
Title:
14. Backward induction: commitment, spies, and firstmover advantages

Description:
Game Theory (ECON 159)
We first apply our big ideabackward inductionto analyze quantity competition between firms when play is sequential, the Stackelberg model. We do this twice: first using intuition and then using calculus. We learn that this game has a firstmover advantage, and that it comes commitment and from information in the game rather than the timing per se. We notice that in some games having more information can hurt you if other players know you will have that information and hence alter their behavior. Finally, we show that, contrary to myth, many games do not have firstmover advantages.
00:00  Chapter 1. Sequential Games: First Mover Advantage in the Stackelberg Model
38:13  Chapter 2. First Mover Advantage: Commitment Strategy
49:25  Chapter 3. First Mover Advantage: Why It Is Not Always an Advantage
55:53  Chapter 4. First and Second Mover Advantage: NIM
Complete course materials are available at the Open Yale Courses website: http://open.yale.edu/courses
This course was recorded in Fall 2007.

Professor Ben Polak: All
right, so today I want to do

something a little bit more
mundane than we did on Monday.

I want to go back and talk
about quantity competition.

So in the first half of the
course we talked about price

competition.
We talked about quantity

competition.
We talked about competition

with differentiated products.
I want to go back and revisit

essentially the Cournot Model.
So this was the Cournot Model:

two firms are producing,
are choosing their quantities

simultaneously.
Firm 1 is choosing Q1 and Firm

2 is choosing Q2.
And all of this is just review

so this is all stuff that's in
your notes already.

This is the demand curve.
It tells us that prices depend

on the total quantity being
produced.

So this is Q1 + Q2 and this is
prices, then the demand curve is

a straight line of slope b.
That's what this tells us.

Here's our slope –b.
And we know that payoffs are

just profits,
which are price times quantity,

revenues,
minus cost times quantity,

costs, we're assuming constant
marginal costs.

We did this model out in full
in maybe the third week of class

and we figured out what the best
response diagram looked like.

And if you remember correctly,
this was the best response for

Firm 1 taking Firm 2's output as
given,

and this is the best response
for Firm 2 taking Firm 1's

output as given,
and there were a few other

details in here.
This was the monopoly quantity,

this was the competitive
quantity and so on,

but this is enough for today.
Actually, we had done a bit

more than that,
we'd actually worked out in

class what the equations were
for these best responses.

Here they are.
I'm not going to rederive

these today, but they're
somewhere in your notes.

We kind of crunched through
some calculus and figured out

what Firm 1's best response
looks like algebraically,

here it is.
So this is the equation of this

line, and similarly for Firm 2,
so this is the equation of this

line.
Finally, we figured out what

the Nash Equilibrium was,
and there's no prizes here:

the Nash Equilibrium in Cournot
was where these best responses

crossed,
and this is the equation for

the Nash Equilibrium.
Have I made a mistake?

The best response, oh thank you.
The best response for Firm 1 is

a function of Q2,
exactly.

Thanks Jake.
So this is all stuff we did

before, I want to go back to
this model now to revisit it in

the context of thinking about
sequential dynamic games.

So what we're going to do
iswe're going to do is,

we're going to imagine that
rather than having these firms

choose their quantities
simultaneously,

one firm gets to move first and
the other firm moves after.

Let's be clear,
we're going to assume that Firm

1s moving first and the other
firmwe'll assume Firm 1's

going to move firstthe other
firm,

Firm 2, is going to get observe
what Firm 1 has chosen and then

get to make her choice.
So we're going to see what

difference it makes when we go
from this classic simultaneous

move game into a sequential move
game.

This model is fairly famous and
I'm almost certainly spelling

this wrong, but it's due to a
guy called Stackelberg.

So what we're looking at now is
the Stackelberg Model.

So how do we want to think
about this?

A natural question to bear in
mind is, assuming we're in this

world of quantity competition,
is it an advantage to get to

move first, to set one's
quantity first?

Or is it an advantage to be
able to wait,

see what the other firm has
done, and then respond?

Is there an advantage in going
first or is there an advantage

in knowing a bit more about the
other firm and being able to

move second?
That's going to be the question

at the back of our minds for
most of today.

So how are we going to think of
this?

How are we going to figure this
out?

There shouldn't be a silence in
the room.

There should be an instance
answer.

How are we going to figure this
out?

We're going to use backward
induction right.

This is going to be an exercise
in backward induction.

We won't be able to draw a tree
here because the game's too

complicated because there's a
continuum of actions,

but nevertheless,
we are going to use backward

induction.
So what does using backward

induction mean here?
Using backward induction means

starting at the end and the end
is what?

The end here is Firm 2.
Firm 1 is going to move first,

Firm 2's going to observe that
choice and then move.

So the end of the game is Firm
2.

So we're going to solve out
Firm 2's problem first.

We're actually going to do this
entire analysis twice.

We're first of all going to do
this analysis a bit intuitively

looking at pictures,
and then I want to go back and

crunch it out in the math.
I want to get used to seeing

that we can actually do it
crunchily.

This board is just review,
so I'm going to get rid of it I

think.
I didn't manage to get rid of

it, never mind.
We're not going to be using

this board.
This is just what we did do in

the simultaneous move game,
so we'll get rid of it.

So in the sequential move game
we're going to start by

analyzing the move of Firm 2.
So imagine yourself as the

manager of Firm 2,
you're coming along to make

your output decision.
The output of Firm 1 is already

set.
So analyze Firm 2 first,

Firm 2 sees Q1 and now I must
choose Q2.

So what is Firm 2 going to do?
So I claim we already know this.

We've already solved this
problem.

When did we solve this problem?
Anyone know when we solved this

problem?
The problem of what Firm 2's

going to do.
Well we already solved it about

a month ago when we looked at
the simultaneous move game,

because what we worked out then
was what is Firm 2's best

response for any particular
choice that Firm 1 makes?

We already solved out that
problem.

It took us a while to solve
out, but basically it was to

maximize Firm 2's payoff,
taking as given Firm 1.

We already know what the
equation looks like and let's

just remind ourselves what the
picture looked like,

just a repeat of the picture we
had before.

