Professor Ben Polak: So

last time we saw this, we saw an example of a mixed

strategy which was to play 1/3, 1/3, 1/3 in our rock,

paper, scissors game. Today, we’re going to be

formal, we’re going to define mixed strategies and we’re going

to talk about them, and it’s going to take a while.

So let’s start with a formal definition: a mixed strategy

(and I’ll develop notation as I’m going along,

so let me call it P_i,

i being the person who’s playing it) P_i is a

randomization over i’s pure strategies.

So in particular, we’re going to use the notation

P_i (si) to be the probability that Player i plays

si given that he’s mixing using P_i.

So P_i(si) is the probability that P_i

assigns to the pure strategy si. Let’s immediately refer that

back to our example. So for example,

if I’m playing 1/3,1/3,1/3 in rock, paper, scissors then

P_i is 1/3,1/3,1/3 and P_i of rock–so

P_i(R)–is a 1/3. So without belaboring it,

that’s all I’m doing here, is developing some notation.

Let’s immediately encounter two things you might have questions

about. So the first is,

that in principle P_i(si) could be zero.

Just because I’m playing a mixed strategy,

it doesn’t mean I have to involve all of my strategies.

I could be playing a mixed strategy on two of my strategies

and leave the other one with zero probability.

So, for example, again in rock,

paper, scissors, we could think of the strategy

1/2,1/2,0. In this strategy I assign–I

play rock half the time, I play paper half the time,

but I never play scissors. So everyone understand that?

And while we’re here let’s look at the other extreme.

The probability assigned by my mixed strategy to a particular

si could be one. It could be that I assign all

of the probability to a particular strategy.

What would we call a mixed strategy that assigns

probability 1 to one of the pure strategies?

What’s a good name for that? That’s a “pure strategy.”

So notice that we can think of pure strategies as the special

case of a mixed strategy that assigns all the weight to a

particular pure strategy. So, for example,

if Pi(R) was 1, that’s equivalent to saying

that I’m playing the pure strategy rock,

i.e. a pure strategy.

So there’s nothing here. I’m just being a little bit

nerdy about developing notation and making sure that everything

is in place, and just to point out again,

one consequence of this is we’ve now got our pure

strategies embedded in our mixed strategies.

When I’ve got a mixed strategy I really am including in those

all of the pure strategies. So let’s proceed.

I’m going to push that up a little high, sorry.

So now I want to think about what are the payoffs that I get

from mixed strategies, and again, I’m going to go a

little slowly because it’s a little tricky at first and we’ll

get used to this, don’t panic,

we’ll get used to this as we go on and as you see them in

homework assignments and in class.

So let’s talk about the payoffs from a mixed strategy.

In particular, what we’re going to worry about

are expected payoffs. So the expected payoff of the

mixed strategy P, let’s be consistent and call it

P_i, the mixed strategy

P_i is what? It’s the weighted average–it’s

a weighted average or a weighted mixture if you like–of the

expected payoffs of each of the pure strategies in the mix.

So this is a long way of saying something again which I think is

a little bit obvious, but let me just say it again.

The way in which we figure out the expected payoff of a mixed

strategy is, we take the appropriately weighted average

of the expected payoffs I would get from the pure strategies

over which I’m mixing. So to make that less abstract

let’s immediately look at an example.

So here’s an example we’ll come back to several times,

but just once today, and this a game you’ve seen

before. Here is the game Battle of the

Sexes, in which Player A can choose–Player I can choose A

and B, and Player II can choose a and

b, and what I want to do is I want to figure out the payoff

from particular strategies. So suppose that P is being

played by Player I and P is let’s say (1/5,4/5).

So what do I mean by that? I mean that Player I is

assigning 1/5 to playing A and 4/5 to playing B.

And suppose that Q–so I am going to use P and Q because

it’s convenient to do so rather than calling them P_1

and P_2. So suppose that Q is the

mixture that Player II is choosing and she’s choosing a

(½, ½), so she’s putting a probability

1/2 on a and a probability 1/2 on b.

Just to notice I switched notation on you a little bit,

for this example to keep life easy,

I’m going to use P to be row’s mixtures and Q to be column’s

mixtures. And the question I want to

answer is what is the expected payoff in this case of P?

What is P’s expected payoff? The way I’m going to do that

is, I’m first of all going to ask what is the expected payoff

of each of the pure strategies that P involves,

the pure strategies involved in P.

So to start off–so the first step is ask what is the expected

payoff for Player I of playing A against Q and what is the

expected payoff for Player I of playing B against Q?

That will be our first question and we’ll come back and

construct the payoff for P. So these are things we can do I

think. So the expected payoff of A

against Q is what? Well, half the time if you play

A you’re going to find your opponent is playing a,

in which case you’ll get 2, and half the time when you play

A you’ll find your opponent is playing b in which case you’ll

get 0. So let’s just write that up.

So I’m going to get 2 with probability 1/2 plus 0 with

probability 1/2. Everyone happy with that?

That gives me 1. Please correct my math in this.

It’s very easy at the board to make mistakes,

but I think that one is right. Conversely, what if I played B?

What’s the expected payoff for the row player of playing B

against Q, where Q is 1/2,1/2? So half the time when I play B,

I’ll meet a Player II playing a and I’ll get 0 and half the time

I’ll find Player II is playing b and I’ll get 1.

So let’s write that up. So I’ll get 0 half the time and

I’ll get 1 half the time for an average of 1/2.

