Original subtitles

so last time we covered a whole bunch of

new ideas and it was 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 and

sequential moves in the same game so

that it's a way that 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

instruction a strategy for each player

gives them an instruction at each of

their information sets then we to robot

some games were leaving aside

technicalities some games were just

games within games and finally we

introduced the idea of sub game perfect

that refines the idea of Nash

equilibrium and what sub game perfection

is going to do is it's going to instruct

the players to play a nash equilibrium

in every sub game what I was saying it

is a sub game equilibrium is a Nash

equilibrium in the in the whole game but

in each sub game it induces Nash play as

well all right 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 the takeaway

from that example all right so last time

with a lot of for more staff today's can

be a lot of examples ok that's our

agenda here's the game here's our first

example

and I called this example that I call

this game don't screw up for reasons

we'll see in a minute alright so this is

a game in which player one 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 all right if we're

looking at that game so why don't we

play this game

we haven't played a game for a while or

play a couple of games today so what I'm

going to do is let's divide the class

into so if I just draw a line down the

middle of the class okay everybody to my

left your right everybody over this side

of the class is a player one okay you're

all player ones all right and everyone

on this side of the class your player

two including you guys hiding from the

camera alright your party's alright okay

so let's let's figure out what we're

going to do

hi everyone time to look at the game

alright so player ones you get to move

first those of you who are going to

choose down raise your hand now raise

your hand with the air keep it up keep

it up to the camera can see you okay and

those of you who are going to choose up

raise your hands all right lots more ups

those of you chose up why don't you all

stand up why should I don't do all the

exercise here so all those who shows up

stand out all right so you can see that

choosing down ends the game so this many

people are still playing the game all

right it's people everyone who's

listening down I want to so sat down

here has exited all right player two's

you get to move now alright so player

two's those of you who choose rights

including the people on this aisle those

people who choose right raise your hand

now one right over there those of you

who choose left raise your hands

all right won't you guys all stand up

just to get you awaken Monday morning

right everyone's everyone's sleepy

otherwise alright let's go back to play

at twos let's go back to player two all

right so those of you who are still

alive so we play one those of you who

are still in the game so those of you

who'd shows up the first time that you

just 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 alright just take it Istanbul

this is get the twos to sit down again

trying to people to see them to choose

to sit down all right those of you

player ones who are still in the game

who are choosing up raise your hand now

I think if everybody it's everybody that

great okay you all sit down okay okay

so let's 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

where we start there okay so by backward

induction we find that if player 1 if

player 1 gets to move a second time then

they're choosing between 4 & 3 and

they're going to choose 4 all right and

player 2 player 2 if they get to move

knowing that player 1 is going to choose

up tomorrow they're going to be choosing

between three if they choose left or 2

if they choose right so they're going to

stay on the game and choose left which

is what most of you did all right and

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 is that right

however not everybody did it some of the

player ones some of the player ones

actually great when you raise your hand

those are people who choose who chose

down the ones who chose down there more

than that can all stand up those who

didn't stand up just now raise your hand

people are hiding now it's okay those

people who chose down they may have a

reason for choosing down and their

reason for choosing down might have been

that they thought that even though they

know they can do backward induction so

even though they know that by backward

induction up gets them the better answer

they might be worried that if they

choose up player 2 will screw up and

choose right

and notice if player 2 chooses right

then player 1 and it gets one where as

down yielded two so in some sense down

was that quote safe thing to do for

player one given that they won't might

be worried that player two might screw

up right does that roughly just 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 all right all right now why

might play it

in facts screw up and shoes rights

because player 2 might themselves think

that player 1 might screw off at this

stage in playoffs if player 1 would have

screw up at the last stage and shoes

down then player 2 by choosing left we'd

only get one and for him the safe option

therefore is right which yields to all

right so to get the backward induction

answer here which most of we did most of

us did to get the backward action here

answer here relies play on player 1

trusting player 2 to play backward

induction and that requires player 1 to

have trust and player 2 trusting player

1 not to screw up at the last stage so

said again player one needs player 2 not

to screw up

and that means player one needs to trust

that player 2 will trust her not to

screw up how everyone sue the game ok ok

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 let's

look at strategies in this game so

player 2 has just one has just has two

strategies left and rights because

player 2 only has one information set

and those is this game is actually a

game of perfect information all right

it's games a game of perfect information

it's gonna 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 at all all right so

this game when we put it in its matrix

form is going to be a 4 by 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 this 4/3

up up right gets us 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 two

one okay we're happy with that right

well I've done is translate the game

into its matrix form and let's look for

Nash equilibria in this game alright

just look for Nash equilibria let me let

me do it at the board since it's quite

easy at this stage to look for Nash

equilibria just wear a pure strategy

equilibria now alright so if player two

was choosing left then player 1s best

response is the up up strategy and if

player 2 is choosing right then player

1s best response is either down up or

down down that's exactly the

conversation we just had if player two

was going to quote screw up and choose

right then player one wants to get out

of the game immediately alright

conversely if player 1 is choosing up up

then player two is happy and he's going

to choose left to the trust trusting

player 1 if player 1 was going to choose

up down however then player 2 let'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 actually that doesn't matter