We said for any particular
choice of Firm 1,

Firm 2's best response can be
drawn on a best response

diagram, and looked like this.
It's exactly the picture we

have up there.
So this is the best response

for Firm 2 taking as given the
choice of Firm 1.

We even know the equation of
itI won't bother rewriting

that.
We already know the equation.

So in some sense Firm 2's
problem is a problem we've

already seen.
We already worked out months

ago what Firm 2 should do,
taking if Firm 1's output is

given and that's exactly the
problem Firm 2 finds herself in.

She wakes up one morning,
Q1 has been set already,

and now she must choose Q2 to
maximize her profits,

so she's going to choose her
best response.

Just to remind you how we read
this picture,

for any particular choice of Q1
we go up to the line and look

across, this tells us what Q2
will respond.

So if Q1 chooses this amount
then Q2 will choose this amount.

If Q1 chooses this amount then
Q2 will choose this amount and

so on.
So there's no mystery here.

We already know what Firm 2's
going to do.

So by definition,
the best response of 2 to Q1

tells us the profit maximizing
output of Firm 2 taking Q1 as

given.
All right, so we've done the

second step for this already,
we already know what Firm 2's

going to do.
Of course, the additional step

here now is that Firm 1 knows
that Firms 2's going to do it.

Firm 1's going to move first
and Firm 1 knows that after she

sets her quantity Q1,
Firm 2 will respond by choosing

her corresponding quantity,
which is the best response to

it.
So if Firm 1 knows that if Firm

1 were to choose this quantity,
then Firm 2 will respond by

choosing this quantity,
and Firm 1 knows that if she

chose this smaller quantity,
then Firm 2 will respond by

choosing this larger quantity.
Is that right?

So Firm 1 can anticipate how
Firm 2 is going respond to each

of these choices.
So let's just make that clear.

So in particular,
if Firm 1 was to choose Q^1,

I'm not suggesting it should,
but if Firm 1 was to choose

this Q^1, then Firm 1 knows that
Firm 2 will produce this

quantity,
which is the best response to

Q^1, and if Firm 1 were to
choose Q^^1, then Firm 2 will

respond by choosing the best
response to Q^^1.

So this is pretty
straightforward so far,

but what we're able to see now,
is the problem facing Firm 1,

which is the interesting
problem.

The problem facing Firm 1 is,
what quantity should Firm 1

choose knowing that this is how
Firm 2 is going to respond?

Before we solve this out
mathematically,

I just want us to think it
through a little bit.

So the first way I want to
think this through is,

is to make the following
observation.

From Firm 1's point of view,
Firm 1 knows that any Q1 she

chooses leads to a response on
this line by Firm 2.

That's what Firm 1 knows.
So Firm 1 is effectively

choosing points on this line.
Let me say it again,

so what's actually happening is
Firm 1 is choosing Q1 and Firm 2

is responding by choosing a Q2
that puts them on this line.

But in effect that means Firm 1
is choosing points on this line.

So you could think of Firm 1's
problem as, choose the joint

output level on this line that
maximizes Firm 1's profits.

Think of Firm 1's problem as
choose the combination of

outputs on this line by choosing
Q1 and then Q2 responds,

choose the combination on this
line that maximizes profits for

Firm 1.
So I'm belaboring this a little

bit because it's a more general
mathematical idea here.

How many of you are in Econ 150
right now?

So for those of you in Econ
150, this should be a very

familiar kind of thing.
This is a constrained

optimization problem and you've
been having constrained

optimization problems rammed
into you for the last month or

so,
so this is an example of a

constrained optimization
problem.

You have to choose a point but
you can't choose a point freely:

you have to choose a point on
the line.

Okay, so let's talk about it a
bit more before we do the math.

Let's actually redraw it again
since I made a mess of this

picture.


So one thing you might want to
ask is, in making this choice

for Firm 1, should Firm 1 choose
more or less or the same as it

used to choose when the problem
was simultaneous?

So let's put in again what it
used to choose when the problem

was simultaneous.
I'll put it in just faintly.

So here's our old Cournot
picturelooked like thisand

this was the quantity that Firm
1 chose in the Cournot game,

so let me call that Q1C.
So certainly one possibility is

that Firm 1 could choose her
Cournot quantity,

she can certainly do that,
and she knows that if she does

that, Firm 2 will respond by
choosing the best response of

Firm 2 to Firm 1 choosing the
Cournot quantity,

but we know what that is.
What's Firm 2's best response

to Firm 1's Cournot quantity?
It's Firm 2's Cournot quantity,

right?
So if Firm 2 does that,

if Firm 1 chooses the Cournot
quantity, then Firm 2 will also

choose the Cournot quantity.
So one thing that Firm 1 could

do is effectively choose the old
equilibrium.

That's certainly something
that's available to Firm 1.

But Firm 1 could also do other
things.

Firm 1 could produce less than
that or Firm 1 could do more

than that.
So who thinks Firm 1 should

playshould choose the old
equilibrium quantity?

Who thinks Firm 1 should choose
more than that?

Who thinks Firm 1 should choose
less than that?

Let's just try it with the
camera on you.

So once again,
how many people think that Firm

1 should choose the old
equilibrium quantity?

A dribbling of hands,
and how about less than that?

A few hands and then they went
down again, and how about more

than that?
There's a majority for

moreturns out more is correct,
so that's good news.

Why?
Why do we think Firm 1 should

produce more than it used to
produce before?

Any takers on this?
Well let's think about it.

As Firm 1 produces more,
or if Firm 1 were to produce

more, then Firm 2my voice is
goingthen Firm 2 would produce

what?
Less.

As Firm 1 produces more than
her Cournot quantity,

Firm 2's response is to produce
less.

Does anyone remember the jargon
for this?

What do we call games where the
more I do of my strategy the

less you do of yours?
"Strategic substitutes," good.

This is a game of strategic
substitutes.

What that means is that as Q1
goes up, Q2, the best response

of Firm 2 to Q1 goes down.
So what?

We can look at that just by
looking at the picture.

Well the "so what" is,
now we're in a sequential game.

If Firm 1 produces more than
her Cournot quantity she induces

Firm 2 to produce less.
That's what we just said and

that's what?
That's good for Firm 1.