That’s the first thing I ask. And now to finish the job,

I now want to figure out what is the expected payoff for

Player I of using P against Q? That was the question I really

wanted to start off with. What’s the way to think about

this? Well P is 1/5 of the

time–according to P, 1/5 of the time Player I is

playing A and 4/5 of the time Player I is playing B,

is that right? So to work out the expected

payoff what we’re going to do is we’re going to take 1/5 of the

time, and at which case he’s playing

A and he’ll get the expected payoff he would have got from

playing A against Q, and 4/5 of the time he’s going

to be playing B in which case he’ll get the expected payoff

from playing B against Q. Now just plugging in some

numbers to that from above, so we’ve got 1/5 of the time

he’s doing the expected payoff from A against Q and that’s this

number we worked out already. So this number here can come

down here, 1. And 4/5 of the time he’s

playing B against Q, in which case his expected

payoff was 1/2, so this 1/2 comes in here.

Everyone okay so far, how I constructed it so far?

Is this podium in the way of you guys, are you okay?

Let me push it slightly. So the total here is what?

It’s going to be 1/5 of 1 plus 4/5 of ½.

4/5 of 1/2 is 2/5, so I’ve got a total of 3/5.

So the total here is 3/5. Everyone understand how I did

that? Now while it’s here let’s

notice something. When I played P,

some of the time I played A and some of the time I played B.

And when I ended up playing A, I got A’s expected payoff.

And when I played B, I got B’s expected payoff.

So the number I ended up with 3/5 must lie between the payoff

I would have got from A which is 1, and the payoff I would have

got from B which is 1/2. Is that right?

So 3/5 lies between 1/2 and 1. Everyone okay with that?

Now that’s a simple but very general and very useful idea it

turns out. The idea here is that the

payoff I’m going to get must lie between the expected payoffs I

would have got from the pure strategies.

Let me say it again. In general, when I play a mixed

strategy the expected payoff I get, is a weighted average of

the expected payoffs of each of the pure strategies in the mix,

and weighted averages always lie inside the payoffs that are

involved in the mix. So let me try and push that

simple idea a little harder. Suppose I was going to take the

average height in the class–average height in this

class. So let me just,

rather than use the class, let me just use some T.A.’s

here. So let me get these three

T.A.’s to stand up a second. Suppose I want to figure out

the average height of these three T.A.’s.

So stand up close together so I can at least see what’s going on

here. So I think, from where I’m

standing, I’ve got that Ale is the tallest and Myrto is the

smallest, is that right? So I don’t know instantaneously

what this average would be, but I claim that any weighted

average of their three heights, is going to give me a number

that’s somewhere between the smallest height of the three,

which is Myrto’s height, and the tallest height of the

three, which is Ale’s height, is that right?

Is that correct? So that’s a pretty general idea.

Thanks guys I’ll come back to you in a second.

Let’s think about this somewhere else,

let’s think about the batting average of a team.

The team batting average in baseball, let’s use the Yankees,

for example. We know that the team batting

average, the average batting average of the Yankee’s–I don’t

know what it is, I didn’t look it up this

morning–but I know it lies somewhere between the player who

has the highest batting average which I’m guessing is Jeter,

I’m guessing, and the lowest,

the person on the team who has the lowest batting average,

who is probably one of the pitchers who played,

who batted a few times in one of those inter-league games.

(It would have been better if I’d used the Mets but I feel I

should take pity on Mets fans this week and not mention them.)

So this is a very simple idea, it’s deceptively simple.

It says averages, weighted averages,

lie between the highest thing over which you’re averaging and

the lowest thing over which you’re averaging.

Everyone okay with that idea? Now this very simple idea is

going to have an enormous consequence, and here’s the

enormous consequence. Simple idea, big consequence.

So there’s going to be a lesson that follows from this

incredibly simple idea and this is the lesson.

If a mixed strategy is a best response, so if a mixed strategy

is the best thing you can be doing,

then each of the pure strategies in the mix–I’m being

a little bit loose here but I mean assigned positive

probability in the mix, for those people who are nerdy

enough to worry about it–each of the pure strategies in the

mix must themselves be best responses.

So, in particular, each must yield the same

expected payoff. So here’s a big conclusion that

follows from that incredibly simple idea about averages lying

between the highest one and the lowest one.

Let’s draw ourselves from that lesson to this big conclusion.

What is the conclusion? The conclusion is if a mixed

strategy is a best response, if the best thing I can do is

to play a mixed strategy, then each of the pure

strategies which I’m playing in that mix, which I’m assigning

positive probability to in that mix,

must themselves be best responses.

In particular, each of them therefore must

yield the same expected payoff. So let’s go back to our example.

Can I steal my three T.A.’s again?

Suppose the game, suppose the thing I’m involved

in–I should have made this easier before,

let me come down a little bit. I’ll stand above here,

this is good. So suppose the game I’m

involved in, the payoff in the game is, a game in which I have

to choose the tallest group of my T.A.’s.

So my payoff is going to be the average height of whichever

subgroup of my T.A.’s I pick and these are my three choices.

So if I pick more than one of them I’m going to get a weighted

average, that’s a mixed strategy.

My aim here is to maximize the height of whatever subgroup I

pick. So in this game,

here’s my three pure strategies: my three pure

strategies are to pick Myrto; Ale;

or Jake. Those are my three pure

strategies. And my mixture,

I could mix these two, I could mix these two,

I could mix all three. But remember my payoff here is

to get the group, the average as high as I can.