they're indifferent and if player 2 is

choosing down down then once again

player 2 is indifferent so they'll get

to move at all alright so from this we

see that there are three Nash equilibria

let me call them one two and three all

right so one Nash equilibrium is up up

left another Nash equilibrium is down up

right that's here and a third

equilibrium is down down right down down

right alright to the three pure strategy

Nash equilibria in this game and let's

just see what they do so the first one

up up left is up left up so it gets us

to here

all right so this one is the same

equilibrium as corresponds to backward

induction is that right this one too the

backward induction equilibrium and the

other two are different down up right

and down down right both end up down

here and exiting the game immediately

right so both of these other equilibria

fail backward induction so let's put

that conduction with a line through it

right there they're failing backward

induction all right and so you may ask

why are they equilibria we've seen that

examples like this before for example in

the entry game we looked at some

examples last time so in this in this in

this game the reason these are

equilibria even though they fail

backward induction is exactly coincides

with that conversation we just had about

worrying about the other person's

growing up all right so in particular in

particular if player one thinks that

player 2 is choosing right say thinks

that player 2 is going to screw up then

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 to all right so

from player ones point of view if player

2 was going to choose right then getting

out getting out of the game doing the

safe thing is the best response for

player 1 and for player 2 point of view

if player 1 is exiting the game it

really doesn't matter what player 2 says

she's going to do right because she

doesn't get to move anyway all right so

that's why these are both equilibria ok

so what we want to do next what we want

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 sub game perfect

which are these Nash equilibria our sub

game perfect all right let me give

myself a room a because I want to keep

this in sight so let's get rid of those

and raise this one alright so the next

question is which of these three Nash

equilibria are sub-game perfect I want

to do that we need to start by

identifying the sub games and force

having just hosted that board up that I

have to always get down again ok so what

are the sub games here with a simpler

sub game the simplest sub game is this

simple sub game at the end in which

player 1 moves alright that's a very

obvious sub game is that right the total

game with 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 sub game alright

so the last sub game here is a somewhat

trivial sub game it looks like this

player 1 is the only mover and they're

choosing either up or down and the

payoffs are 4 3 & 3 1 and frankly we

don't really care at this point

what player two's payoffs are because

player 1 is the only person who's

playing at this sub game alright but

nevertheless let's put them there and if

we write this up as a matrix here it is

as a matrix its player 1 is the only

mover they're choosing between up and

down and the payoffs are 4 3 & 3 1 and

of course player 2 doesn't get to move

so player 2 is open it player 2 is

relevant here and clearly clearly the

only Nash equilibrium in this game with

the only Nash equilibrium in this game

is for player 1 to choose up right

player 2 doesn't matter what they choose

I think they can do about it anyway

but for player 4 player one to choose up

is the Nash equilibrium so the Nash

equilibria in this trivial sub game is

one just chooses up is that right so

let's look at the play induced by our

three candidate

Nash equilibria in this sub game all

right

so each of our candidate Nash equilibria

here they are this one this one and this

one in

have an instruction of how player one

should play on this sub game and let me

just pause a second the reason that

these three equilibria have an

instruction for how player one should

play in the sub game is because of our

definition of a strategy each strategy

tells the player how they should move at

every information set of every

information set of that player so even

if the strategy is such that that is is

such that that information set will be

reached the strategy still has to tell

you what you would do when you got there

and now for the first time maps we're

going to see why that redundancy helps

us alright so let's look at the

instructions each of them gives an

instruction the first one the first one

tells us to play up in the sub game

first once it's up the second one the

second one says up again and notice this

was redundant writing it once you've

chosen down you know you're not going to

get to make a choice at the third at the

third note or your second node but

nevertheless there's the instruction and

it says up right and the third one and

the third one says down alright says

down so this is the instructions of

these three equilibria in this little

sub-game and is this is the play

prescribed by these three equilibria in

this sub game alright two of them say up

and that those ones are going to induce

the Nash equilibrium in the sub-game but

the third one does not the third one

says down and that's not allowed that's

not allowed in a sub game perfect

equilibrium because in a sub game

equilibrium a sub game perfect

equilibrium has to a prescribed play in

every sub game that's Nash and here the

third equilibrium is telling player one

to choose down which is not a Nash

equilibrium the son game all right so

three so what are we doing here let's

make it clear we're finding

the sub-game perfect equilibria and what

we've done is number three is eliminated

because it induces play in this sub game

that is not Nash equilibrium not a Nash

equilibria the sub game

alright we're already putting stuff

together now this really used to be able

to draw this conclusion we really used

the fact that strategy three contained a

