
Title:
19. Subgame perfect equilibrium: matchmaking and strategic investments

Description:
Game Theory (ECON 159)
We analyze three games using our new solution concept, subgame perfect equilibrium (SPE). The first game involves players' trusting that others will not make mistakes. It has three Nash equilibria but only one is consistent with backward induction. We show the other two Nash equilibria are not subgame perfect: each fails to induce Nash in a subgame. The second game involves a matchmaker sending a couple on a date. There are three Nash equilibria in the dating subgame. We construct three corresponding subgame perfect equilibria of the whole game by rolling back each of the equilibrium payoffs from the subgame. Finally, we analyze a game in which a firm has to decide whether to invest in a machine that will reduce its costs of production. We learn that the strategic effects of this decisionits effect on the choices of other competing firmscan be large, and if we ignore them we will make mistakes.
00:00  Chapter 1. Subgame Perfect Equilibria: Example
27:22  Chapter 2. Subgame Perfect Equilibria: Matchmaking
34:31  Chapter 3. Matchmaking: SPEs of the Game
49:37  Chapter 4. Subgame Perfect Equilibria: Strategic Investments
01:13:15  Chapter 5. Strategic Investments: Discussion
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:
So last time we covered a

whole bunch of new ideas,
and it was really quite a lot

of ideas for one class.
Here's some of the ideas we

covered.
We talked about information

sets, and these were ways to
allow us to model imperfect

information.
So what's imperfect information?

It's a way of being able to
capture both simultaneity in

moves and sequential moves in
the same game.

So it's a way that's going to
allow us to meld the lessons

from before the midterm and
after the midterm.

Then we talked about what
strategies meant in this

context, and the basic idea is
strategies are

instructionsstrategies for
each playergive them an

instruction at each of their
information sets.

Then we talked about what
subgames were,

and, leaving aside
technicalities,

subgames were just games
within games.

And finally we introduced the
idea of subgame perfection

which is our new solution
concept that refines the idea of

Nash equilibrium.
What subgame perfection is

going to do is it's going to
instruct the players to play a

Nash equilibrium in every
subgame.

Another way of saying it is,
a subgame equilibrium is a

Nash equilibrium in the whole
game, but in each subgame it

induces Nash play as well.
Now, we're going to see today

examples.
If we have time I'll go through

three different examples,
and I'll tell you at the end of

each example what it is I'm
hoping to be able to take away

from that example.
So, last time was a lot of

formal stuff.
Today is going to be a lot of

examples.
Okay, that's our agenda.

Here's a game.
Here's our first example.

And I call this example,
I call this game,

"don't screw up," for reasons
we'll see in a minute.

So this is a game in which
Player 1 has to choose between

Up and Down.
If Player 1 chooses Up then

Player 2 gets to move and
chooses between left and right.

And if Player 2 chooses left
then Player 1 gets to move again

and Player 1 chooses between up
or down.

Everyone looking at that game?
So why don't we play this game

since we haven't played a game
for a while.

We'll play a couple of games
today.

So what I'm going to do is
let's divide the class in two.

So if I just draw a line down
the middle of the class,

everybody to my left (to your
right), everybody on this side

of the class is a Player 1.
Okay you're all Player 1's.

And everyone on this side of
class you're Player 2,

including you guys hiding from
the camera, you're Player 2's.

Okay, so let's figure out what
we're going to do.

Everyone had the time to look
at the game?

So Player 1's you get to move
first, those of you who are

going to choose Down raise your
hand now.

Raise your hand.
Wave it in the air.

Keep it up so the camera can
see you.

And those of you who are going
to choose Up raise your hands.

Lots more Ups.
Those of you who chose Up why

don't you all stand up.
I don't want to do all the

exercise here,
so all those who chose Up,

stand up.


So you can see that choosing
down ends the game,

so this many people are still
playing the game.

Everyone who is still sitting
down, everyone who sat down here

has exited.
All right, Player 2's you get

to move now.
So Player 2's,

those of you who choose right,
including the people on this

aisle,
those people who choose right

raise your hand nowone right
over there.

Those of you who choose left
raise your hands.

Why don't you guys all stand
up, just to get you awake on a

Monday morning,
everyone's sleepy otherwise.

Let's go back to Player
2'ssorry Player 1those of

you who are still in the game.
So those of you who chose Up

the first time,
how many of you now choose

down?
Raise your hand if you choose

down;
and raise your hand if you

choose up.
Just to get a sample of this,

let's get the 2's to sit down
again so people can see them.

So 2's sit down.
Those of you Player 1's who are

still in the game who were
choosing up raise your hand now.

I think that's everybody,
is that correct?

Okay, you can all sit down.
So let's just talk about this

game for a while and then we'll
analyze it.

Now, this is not a difficult
game from the point of view of

stuff we've done since the
midterm.

It's pretty clear what we
should do in this game by

backward induction.
So why don't we start there.

Okay, so by backward induction,
we find that if Player 1 gets

to move a second time then
they're choosing between 4 and

3,
and they're going to choose 4.

Player 2, if they get to move,
knowing that Player 1 is going

to choose up tomorrow,
they're going to be choosing

between 3 if they choose left or
2 if they choose right.

So they're going to stay in the
game and choose left,

which is what most of you did.


Finally, Player 1 at the
beginning of the game,

knows that Player 2 is going to
choose left whereupon she's

going to choose up,
so if she chooses Up she's

going to end up getting 4,
and if she chooses Down she's

going to get 2,
so she's going to choose Up.

So it's clear what backward
induction does in this game and

that's what most people did in
the game.

Is that right?
However, not everybody did it.

Some of the Player 1's
actually, why don't you raise

your hand, those people who
chose Downthe ones who chose

down.
There were more than that.

You can all stand up.
Those of you who didn't stand

up just now raise your hands.
People are hiding now,

but that's okay.
Those people who chose Down,

they may have had a reason for
choosing Down,

and their reason for choosing
Down might have been that they

thoughteven though they can do
backward inductionso even

though they know that by
backward induction Up gets them

the better answerthey might be
worried that if they choose Up,

Player 2 will screw up and
choose right.

Notice that if Player 2 chooses
right then Player 1 only gets 1,

whereas Down yielded 2.
So in some sense Down was the

"safe" thing to do for Player 1
given that they might be worried

that Player 2 might screw up.
Does that roughlyjust nod if

this is the case:
for those people who chose down

is that kind of what you were
thinking?

Some people are shaking their
heads, but some people are

nodding.
That's a good sign.

Now why might Player 2 in fact
screw up and choose right.

Because Player 2 might,
themselves, think that Player 1

might screw up at this stage.
If Player 1 were to screw up at

the last stage and choose down,
then Player 2 by choosing left

would only get 1,
and for him the safe option

therefore is right which yields
2.