So how am I going to get the average as high as I can?

I get the average as high I as I can, I’m going to kick out

Myrto for a start because Myrto’s just bringing down the

average, is that right? Average height I should say,

there’s nothing–and actually I think I’m going to kick out Jake

as well I think, I’m probably going to kick out

Jake as well because that way I just have Ale.

So if it was the case that I was picking both of them,

it would have to be they were equally tall but since they’re

not equally tall, I should just pick the best one.

Let’s go back to my Yankee’s example, if I want to pick a

sub-team of the Yankee’s, I’m allowed to pick any number

of people, to have the highest average, batting average,

in that sub-team. The way to do it is to find the

Yankee who has the highest batting average and just pick

him. Let’s do one more example.

Let me use the front row of students here,

so here’s my, can I get this front of

students to stand up a second? This is a part of the row.

And suppose my aim in life is to construct the highest average

GPA. I’m not going to embarrass

these guys and ask them what their GPA’s are.

So my aim in life here is to pick some sub-group of these

one, two, three, four,

five, six, seven, eight students,

such that the average GPA of that sub-group is as high as I

can make it. So what will I do here?

So this being Yale I’ll just find the people who have the 4.0

GPA’s and just pick them. Is that right?

You might think well why not include somebody who has a 3.9

GPA? That’s pretty good.

So why not? Because if there’s anybody in

this group who has a 4.0 GPA, I’d do better just to pick that

person. The 3.9 person would just be

pulling down the average. Now suppose there’s nobody with

a 4.0 GPA and suppose it’s the case that three of these people,

let’s say these three people have a 3.9 GPA.

So these three have 3.9 GPA, imagine that,

and these other people they’ve got horrible grades like B+

somewhere. These are our future law school

students and these are the people–who knows what they’re

going to end up doing–being President probably.

So to construct the group with the highest average GPA,

what am I going to do? Well first I’ll throw out all

these guys with low GPA’s, so they can all sit down and

I’ll look at these last three and these last three,

if they’re all in the group they better all have the same

GPA. Why on earth?

If I’m trying to maximize the average of my group,

if any of them had a lower GPA I should kick them out,

and if one of them has a higher GPA than the other two,

I should kick out both the other two.

So if I’m including all three of them, in my constructing of

the average all of them must have the same GPA,

which I’m going to assume is 3.9, to assume you can still

make into law school. Everyone understand that?

Yeah? Okay, thanks guys.

So that’s the way I want to think about this.

So the idea here is if I’m using a mixed strategy as a best

response, it must be the case that everything on which I’m

mixing is itself best. And the reason is,

if it wasn’t, kick out the thing that isn’t

best and my average will go up. So that leads us to the next

idea, but before I do just for formality, let me add a

definition. The definition is this,

a mixed strategy profile–what I’m going to do now is I’m going

to define Nash Equilibrium again,

just so we have it in our notes somewhere.

So a mixed strategy profile–there should be a

hyphen there–(P_1*, P_2*,

…all the way up to P_N*),

is a mixed strategy Nash Equilibrium if for each Player

i–so for each Player i–that player’s mixed strategy

P_i* is a best response for Player i to the

strategies everyone else is picking P _-i*.

So I’m exploiting, by now, a well developed

notation for player strategies. So this definition of Nash

Equilibrium, it’s exactly the same as the definition of Nash

Equilibrium we’ve been using now for several weeks,

except everywhere where before we saw a pure strategy,

which was an S, I have replaced it with a P.

So the same definition except I’m using mixed strategies

instead of pure strategies. But an implication of our

lesson is what? It’s that if P_i* is

part of a Nash Equilibrium–so if Pi* is a best response to

what everyone else is doing, P_-i* –,

then each of the pure strategies involved in

P_i* must itself be a best response.

So an implication of the lesson is, the lesson implies the

following. If P_i* of a

particular strategy is positive, so in other words,

I’m using this strategy in my mix,

then that strategy is also a best response to what everyone

else is doing. Okay, so from a math point of

view this is the big idea of the day, this board.

If you’re having trouble reading this at the back,

trust me I’ve written that up on the handout that will appear

magically on the computer, at the end of class.

At the moment you’re staring at this, it’s all a bit new,

and as well as being new, you’re saying,

okay but so what, why do I care about this

seemingly mundane fact? The reason we’re going to turn

out to care about this seemingly mundane fact,

is that this fact is going to make it remarkably easy to find

Nash Equilibria. This fact, this lesson,

this idea that if I’m playing a pure strategy as part of the

mix, it must itself be a best

response, that’s going to be the trick we’re going to use in

finding mixed strategy Nash Equilibria.

The only way I can illustrate that to you is to do it,

so I’m going to spend the rest of today just doing that.

I’m going to look at a game and we’re going to go through this

game. We’ll discuss it a little bit

because it’s a fun game, and we’re going to find the

mixed-strategy equilibria of this game.

Everyone know where we’re going? I want to make sure before I go

on, are people looking very sort of deer in the headlamps?

That was a lot of formality to get through in a short period of

time. Does anyone want to ask a

question at this point? Are you okay?

Okay to go on? So just remember that the

conclusion here comes from this very simple idea.

The simple idea is, the payoff to a weighted

average must lie between the best and worst thing involved in

the average, and therefore if I’m including

things in there as part of a best response,

they must all be good. That’s the simple idea,

this is the dramatic conclusion.

So the only way to prove this to you and the only way to prove

to you that this is useful is to go ahead and do it.