redundant instruction and instruction

down at this node that was never reached

but that helped us get rid of it

alright so that one's gone that one's

got three is gone all right

let's proceed I'm going to run out of

board space here have people got this

one now who's got this down all right

I'll bring it back in a second

almost did we give it a second let me

give it a second all right what I want

to do now is look at the next sub game

all right

maybe what I can do if I just remove

this comment I can work on the right

hand board that allows us to look at it

so all that comment said was three is

eliminated and let me work now on the

right hand half of this board that allow

it to be up there a bit longer okay so

let's now look at the next sub game and

again we're going to work from the back

so the next sub game the sub game back

is the sub game that starts from this

node all right the sub game that starts

from that node all right so that that's

identify that in a different color

I use blues let me use pink

so now we're going to look at this

sub-game this big pink sub game alright

and again in this pink sub game that

this game look like 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 + 3 1 alright once again let's

look at the matrix form of this sub game

and this is a little 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 is a

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

alright and it's 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

all right now everyone happy without

hill just putting the putting the

payoffs in and let's just look again

just a pure strategy Nash equilibria

here ok there are actually mixed ones

let's just worry about pure ones for now

alright so the pure Nash equilibria here

in this little sub-game Oh what well

it's to see if two chose left then one

wants to choose up or two to chooses

left and once to choose up if two shoes

is right it doesn't really matter what

one chooses because she isn't getting it

to move anyway all right

conversely if one chooses up then two

wants to choose left that's the example

of one not screwing up so two wants to

stay in the game but if one was to

choose down then player 2

would like to get out of the game so if

player two things one is going to screw

up

she wants to exit the game all right and

very quickly we can see there are two

equilibria here one of them is up left

and the other one is down right all

right they correspond to anything

playing back down this way that's

downright and up left playing up this

way all right

and once again let's look at our three

equilibria in the parent game here's our

three equilibria in the parent game

three and let's see what play they

induce in this little sub-game so

exactly what we did before so one two

and three these are our three equilibria

from above and equilibrium number one up

up left in this game prescribes up left

that right and equilibrium number two

down up right down up right here

prescribes up right and equilibrium

number three down down right here

prescribes down right so which of these

are prescribing Nash equilibria in the

sub-game 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 three is okay in this sub

game all right

but up right up right is not up right is

not a Nash equilibrium all right so in

this in this sub game strata Nash

equilibrium number two the down up right

equilibrium is prescribing play that is

not a Nash equilibrium in the sub-game

so it's eliminated it can't be a sub

game perfect equilibrium so here too

is eliminated since it induces non Nash

equilibrium play in this subject alright

and now we're done we know what the

whole games what we did here we started

with the whole game we found all that we

found they were three Nash equilibria we

found that only one of them agreed with

backward induction we then looked at the

sub games we first of all looked at that

blue sub game and we found that one of

the equilibria equilibrium number three

was eliminated the equal number three is

not prescribing Nash behavior in this

sub game right then we looked at the

slightly more complicated sub game the

pink sub game and we found that equal

number two prescribes the Paiva up right

which is not Nash in this sub game right

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

one we're left with the only sub game

perfect equilibrium the only equilibrium

that wasn't eliminated by the fact it

would prescribe bad behavior and sub

games the only SPE is number one which

is up up and left and what do we notice

we noticed that that the equilibrium

that's the play that backward induction

would have selected all right so notice

notice

this is the backward induction

prediction all right so what are the

lessons here the lessons here are that

our new idea our idea of sub-game

perfect equilibrium is pretty easy to go

about finding you just look at subgames

and check that the play in each sub game

has to be Nash way all right now if you

start at the back you'll construct it by

rolling backwards much like we did

backward induction right start at the

last sub game and work backwards and the

second thing is not surprisingly given

given that remark not surprisingly we're

backward induction applies for example

in this game the sub game perfect

equilibrium will find the equilibrium

that is consistent with backward

induction and remember that's what that

was our aim last week we wanted a way of

refining their sheeple of rim to throw

away those Nash equilibria that were

inconsistent with backward induction so

sub game perfect equilibrium has done

that it tells us now if backward

induction applies the Nash equilibria

should focus on are the sub game perfect

equilibria and indeed most people in the

class played that equilibrium just now

all right ok so that's really all I want

to say about this example but let me

make a remark impasse I made this mark

in the middle to me take it again when

we write down strategies those

strategies tell us what seem to be

redundant moves but that 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 beyond to write