So, to get the backward
induction answer herewhich

most of us didto get the
backward induction answer here

relies on Player 1 trusting
Player 2 to play backward

induction,
and that requires Player 1 to

have trust in Player 2 trusting
Player 1 not to screw up in the

last stage.
So say it again,

Player 1 needs Player 2 not to
screw up, and that means Player

1 needs to trust that Player 2
will trust her not to screw up.

Everyone see the game?
Okay, so let's try and analyze

this game using what we learned
last time and see what we find.

So the first thing to do is
let's look at strategies in this

game.
So Player 2 just has two

strategies, left and right,
because Player 2 only has one

information setand notice this
game is actually a game of

perfect information so it's
going to be very easy.

Player 1 has two information
sets, this information set and

that information set.
At each of them Player 1 has

two choices so she must have
four strategies in all.

So this game when we put it in
its matrix form is going to be a

4 x 2 game.
Here it is.

And the strategies for Player 1
are (Up, up),

(Up, down), (Down,
up), and (Down,

down).
And the strategies for Player 2

are just left and right.
And now we can put the payoffs

in.
So ((Up, up),

left) gets us (4,3).
((Up, up), right) gets us (1,2).

((Up, down),
left) gets us (3,1).

((Up, up, right) gets us (1,2)
again: we end up exiting the

game here.
(Down, up) is easy because it's

just exiting the game at the
first stage, so all of these are

going to be just (2,1).
Everyone happy with that?

So what I've done is translated
the game into its matrix form.

And let's look for Nash
equilibria in this game.

Let me do it at the board since
it's quite easy at this stage.

So to look for Nash equilibria,
let's just worry about pure

strategy equilibria for now.
So if Player 2 was choosing

left then Player 1's best
response is the (Up,

up) strategy.
And if Player 2 is choosing

right, then Player 1's best
response is either (Down,

up) or (Down,
down).

That's exactly the conversation
we just had.

If Player 2 was going to "screw
up" and choose right then Player

1 wants to get out of the game
immediately.

Conversely, if Player 1 is
choosing (Up,

up) then Player 2 is happy and
is going to choose left,

trusting Player 1.
If Player 1 was going to choose

(Up, down), however,
that's Player 1 screwing up at

the second stage.
So in that case Player 2 wants

to get out of the game and
choose right.

If Player 1 is choosing (Down,
up) then Player 2 is

actuallyit doesn't matter
they're indifferent.

And if Player 2 is choosing
(Down, down) then once again

Player 2 is indifferent since
they don't get to move at all.

So from this we see that there
are three Nash equilibria,

let me call them 1,2,
and 3.

So one Nash equilibrium is
((Up, up), left).

Another Nash equilibrium is
((Down, up, right):

that's here.
And the third equilibrium is

((Down, down),
right).


So there are three pure
strategy Nash equilibria in this

game.
Let's just see what they do.

So the first one ((Up,
up), left) is Up,

left, up so it gets us to here.
So this one is the same

equilibrium as corresponds to
backward induction.

Is that right?
This one's the backward

induction equilibrium,
and the other two are

different.
((Down, up),

right) and ((Down,
down), right) both end up down

here exiting the game
immediately,

so both of these other
equilibria fail backward

induction.
So let's put backward induction

with a line through it:
they're failing backward

induction.
So you might ask why are they

equilibria?
We've seen examples like this

before, for example,
in the entry game.

We looked at some examples last
time.

So, in this game,
the reason these are equilibria

even though they fail backward
induction exactly coincides with

that conversation we just had
about worrying about the other

person screwing up.
So, in particular,

if Player 1 thinks that Player
2 is choosing rightthat is to

say,
thinks that Player 2 is going

to screw upthen Player 1
doesn't want to travel up the

tree because she knows she'll be
carried down here,

and instead she just chooses
the safe option and gets 2.

So from Player 1's point of
view, if Player 2 was going to

choose right,
then getting out of the game

doing the safe thing is the best
response for Player 1.

And for Player 2's point of
view, if Player 1's exiting the

game it really doesn't matter
what Player 2 says she's going

to do,
because she doesn't get to move

anyway so that's why these are
both equilibria.

Okay, so what we're going to do
next, we've translated this into

a tree.
We've written down the

strategies.
We want to actually see which

of these Nash equilibria are
subgame perfect?


Let me give myself a bit more
room here because I want to keep

this in sight.
So let's get rid of this and

raise this one.


So the next question is which
of these three Nash equilibria

are subgame perfect?
To do that we need to start by

identifying the subgames.
And, of course,

just having hoisted that board
up there, I have to hoist it

down again.
Okay, so what are the subgames

here?
Well the simplest subgame is

this simple subgame at the end,
in which Player 1 moves.

That's a very obvious subgame,
is that right?

That's a little game within a
game, it's a rather trivial game

because it's a one player game,
but it is a game.

So let's examine that one
first, that's the last subgame.

So the last subgame here is a
somewhat trivial subgame.

It looks like this.
Player 1 is the only mover and

they're choosing either up or
down and the payoffs are (4,3)

and (3,1): and frankly we don't
really care at this point what

Player 2's payoffs are because
Player 1 is the only person

who's playing in this subgame.
But nevertheless,

let's put them there.
And if we write this up as a

matrixhere it is as a matrix.
Since Player 1 is the only

mover they're choosing between
up and down, and the payoffs are

(4,3) and (3,1),
and of course Player 2 doesn't

get to move so Player 2 is
irrelevant here.

And clearly the only Nash
equilibria in this game is for

Player 1 to choose up.
Player 2, it doesn't really

matter what they choose,
there's nothing they can do

about it anyway,
but for Player 1 to choose up

is the Nash equilibria.
So the Nash equilibria in this

trivial subgame is 1 just
chooses up.

Is that right?
So let's look at the play

induced by our three candidate
Nash equilibria in this

subgame.
So each of our candidate Nash

equilibriahere they are,
this one, this one,

and this onehave an
instruction of how Player 1

should play in this subgame.
And let me just pause a second.

The reason that these three
equilibria have an instruction

for how Player 1 should play in
the subgame is because of our

definition of a strategy.
Each strategy tells the player

how they should move at every
information set of that player.

So even if the strategy is such
that that information set won't

be reached, the strategy still
has to tell you what you would

do when you got there.
And now for the first time

perhaps we're going to see why
that redundancy helps us.

So let's look at the
instructions.

Each of them gives an
instruction.

The first one tells us to play
up in this subgame.

The second one says up again,
and notice this was redundant.

Once you've chosen Down you
know you're not going to get to

make a choice at the third node,
or your second node,

but nevertheless there's the
instruction and it says up.