So what I’m going to do is I’m going to clean these boards and

I’m going to start showing an example.

Again don’t panic, I think a lot of people at this

part of the class have a tendency to panic,

because it’s a new idea, it seems like a lot of math

around. None of it’s very hard math,

it’s all kind of arithmetic. It’s just this idea of not

panicking. So the example I want to look

at is going to be from tennis, and I’m going to consider a

game within a game, played by two tennis players,

and let’s call them Venus and Serena Williams.

So a couple of years ago we used to use Venus and Serena

Williams for this example, and then for a while I worried,

that you wouldn’t even remember who Venus and Serena Williams

were, and so we picked any two random

Russians, but now we’re back. Seems like we’re back to

picking Venus and Serena. So the game within the game is

this, suppose that they’re playing and Serena is at the net

and the ball is on Venus’ court, and Venus has reached the ball

and Venus has to decide whether to try to hit a passing shot

past Serena on Serena’s left or on Serena’s right.

Notice I’m going to exclude the possibility of throwing up a lob

for now, just to make this manageable.

So basically the choice facing Venus is should she try to pass

Serena to Serena’s left, which is Serena’s backhand side

or to Serena’s right, which is Serena’s forehand

side. People are familiar enough with

tennis to understand what I’m talking about?

So we’re going to assume this is Wimbledon,

otherwise no one would be at the net to start with I guess.

So this is at Wimbledon. Let’s try and put up some

payoffs here. So these are going to be the

payoffs. I think that this example is

originally due to Dixit, but it’s not a big deal.

I think this example is due to Dixit and Skeath.

So here’s some numbers and I’ll explain the numbers in a minute.

So this is 50,50, 80,20, 90,10 and 20,80.

So what are these numbers? So first of all let me just

explain what the strategies are, so I’m assuming the row player

is Venus and the column player is Serena.

I’m assuming that if Venus chooses L that means she

attempts to pass Serena to Serena’s left,

we’ll orient things from Serena’s point of view,

and if she hits right that means she’s attempting to pass

Serena on Serena’s right. If Serena chooses L that means

she cheats slightly towards her left: not cheats in the sense of

breaking the rules, but cheats in terms of where

she’s standing or leaning. And if she chooses right that

means she cheats slightly towards her right.

So this is cheating towards her backhand and this is cheating

towards her forehand, assuming she’s right handed,

which she in fact is. Okay, what do these numbers

mean? So let’s start with the easy

ones. So if Venus chooses left and

Serena chooses right, then Serena has guessed wrong.

Is that correct? In which case Venus wins the

points 80% of the time and Serena wins it 20% of the time.

Conversely, if Venus chooses right and Serena chooses left,

then again, Serena has guessed wrong and this time Venus wins

the points 90% of the time and Serena wins the points 10% of

the time. This should be a familiar idea

by now, but why is it the case these nineties and eighties are

not a 100%? Why is it the case that if

Serena guesses wrong Venus doesn’t win 100% of the time?

Anybody? Perhaps we can get a show of

hands, get some mikes up. Why isn’t it 100% here?

Somebody? Patrick?

Wait for the mike. Student: Sometimes she

hits it out of bounds when she serves.

Professor Ben Polak: Right, this isn’t even a serve,

this is a passing shot but the same is true.

So sometimes you’re successfully going to hit it

past Serena but the ball is going to sail out.

So that happens 10% of the time here and 20% of the time here.

Look at the other two boxes, if Venus hits to Serena’s left

and Serena guesses left, then we’re going to assume that

Serena’s going to reach the ball and make a volley,

but her volley only manages to go in–go over the net and go

in–half the time, so the payoffs are (50,50).

Half the time Venus wins the point and half the time Serena

wins the point. Conversely, if Venus hits the

ball to Serena’s right and Serena guesses correctly and

chooses right, then we’re in this box.

Once again, Serena has guessed correctly and she’s going to

successfully reach the volley and this time she gets it in 80%

of the time, so Venus wins the point 20% of

the time and Serena wins it 80% of the time.

So just to finish up the description of the game here,

notice that we’re assuming that Serena is a little better at

volleying to her right than she is volleying to her left.

So this is her forehand volley and we’re going to assume that

that’s stronger than her backhand volley.

Conversely, we’re assuming that Venus’ passing shot is a little

better when she shoots it to Serena’s left than when she

shoots it to Serena’s right. This is her cross court passing

shot and this is her down the line passing shot.

So none of that fine detail matters a great deal,

but just if you’re interested that’s where the numbers come

from. I’m not claiming this is true

data by the way, I made up these numbers.

Actually I think Dixit made up these numbers,

I forget where I got them from. So okay, everyone understand

the game? So now imagine,

either imagine you are Venus or Serena, or imagine perhaps more

realistically, that you’ve become Venus or

Serena’s coach. Do I have any members of the

tennis team here? No.

Well imagine you’ve become their coach, so you take this

class and then you apply to replace their father as being

their coach. That’s a tough assignment I

would think. So an obvious question is,

you’re coaching Venus before Wimbledon, you know this

situation’s going to arise and you might want to coach Venus on

what should she do here? Should she try and pass Serena

down the line or she should try and hit the cross court volley,

cross court passing shot? Notice that this is a question

of should you, Venus, play to your strength

which is the cross court passing shot,

or should you play to Serena’s weakness, which would be to hit

it to Serena’s backhand. Playing to your strength is to

choose right and playing to Serena’s weakness is to choose

left. Conversely, for Serena,

should you lean towards your strength, which I guess is

leaning to the right or should you lean towards Venus’

weakness, which I guess is leaning left?