down everything in a strategy allows us

to have to have everything in front of

us and makes that analysis simple and

that's exactly what we did here all

right 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 all right so new

example you clean this off

and once again I want to play this

example what I'd really like to do I'm

going to call this game the matchmaker

game the matchmaker game and what I'd

actually like to find out is I do we

still have our couple we tried to send

on a date our hapless couple which I

descend on a date about week three are

you guys still here there's the guy

what's your name again David and what

was the she hiding oh there she is oh

thank you thank you and your name was

nina

neither and david good good can we get

some mics - nina and david actually let

me let me do it and i'll go on talking

while I'm doing this so let me it's okay

where's David thank you and let me grab

them

and I can go get Matt Meg - Nina that'd

be great thank you all right so four

weeks we've been trying to get this

couple to go on a date it's our it's our

attempt to get economics majors to

become real people it's the hard thing

to do and they're kind of the hapless

couple because we we first wish

something to the movies and they're not

going to different movies and then we

sent them off for a romantic weekend in

New England and they end up doing

different things and one went to the

theater another went 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

I'll write down what the game is all

right so the game at this point him to

look like this player one is the

matchmaker we can compare aim if you

like and he has a choice he or she has a

choice he/she could not send the couple

out on the dais or she could send the

couple out on the date all right but

being a better matchmaker than me if she

sends them out on the date she's going

to stake some money no she'll pay for

the date she'll pay for the date all

right and

in the days once again they're trying to

meet and once again unfortunately they

haven't figured out where to meet all

right we'll put the payoffs and then a

second or two look the strategies are

alright so what so what we're going to

assume here is I'm going to let Jake RTA

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 Jewish he's not

Jewish he's not my mother

but he is the CA who's responsible for

bringing me some drink every day in case

I pass out in the lecture so that's

that's nearest thing I can think up to

Jake's gonna be our Jewish mother and

Jake's gonna 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 really he knows Yale students but I

do and he does send them really some

more 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 sat next to other people some

point so that's that seems like a good

idea so the classes he thinks of sending

them Souza's go to a large lecture class

so they're either going to go to the

Gaddis class or which is called Cold War

order the Spence class which is called

China everyone know about these classes

right right and these are like these are

like reasonable classes to go to to meet

to meet your dummy did have a day to

meet somebody I mean the Cold War again

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

prediction of the future relationship

but Cold War simply right and China's up

by all accounts as a fantastic class it

involves you know something involving

twenty million people most of whom were

in the class they gather so it's pretty

big class so let's do that so we'll call

this gap 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 will call them players two

and three 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 what if Jake doesn't

send them everybody gets nothing all

right

and if Jake does send them

they coordinate Jake makes one because

he feels really happy about this

after all I may not be some motivation

for people matchmaking and if they

coordinate down here Jake gets one as

well but if they fail to coordinate Jake

feels rotten about it particularly

because you know he paid for them to go

to this class whatever the cost of a

class at Yale is which is probably quite

a lot actually I don't it all right all

right so okay we'll call it Wow

all right 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 two one here and zero zero if they

failed to coordinate and zero zero here

if they failed to coordinate and one two

one two here all right or the

implication of this is that player two

who will assume is is David is that

right that 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 all

right all right so this is our game all

right and we're going to analyze this

game but before we analyze it let's try

and play it so so what we need to do is

first of all let's make sure things I

work smoothly

let's have David write down which class

he's going to choose and mean the write

down which class she's going to choose

I've lost sight of Nina does something

point out she is right write down with

partners choose okay something written

down okay and Jake got you've got your

mic there so right Jake you're going to

you going to are you going to send this

hapless couple or not so so I have Dave

in my section actually and I hear how

much he's been talking about Nina so I'm

gonna roll the dice and send them all

right understand okay good good all

right so we have them going off to this

class all right and let us see what they

wrote down so Dave what did you write

down I'm gonna give it an good at China

you're gonna go to China and neither I

chose s you totally managed to meet so

there's the successful date so it's

giving a round of applause all right

I hear it's a great class - in fact I

think it's gonna happen forever because

I think he must get approaching

retirement it seems a pretty good choice

so so good okay that worked very well

let's have a look now at this cadets

analyze this game and see what we can do

with this game all right

so how we're going to analyze this game

to no surprise we're going to use the

idea of a sub game perfect equilibrium

I'll collect the mics later des moines

alright so we're going to use the idea

of a sub game perfect equilibrium so how

do we how do we figure out how to work

out with a sub game perfect equilibrium

is we're going to use the same basic

idea that we use it when we what we've

been using all along in backward

induction and the conveys 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 you