The third one says down.
So this is the instructions of

these three equilibria in this
little subgame.

This is the play prescribed by
these three equilibria in this

subgame.
Two of them say up and those

ones are going to induce the
Nash equilibrium in this

subgame, but the third one does
not.

The third one says down and
that's not allowed.

That's not allowed in a
subgame perfect equilibrium

because a subgame perfect
equilibrium has to prescribe

play in every subgame that's
Nash,

and here the third equilibrium
is telling Player 1 to choose

down which is not a Nash
equilibrium in the subgame.

So what are we doing here?
Let's make it clear.

We're finding the subgame
perfect equilibria.

And what we've done is:
number 3 is eliminated because

it induces play in this subgame
that is not Nash equilibrium,

not a Nash equilibrium in the
subgame.

We're really putting stuff
together now:

to be able to draw this
conclusion,

we really used the fact that
strategy 3 contained a redundant

instruction, an instruction down
at this node that was never

reached.
But that helped us get rid of

it.
So that one's gone: 3 is gone.

Let's proceed.
I'm going to run out of board

space here.
Have people got this one down?

I'll bring it back in a second,
almost, let me give it a

second.
What I want to do now is look

at the next subgame.
Maybe what I can do if I just

remove this comment,
I can work on the right hand

board, that'll allow you to look
at it.

So all that comment said was 3
is eliminated.

Let me work now on the right
hand half of this board,

that'll allow it to be up there
a bit longer.

Okay, so let's now look at the
next subgame,

and again, we're going to work
from the back.

So the next subgame back is
the subgame that starts from

this node, the subgame that
starts from that node.

So let's identify that in a
different color.

I used blue so let me use pink.


So now we're going to look at
this subgame,

this big pink subgame,
and again in this pink subgame

this game looks like this.
It starts with Player 2

choosing between left and right,
and then Player 1 has to choose

up or down, and the payoffs are
(1,2), (4,3) and (3,1).

Once again let's look at the
matrix form of this subgame,

and this is a little bit less
trivial than the last one

because now there are really two
players playing.

So here's the matrix that goes
along with this.

Player 1 is choosing between up
or down and Player 2 is choosing

between left or right.
And this is slightly,

slightly cheating because in
fact, Player 1 of course knows

what Player 2 is going to have
done by the time she moves,

but never mind it'll do for now.
Let's just put the payoffs in.

So (up, left) is (4,3) and
(down, left) is (3,1) and (up,

right) is (1,2) and this must
also be (1,2).

Everyone happy with that?
Just putting the payoffs in.

And let's just look again at
purestrategy Nash equilibria

here.
There are actually mixed ones,

but let's just worry about pure
ones for now.

So the pure Nash equilibria
here in this little subgame are

what?
Well, let's just see.

If 2 chose left then 1 wants to
choose up.

If 2 chooses right it doesn't
really matter what 1 chooses

because she isn't going to get
to move anyway.

Conversely, if 1 chooses up
then 2 wants to choose left.

That's the example of 1 not
screwing up, so 2 wants to stay

in the game.
But if 1 was to choose down

then Player 2 would like to get
out of the game,

so if Player 2 thinks 1 is
going to screw up she wants to

exit the game.
Very quickly we can see there

are two equilibria here.
One of them is (up,

left) and the other one is
(down, right).

They correspond to playing down
this way, that's (down,

right) and (up,
left) playing up this way.

Once again, let's look at our
three equilibria in the parent

game.
Here's our three equilibria in

the parent game,
and let's see what play they

induced in this little subgame.
So we'll do exactly what we did

before.
So 1,2, and 3:

these are our three equilibria
from above.

And equilibrium number 1,
((Up, up), left) in this game

prescribes (up,
left).

Is that right?
Equilibrium number 2,

((Down, up),
right) here prescribes up,

right.
And equilibrium number 3,

((Down, down),
right) here prescribes (down,

right).


So which of these are
prescribing Nash equilibria in

the subgame?
Well (up, left) is an

equilibrium.
It's that one.

(up, left) is an equilibrium so
this is okay.

And (down, right) was an
equilibrium, so 3 is okay in

this subgame.
But (up, right) is not a Nash

equilibrium.
So in this subgame,

Nash equilibrium number 2,
the ((Down, up),

right) equilibrium is
prescribing play that is not a

Nash equilibrium in the
subgame: so it's eliminated.

It can't be a subgame perfect
equilibrium.

So here 2 is eliminated since
it induces nonNash equilibrium

play in this subgame.


Now we're done,
we know about the whole thing.

So what we did here we started
with the whole game,

we found there were three Nash
equilibria,

we found that only one of them
agreed with backward induction.

We then looked at the subgames.
We first of all looked at that

blue subgame and we found that
one of the equilibria,

equilibrium number 3,
was eliminated.

Equilibrium number 3 is not
prescribing Nash behavior in

this subgame.
Then we looked at the slightly

more complicated subgame,
the pink subgame,

and we found that equilibrium
number 2 prescribes the behavior

(up, right) which is not Nash in
this subgame.

At this stage we've eliminated
two of the three equilibria and

we're just left with one.
And the one we're left with,

the only subgame perfect
equilibrium, the only

equilibrium that wasn't
eliminated by the fact that it

would prescribe bad behavior in
subgames,

the only SPE is number 1:
which is (Up,

up) and left.
What do we notice?

We notice that that's the
equilibrium, that's the play

that backward induction would
have selected.

So notice this is the backward
induction prediction.


So what are the lessons here?
The lessons here are that our

new idea, the idea of subgame
perfect equilibrium is pretty

easy to go about finding.
You just look at subgames and

check that the play in each
subgame has to be Nash play.

If you start at the back,
you construct it by rolling

backwards, much like we did
backward induction.

Start at the last subgame and
work backwards.

The second thing isnot
surprisingly given that

remarknot surprisingly,
where backward induction

applies,
for example in this game,

the subgame perfect
equilibrium will find the

equilibrium that is consistent
with backward induction.

Remember that was our aim last
week.

We wanted a way of refining
Nash equilibrium to throw away

those Nash equilibria that were
inconsistent with backward

induction.
So subgame perfect equilibrium

has done that.
It tells us now if backward

induction applies,
the Nash equilibria you should

focus on are the subgame
perfect equilibria.

Indeed, most people in the
class played that equilibrium

just now.
Okay so that was really what I

wanted to say about this
example, but let me just make a

remark in passing.
I made this remark in the

middle, so let me just make it
again.

When we write down strategies,
those strategies tell us what

seem to redundant moves.
But being forced to write down

those redundant moves is useful
because it allows us to model

what other people think you
would have done at those later

nodes.
And sometimes I have to think

what you think I would have done
at this later node before I

decide not to go down that
branch of the tree.