My producing more inducing you
to produce less is good for me.

It's going to keep prices
higher in the market.

Is that right?
So let's just think it through

again.
In the Cournot equilibrium,

the choice of Firm 1 was the
best choice for Firm 1,

taking the choice of Firm 2 as
given.

That was the old Cournot
quantity.

But now in the Stackelberg
setting, the sequential setting,

there's an additional feature.
Firm 1 doesn't have to take

Firm 2's output as given.
There's an additional reason

for producing at the margin,
which is, at the margin if I

produced some more units of
output,

that leads you to produce less
which is good for me.

So that suggests that I'm going
to produce than I used to

produce under the old
assumption.

So this suggests that Firm 1
should set Q1 bigger than Q1^(C)

to induce Q2 to be less than
Q2^(C).

So the first thing we've
learnedwe'll see this in the

math lateris that Firm 1 will
in fact produce more than they

used to under Cournot,
and that will result in Firm 2

producing less than Cournot.
Now we've already got a lot on

the board now,
we can actually solve out

intuitively the problem.
Do we think that Firm 1's

profits, by this procedure,
are the same as they were under

Cournot?
Are they less than they were

under Cournot,
or are they more than they were

under Cournot?
So just let me say it again.

Firm 1 is going first now.
We've argued that Firm 1's

going to produce more.
Do we think that Firm 1's

profits at the end of the day
are going to be the same as they

were under Cournot,
higher than they were under

Cournot, or lower than they were
under Cournot?

So let's have a poll again,
let me get the camera on you

guys.
So who thinks their profits are

going to be the same as they
were under Cournot?

Who thinks the profits have
gone up?

Who thinks the profits have
gone down?

We're in good shape here
because indeed the profits have

gone up.
There's a very simple argument

why the profits have to have
gone up.

How do we know the profits must
have gone up?

Let me actuallyit's simple
enoughlet me grab a mike on

this.
How do we know the profits just

must have gone up?
There was a hand in the back,

was there a hand in the back?
Yes, way at the back.

How do we know profits must
have gone up here?

Way at the back.
Student: If Firm 1 was

going to lower their profits
they wouldn't have chosen to

produce more.
Professor Ben Polak: All

right, good exactly.
The fact that Firm 1 has

changed their output,
and in particular,

are producing more,
tells you they must be able to

increase their profits by this
maneuver.

Let's just think that through.
One option that was available

to Firm 1 before was to set
output at the Cournot level.

If Firm 1 had set output at the
Cournot level,

that would have led Firm 2 to
set output at the Cournot level,

and in that case,
profits would have been exactly

the same as before.
The fact that Firm 1 has moved

away from that must mean there
are higher profits available.

Say it another way,
Firm 1 could have had exactly

what it had before,
so it must be doing at least as

well as it was doing before,
and the fact it has changed

means it must be doing better
than it was doing before.

So indeed, Firm 1's profits
have gone up.

We don't even need any math to
prove that: it just must be the

case logically.
What must have happened to Firm

2's profits?
What do you think has happened

to Firm 2's profits?
That's not so immediately

obvious.
It's obvious,

I think, that Firm 1's profits
have gone up here because Firm 1

could have had the same old
profits and has chosen something

else.
But it's not immediately

obvious what happened to Firm
2's profits, is that right?

Before we get to what's
happened to Firm 2's profits

let's go through an intermediate
step.

Let's try and ask what must
have happened to total output in

the market in this examplein
this nice simple example.

That's not immediately obvious
either.

Why?
We've argued that Firm 1's

output went up but Firm 2's
output went down relative to

Cournot.
So it's not immediately obvious

whether the sum of those two Q1
+ Q2 went up or down.

We'd like to know what happened
to Q1 + Q2, total output,

in the market.
By the way, one particular

reason we might care about this
is of course consumers would

like it to have gone up.
Because if their total output

has gone up, prices have gone
down, and that's good for

consumers.
So if you're the regulator,

if you're designing this
industryif you're working for

the Justice Department,
or if you're working for

European Commissionyou're
going to want to know the answer

when we switch from a
simultaneous setting to an

asymmetric setting where there's
a leader firm and a follower

firm,
is that going to be good for

consumers or bad for consumers?
Well let's have a look.

Well we know that Firm 1's
output went up and we know that

Firm 2's output went down,
but can anyone tell me what

happened to the total output and
why?

Let's have a poll again.
Who thinks total output went

down?
Who thinks total output stayed

the same?
Who thinks total output went up?

There's lots of abstentions.
Let's try that again because

too many abstentions.
Who thinks total output went

down?
Who thinks total output stayed

the same?
Who thinks total output went up?

That's pretty split.
So I think total outputI know

actuallythat total output went
up, and I claim I can see it on

the picture.
I claim if you stare at that

picture you can actually see
that total output must have gone

up.
Who's good at looking at a

picture?
Let me get the mike in here.

The picture's there.
Let me try this person.

Your name is?
Student: Andy.

Professor Ben Polak:
Andy go ahead.

Student: Judging by the
slope of the line you know that

as it moves the amount that Q1
changes will be greater than the

amount that Q2 goes down.
Professor Ben Polak:

Good, so what Andy said,
what Andy's saying is look at

the slope of the line along
which we slid.

When we went from Cournot to
the Stackelberg equilibrium,

we started here and we slid in
this direction down this line:

Q1 went up and Q2 went down.
And what Andy's pointing out is

we can see from the slope of
this line that Q1 goes up more

than one unit for every unit of
reduction of Q2.

Let me say it again,
for every unit of increase of

Q1 there's less than a
proportionsorry,

say it again.
For every unit of increase of

Q1 there's less than a unit of
decrease of Q2.

Another way of saying it is the
slope of this line is less than

1.
Everyone see that?

So we know by the slope of the
linewe know that Q1 had to go

up more than Q2 went down,
which means total output went

up, which means that prices
wentwhat happened to prices as

total output went up?
That shouldn't be hard,

everyone's taken 115 here,
right?

So when total output went up,
prices went down,

good.
Demand curves sloping down is

not as important as backward
induction but it's still quite

important.
So prices went down.