can't do that anymore and work backwards

instead of doing that we're going to

start from the last sub game and work

backwards and in this example it's

pretty obvious what the last sub game is

right the last sub game the game within

a game there is only really one the game

within the game is this object here that

right this is the game within the game

now I could at this stage at KU do

something else I could write down the

whole matrix for the whole game and have

Jake choose matrix and and dave and nina

choose the row and the column but that's

going to take notes gonna get us astray

so we could do that it's not worry about

that let's just let's just start doing

things backwards so we do things

backwards we'll start at the last sub

game and that last sub game is an old

friend of ours it looks exactly like

this it's a spread in all right so

here's it involves players 2 & 3 all

right

and 2 is choosing between Gaddis and

spence

and 3 was choosing between Gaddis and

spence and their payoffs were liberal

space here so it's 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

when we put in Jake's payoffs as well so

Jake's payoffs were one here one here

minus one here

and minus one there all right there new

relevant players here are players two

and three but I've put player ones

payoffs in as well because why not

my let's keep track of them okay and

we're happy that that's that's exactly

describes this little game all right

that will turn some purposes we can

forget the first payoff but there it is

alright and 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 just

underline best responses so if if

neither is choosing Gaddis then dave

chooses Gaddis if neither is choosing

spence then dave would like to choose

spence

and conversely all right I'll just

underline the best responses for the

players who actually involve them again

I haven't bothered underlining anything

for G because he isn't a player in this

game that makes sense all right so the

pure Nash equilibria in this game the

pure Nash equilibria in this game are

essentially gaddis gaddis or spence

spence all right pretty easy all right

and from Jake's point of view each of

these pure strategy Nash equilibria

yield a payoff of here for him of what

what does he get if they put if they go

to gaddis gaddis he's happy that they

met and he gets one and if they choose

spence spence he's happy that they met

he gets one Jake doesn't really matter

Jake himself doesn't really mind whether

dave and nina learn about china

or learn about the Cold War he's one of

the meet alright so both of these yields

while for Jake both yield a values call

as value of one for player one for

player one who's Jake all right so from

Jake's point of view from Jake's point

of view going back a stage and we're

going to do now just a bit of just me

with bat conduction we're going to roll

the game back so we started by analyzing

this sub game and then we're going to

roll back a stage just as we did with

rapid induction so when we roll back

Jake is moving here

if jake chooses not to send them not to

send them then we get zero zero zero but

the key part of this is the first sir

that's Jake's payoff and if jake sends

them if jake sends them then what Jake

gets is the value to Jake oh okay I'll

put it Jake but gladly to play a while

okay value to Jake of the Nash

equilibria in this sub game all right

another big thing to right but that's

what Jake gets and the others do to

always get that as well and in this case

in this case rather than writing that

long piece this is just equal to one

that right all right so if Jake sends

them he knows that they're going to play

a nash equilibrium in this son game i

believes they're going to play a nash

equilibria in this sub game and either

of those two Nash equilibria in the

sub-game hang a second either the two

Nash equilibria the sub game yield a

payoff to Jake of one alright so

actually the one of them is one one of

them yields payoffs of one one two and

the other yields payoffs of one to one

but since Jake is the only mover here

lets us focus on Jake all right so from

Jake's point of view he's really

choosing between zero and one so he's

going to choose sent alright so the Nash

it that so it's with a sub game perfect

equilibrium the sub game perfect

equilibrium therefore there are actually

two of them here one is send spence

spence that's what actually happened but

there's another one which is send gaddis

gaddis all right that would also been a

pure strategy it's like game perfect

equilibria

all right to neither case what we did is

the rock dust remind ourselves we first

of all solve the equilibrium down the

sub game the equilibrium in this blue

sub game we figured out how much that

equilibrium was worst for everybody but

in particular for Jake but for everybody

and then we roll that payoff back and

looks at Jakes

choice right in this particular case

that gave us too full of rare sense

fence vents and send gaddis gaddis

however some of you must be suspecting

at this point that there's actually

another sub-game perfect equilibrium

here how do we know that what does think

about this game I mean 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 how that they

haven't on a date all semester so there

must be some possible to see that they

would fail to coordinate and it'll be a

pretty weird notion of equilibrium that

concluded that they always managed 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 sub-game the reason

is that the reason there's another

equilibrium so the reason is only

equilibrium in the whole game there

isn't there's another sub game perfect

equilibrium is that there's another Nash

equilibrium in the sub-game what's the

other Nash equilibrium in the sub-game

they could mix right they could mix

right the key right so it turns out that

in the sub game in the sub game here it

is there's also a third mixed

equilibrium there is a mixed Nash

equilibrium all right now we can we know

how to work that out we could we could

write down a P and a Q and we could look

at those indifference conditions and

solve it out but this is a sub game so

this is a game this this sub game

corresponds for a game we've seen many

times in this part so far and I think we

probably remember what that equilibrium

is that right I do anyway so if it's C

if you remember as well I'll write down

or C if you look alarmed so I claim the

equilibrium at the other equilibrium has

Dave playing with probability 2/3 1/3

and has Nina playing with probability

1/3 2/3 right so this is another

equilibrium in the sub-game people

remember that this was an equilibrium

Battle of sexes yeah yeah people nodding

at me yeah ok so so it isn't unintuitive

we could we all know how we'd work it

out we could go back and put on 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 our things be