So being able to write down
everything in a strategy allows

us to have everything in front
of us and makes that analysis

simple;
and that's exactly what we did

here.
So this was a fairly mundane

example, because in particular
we didn't use any kind of

information set.
So next let's look at an

example that does use some
information sets.

So: new example.
Let me clean this off.


Once again, I want to play this
example.

But what I'd really like to
dowe're going to call this

game the matchmaker gameand
what I'd actually like to find

out is: do we still have our
couple we tried to send on a

date?
Our hapless couple we tried to

send on a date at about week
three.

Are you guys still here?
There's the guy,

what's your name again?
Student: David.

Professor Ben Polak:
David, and what was theis

she hiding?
There she is thank you.

Your name was?
Student: Nina.

Professor Ben Polak:
Nina and David.

Good, can we get some mikes to
Nina and David actually?

Let me do it.
I'll go on talking while I'm

doing this–.
Where's David,

and Ale can you get a mike to
Nina?

That would be great thank you.
All right, so for weeks we've

been trying to get this couple
to go on a date.

It's our attempt to get
economics majors to become real

people.
It's a hard thing to do and

they're kind of the hapless
couple because first we sent

them to the movies and they end
up going to different movies;

and then we sent them off for a
romantic weekend in New England

and they end up doing different
things,

one went to the theatre and
another one apple picking,

I forget which way around it
was.

And at this point,
I figure I'm a pretty bad

matchmaker so what I'm going to
do is.

I'm going to introduce a third
player into the game as the

matchmaker.
So first of all I'll write down

what the game is.
So the game at this pointthe

game is going to look like this.
Player 1 is the matchmaker,

we can call him Player M if you
like, and he has a choice,

he or she has a choice.
She could not send the couple

out on a date or she could send
the couple out on a date.

But being a better matchmaker
than me, if she sends them out

on a date, she's going to stake
some money, she'll pay for the

date.
And in the date once again

they're trying to meet,
and once again,

unfortunately,
they haven't figured out where

to meet.
We'll put the payoffs in,

in a second and we'll tell you
what the strategies are.

So what I'm going to assume
here is I'm going to let Jake

our T.A.
be our matchmaker.

And the reason I'm choosing
Jake is I think he's the nearest

thing I have in mind in this
class to being a Jewish mother.

I mean: he's neither JewishI
think he's not Jewish and he's

not my mother,
but he is the T.A.,

who's responsible for bringing
some drink everyday in case I

pass out in the lecture.
So that's the nearest thing I

can think of.
So Jake's going to be our

Jewish mother,
and Jake's going to either send

these guys on a date or not.
And Jake's smarter than me at

this: he's actually good at
matchmaking.

And what he's going to do is,
he's going to send them

somewhere where they reallyhe
knows Yale students better than

I doand he's going to send
them really somewhere where

they're going to meet.
So he's going to send them to

go to the same lecture class
next year, and then they'll be

sitting in the aisles in this
huge lecture class and they're

bound to meet.
All of you have sat next to

other people at some point.
So that seems like a good idea.

So the classes he thinks of
sending them tohe says go to a

large lecture class.
So they're either going to go

to the Gaddis class which is
called "Cold War";

or to the Spence class which is
called "China."

Everyone know a little bit
about these classes?

These seem like reasonable
classes to go to meet yourto

have a dateto meet somebody.
I mean the Cold War can be a

fun class, I mean,
you hope it isn't a prediction

of the future relationship but
the Cold War seems all right.

And China is,
by all accounts,

a fantastic class.
It involves,

something involving 20 million
people, most of them were in the

class together,
so it's a pretty big class.

So let's do that.
Unfortunately Jake makes the

same mistake I do,
he's not going to tell them

which class to go to.
So they have to decide whether

to take Gaddis or Spence,
and once again they're

coordinating.
We'll call them Players 2 and 3.

So here they are trying to
coordinate, and the payoffs are

as follows.
So let's put in Jake's payoffs

first of all.
So if they manage to

coordinate, first of all,
if Jake doesn't send them,

everybody gets nothing.
And if Jake does send them and

they coordinate Jake makes 1
because he feels really happy

about this.
After all there must be some

motivation for people
matchmaking.

So if they coordinate down here
Jake gets 1 as well,

but if they fail to coordinate,
Jake feels rotten about it,

particularly because he paid
for them to go this

classwhatever the cost of a
class at Yale iswhich is

probably quite a lot actually.
So okay we'll call it 1 though

and otherwise the payoffs are
exactly the same as the payoffs

we used when we looked at this
game earlier on in the course.

So the payoffs are going to be
(2,1) here and (0,0) if they

fail to coordinate;
and (0,0) here if they fail to

coordinate;
and (1,2), (1,2) here.

So the implication of this is
that Player 2 who we'll assume

is David, so David would like to
meet Nina but all other things

being equal,
he'd like to meet her at the

Cold War.
And Nina would like to meet

David, but all things being
equal, she would like to meet in

China.
Not literally in China,

but in the class.
So this is our game and we're

going to analyze this game,
but before we analyze it let's

try and play it.
So what we need to do is first

of all let's make sure things
work smoothly,

let's have David write down
which class he's going to

choose,
and Nina write down which class

she's going to choose.
I've lost sight of Nina.

Somebody has to point
outthere she is.

Write down which class you are
going to choose.

Something been written down?
Jake you got your mikethere,

so Jake are you going to send
this hapless couple or not?

Student: So I have Dave
in my section actually and I

hear how much he's been talking
about Nina, so I'm going to roll

the dice and send them.
Professor Ben Polak:

He's going to send them
good, so we have them going off

to this class and now let's see
what they wrote down.

So Dave what did you write down?
Student: I'm going to

give in and go to China.
Professor Ben Polak:

You're going to go to China
and Nina?

Student: I chose S.
Professor Ben Polak:

Great, so they managed to
meet, so it's a successful date.

So let's give them a round of
applause.


I hear it's a great class too.
And in fact,

I don't think it's going to
happen forever because I think

he must be approaching
retirement.

So that seems a pretty good
choice.

So good: that worked very well.
Let's have a look now at this.

Let's analyze this game and see
what we can do with this game.

So how are we going to analyze
this game?

So no surprise we're going to
use the idea of a subgame

perfect equilibrium,
I'll collect the mikes later

don't worry.
So we're going to use the idea

of a subgame perfect
equilibrium.

So how do we figure out how to
work out what a subgame perfect

equilibrium is?
We're going to use the same

basic idea that we usedwhat
we've been using all along in

backward induction.
It's the same idea in the game

we just looked at just now.
What we're going to do is,

rather than start from the last
decision node (we can't do that

anymore) and work backwards,
instead of doing that,

we're going to start from the
last subgame and work

backwards.
In this example it's pretty

obvious what the last subgame
is.