So therefore,
we now are ready to say what

happened to Firm 2's profits.
Firm 2 is producing less than

before.
Firm 2's costs are the same,

and prices have gone down,
so what's happened to Firm 2's

profits?
They must have gone down as

well.
So Firm 2's profit has gone

down, and we know that consumer
surplus, CS, consumer surplus,

has gone up.
For those people who remember

their 115, prices have gone
down, quantities have gone up,

so consumer surplus here has
gone up.

So we've analyzed everything
qualitatively I can think of in

this game without reference to
any math at all.

Is that right?
We really haven't done any math

there.
We talked about slope of the

line.
I guess that's math in junior

high or something,
but we haven't really done any

math here, right?
Is that fair?

We already have a pretty good
intuition for what's going to go

on in this market.
We think Q1's going to go up.

We think Q2's going to go down,
we think that Firm 1's profits

are going up.
We think Firm 2's profits are

going down.
We think total quantity is

going up.
Now let's see if we're right.

Let's go back and do the math.
So I want to spend a bit of

time grinding this out.


So I don't claim that doing the
math is fun, but I want to prove

that we can do it,
because otherwise everyone's

going to either think that this
was all kind of just blah,

blah, blah, and/or people are
going to be scared to do the

math when it arrives on a
homework assignment.

So for now we're going tomore
or less for the next few

minuteswe're going to forget
we're economists and we're going

to turn into nerds.
That isn't a huge transition,

but we'll do it anyway.
I've got the demand curve up

there, so the demand the curve
is still there it's P = A  B

[Q1 + Q2]
and I've got profits written up

there,
but let's put them somewhere

more convenient anyway.
So P = A  B [Q1 + Q2]

and profit is equal toprofit
for Firm i is equal to P QiC

Qi.
And what we're told to do in

backward induction is what?
First of all,

solve things out for Firm 2
taking Firm 1 as given,

and then go back and solve for
Firm 1.

So exactly the discussion we've
just had informally,

we're now going to do more
formally.

So backward induction tells us,
solve for Firm 2 first,

taking Q1 as given.
What is that problem?

It's this sort of boring math
problem, it says maximize by

choosing Q2 the profits of Firm
1, so that's going to be A  B

Q1B Q2.
That's the price times the

quantity Q2,  C Q2.
So this bit here is the price.

These two terms together are
revenues, and this term is

costs.
Now I could do that,

I could grind that bit out,
but we already ground out that

bit out three or four weeks ago
right,

so I'm not going to grind it
out again, we know how to do

that.
But by the way,

let's just remind ourselves
what we did.

We differentiated with respect
to Q2.

We set the thing we found,
the derivative,

equal to 0.
That was our first order

condition.
And then we solved for Q2.

Is that right?
So when we did that,

we went through the first order
condition and then we solved it,

we know what we actually got.
So what we actually got was

that Q2*, if you likeor Q2
let's just call itis equal to

[A  C]
/ 2B  Q/2.

In fact, I've already given it
to you up there,

it's up on that top board.
It's the best response for Firm

2.
So again, I could do this

again, but since we did it a few
weeks ago I don't want to redo

it.


Now the more interesting part,
not thrilling,

but a little bit more
interesting.

Now let's solve for Firm 1.
So what is Firm 1 doing?

Firm 1 is also trying to
maximize profits.

Firm 1 is choosing Q1 and Firm
1at least initially it looks

like the same problem.
It's A B Q1B Q2 Q1C Q1,

this is the same line we had
before but now whereas Firm 2

was taking Q1 as given,
Firm 1 knows that Q2 is given

by this formula here.
So what we're going to do is

we're going to plug this Q2 into
there.

Now again, for those of you in
115, this isn't the only way to

do itI'm sorry 150this isn't
the only way that we could do

it,
we could also set up a

Lagrangian equation,
but for those people who don't

know what that is,
don't worry we're going to plug

in today.
So we're going to plug this in,

and when we plug it in,
we get a right old mess but

let's do it anyway.
So we're going to get max Q1,

[A  B Q1B [[A  C]/2B 
Q1/2]

 C]
Q1.

Everyone okay with that?
I'm doing algebra on the board

which is not fun but it's useful
to do occasionally.

So eventually what are we going
to do?

Eventually we're going to
differentiate this thing,

set it equal to 0,
look at our first order

condition and so on,
just as we normally would.

So eventually we're going to
use basically 112 level calculus

to solve this thing.
Everyone remember how to solve

a maximization problem?
Yeah?

But before we do that let's
tidy up the algebra a bit.

So this thing islet's just
tidy it up.

So this is equal to max with
respect to Q1and notice I've

got an A  C here and once I
take this B inside the brackets

I've got a[[A C]/2]
here.

So I've got an [A  C][[A 
C]/2]

so that's going to give me an
[A  C]/2and I'm really going

to pray that the T.A.'s are
watching me carefully and are

going to catch my errors here.
Okay, so I think I'm okay so

far but please catch me.
What else have I got?

I've got a B Q1 here and from
in this bracket I've got a  

that's a + B Q1/2.
So I have a –B Q1 + B

Q1/2 so that's a –B Q1/2.
So far so good and that whole

thing is multiplied by Q1.
Okay so far?

Let's multiply out the bracket
because otherwise I'll make a

mistake.
So this is the same as saying

[A  C]/2 Q1B Q1²/2.
So far so good?

Now we're at the level where
even my very rusty memory of

calculus will get us through,
so let's try and do it.

So what we're going to do is
we're going to differentiate

this thing with respect to Q1.
So differentiating with respect

to Q1let's do it up here ,
we get [A C]/2 from this term

and from the –B Q1²/2
we're going to getthe two's

are going to cancel rather
pleasantlyso we're going to

get B Q1 from that term.
Everyone with me so far?

I'm going through this in sort
of slow steps,

I agree it's not exciting but I
want to make sure I don't make a

mistake.
So to turn this into a first

order condition what must be
true about this derivative?

At the maximum,
what must be true about this

derivative?
Should be equal to 0,

good, and we should just check
the second order condition.

How do I check the second order
condition?

I differentiate again and check
it's negative,

but if I differentiate again
I'm just going to get B.