equal and they do so in just such a way

as to make each of them in different all

right now this sub-game induces a

different value for Jake

all right so suppose Jake thinks I trust

Dave and Nina to play a nash equilibrium

in their sub game but i don't know which

one it is and i think maybe they're

going to play this one

all right so suppose that suppose

thinks that this is the equilibrium that

Dave and Nene are they going to play so

now should Jake send them or not well

let's work it out let's work it out so

now now if he sends them if Jake sends

dave and nina or more anonymously if

player one sends players two and three

all right

then with what probability will they

meet and what probability will they meet

well there's 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

and Nina is playing 2/3 1/3 all right

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 can meet at

spence that right so the property of

this box is 1/3 times 2/3 so this box

has probably two ninths and the property

of this box is 2/3 times 1/3 so this

probability had this box has probability

two ninths so the probability of their

meeting is 2/9 plus 2/9 that makes four

nights right over okay with that

all right so if Jake sends dave and nina

and they play this mixture actually

equal a rumor then they meet with

probability 2/9 plus 2/9 two knives at

Gaddis 2/9 expense for a total of 4/9

right which means they fail to meet

and hence fail to meet with probability

well if I'm meeting with 40 4/9 what

must be the property that they're

failing to meet 5 9 thank you so they

failed who with poverty 5/9 all right

so Jake's expected payoff if he sends

them the value for Jake of this

equilibrium is what so the value the

value to Jake of this Nash equilibrium

if he sends them is four ninths times

one plus five ninths times minus one for

a total of minus 1/9 and we're okay with

that right so if Jake sends them they

fail to meet five months at the time and

he gets minus one each of those times

they succeed in meeting 4/9 at the time

he gets plus one each of those times

so his expected payoff is expected value

from sending dave and nina on the date

is minus 1/9 all right so from Jake's

point of view what this game looks like

if you think this is the if thinks that

this is the Nash equilibrium being

played if he doesn't send you get 0 and

if he does he gets the value of this

Nash equilibrium which in this case is

minus 1/9 so he's not going to send all

right and the SPE here is not send mix

mix all right where this where this is

the mix all right 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 selling them on the

date you guys were lucky because Jake

chose the other equilibrium figured

you're playing the old equilibrium which

it turned out that you were all right so

in this

aim there were three sub-game perfect

equilibria one for each of the Nash

equilibria in the sub-game as it turned

out there was one in which Jake sent

them and they coordinated on the pure

strategy equilibria in the game SS there

was one in which Jake send them and they

coordinated on the pure strategy

equilibrium in the sub-game GG and

there's one in which Jake didn't send

them but had he in fact send them they

would have both mixed and hence from all

the time

failed to coordinate all right and

what's the big lesson here at the big

lesson of the first game we saw this

morning was that sub game perfect

equilibrium implies backward induction

all right the big lesson of this game

other than the fact that we're getting

close to the getting Dave and Mina on

their date the big lesson of this game

is to show that define sub game perfect

equilibria all you have to do is keep

your head and solve out the Nash

equilibria in each of the sub games roll

the payoffs back up and then look for

behavior up the tree all right once

again you look for this Nash equilibria

in each of these sub games roll the

payoff back up and then see how behavior

is to what the optional moves are higher

up the tree all right so we have time to

do one more example all right and the

third example I want to do is more of an

application all right so so far we've

seen two fairly simple examples now I

want to do an application

and the application I want to do is kind

of a classic business school case if you

like on mini case involving strategic

investment all right and the game is

this

well the setting is this there are two

firms we'll call them a and B and these

two firms initially before we start

considering the four before you start

initially 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 right

imagining that they're producing

fertilizer all right and let's be

specific here let's assume that the

prices in this market are given by the

following demand curve 2 minus 1/3 times

QA plus QB

all right this is the demand curve that

they face all right and we'll assume

that costs marginal costs C is equal to

$1 a tub all right all right so this is

this is that this is the price in

dollars per ton price per ton in dollars

and the costs are $1 per ton right 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 all right

it's a bit of a review all right it's a

while since we've seen corners let's

remind ourselves so I claim I claim that

the quantity the cournot quantity chosen

Q star has the formula a minus C over 3

B is that right right if you go back in

your notes you'll find it and I'm not

going to resolve it here with

many times so a minus C minus 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 so we