The last subgame,
the game within a gamethere

is only really onethe game
within the game is this object

here, is that right?
This is the game within the

game.
Now, I could at this stage,

I could do something else.
I could write down the whole

matrix for the whole game and
have Jake choose the matrix,

and Dave and Nina choose the
row and the column,

but that's going to get us
astray.

So I mean we could do that but
let's not worry about that.

Let's just start doing things
backwards.

So when we do things backwards
we'll start at the last subgame

and that last subgame is an old
friend of ours,

it looks exactly like this.
Let's just write it in.

So it involves Players 2 and 3.
And 2 was choosing between

Gaddis and Spence.
And 3 was choosing between

Gaddis and Spence.
And their payoffs werelet me

leave a space hereso it was
(2,1), (1,2),

(0,0), (0,0).
So here are the payoffs of the

relevant players in the game but
while we're here why don't we

put in Jake's payoffs as well.
So Jake's payoffs were 1 here,

1 here, 1 here,
and 1 there.

So the only relevant players
here are Players 2 and 3,

but I've put Player 1's payoffs
in as well because:

why not, why not just keep
track of it.

Everyone happy that that
exactly describes this little

game?
For all intents and purposes we

can forget the first payoff but
there it is.

This is a game we've seen many
times so far.

It's the battle of the sexes,
or the battle of Dave and Nina,

and in this game we already
know what the equilibria are.

So the equilibria here,
let me just underline the best

responses.
So if Nina is choosing Gaddis

then Dave chooses Gaddis.
If Nina is choosing Spence then

Dave would like to choose Spence
and conversely.

So I've just underlined the
best responses for the players

who are actually involved in the
game and haven't bothered

underlining anything for Jake
because he isn't a player in

this game.
Does that make sense?

So the pure Nash equilibria in
this game are essentially

(Gaddis, Gaddis) or (Spence,
Spence).

That's pretty easy.
From Jake's point of view,

each of these pure strategy
Nash equilibria yield a payoff

for him of what?
What does he get?

If they go to (Gaddis,
Gaddis), he's happy that they

met and he gets 1 and if they
choose (Spence,

Spence), he's happy that they
met and he gets 1.

Jake himself doesn't really
mind whether Dave and Nina learn

about China or learn about the
Cold War.

He just wants them to meet.
So, both of these yield 1 for

Jake.
They both yield a value of 1

for Player 1,
who is Jake.

So from Jake's point of view,
going back a stagewhat we're

going to do now,
just as we did with backward

induction, we're going to roll
the game back.

So we started by analyzing this
subgame and now we're going to

roll it back a stage,
just as we did with backward

induction.
So when we roll back Jake is

moving here, if Jake chooses not
to send them then we get (0,0,

0) but the key part of this is
the first 0, that's Jake's

payoff.
And if Jake sends them,

then what Jake gets is the
value to JakeI'll put in Jake

but value to Player 1okay
value to Jake of the Nash

equilibria in this subgame.
Not a big thing to write,

but that's what Jake gets.
And the others do too:

the others get that as well.
In this case,

rather than writing that long
piece, this is just equal to 1.

Is that right?
So if Jake sends them he knows

that they're going to play a
Nash equilibrium in this

subgame,
or he believes they're going to

play a Nash equilibrium in this
subgame.

And either of those two Nash
equilibria in the subgame yield

a payoff to Jake of 1.
So actually,

one of them is 1,
one of them yields payoffs of

(1,1,2);
and the other yields payoffs of

(1,2,1).
But since Jake is the only

mover here, let's just focus on
Jake.

So from Jake's point of view
he's really choosing between 0

and 1, so he's going to choose
send.

So the subgame perfect
equilibrium thereforethere are

actually two of them hereone
is (send, Spence,

Spence), that's what actually
happened.

But there's another one which
is (send, Gaddis,

Gaddis) that would also have
been a pure strategy subgame

perfect equilibrium.
So in either case,

what we did,
just to remind ourselves,

we first of all solved the
equilibrium down the subgame,

the equilibrium in this blue
subgame.

We figured how much that
equilibrium was worth for

everybodybut in particular for
Jake,

but for everybodyand then we
rolled that payoff back and

looked at Jake's choice.
In this particular case that

game has two equilibria,
(send, Spence,

Spence) and (send,
Gaddis, Gaddis).

However, some of you must be
suspecting at this point that

there's actually another
subgame perfect equilibrium

here.
How do we know that?

Well let's just think about
this game.

We've been trying to send this
couple on a date all semester.

They haven't gone on a date all
semester.

I'm embarrassing them,
but they haven't gone on a date

all semester.
So there must be some

possibility that they would fail
to coordinate.

It would be a pretty weird
notion of equilibrium that

concluded that they always
manage to coordinate and hence

Jake always wants to send them.
Is that right?

So let's also look at the other
equilibria here.


Now the reason there's another
equilibrium in this

subgamesorry,
the reason there's another

equilibrium in the whole
gamethe reason there's another

subgame perfect equilibrium in
the whole game is that there's

another Nash equilibrium in the
subgame.

What's the other Nash
equilibrium in the subgame?

They could mix.
So it turns out that in the

subgame (here it is) there's
also a third mixed equilibrium.


There is a mixed Nash
equilibrium.

Now we know how to work that
out.

We could write down a P and a
Q, and we could look for those

indifference conditions and
solve it out.

But this is a subgame,
sorry this is a game,

this subgame corresponds to a
game we've seen many times in

this class so far,
and I think we probably

remember what that equilibrium
is.

Is that right?
I do anyway,

so let's see if you remember it
as well.

I'll write it down and we'll
see if you all looked alarmed.

So I claim the equilibrium,
the other equilibrium has Dave

playing with probability
(2/3,1/3), and has Nina playing

with probability (1/3,2/3).
So this is another equilibrium

in the subgame.
People remember that this was

an equilibrium in battle of the
sexes?

Yeah, people are nodding at me,
yeah okay.

So it isn't too unintuitive.
We all know how we'd work it

out.
We could go back and put in the

P and the Q, but it isn't too
unintuitive, it has Dave going

more often to the lecture course
that he would prefer all other

things being equal;
and it has Nina going more

often to the lecture course that
she would prefer all other

things being equal.
And they do so in just such a

way as to make each of them
indifferent.

Now, this subgame induces a
different value for Jake.

So suppose Jake thinks:
"I trust Dave and Nina to play

a Nash equilibrium in their
subgame but I don't know which

one it is and I think maybe
they're going to play this one."