So B is certainly negative all
right, so second order condition

is okay.
So let's solve it out.

Solving this for Q1,
I get Q1 = [A  C]/2B.

So we're leading in sheer
boredom, it has to be done

occasionally,
Q1 = [A  C]/2B.

We're not done yet.
Now we want to go back and

solve out algebraically for Q2.
I know what Q1 is now,

Q1 is [A  C]/2B.
How do I find Q2?

Somebody?
Shout it out, how do I find Q2?

I've got to plug it in.
I'm going to go back and plug

this Q1 back into this
expression here,

so I plug it back in,
I'll get Q2 = [A  C]/2B  1/2

[A  C]/2B for a total of [A 
C]/4B.

Is that right?
That's what I have in my notes.

This looks good.
So I'm now done.

I've now found the equilibrium.
I've found that in this

leaderfollower game,
this Stackelberg version of

quantity competition,
Q1 is given by [A  C]/2B and

Q2 is given by [A  C]/4B.
Let's see how it matches up

with the intuition we developed
before without using any boring

math.
So first of all we're comparing

Q1 and Q2 with what they used to
produce, and what they used to

produce is on the top board.
What they used to produceI

can use this to guide the camera
as wellwhat they used to

produce is [A  C]/3B,
is that right?

So our claim was that we think
the new Q1 is bigger than the

old Cournot quantity.
So now Firm 1 is producing [A 

C]/2B, previously it was
producing [A  C]/3B,

so that is indeed bigger,
that's good news.

So this is indeed bigger than Q
Cournot.

And our claim was that Firm 2
will produce less than the old

Cournot quantity.
So Firm 2 used to produce [A 

C]/3B and now it's producing [A
 C]/4B, and that is indeed less

than the old Cournot quantity.
So far so good.

What about total output?
How do I solve for total output?

Add the two outputs together,
that's not too hard.

So Q1 + Q2 = [A  C]/2B + [A 
C]/4B which is in fact

3[AC]/4B.
Is that right?

So it's going to be 3[AC]/4B.
So I've justactually,

for the first time today,
I skipped a step,

but is that okay?
A half plus a quarter is

threequarters.
So total output is 3[A  C]/4B.

What was total output before?
It used to be the Cournot total

output, let's put it here
somewhere, this is bigger than

2[A  C]/3B which is equal to
the Cournot quantity Q1^(C) +

Q2^(C).
So everything we've predicted,

just by looking at the picture,
and thinking about the

economics works out in the math.
That's a good thing.

We should feel a little bit
relieved.

I'm feeling a little bit
relieved.

Everything we thought out
intuitively, just using the

economics, the logic of the
situation,

when we grind out the algebra
we get the right answers:

we get confirming answers.
Everyone okay?

That isn't a particularly fun
exercise per se,

but I want to do it just to
show that you can use backward

induction to solve out problems
exactly.

Backward induction and a little
bit of what you learned in high

school and/or freshman calculus
can get you the answer.

So now I want to leave aside
the math and go back to the

economics again.
So we started off with a

question, who would you rather
be Firm 1 or Firm 2,

and we know the answer now.
Who would you rather be Firm 1

or Firm 2?
Firm 1 because Firm 1's profits

went up and Firm 2's profits
went down.

Let's just talk about this a
little bitabout what's going

on here.
So previously Firm 1 and 2 were

just setting quantities
simultaneously.

We know now there's an
advantage in going first.

So suppose we change the game
from the simultaneous move game,

to a game in which Firm 1 and
Firm 2 can make announcements.

They can announce how much
they're going to produce.

So Firm 1 comes in one day and
says, I'm going to produce this

much and Firm 2 comes in and can
see Firm 1's announcement.

So I've changed it into a
sequential game,

Firm 1 has announced how much
it's going to produce,

Firm 2 is going to go
afterwards.

Is that really a sequential
game?

Is that going to make a
difference?

Why is that,
I claim that's not really

enough.
Let me say it again.

We start from the simultaneous
move again.

If we just change it by simply
allowing Firm 1 to announce what

they're going to producedon't
actually produce it but just

announce what they're going to
produce ,

you might think that's a
sequential move game and you

might think that Firm 1 now has
an advantage.

But I'm claiming that's not
enough really to give Firm 1 an

advantage.
Why?

Why is that not enough?
Let's try and get some ideas

here.
Let me come around to this side.

So Patrick why is that not
enough?

Student: There's no
credible commitment that you're

going to produce at that level.
Professor Ben Polak:

Good, so imagine these two firms
are, let's say that they're

newspaper producing firms and
one is owned by NBC's parent

company and one is owned by
Rupert Murdoch.

And they're moving into this
new market and the market is a

town.
Both of them are going to issue

newspapers in this market that
currently doesn't have any

newspapers,
and Murdoch simply says I'm

going to produce lots of
newspapers.

There's no reason for NBC to
believe that.

So moving first here,
it isn't enough to say you're

going to move first,
it isn't enough even to make a

decision that's reversible.
Even if Q1 moved but that

decision could be undone that
isn't enough.

What we need,
and Patrick gave us the key

word, what we need is
commitment: a word that came up

last week.
So for moving first to help you

here there really has to be
commitment.

Let's just get some of this
down.


There really needs to be
commitment for this to work.

So going back to the example of
Murdoch and his competitor,

Murdoch actually has to build
the plant.

There actually has to be a
factory that he's built in,

and that factory can't just be
sold for scrap.

So what creates the commitment,
in the case when you've built

the plant is what?
Because having built it,

it's a sunk cost,
it's there.

So sunk costs can help here.
It can help making you

committed.
Once that money's gone,

you can't get it back,
so you're really committed to

that scale.
Does everyone know what I mean

by sunk costs,
by the way?

So here's a case where a
strategic move,

entering first and sinking some
investments can help you in the

marketplace.
Let's also look at this another

way.
Let's again go back to the

simultaneous move game that we
had before, where Murdoch and

his competitor,
the NBC parent corporation,

are in fact,
going to move simultaneously.

