number a here, and you have different numbers b,

c, d for other combinational strategies, okay?

So first task for

you to analyze mixed strategy equilibrium to, is to get those numbers.

Probabilities that kicker wins in different situations, okay?

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On the other hand, the goal keeper,

goalie, is trying to minimize the probability that kicker wins.

So minus of those numbers can be regarded as goalie's payoff, okay.

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And if you get, if you get a good set of data, those four numbers a,

b, c,and d can be estimated from, from the data, okay.

And there is actually an empirical study conducted by a professor now at the LSE,

Ignacio Palacios Huerta, and he collected a large number of penalty kick data.

It's, the data set contains 1,417 penalty

kicks in professional soccer games in Europe, Spain, Italy, UK,

and other countries between September 1995 and June 2000.

Okay, so by looking at those large number of penalty kicks you can estimate

those four numbers, probability of winning of the kicker.

So the data set shows that left, left, I have to be more precise.

And left means from the point of view of goalkeeper.

So goalkeeper's left side is the left, okay.

So if kicker kicks the ball to the left of side of goalkeeper, and the goalie

jump to the, the same side, it's likely that goalkeeper catches the ball.

So the winning rate of the kicker is not so high, okay?

So winning rate is 58%.

58.30%.

One the other hand, if kicker kicks the ball to the left and

goalie jumps to the wrong side, then kicker can win.

But, it's not 100% because, the, the ball may not go into, into the goal.

The ball may not go into the goal.

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And as you can see, this payoff table is not symmetric, okay?

So numbers here and here are large but they are not equal.

Numbers here and here are smaller, but they are not equal.

So unlike rock, paper, and scissors game, situation is asymmetric.

And in rock, paper, and

scissors, paper, people are equally mixing rock and paper and scissors.

Equal distribution, equal randomization was optimal in rock, paper, and scissors.

But in this penalty kick game,

mixed strategy equilibrium may assign different probability for left and right.

So, mixed strategy equilibrium is very likely to

be different from equal mixture of left and right.

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So let's try to determine goalie's probability of

kicking jumping to left probability p.

And jumping to the right, probability 1 minus p, okay?

In mixed strategy equilibrium, everybody is acting randomly,

so Kicker is randomly choosing left or right.

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So let's as in rock, paper, scissors game,

given that Goalie is choosing left and right with this probability,

let's examine the kicker's expected payoff when he kicks to left.

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His winning rate is 94.97 so

overall probability of winning is equal to this expression, 'kay?

So given Goalie is mixing left and right with this probability,

if kicker kicks to the left, this is his winning rate, okay?

So similarly, if he kicks to the right side,

his winning probability is given by a similar formula.

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And since kicker is mixing left and

right, that means, you know, left and right are equally good for him.

That's why he's mixing.

If left is better, he should choose left with probability 1.

So given that kicker is mixing between left and

right, so both strategies should be equally good.

So that means the winning rate here should be exactly equal to the winning rate here.

9:18

Okay, so, given the pay-off table, Goalie has a higher chance of

jumping to right, and Kicker also has a higher chance of kicking to right.

So, this is the prediction given by game theory.

So let's compare this equilibrium prediction about the,

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Okay, so game theoretic prediction worked surprisingly well,

at least in this data set.

So, just as in rock, paper, and scissors game, goalies and kickers in soccer game

are trying to be unpredictable, and they are playing, mixed strategy equilibrium.