So suppose Jake thinks that
this is the equilibrium that

Dave and Nina are going to play.
So now should Jake send them or

not?
Well let's work it out.

So now if he sends them,
if Jake sends Dave and Nina,

or more anonymously if Player 1
sends Players 2 and 3,

then with what probability will
they meet?


Well this is just a little math
exercise, let's have a look at

the game again.
So Dave is playing 2/3,1/3,

is that right?
Nina is playing 2/3,1/3,

is that right?
So the probability of their

meeting is the probability of
this box, they could meet at

Gaddis,
plus the probability of this

box, they could meet at Spence,
is that right?

So the probability of this box
is 1/3 x 2/3 so this box has

probability 2/9 and the
probability of this box is 2/3 x

1/3 so this box has probability
2/9.

So the probability of their
meeting is 2/9 + 2/9 that makes

4/9.
Everyone okay with that?

So if Jake sends Dave and Nina
and they play this mixed

strategy equilibrium,
then they meet with probability

2/9 + 2/9,2/9 at Gaddis,
2/9 at Spence for a total of

4/9,
which means they failed to meet

with probabilitywell if
they're meeting with probability

of 4/9 what must be the
probability that they're failing

to meet?
5/9, thank you.

So they fail to meet with
probability of 5/9.

So Jake's expected payoff if he
sends them, the value for Jake

of this equilibrium is what?
So the value to Jake of this

Nash equilibrium,
if he sends them,

is 4/9 x 1 + 5/9 x 1 for a
total of 1/9.

Everyone okay with that?
So if Jake sends them they fail

to meet 5/9 of the time and he
gets 1 each of those times.

They succeed in meeting 4/9 of
the time, he gets +1 each of

those times, so his expected
payoff,

his expected value from sending
Dave and Nina on the date is

1/9.
So from Jake's point of view,

what this game looks like,
if he thinks that this is the

Nash equilibrium being played:
if he doesn't send he gets 0

and if he does,
he gets the value of this Nash

equilibrium, which in this case
is 1/9.

So he's not going to send and
the SPE here is (not send,

"mix," "mix") where this is the
mix.

So there's a third equilibrium
here in which our matchmaker

says this hapless couple is just
too hapless: they're going to

play the mixed strategy
equilibrium in which case it

isn't worth my while sending
them on the date.

You guys were lucky because
Jake chose the other

equilibrium.
He figured you were playing the

other equilibrium,
which it turned out that you

were.
So in this game there were

three subgame perfect
equilibria, one for each of the

Nash equilibria in the subgame,
as it turned out.

There was one in which Jake
sent them and they coordinated

on the pure strategy equilibrium
in the game (S,S).

There was one in which Jake
sent them and they coordinated

on the pure strategy equilibrium
in the subgame (G,G).

And there's one in which Jake
didn't send them,

but had he had in fact sent
them,

they would have both mixed,
and hence, for a lot of the

time, failed to coordinate.
Now, what's the big lesson here?

The big lesson of the first
game we saw this morning was

that subgame perfect
equilibrium implies backward

induction.
The big lesson of this

gameother than the fact that
we're getting closer to getting

Dave and Nina on their datethe
big lesson of this game is to

show that to find subgame
perfect equilibria,

all you have to do is keep your
head and solve out the Nash

equilibria in each of the
subgames,

roll the payoffs back up,
and then look for behavior up

the tree.
Once again, you look for the

Nash equilibria in each of these
subgames, roll the payoffs back

up,
and then see what the optimal

moves are higher up the tree.
So we have time to do one more

example, and the third example I
want to do is more of an

application.


So far we've seen some fairly
simple examples.

Now I want to do an application.


The application I want to do is
kind of a classic business

school case if you like,
or a mini case involving

strategic investment.


The game is this,
or the setting is this.

There are two firms,
we'll call them A and B.

And these two firms,
initially, before we start

considering what we're actually
going to talk about,

initially they are playing
Cournot competition.

So two firms and they're
playing Cournot competition.

And we can imagine that they're
producing fertilizer.

And let's be specific here,
let's assume that the prices in

this market are given by the
following demand curve 2  1/3 x

[qA + qB],
so this is the demand curve

that they face.
We'll assume that costs,

marginal costs,
c is equal to $1 a ton.

So this is the price in dollars
per ton, and the costs are $1

per ton.
In a minute what we're going to

do is we're going to consider a
change in this game,

but before we do that let's
just remind ourselves what the

Cournot equilibrium of this game
would look like.

Let's do a bit of a review.
So it's been a while since

we've seen Cournot,
so let's remind ourselves.

So I claim that the quantity,
the Cournot quantity chosen Q*

has the formula [ac]/ 3b.
Is that right?

If you go back in your notes
you'll find it.

I'm not going to resolve it
here.

We've done it many times.
So [ac]/3b trust me,

is what came out of our
calculation before the midterm.

What I want to do is,
I just want to make sure we can

translate that into numbers
here.

Sorry for having it in letters,
but let's translate it into

numbers.
So in particular,

this a is this 2,
is that right?

This c is this 1 and this b is
this 1/3, is that right?

So let's just put that down.
So in this case this is

[21]/[3 x 1/3],
so this says that this is a

million tons.
So the quantity here,

the Cournot quantity is a
million tons each.

So: one each.
So in this equilibrium,

each of these two firms is
producing a million tons of

fertilizer.
What else do we know?

We know therefore what prices
must be.

Let's just do that before we
even get started.

So prices must be [2  1/3]
times the quantity that the

first firm produces plus the
quantity that the second firm

produces.
So that's 2  2/3 so that

should be 4/3,
if I've got that right,

or one and a third.
So prices here are $1.33 per

ton.
Finally profits.

So profit for each firm here,
in this equilibrium,

before we even start the game,
or start the more interesting

part of the game,
profit is what?

So they're going to get $1 and
1/3 for every ton they produce.

It's going to cost them $1 to
produce each ton and they're

producing one million of these
things.

So their profits are 1/3,
if these are millions,

their profits are 1/3 of $1
million dollars.

So this is their per period
profit, in each period they're

doing this, each year they're
doing this and this is their

profits in each period.
So this is a simple model that

we've done many times before.
This is Cournot,

and now we're going to make it
more interesting.


If the algebra here was a bit
quick don't worry about it,

check it at home,
it's just basic,

basic algebra.
So now suppose that you are the

manager of Firm A.
So it's a classic business

school case.
I'm looking at my

businessschool students in the
balcony.

You're the manager of Firm A
and you have to choose whether

to accept an offer to rent a new
machine.

So this new machine has two
features.


The firstwell three
featuresthe first feature is

it only works for A.
So this machine is being

offered to you.
It wouldn't fit in to Firm B's

technology, so this is only
being offered for A.