When you look at coaching manuals on this stuff,

or you listen to the terrible guys who commentate on tennis

for ESPN–oh no I’m getting in trouble again–very nice guys

who commentate on tennis for ESPN,

they say just incredibly dumb things at this point.

They say things like, you should always play to your

strengths and don’t worry about the other person’s weakness.

I think it won’t take much time today to figure out that’s not

great advice. But can people at least see

that this is a difficult problem, this is not an

immediately obvious problem, is that correct?

One reason it’s not immediately obvious is not only is no

strategy dominated here, but there is no pure strategy

Nash Equilibrium in this game, in this little sub game.

There is no pure strategy Nash Equilibrium–and notice that I

added the qualifier now. Previously I would just have

said Nash Equilibrium, but now that we have mixed

strategies in the picture, I’m going to talk about pure

strategy Nash Equilibria to be those that are the only

involving pure strategies. Okay, so why is there no pure

strategy Nash Equilibrium? Well let’s have a look.

So if Venus–If Serena thought that Venus was going to choose

left then her best response, not surprisingly,

is to lean left and if Serena thought that Venus was going to

choose right, then her best response is to

cheat to the right, so 50 is bigger than 20,

and 80 is bigger than 10. And conversely,

if Venus thought that Serena was cheating a bit to the left

then her best response is to hit it to Serena’s right,

and if Venus thought Serena was leaning to the right then Venus’

best response is to hit it to Serena’s left.

So I think that’s not at all surprising when you think about

it, not at all surprising, you’re going to get this little

cycle like this, but we can see immediately that

these best responses never coincide,

so there is no pure strategy equilibrium.

So that leaves us a bit stuck except I guess you know what the

next question’s going to be, and I shouldn’t leave it in too

much suspense. The next question’s going to

be, okay there’s not pure strategy Nash Equilibrium,

but we’ve just introduced a new idea which was what?

It was Nash Equilibrium in mixed strategies.

Maybe there’s going to be a mixed strategy Nash Equilibrium.

In fact, there is, there is going to be one.

So our exercise now is, let’s find a mixed strategy

Nash Equilibrium, and before we find it,

let’s just interpret what it’s going to mean.

A mixed strategy Nash Equilibrium in this game,

is going to be a mix for Venus between hitting the ball to

Serena’s left and Serena’s right,

and a mix for Serena between leaning left and leaning right,

such that each person’s mix, each person’s randomization is

a best response to the other person’s randomization.

Since these players are sisters and have played each other many,

many times, not just in competition but

probably in practice, it seems like a reasonable idea

that they might have arrived in playing each other,

at a mixed strategy Nash Equilibrium.

That’s what we’re going to try and do, now how are we going to

do that? So what we’re going to do is

we’re going to exploit the trick that we have here,

the lesson here. The lesson we have here says if

players are playing a mixed strategy as part of a Nash

Equilibrium, each of the pure strategies

involved in the mix, each of their pure strategies

must itself be a best response. We’re going to use that idea.

So let’s try and do that. So I’m hoping that by doing

this, I’m going to illustrate to you immediately,

that this idea is actually useful, at least useful if you

end up coaching the Williams sisters.

Alright, I want to keep this so you can still read it.

Ill bring it down a bit. Can people still read it?

Okay, so what I want to do is, I want to find a mixture for

Serena and a mixture for Venus that are equilibrium.

Having put it up there let me bring it down again.

This was not so intelligent of me.

I actually want to bring in some notation,

so as before, let’s assume that Serena’s mix

is, let’s use Q and (1-Q) to be

Serena’s mix and let’s use P and (1-P) to be Venus’ mix.

Let’s establish that notation. So here’s the trick,

So this is the slightly magic bit of the class,

so pay attention, I’m about to pull a rabbit out

of a hat. Trick, what should I do first,

to find Serena’s Nash Equilibrium mix,

so that’s (Q, (1-Q)), what I’m going to do is

I’m going to look at Venus’ payoffs.

So to find Serena’s Nash Equilibrium mix the trick is to

look at Venus’ payoffs, that’s going to be my magic

trick. Let’s try and see why.

So let’s look at Venus’ payoffs, Venus’ payoffs against

Q. So if Serena is choosing (Q,

1-Q), what are Venus’ payoffs? So if she chooses left then her

payoff is 50 with probability Q–and I’m going to use the

pointer here, and hope that the camera can

see this too. She gets 50 with probability Q

and she gets 80 with probability 1-Q.

If she chooses right then she gets 90 with probability Q and

she gets 20 with probability of 1-Q.

I meant to point to that. So what?

So what is this: we’re looking for a mixed

strategy Nash Equilibrium, so in particular,

not only Serena is mixing but in this case what we’re claiming

is, Venus is mixing as well. So if Venus is mixing as well,

that means that Venus is using the strategy left with some

probability P and using the strategy right with some

probability 1-P. Since Venus sometimes chooses

left and sometimes chooses right as her best response to Q,

her best response to Serena, what must be true of the payoff

to left and the payoff to right? Let’s go through it again,

so we’re going to assume that Venus is mixing.

So sometimes she chooses left and sometimes she chooses right

and she’s going to be, she’s in a Nash Equilibrium,

so she’s choosing a best response.

So whatever that mix P, 1-P is, it’s a best response.