Both of them are in the
business of discussing how big a

newspaper plant to put in this
new town which hasn't got a

newspaper yet,
somewhere in Alabama or

something.
Suppose that there are two

boardrooms, both of which are
avidly just trying to discuss

how big a newspaper plant to
build.

So suppose that one of the
boardrooms, these four people in

this row, this is the NBC parent
company boardroom and they're

trying to decide how big a plant
to build.

And over on the other side of
the room is our Murdoch group

which is in factlet's take the
row parallelso these guys over

there are Murdoch,
are News International Group.

And it is a simultaneous move
game, so basically we're in

Cournot.
Now suppose that Murdoch,

just to pick a name out of a
hat, might not be the most moral

gentleman in the worldwho
knows?

and suppose that he in fact
has hired one of the people in

the NBC boardroom,
in fact this guy,

what's his name?
Student: Ryan.

Professor Ben Polak:
He's hired Ryan to be a spy.

So Murdoch has a spy in the NBC
boardroom.

So Murdoch has a little
advantage here,

information wise.
Why?

Because the NBC boardroom
doesn't know what's going on in

the Murdoch boardroom,
but the Murdoch boardroom is

going to know what's going on in
the NBC boardroom.

But to make the problem more
interesting, suppose that

somebody tells NBCs parent
company, that in fact,

there is a spy in their
boardroom.

So these guys know they have a
spy, they don't know who it is.

If they knew who it was they'd
beat him up or fire him or

something or maybe notbut
they know that someone's there.

Maybe they even suspect it's
Ryan, so what should NBC do

here?
What decision should NBC make?

One thing they could do
isthey don't know it's Ryan

but Ryan has this sort of
Murdoch like face,

so they might just fire him
because he might be a spyor

what should they do here?
Any takers?

You're in the Murdoch,
you're in the NBC boardroom.

You know that Murdoch has some
spy in the camp.

What should you do?
Student: You can come up

with a fake plan and see if it
goes back to Murdoch.

Professor Ben Polak: So
Chris is suggesting come up with

a fake plan to feed it back to
Murdoch, so Chris has been

reading spy novels.
So if this was a John Le

Carré novel,
that's certainly what you would

do.
You would create a whole bunch

of fake information to feed back
to the Russians,

through this spy who in fact
you've discovered.

That isn't a bad idea:
that might be a good thing to

do.
It's pretty hard to do,

right, because ultimately these
actual decisions have to be made

in the boardroom,
contracts have to be signed and

so on, but I think Chris is onto
the right idea here.

So Chris' idea is create a fake
plan to feed back to Murdoch,

to give Murdoch some
misinformation.

But there's another thing you
could do, let me get someone who

hasn't contributed yet,
anyone else?

Yeah, what's your name?
Student: Usman.

Professor Ben Polak: So
Usman what would you do?

Student: These guys get
the first move now effectively

because they can just decide
they know.

Professor Ben Polak: Say
what you just said but shout it

out so everyone can hear you.
Student: NBC now gets

the first move because they can
decide and they know the other

people are going to respond to
it.

Professor Ben Polak:
Good, so what Usman is

suggesting is maybe you don't
feed Murdoch a fake plan,

you feed Murdoch the true plan.
Effectively,

what's going to happen now is
if NBC decided to build a large

plant,
this information will be fed

back to Murdoch,
and Murdoch is now in the

position of being the second
mover.

When Murdoch moves,
he or she knows what NBC is

doing, and NBC knows that
Murdoch is going to choose a

best response to that.
So it's as if NBC has been put

in the position of Firm 1 and
Murdoch has been put in the

position of Firm 2.
Even with the correct planso

the correct thing to do here for
NBC is not necessarily to

mislead Murdoch,
but just go ahead and build a

big plant.
Have that information be fed to

Murdoch and let Murdoch respond
to it.

So notice here's a slightly
paradoxical thing.

You might think that having a
spy in the camp of the other

team, you might think having a
spy would help you.

But here having a spyor
having more information if you

likecan actually end up
hurting you.

Everyone see that?
Paradoxically,

Murdoch ends up losing by the
fact that he was able to predict

what NBC was going to do.
Now there's a key to this of

course.
It was crucial to the argument.

What was crucial to the
argument?

It was crucial to the argument
that NBC knew that Murdoch had a

spy.
The key here is that the other

side, the other players,
knew you had or were going to

have more information.
So what's the bigger idea here?

There are two bigger ideas.
Bigger idea number one is,

games being simultaneous or
sequential is not really about

timing per se,
it's about information.

It's about who knows what,
and who knows that who's going

to know what.
In a situation where Firm 1,

our boardroom over here,
knows that Murdoch is going to

have this information before
Murdoch moves,

that's actually a sequential
game.

The timing is somewhat
irrelevant.

So that's the first
observation, and the second

observation is already on the
board.

What we have learned is
sometimes in strategic

settingssometimes not
alwaysmore information can

hurt you.
Sometimes more information can

hurt.
We have to be careful here

because that's not always true
but sometimes it's true.

And the reason that's true
isthe reason is it can lead

other players to take actions,
in this case,

to create a large plant,
that hurt you.

Now, if you put Monday's
lecture together with today's

lecture, we've seen
somethingtwo very similar

things arose.
On Monday's lecture having

fewer options,
burning your boats,

ended up helping you or having
lower payoffs by putting

collateral down ended up helping
you.

And that might seem like a
paradox but it isn't really a

paradox because what happened
was provided the other side

knows you have fewer options,
they know you've burnt your

boats, or they know you'll
suffer if you default on the

loan,
they know you've posted

collateral, it will lead them to
take behavior that helps you.

In the case of the loan,
it led to the lender giving you

a bigger loan.
In the case of the Saxon Army,

you at least hope at least that
the Saxon Army's running away.

Today, we see that more
information can hurt you,

and once again,
it isn't really a paradox,

it's the same kind of argument.
The fact that the other side

know you're going to have this
extra information,

leads them to take actions that
end up hurting you.

So in games,
unlike in standard single

person decision problems,
more information can hurt and

more options can hurt and here's
the reason.