The second feature of this
machine is it costs $0.7 million

dollars in rental.
So each year you rent this

machine, you'd have to pay $0.7
million dollars.

But that's the bad news.
The good news is it will lower

A's costs to $0.50 a ton.
So classic businessschool

situation.
You're the manager of a firm.

You're involved in competition
with another firm,

B.
And suddenly an opportunity

comes along to rent some new
technology.

It's going to cost you $0.7 a
year to rent this machine,

but it will lower your costs by
$.50 a ton.

So this is a classic thing that
you might be asked in your

interview for Morgan Stanley
next week.

How many of you are
interviewing with investment

banks?
No one's going to admit it.

In a couple of years,
when you're interviewing with

these guys.
So what's the obvious question?

The obvious question is,
should you go ahead and rent

this new technology or not?
Should you rent it or not?

To rent or not to rent?


Less dramatic than it's
equivalent question in the

English class,
but important nevertheless.

This board is stuck
unfortunately,

so I'll have to write there a
bit more.

So what I want to do is I want
to analyze this three times and

each time I analyze it,
I want us to see what I'm

doingwhat mistakes I'm
makingbecause I want you guys,

when you interview with Morgan
Stanley about this kind of

thing, to impress them so that
they tell lots of people to come

to Yale and preferably give lots
of money to Yale.

So we're going to look at this
way three different times.

And the first thing we're going
to do, the first way we're going

to look at this is look at it as
if we were accountants.


We're going to look at the
accountants' answer to this

question, and some of you may
decide you don't want to

interview with Morgan Stanley or
McKinsey,

you might want to interview
with some accounting firm when

you leave Yale.
God forbid, but you might.

So let's have a look at how the
accountants would answer this

question.
So I think the accountants

would do this.
They would saybut before we

do this let's have a poll.
How many people think you

should rent?
You've had some time to think

about it now.
So how many people think you

should rent the new machine?
How many people think you

should not rent the new machine?
You're not allowed abstensions

here.
Let's try it again:

no abstentions right.
You can't abstain in an

interview.
So you're on the spot,

you're in the boardroom,
how many think you should rent

the new machine?
Raise your hands.

Wave them in the air.
How many people think you

should not rent the new machine?
So we're split kind of down the

middle, I'm looking at my MBA
students to see which they

voted.
Which did you guys vote,

rent or not rent?
Rent, the MBA's seem to think

rent.
We'll see if that's right.

So let's move forward,
so accountant's answer.

So I think what the
accountant's going to say is

this.
They're going to say,

right now you're producing a
million tons a year.

The new machine saves youso,
let's put per annumlet's try

and be fancyso a million tons
per annum.

The new machine saves you $.50
per ton.

So if you rent this new
machine, you're producing a

million tons a year,
it's going to save you $.50 a

ton.
So it's going to save you half

of one million a year in
variable cost.

Those people who were in 115 or
150 will know what I mean by

variable cost.
It's the cost you're going to

save in the actual production of
your fertilizer.

So it saves you half a million
a year.

Unfortunately,
it costs you the cost of the

machine, which is a fixed cost
of $0.7 million a year.

And .7 is bigger than .5,
so you should not rent.


So .5 is less than .7 so don't
rent the machine.

How many of you said no?
That's kind of the back of the

envelope calculation you were
doing, is that right?

Kind of the back of the
envelope calculation that

accountants do.
So what's going on here?

Our business school student up
in the balcony says you should

rent.
He took accounting.

I know he did that because you
have to take accounting at

business school.
So what's wrong?

Did he fail accounting,
or is this answer wrong?

The answer is wrong.
There's two things you need to

know about accountants.
One is that they're usually

boring, and the other is that
they're often wrong.

They're more often boring than
wrong, but they're almost always

boring.
So this answer is kind of

boring, and it happens also to
be wrong.

Why is it wrong?
It's wrong because we made an

assumption here that's not a
good assumption.

We made the assumption that
you're going to go on producing

the same amount per year after
you've invested in the new

machine that lowers marginal
costs as you are producing

beforehand.
We made the assumption that it

would lowerwe know it lowers
your costsand we assumed

implicitly that you would go on
producing a million tons a year,

but that's not right.
So let's try and have a more

sophisticated answer,
and if you want to be more

sophisticated and less boring
than accounting,

what class would you want to
take?

Economics probably right,
so let's have a look at an

Economics answer.
Let's look at an Economics 115

answer.


How many of you have taken
Economics 115?

How many of you are in 115 at
the moment?

Quite a few,
okay so let's have a look at

the Economics answer.
Let's see why that previous

answer was wrong.
So here is qA and here is cost

of a $1, your new cost will be
$.50.

So I'm putting prices and costs
on this axis.

And here is your residual
demand curve.

This is the demand curve you
face after the other guy has

finished producing,
so this is your residual demand

curve.
It's the demand curve on that

part of the market you're
supplying, or not being supplied

by the other side of the market.
And to figure out your optimal

quantity on your residual demand
curve what you should do.

It's like you're a monopolist
on this residual demand curve,

so you should set what?
If the answer isn't backward

induction it's probably marginal
revenue equals marginal cost

right?
So let's try that.

So here's the marginal revenue
curve roughly speaking,

should be twice as steep.
This is residual marginal

revenue and here's what you used
to produce.

So we know what this is:
this was a million tons.

This was marginal revenue
hitting marginal cost.

Now your costs have gone down
so notice that your quantity,

your new quantity has gone up.
Your new quantity has gone up

because you slid down the
marginal revenue curve as the

marginal cost curve dropped.
Is that clear to everybody?

So this is the kind of picture
that you probably saw a lot of

in 115, is that right?
Or in 150 for that matter,

is that correct?
So notice in this picture we

can actually see the
accountant's answer,

the boring answer.
The boring answer is this

rectangle.
This rectangle,

this is the accounting answer.
This rectangle is 0.5 x 1,

so it comes out as a 1/2 and
that was the accountant's

answer.
And what did they miss.

What did the accountant's miss?
They missed the triangle.

So I told you they were boring,
they were a little bit square,

so they tend to miss triangles.
So here's the triangle that

they missed.
So we missed this triangle.


And how big is this triangle?
Well we could do it at home,

it's 1/2 base x height.
So we could figure this out.

We know the slope of this line.
We know the slope of this line

is 1/3.
We know the slope of this line

is 2/3.
We know that the height of this

triangle is 1/2.
We could figure out what the

width is as well,
therefore we could do half base

times height.
Turns out that this has areaI

did this at home so let me just
write it downit has area 3/16.

So again, everyone could figure
out the area of a triangle at

home, is that right?
You all know that from your

probably junior high school
geometry.