Since she’s playing a best response of P and that sometimes

involves choosing left and sometimes involves choosing

right, it must be the case that what?

It must be the case that both left itself and right itself are

both themselves best response. If she’s mixing between them,

it must be that both choosing left or choosing right are

themselves best responses. If they weren’t she should just

drop them out of the mix, that would raise her average

payoff. Right, just like we dropped out

the short T.A.’s to get a high height and we dropped out the

failing Yale students to get a high GPA.

So if Venus is mixing in this Nash Equilibrium then the payoff

to left and to right must be equal,

they must both be best responses, both left and right

must be a best response, so in particular,

the expected payoffs must be the same.

Is that right, is that correct? So what does that allow me to

do? It allows me to put an equals

sign in here. Since left is a best response

and right is a best response, since they’re both best

responses, they must yield the same expected payoff.

Here’s their expected payoffs, they must be the same.

Now, I’ve got one equation and one unknown, and now I’m down to

algebra. So let me do the algebra.

I claim this expression is equal to that expression,

so simplifying a bit I’m going to get–you should just watch to

make sure I don’t get this wrong–I’m going to get 40Q,

so this implies 40Q is equal to 60(1-Q).

So I took this 50 onto this side and this 20 onto that side,

so I have 40Q is equal to 60(1- Q) and that implies that Q is

equal to .6. So those last two steps were

just algebra. So what was the trick here?

The trick was I found Q, which is how Serena is mixing

by looking at Venus’ payoffs, knowing that Venus is mixing

and hence I can set Venus’ payoffs equal to one another.

Say that again, I found the way in which Serena

is mixing by knowing that if Venus is mixing,

her expected payoffs must be equal and I solved out for

Serena’s mix, this is Serena’s mix.

Let’s do it again. Here I’m wishing I had another

board. I don’t want to lose those

numbers entirely, so I’m going to try and squeeze

in a bit. I know what I can do.

Let’s get rid of this one entirely.

There we go, that works. Let’s get rid of this one

entirely. I can still see my numbers.

Let’s do the converse. Let’s do the trick again,

this time what I’m going to do is I’m going to figure out how

Venus is mixing. I know how Serena is mixing

now, so now I’m going to work out how Venus is mixing.

Now, to figure out how Serena was mixing, I used Venus’

payoffs. So to find out how Venus is

mixing what am I going to do? I’m going to use Serena’s

payoffs. So to find Venus’ mix,

which is P, 1-P, –let’s be careful it’s her

Nash Equilibrium mix–use Serena’s payoffs.

Here we go, so if Serena chooses, this is S’s payoffs,

if Serena chooses L then her payoffs will be what?

So again, just watch to make sure I don’t get this wrong and

I’ll point to the things to try and help myself a bit.

So with probability P she’ll get 50.

So 50 with probability P, and with probability 1-P she’ll

get 10. And if she chooses to lean to

the right, to lean towards her forehand, then with probability

P she’ll get 20 and with probability 1-P she’ll get 80.

We know that Serena is mixing, so since Serena is mixing what

must be true of these two payoffs?

What must be true of the two payoffs?

The payoff to l and the payoff to r, what must be true about

them since Serena is using a mixture of these two strategies

in Nash Equilibrium? It must be the case that both l

is a best response and r is a best response,

in which case the payoff must be, someone shout it out,

equal, thank you. They must be equal,

these must be equal. They must be equal since Serena

is indifferent between choosing left or right and hence is

mixing over them. So again, using the fact that

they’re equal reduces this to algebra, and again,

I’ll probably get this wrong but let me try.

So I claim, let’s take 20 away from here, I’ve got 30P equals

70(1-P). I hope that’s right,

that looks right. Again, this is just algebra at

this point. So I took 20 away from here and

10 away from there, and this implies that P equals

.7. So I claim I have now found the

mixed strategy Nash Equilibrium. Here it is.

The Nash Equilibrium is as follows.

Let’s be careful, this is Venus’ mix.

So if Venus is mixing .7, .3, .7 on left and .3 on right,

and Serena is mixing .6, .4, so this is Venus’ mix and

this Serena’s mix. Venus is shooting to the left

of Serena with probability of .7 and Serena is leaning that way

with probability of .6. So we were able to find this

Nash Equilibrium by using the trick before.

Now let’s just reinforce this a little bit by talking about it.

So suppose it were the case that Serena, instead of leaning

to the left .6 of the time leant to the left more than .6 of the

time. So suppose you’re Venus’ coach,

and suppose you know that Serena leans to the left more

than .6 of the time, what would you advise Venus to

do? Let me try it again.

So suppose your Venus’ coach and suppose you’ve observed the

fact that Serena leans to the left more than .6 of the time,

what would you advise Venus to do?

Pass to the right, shout out. Student: Pass to the

right. Professor Ben Polak:

Pass to the right, exactly.

So if Serena cheats to the left more than .6 of the time,

then Venus’ best response is always to shoot to the right.

That maximizes her chance of winning the point.

Conversely, if Serena leans to the left less than .6 of the

time, then Venus should do what? Shoot to the left all the time.

So if Serena doesn’t choose exactly this mix,

then Venus’ best response is actually a pure strategy.

Say it again, if Serena leans to the left too

often, more than .6, then Venus should just go right

and if Serena leans to the left too little,

then Venus should always go left.

We can do exactly the same the other way around.

If Venus shoots to the right, so that’s her cross hand

passing shot more than .7 of the time,

and you’re Serena’s coach, what should you tell Serena to

do? Go that way all the time.