are having in letters but let's

translate it into numbers so in

particular this a this a is this to that

right

this C is this one and this B is this

third that right so let's just put that

down

so what this in this case this is 2

minus 1 over 3 times 1/3 right 2 minus 1

times 3 minus 1/3 so this says that the

quantity is what this is million tons so

the quantity here the cournot quantity

is a million tons each all right so one

each so each in this equilibrium each of

these two firms is producing a million

tons of fertilizer all right

and what else do we know we know

therefore what prices must be Nets has

to do that before we can get started

so prices must be two minus 1/3 times

the quantity that the first firm

produces plus the quantity that the

second firm produces that's 2 minus 2/3

so that should be 4/3 I've got that

right or 1/3 so prices here are $1 and

33 cents per ton all right and finally

profits so profit for each firm here in

this equilibrium before we even start

the game products are the more

interesting part of the game profit is

what so they're going to get $1 and 1/3

for every time they produce it's going

to cost them $1.00 to produce each ton

and they're producing 1 million of these

things all right so their profits are

to be a dollar sign here as well their

profits are one-third if there's a

millions they're producing one-third

their profits are one third of a million

dollars all right all right so this is

their period profit or in each period

they're doing this each year they're

doing this and this is their profits in

each period all right this is a simple

model we've done many times before this

is cournot and now we're going to make

it more interesting or everything if the

algebra here is a bit quick don't worry

about it check it at home it's very it

just basic basic algebra all right so

now suppose that you are the manager of

firm a all right classic known as a

classic business school K are looking up

my business school students in them up

in the balcony right you're the manager

of firm a and you have to choose whether

to accept an offer to rent a new machine

all right this new machine has two

features right the first well three

features the first feature is it only

works for a right - 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 point seven

million dollars in rental so each year

you rent this machine you'd have to pay

point seven million dollars all right

but that's the bad news the good news is

it will lower phase costs to 50 cents a

ton

all right a plot classic business cool

situation you're the manager affirm

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 point

seven a year to rent this machine but it

will lower your costs by 50 cents a ton

all right 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

listen no one's going to admit it in a

couple of years when you're in bet when

you're interviewing with it with these

guys all right 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 surest to rent an

alternate less dramatically it's a

coolant question of the English class

but important nevertheless all right

this board is stuck unfortunately I

don't have to write that a bit more

there we go all right 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 doing what mistakes I'm

making because I want you guys when you

interview with Morgan Stanley about this

kind of thing to impress them so that

they tell us we will come to Gail and

preferably give lots of money to Gail

all right 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

can 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

some awesome accounting firm when you

leave Yale god forbid but you might all

right so let's have a look at how the

accountants would answer this question

all right so I think the accountants

would do this they would say probably do

this that's of a poll how many people

think you should rent you bought some

think about it now so how many people

think you should rent the new machine I

don't think you should not

rent the new machine allowing

abstentions yeah it's try it again all

right without no aesthetic you can't

have stain in the interview all right so

you're on the spot you're in the

boardroom all right how many think you

should rent the new machine

raise your hands wave them in the air

how many all think you did not rent the

new machine all right so we're split

kind of learn the middle I'm looking at

my MBA students see which they voted

which do you guys brette rent or not

vote rent okay the MBA scene think rent

we'll see that's right all right so it's

move forward

all right to accounting accountants

answer so I think what the accountants

going to say is this they're going to

say produce right now you're producing a

million tons a year all right the new

machine saves you yes as I put per annum

extra let's try and be fast against a

million tons per annum the new machine

saves you 50 cents per ton so if you

rent this new machine you're producing a

million tons a year it's going to save

you 50 cents a ton so it's going to save

you 1 million a year in variable cost

all right those people own 115 or 150

now let me my variable cost is the cost

you're going to save input in actual

production of your fertilizer so it

saves you half a million a year all

right unfortunately it costs you cost of

machine which is a fixed cost

there's a fixed cost of 0.7 million a

year all right and point seven is bigger

than 0.5 so you should not rent all

right

so point five is less than 0.7 so don't

rent the machine right how many of you

said nose it's kind of its kind of the

back of the envelope calculation you

were doing is that right right it's kind

of back an envelope calculation but

accountants do so what's going on here

not our business school student up in

the balcony says you should rent he took

accounting I know he did that because he

have to accounting at business school so

what's wrong did he fail accounting or

as this answer wrong the answer is wrong

all right 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

right they're more often boring than

wrong but they're all they're almost

always boring but all right so this is

this answer is kind of boring right 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 invested

in the new machine that lowers marginal

costs as you are producing beforehand

all right all right we made the

assumption that it would lower your we

know it lowers your cost and we assumed

implicitly you'd 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

all right so let's look at an economics

officer let's look at an economics 115

answer how many of you've taken

economics 115 how many 115 at the moment

quite a few okay so let's have a look at

an economics also let's see why that

previous answer was wrong

so here is QA and here is cost of $1

your new cost will be 50 cents all right

I'm putting prices and cost on this axis

all right 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 I'm 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 yes it again Chad yeah 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 right

roughly speaking should be twice as

steep this is residual marginal revenue

all right and here's what you used to

produce right we're not this is this was

a million tons when it was marginal

revenue hitting marginal cost now your

costs have gone down so noticed 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 it's