Now, one other thing to say
about this, the game we just

looked at, the Stackelberg game
we just looked at is an example

of something pretty famous.
It's an example of firstmover

advantage.
It's an example of a game with

a firstmover advantage.
Now how many of you have

heardhow many of you in the
room have heard the term first

mover advantage before?
As few as that, seriously?

The rest of you,
how many of you have not heard

the term first mover advantage
before, one or two?

So I'm always a bit weary of
first mover advantage as a term.

If you ask the students in the
business school how many of them

have heard the term,
they've all heard it.

It's a very popular business
school term.

It's a very popular term that
you see in bad books.

So let me just warn you a
little bit about this.

So if you go to the airport,
let's say the Hartford Airport,

and the flight you want to get
on is late (which is usually the

case),
and hence you end up in the

bookstore.
And you find yourself on the

economics and business shelves.
You start looking at strategy

books and typically the kind of
book you find at the airports on

business,
or strategy,

or economics is a pretty bad
book.

So it has some embossed cover
on it and says 'Strategy for

Dummies', or 'My Boring Life' by
a famous CEO.

I worry about these books
because you end up buying these

books.
You've got time on your hands,

and you read them,
and they give you absolutely

terrible advice.
So not always,

but almost always,
and the kind of things they'll

say is: "it's always a good idea
to move first because that way

you'll have a first mover
advantage."

This sounds right and I'm
always worried about things that

sound right.
It may mean they're right,

but the problem is that if
they're wrong,

they're tempting,
they sort of lure you in.

So it's good to move first
because that way you have a

first mover advantage sounds
right, but it's nonsense.

There are situations of which
this is onethere are

situations where it's good to
move first.

In quantity competition it's
good to set your quantity,

to be committed and that will
lead the other side to producing

smaller quantity which helps
you.

So sure, there are situations,
there are games in which you

want to move first,
but there are also games in

which you'd rather move second.
Let me give you an example.

This is an example we've seen
in this class before rock,

paper, scissors.
If anybody is going to read

those books and believe them,
so if anyone's going to read

'How to Discover Your Inner Bill
Gates' and base their life on

it,
and therefore,

think there's a first move
advantage, I want to play rock,

paper,
scissors with that person.

Everyone happy that you'd
rather go second in rock,

paper, scissors?
Do I need to prove that or is

that obvious?
Okay good.

Let me just expand a little bit
more, into a more real world

setting.
There are plenty of settings in

the real world where the
advantage of moving second isn't

because as in,
rock, paper,

scissors you just get to crush
the other guy,

it's simply that you learn from
their mistakes.

So, for example,
in the game of buying new

equipment for the office or home
it's great to move second.

The other guy goes out and
samples some new piece of

equipment, I wait to see if it
works, and then buy it if it

does.
Or if I'm setting up a Firm in

a new expanding market,
let's say in a new part of the

former Soviet Union,
I'm quite happy to let some

other firms go in there first
and then watch what they did and

try and learn from their
mistakes.

If I'm setting a new curriculum
for a university,

I'm quite happy to let other
universities,

Duke and Cornell and so on,
move first and then I can go in

as Yale, see what they did,
and for sure they'll have made

mistakes and I'll learn from
them.

So there's plenty of obvious
situations where moving second

helps you, for the obvious
reason that information is often

very useful.
We argued here,

information can hurt you,
but there's plenty of other

perfectly natural situations
where information is to your

advantage.
So there are games with first

mover advantages,
but there also games with

second mover advantages.
And let me give you one example

of a game that neither has a
first mover advantage or a

second mover advantage,
just to convince you that can

happen as well.
So when you were a child you

probably occasionally had to
divide a cake and/or candy bar

with your sibling.
Anyone been in this situation?

There was some candy bar or
cake and you had to divide this

thing between you and your
brother and/or sister,

is that right?
There's a way in which,

there's a typical way in which
we divide things among siblings

in that setting.
There's a game we play to

divide it.
What's the game we play anyone?

I'll cut and you choose,
or vice versa,

right.
So I'll cut and you choose

neither has first a mover
advantage or a second mover

advantageassuming you can cut
accurately,

neither has a first mover
advantage or a second mover

advantage, which is precisely
why it's a good way to divide

the candy bar.
Now, to drive this point home

and because we've had a very dry
lecture up to now,

let's play a game,
so everyone can wake up now.

Math is over.
I want to play a game and the

game I want to play for the rest
of today is called NIM.

It's not going to surprise you
that one of the things we're

going to learn in this game is
that sometimes games have a

first mover advantage and
sometimes games have a second

mover advantage,
so I'm giving away the punch

line.
How many of you have played NIM

before?
If you've played it before you

can't play now,
so don't shout out.

So this is the game.
There are two players and there

are two piles of stones.
We'll make the piles of stones

into just chalk lines on the
board.

The players are going to move
sequentially.

In each turn,
the player whose turn it is to

move, picks one of the two piles
and removes some of the stones,

which will just be lines on the
board.

So they decide how many of
those lines to delete.

Each time it's your turn you
get to move again,

you can choose the other pile
this time.

You move sequentially,
and the only other rule that

matters is, the person who gets
the last stone wins.

So, for example,
here's a case where there's

one, two, three,
four, five, six,

seven stones on this pile and
one, two, three,

four stones on this pile.
So I'm going to get some

volunteers to come on the stage
and do this, but I was told I

should choose a volunteer.
So is LeeShing Chang here?

So LeeShing Chang is going to
volunteer for this.

I'm volunteering LeeShing
Chang because it's his birthday,

so a round of applause for
LeeShing Chang,

come on the stage.
Who wants to play against

LeeShing Chang?
Anyone else want to play?

Who hasn't played a game yet?
How about the guy with the Yale

hat on back there,
the white Yale hat.

I'm staring right at the guy,
yeah you, do you want to come

up?
All right?

What's your name?
Student: Evan.

Professor Ben Polak:
Evan, so we've got Evan and

Leesing, is it LeeShing?
LeeShing and Evan,

come up on the stage,
there's a step up here,

come on.
So we'll let LeeShing go first

and we'll let Evan go second.
Have you played this game

before?
Student: I didn't hear

what the game is.
Professor Ben Polak:

Okay, he didn't hear what the
game is, so we'll explain the

rules again for those people who
are sleeping.