So assuming I did it correctly
at home, this is 3/16 which is

approximately 0.19.
So we missed this 0.19.

So how are we doing now?
So we know from the accounting

answer we had a 1/2 in savings.
We know from the Economics

answer that we should add
another 0.19 to thisthat's the

trianglefor a total of 0.69
but unfortunately this is still

less than .7 which is the cost
of the machine,

the per annum cost of the
machine.

So it looks like we should
still not rent.


So even after taking Economics
115, which is a good thing to

doit is much less boring than
accounting and will get you the

accounting answer anyway if you
do things carefullywe still

end up concluding you shouldn't
rent.

But our guy from the business
school said you should rent,

right?
So did he fail Economics as

well as accounting,
or is this answer wrong?

This answer is still wrong.


[I didn't really want to delete
that.

That's a shame.
Oh well, I've done it now.

People've got those numbers
somewhere.

I hope they are in my notes.]
This answer is still wrong.

We need to get the right answer.
So what we need is a third

answer which is the Game Theory
answer, which is also known as

the right answer.


What are we missing?
What's wrong with the Economics

answer?
Somebody?

Everyone knew it was wrong,
why is it wrong?

It looked pretty good,
what's wrong with it?

Let's bounce down here.
Somebody in the front row will

help me.
What did I do wrong?

Student: Firm B also
changes its quantity.

Professor Ben Polak:
Right, in the accounting

answer, we assumed that Firm A
kept its quantity fixed,

that we kept our quantity
fixed, and that was wrong.

But in addition,
Firm B is going to change its

quantity, isn't it?
Firm B is going to change its

quantity.
Let's remind ourselves why.

We're still playing Cournot
competition, here's our Cournot

diagram with qA and qB.
So prior to making this

investment the model is
symmetric.

Here it is.
And this is the old best

response of Firm A,
and this is the best response

of Firm B.
What we learned just now in the

Economics answer is what?
We learned that Firm A,

as its costs go down,
will produce more for each

possible quantity that Firm B
produces.

So regardless of what generated
this residual demand curve as

the costs go down for Firm A,
it increases its quantity.

So we know that.
So what's that telling us?

It's telling us that the new
best response of Firm A is

shifted to the right.
This is the new best response

for Firm A.
It's shifted to the right.

It now produces more for any
given quantity that Firm B

produces.
So qA has gone up.

That was our Economics answer.


But that leads Firm B to do
what?

To produce less.
That leads to qB producing less.

Notice we slid down Firm B's
best response line from the old

equilibrium to the new
equilibrium,

and at the new equilibrium Firm
B's production has gone down.

By the way, what kind of game
is it where as Firm A increases

its strategy,
Firm B decreases its strategy

in response?
Strategic substitutes,

right.
So because this is a game of

strategic substitutesgood
interview word,

good word to mention in an
interviewbecause this is a

game of strategic substitutes,
we know that Firm B reduces its

quantity.
As Firm B reduces its quantity

is that good for A or bad for A?
It's good for A, right?

This is good for Firm A,
it softens competition.

As a consequence,
it leads to an increase in

profit.
Againwe don't have time

todaybut what we could do is
we could go back and we could

recalculate the new Cournot
equilibrium.

We could calculate thiswe
could as a homework exercise try

itcalculate the new Nash
equilibrium and notice this is a

Nash equilibrium in a subgame.
Why is this a subgame?

Because Firm A made its
decision whether to buy this new

machine or not,
and then they played Cournot.

So the Cournot game,
the game up here,

what is this?
This is the subgame.

This is the diagram of the
subgame, if you like.

It's the best responses of the
subgame.


So what subgame perfection
tells us to do here is first of

all, work out the new
equilibrium in the subgame,

work out how much that new
equilibrium is worth for Firm A

and then roll it back to the
investment decision.

It turns out when we do that we
get an extra $.31 million.

So we could do it at home,
we get an extra $.31 million.

So it turns out that our MBA
student was right,

good.
It turns out that if you add

this .31 to the .69 we had
already we get 1,

which of course is much bigger
than .7 and indeed you should

rent the machine.
Now, I want you to have two

take away lessons from this
game.

The first take away lesson is
this.

When you're analyzing a game
like this, be it in the real

world or in a job interview,
the first thing you want to do

is what?
You want to look at the

subgame.
You want to look at what would

happen if you did invest and
solve out the new Nash

equilibrium in that subgame.
Then you want to roll back the

value of that subgame back into
the initial decision which is

the strategic investment
decision whether to rent this

machine or not.
So schematically,

the game looks like this:
rent or not rent,

and in either case you play
Cournot.

There's a subgame in each case.
In this case you play symmetric

Cournot, when you both have the
same costs;

and here you play asymmetric
Cournot, where you have

different costs.
And the way we analyze this

game is, we solve out the
symmetric Cournot,

we actually did that up front.
We now solve out the new

equilibrium in this asymmetric
Cournot game,

this one here.
This is the old one and this is

the new one.
Solve it out.

Work out how much profit you're
going to get.

And roll that back remembering
that it costs you $.7 million to

make this step.
So that's the first take away

lesson.
But the second take away lesson

is more general,
so let me just pause to get

everyone to wake up again so I
make it.

The second take away lesson is
this., What tips the balance

here from the Economics answer
and the accounting answer,

were the strategic effects.
It was the strategic effect.

This is a strategic effect.


It was the effect of the other
firm or other players changing

their behavior.
And the most common mistake to

make when you're thinking about
strategic decisions is what?

It's to forget that they're
strategic.

It's to forget that the other
players are going to change

their behavior.
In this example,

the other firm cuts back its
production so much as to make

that investment profitable.
But let me give you two other

examples.
Example number one.

You're designing a tax policy
for the U.S..

The dumb way to analyze this is
to say look at what people are

doing now, push through the new
tax numbers,

and act like an accountant and
crunch out how much money the

government's going to make.
Why is that wrong?

Because you're forgetting that
as you change the tax code

people's behavior changes.
Incentives change and people's

behavior changes.
It leads to a mistake in

designing the tax code.
You need to take into account

strategic effects:
how behavior changes.

Example number two,
closer to home.

You're designing a new
curriculum for Yale.

So you change the rules of the
curriculum and when analyzing it

you sayI wouldn't say this but
some people on the committee

might say thisunder these new
rules,

if we look at what people used
to do, they will now do more of

this and less of that,
and they'll learn this and

learn that.
What are you missing?

You're missing that students
are players and students change

their behavior as you change the
curriculum rules.

So the biggest lesson of
today's class is don't be like

an accountant,
partly because it's boring and

you won't go on your date,
and partly because you'll miss

out on these important strategic
effects.

We'll come back and look at
more on Wednesday.