that clear to everybody all right so

this is the kind of picture that you

probably saw a lot of in 115 is that

right or 150 for that matter that

correct so notice in this picture we can

actually see the accountant answer the

boring answer the boring answer is this

rectangle right this rectangle this is

the accounting answer

this rectangle is 0.5 times 1 so it

comes out as a half and there's that's

think that were the countin times there

and what did they miss what are the

accountants miss they miss the triangle

right they miss the triangle they miss

this triangle so I told you they were

boring they're little bit square so they

tend to miss triangles all right so

here's the triangle that they missed all

right and we missed this triangle all

right and how big is this triangle well

we could do it at home its 1/2 base

times height

all right 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

know we could figure out what the width

is as well therefore we could do 1/2

base times height turns out that this

has area I did this at homes let me just

write it down

it has area 3/16 all right

so again ever 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

assume they did it correctly at home

this is 3/16 which is approximately 0.19

all right so we missed this point 1 9

you missed this point 1 9 all right so

how are we doing now so we know from the

accounting answer we had a half in

savings we know from the economics

answer that we should add another point

one nine to this that's the triangle for

a total of point six nine but

unfortunately this is still less than

point seven which is the cost of the

machine

the pranam 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 do and is much less

boring than accounting and we'll get you

the counting answer anyway if you do

things carefully we still end up

including you shouldn't rent but I'll go

from the Business School said you should

rent right so did he feel economics as

well as accounting or is this answer

wrong this answer is still wrong

this answer is still wrong oh we're done

okay all right there we go I think that

board is broken

all right this answer is still wrong

don't want to delete that as a shame I

don't know all right you've all got

those numbers somewhere I hope they're

in my notes

this answer is still wrong we need to

get the right answer all right it's what

we need is a third answer which is the

game theory answer which is also known

as the right answer all right what are

we missing

what's wrong with the economics answer

somebody what's right everyone you was

wrong why is it wrong it look pretty

good what's wrong with it what's wrong

with it I said let's bounce down here

something in the front row will help me

why is this what did I do wrong for B

also changes its quantity right we

assumed not only in the accounting

answer we assumed that firm a kept its

quantity fix that we kept our ponte fix

and that was wrong but in addition firm

B is going to change its quantity isn't

it firm be easy to change its quantity

let's have it 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 V

all right and what we've learned what we

learnt 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 to be

produces so regardless of what generated

this residual demand curve as the cost

go down for firm a it increases its

quantity right we know that you 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 of firm a it shifts to the

right now produces more for any given

quantity that firm B produces all right

so QA has gone up that was our economics

answer all right 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 what kind of game is it

whereas firm a increases its strategy

firm B decreases its strategy in

response strategic substitutes right so

because this is a game of strategic

services use the strategic substitutes

good interview word right good word to

mention an interview because 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 a this is good

for firm a it softens competition it

softens competition right and as a

consequence it leads to an increase in

profit it again we could we don't have

time today what we could do is we could

go back and we can recalculate the new

cournot equilibrium right we could

calculate this we could there's a

homework exercise try it how you're late

the new Nash equilibrium and notice this

is a Nash equilibrium in a sub-game all

right why is this a sub game because

firm a made its decision whether to buy

this new machine or not and then they

played cournot though so the cournot

game the game up here what is this this

is the sub game right this is the

diagram of the sub game if you like

right it's the best response it's the

best response isn't the subject so what

sub game perfection tells us to do here

is first of all work out the new

equilibrium in the sub-game work out how

much that new equilibrium is worth for

firm a and then roll it back to the

investment decision right it turns out

it turns out when we do that we get an

extra point 3 1 million dollars all

right so we do it at home we can extra

point 3 1 million dollars so it turns

out that our MBA student was right good

all right it turns out that you add this

point 3 1 to the point 6 9 we had

already we get 1

which of course is much bigger than 0.7

and indeed you should rent the machine

all right

now I want you to have what to take away

lessons from this game

the first takeaway 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 sub game you

would look what would happen if you did

invest and solve out the new Nash

equilibrium in that sub game then you

want to roll back the value of that sub

game back into the initial decision

which is the strategic investment

decision whether to buy this whether to

rent this machine or not

alright so schematically the game looks

like this rent or not rent and in either

case you play cournot like there's a sub

game 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 sold out the symmetric

cournot we actually did that upfront we

now sold out the new equilibrium in this

asymmetric cournot game this one here

right this is the old one and this is

the new one solve an out work out how

much profit you're going to get and roll

that back remembering that it costs you

point seven million to make this step so

that's the first takeaway lesson but the

second takeaway lesson is more general

so let me just pause to get everyone to

wake up again so I make it the second

limit takeaway lesson is this what tips

the balance here from the economics

answer and the accounting answer were

the strategic effects

it was the strategic effects this was 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

their strategic is 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 US 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 right why is that wrong because

you're forgiving that as you change the

tax code people's behavior changes

incentives change and people's behavior

change it leads to a mistake in

designing the tax code you need to take

an account strategic affects 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 say well I wouldn't say

this but some people on our committee

might say this under 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 alright so the biggest

lesson of today's fast is don't be like

an accountant part because it's boring

and you won't get you worked on your

dates and partly because you'll miss out

on these important strategic effects

we'll come back and look at more on

Wednesday