If you were sleeping the game
is backward induction.

All right so the rules of the
game are this.

They're going to take turns.
In each turn,

they're going to pick one of
these two piles.

This is Pile A and this is Pile
B.

And they're going to tell me
how many of these chalk lines to

delete, and I'm going to delete
those lines.

We're going to go on playing
until somebody gets the last

line.
The person who gets the last

line wins.
You can get the last line,

ten at a go,
it doesn't have to be that

there's only one on the board
when you get the last line,

but the person who gets the
last line wins.

So you have to pick a pile each
go, and tell me how many lines

to remove.
So LeeShing why don't you go

first, you can choose Pile A or
Pile B and tell me how many

lines to remove.
Any advice from the audience?

No advice: you're on your own
here.

Student: Can you remove
three from Pile A?

Professor Ben Polak:
Three from Pile A,

so three from Pile A.
Evan your turn.

Stand nearer to the board.


Student: Can you remove
two from Pile A.

Professor Ben Polak: Two
from Pile Aokay so Pile A is

the popular pile here.


Student: Two from Pile B.
Professor Ben Polak: Two

from Pile B, suspense is
building here.


Student: One from Pile B.
Professor Ben Polak: One

from Pile B, all right.
It should go fast at this

stage, go on.
Student: Two from Pile A.

Professor Ben Polak:
Last stone wins,

last stone wins.
Student: Oh wait.

Professor Ben Polak:
Last stone wins,

be careful here.
It's his birthday come on.

Last stone wins.
Student: As in who?

Professor Ben Polak:
Person who gets the last stone

wins.
Student: Oh okay,

so one from A.
Professor Ben Polak: All

right, one from A,
can't abstain.

Student: One from B.
Professor Ben Polak: One

from B, LeeShing is the winner
here.


Very good, let's get two more
volunteers.

Everyone understand the game
now?

Thank you very much gentlemen.
Two more volunteers.

Have you played before?
Come on up.

I want to get some female
volunteers.

It can't be a male only class.
There we go thank you.


This may not be as important
economics as doing the

Stackelberg model but it's
probably a little bit more fun.

Your names are?
Student: John.

Student: Christine.
Professor Ben Polak:

John and Christine,
okay.

So let's make it a bit more
complicated this time,

so one, two,
three, four,

five, six, seven,
eight, nine,

ten, eleven,
twelve, thirteen and one,

two, three,
four, five, six, seven.

So we'll pick two prime
numbers, and see if that's

important.
Let's see what happens.

Why don't we say ladies first,
so Christine your choice.

Student: That one has?
Professor Ben Polak:

This one started off at 13 I
believe and this one started off

at seven, assuming I counted
correctly.

Let me get out of the way.
Student: Six from Pile A.

Professor Ben Polak: Six
from Pile A, one,

two, three, four,
five, six, is that right?

Okay.
Student: Three from Pile

B.
Professor Ben Polak:

Three from Pile B.
Student: Three from Pile

A.
Professor Ben Polak:

Three from Pile A.
Student: One from Pile B.

Professor Ben Polak: One
from Pile B.

Student: One from Pile A.
Professor Ben Polak: All

right, so we're onto this.
Student: One from Pile B.

Student: One from Pile A.
Professor Ben Polak: All

right.
Student: One from Pile B.

Professor Ben Polak: All
right.

Student: One from Pile A.
Professor Ben Polak: All

right.
Student: One from Pile A.

Professor Ben Polak:
Switch it.

Student: One from Pile B.
Professor Ben Polak: All

right, so Christine is the
winner, okay.

So a round of applause for
these two.

So I think everyone's figured
out now how to play this game,

is that right?
Has everyone figured it out?

Let me take one of these mikes
and go down and just to make

sure.
Lot's of people have figured

this out.
So what is the rule for how to

play this game?
What's the rule for how to play

this game?
People haven't figured it out,

we should play again.
So let's try over here,

someone has an answer,
how should you play this game?

Student: It depends on
whether the piles have the same

number in them.
If they have the same number

you want to be second player,
and if they have different

numbers you want to be first
player.

Professor Ben Polak:
Good, what should you do?

Student: You want to 
Professor Ben Polak:

Let's assume they have different
numbers.

Student: If they have
different numbers you want to

make them equal.
Professor Ben Polak: You

want to make them equal.
So the trick to playing this

game isthank you,
very goodso the trick to

playing this game is if the
piles are uneven then you want

to make them evenyou want to
make them equal.

Everyone see that?
So if you start off with the

piles being unequal as we did
both times, for example 3 and 2,

then you want to be Player 1.
There's a first mover advantage

and the correct tactic is to
equalize the piles.

What you'll notice now is that
player 2 can't do anything.

If they take two from here,
you'll win by taking both of

those two.
If they take one from here

you'll equalize the piles again.
And if they take then one from

here you've won.
So the way to play this game is

to equalize the piles.
What does that mean?

It means if the initial
position has unequal piles,

uneven piles,
then you would rather be Player

1: it has a first mover
advantage.

But if the initial position has
even numbers in the piles then

you'd rather be Player 2.
Is that right?

Because if you start off with
an even [correction:

equal]
number in each pile,

the person moving first is
going to make them unequal,

and thereafter the next
person's in a winning position.

So I want us to notice two
things from this game.

First notice in this game that
from any initial position we can

very quickly tell who is going
to win and who is going to lose,

assuming they play well.
Three things actually.

Second, we didn't actually use
backward induction here,

but it's pretty obvious you do
want to use backward induction

here.
You want to figure out what the

end game is going to look like,
is that right?

It's very easy to see this game
if you look at the end game.

The third lesson is what we
just said, in this game,

sometimes there's a first mover
advantage, sometimes there's a

second mover advantage.
So sometimes there's a first

mover advantage in this game,
sometimes there's a second

mover advantage in this game.
So I'm just illustrating the

point we made earlier that it's
not always the case that you

want to move first.
Sometimes you want to move

second.
Now today I set up the piles

with unequal piles so there' was
a first mover advantage,

but that was just to give an
advantage to the guy whose

birthday it was.
I could have set things up with

equal lines in each pile,
and made the other guy go

first.