So if Venus is hitting it to Serena’s left more than .7 of

the time, Serena should just always go to her left,

and if Venus is hitting to the left less than .7 of the time,

so to the right more than .3 of the time,

then Serena should always go to the right.

So that’s how this kind of comes back into the sort of the

coaching manuals if you like. Okay, so how am I doing so far?

Have I lost everyone yet or are people still with me?

How many of you play tennis, ever?

So all your tennis is going to dramatically improve after

today, right? So now let’s make life more

interesting. Let’s go back to the start.

We’ve figured out this is an equilibrium, this is how Venus

and Serena play, Venus and Serena know each

other perfectly well, they know that they mix this

way, they’re going to best respond

to it, this is going to be where they end up.

But in the meantime, Serena hires a new coach and

Serena’s new coach is just very, very good at teaching Serena

how to play at the net, and in particular,

how to hit the backhand volley. So Serena’s new coach,

let’s say it’s Tony Roche or somebody, it’s just a brilliant

coach and Tony Roche is able to improve Serena’s backhand volley

and that changes these payoffs. So you should rewrite the whole

matrix but I’m going to cheat. So the new game is exactly the

same as it was everywhere else, except for now when Serena gets

to the backhand volley, she gets in it 70% of the time.

So there used to 50,50 in that box and now it’s 30,70.

So the game has changed because Serena has got better at hitting

backhand volleys. We want to figure out how is

this going to affect play at Wimbledon?

Now it doesn’t take much to check that there is still no

pure strategy Nash Equilibrium. It’s still the case,

in fact even more so, that Serena’s best response to

Venus choosing left is to lean to the left.

So it’s still the case that the best responses do not coincide,

there is still no pure strategy equilibrium.

What we’re going to do of course is we’re going to find a

mixed strategy equilibrium, but before we do so,

let’s think about this intuitively.

Let’s see if we can intuit an answer.

I’m guessing we can’t, but let’s see if we can intuit

an answer. So Serena has improved her

backhand volley, and hence when she reaches it

she gets it in more often. So one effect,

you might think, is what we might want to call a

direct effect and I think there’s two effects here.

There are two effects, one of these I’m going to call

the direct effect, and by effect,

I mean in particular an effect on how Serena should play the

game. So since Serena has improved

her backhand volley, when she reaches that volley

she gets it in more often, so one might say in that

case–your Serena’s coach–in that case you should lean to the

left more often than you did before,

because at least when you get that backhand volley you’re

going to get it in more often. So the direct effect says

Serena should lean left more, in other words,

Q should go up. Is that right?

So Serena’s now better at playing this backhand volley,

so she may as well favor it a bit more and hence Q will go up.

So that’s the direct effect, but of course there’s a “but”

coming. What’s the but?

Again, let’s see my tennis players here,

raise your hands if you play tennis.

Suddenly nobody plays tennis, come on raise your hands okay.

What’s the but here? We think Serena’s backhand has

improved so she might be tempted to play towards her backhand a

bit more often, what’s the but?

So I claim the but is this–you tell me if I’m wrong–the but is

that Venus (she’s her sister after all,

right, so Venus knows that Serena’s backhand has improved)

so Venus is going to hit it to Serena’s left less often than

before. Is that right?

So since Serena’s backhand has improved, Venus is going to hit

it to Serena’s backhand less often than before,

and that might make Serena less inclined to cheat towards her

backhand because the ball is coming that way less often.

So this is a indirect or a strategic effect.

The strategic effect is Venus hits L less often,

so Serena should reduce the number of times that she leans

to the left because the ball is coming that way fewer times.

Now notice that these two effects go in opposite

directions, is that right? One of them tends to argue that

Q would go up, that’s the direct effect and

the other one is more subtle, it says we now think about not

just how my play has improved, but also how the other person’s

going to respond to knowing that my play has improved,

that’s the more subtle effect and that’s going to push Q down.

That’s going to make it less likely, that’s an argument

against leaning to the left. So imagine you’re going to be

Serena’s coach, which of these effects do you

think is going to win, let’s have a poll.

Which of these effects do you think is going to win?

The direct effect or the indirect effect?

The direct effect or the strategic effect?

Who thinks the direct effect? Who thinks Serena,

who’d advise Serena to play to her strength a bit more and lean

left a bit more, who thinks the direct effect?

Raise your hands, let’s have a poll.

Who thinks the indirect effect, the effect of Serena hitting it

that way less often is going to win?

Who’s abstaining and basically refusing to be a coach?

Quite a number of you, all right.

Well we’re going to find out by re-solving for the Nash

Equilibrium. What we’re going to do is redo

the calculation we did before starting with Serena.

So to find Serena’s mix, to find Serena’s new

equilibrium mix, what do we have to do?

The question is, in equilibrium,

is Serena going to lean to the left more (so Q is going go up)

or less (so Q’s going to do down).

So I need to find out what is Serena’s new equilibrium mix.

What’s the new Q? How do I go about finding

Serena’s equilibrium Q, what’s the trick here?

Shout it out. Use Venus’ payoffs.

So to find the new Q for Serena, use Venus’ payoffs.

Now let’s do that. So from Venus’ point of view,

if she chooses left then her payoffs are now,

and again I should use the pointer,

30 with probability Q, this is the new Q and 80 with

probability 1-Q, 30 with probability Q plus 80

with probability 1-Q. Again, this is the new Q,

I should really give it, put Q prime or something but I

won’t. If she chooses right then her

payoff is what? It’s going to be 90 with

probability Q and 20 with probability 1-Q.

What do we know about these two payoffs if Venus is mixing in

equilibrium? We know she’s mixing in

equilibrium because we saw there was no pure strategy

equilibrium, so what we do know about these

two payoffs since Venus is using both these strategies in

equilibrium? They must be the same.

Since she’s using both these strategies, these strategies

must be equally good. They must both be best

responses so these two payoffs are equal.

Since they’re equal all I have to do is solve out for Q,

so let’s do it. So I’m going to get 90 minus 30

is 60Q, is equal to 80 minus 20 which is 60(1-Q),

so Q equals .5. If I did the algebra too

quickly just trust me, I think I got it right.

From here on in, it was just algebra.

So what have I found out? Did Q go up or go down?

Well it used to be, Q used to be what?

.6 and now its .5, so let me ask what I think is

an easy question, did it go up or down?

It went down. Q went down,

the equilibrium Q went down. So which effect turned out to

be bigger? The direct effect of playing

more to your strength or the indirect effect of taking into

account that your opponent is going to play less often to your

strength. Which effect turned out to be

the bigger effect? The indirect effect,

the strategic effect. Of course I really did want the

strategic effect to be bigger because this is a course about

strategy, but the strategic effect actually won here.

The strategic effect, the indirect effect is bigger.

That’s good news for me because it says the slightly dumb coach

who didn’t bother to take Game Theory would have stopped at

this direct effect and they’d have told Serena to go the wrong

way, but the smart coach who takes

my class, and therefore somehow contributes to my salary,

in an extraordinarily indirect way, gets it right.

Now we can also solve out for Venus’ new mix and we’ll do it

in a second. But before I do it,

let me just point out that we actually, we really can now

intuit Venus’ effect. It may not be exact numbers but

we can intuit here. As I claim, I claim if we think

this through carefully, we know whether Venus is

shooting more to the left, than she was before,

or less to the left, than she was before.

Notice that in the new equilibrium Serena is going less

often to her left even though she’s better at hitting the

backhand, she’s better at hitting the

ball when she gets there. So since Serena is leaning left

less often what must be true about Venus in this new

equilibrium? It must be the case that Venus

is hitting the ball to the left less often.

Does that make sense? We have enough information

already on the board to tell us that, nevertheless,

let’s do the math. Let’s go and retrieve a board

to do the math. Just to complete this,

let’s figure out exactly what Venus does do.

So to figure out what Venus is going to do, what’s our trick?

I want to figure out how Venus is going to mix.

I’m going to find out Venus’ new P, how do I find out Venus’

new equilibrium mix? I look at Serena’s payoffs.

So if Serena chooses left, her payoff is,

and I’ll read it off quickly this time,

is 70P plus 10(1-P) and if Serena chooses right her payoff

is 20P plus 80(1-P) and I’m praying that the T.A.’s are

going to catch me if I make a mistake here,

and I know these have to be equal because I know that in

fact Venus is mixing–sorry, I know that Serena is mixing,

so I know these must be equal. So since they’re equal I can

solve out and hope that I’ve got this right, so I’ve got 50P

equals 70(1-P), so P is equal to 7/12.

So again, that’s just algebra, I rushed it a bit,

it’s just algebra. Same idea, just algebra.

So 7/12 is indeed smaller than what it used to be,

because it used to be 7/10, so that confirms our result.

So the strategic effect dominated.

Venus shot to Serena’s backhand less often, and as a

consequence, so much so, that Serena actually found it

worthwhile going more to the right than she used to before.

Now let’s just talk this through one more time.

This was a comparative statics exercise.

We looked at a game, we found an equilibrium,

we changed something fundamental about the game,

and we looked again to look at the new equilibrium,

that’s called comparative statics.

Let’s talk through the intuition.

Before we made any changes Venus was indifferent.

She was indifferent between shooting to the left and

shooting to the right. Then we improved Serena’s

ability to hit the volley to her left, we improved her backhand

volley. If we had not changed the way

Serena played then what would Venus have done?

So suppose in fact Serena’s Q had not changed.

If Serena’s Q had not changed, remembering that Venus was

indifferent before, how would Venus have changed

her play? Somebody?

If we started from the old Q and then we improved Serena’s

ability to play the backhand volley, and if Q didn’t change,

what would Venus have done? She’d never,

ever have shot to the left anymore, she’d only have shot to

the right which can’t possibly be an equilibrium.

So something about Serena’s play has to bring Venus back

into equilibrium, it brings Venus back into being

indifferent, and what was it? It was Serena moving to the

left less often and moving to the right more often.

To say it again, if we didn’t change Q,

Venus would only go to the right, so we need to reduce Q,

have Serena go to the right, to bring Venus back into

equilibrium. Conversely, if Venus hadn’t

changed her behavior, if Venus had gone on shooting

exactly the same as she was, P and 1-P as before,

then Serena would have only gone to the left and that can’t

be an equilibrium. So it must be something about

Venus’ play that brings Serena back into equilibrium,

and what is it? It’s that Venus starts shooting

to the right more often. So just two reminders,

before you leave two reminders. Wait, wait, wait.

First, in about five minutes time a handout will magically

appear on the website that goes through these arguments again,

all of them in two other games, so you can have a look at the

handout. Second thing,

a problem set has already appeared by magic on that

website that gives you lots of examples like this to work on.

Play tennis over the weekend for practice and we’ll see you

on Monday.

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