All language subtitles for English (auto-generated)_en_0_6. Nash equilibrium_ dating and Cournot(360P)

all right so last time we were talking

about the investor game and this was a

coordination

game and we learned some things I just

want to recall some of the things we

learned so I can highlight them a bit

one thing we learned was that

communication can help in a coordination

game so I forget who it was but someone

down here who's disappeared uh was was

our Jimmy Stewart character and helped

coordinate you on a better equilibrium

uh simply by suggesting what you should

do and this uh there's two lessons here

one is this is very different from the

prisoners

dilemma so we leared the very first time

that in the prisoner's dilemma

communication per se won't help but in a

coordination problem which could be just

as serious socially as a prisoner's

dilemma in a coordination problem it may

well help and the reason it helps is

you're trying to coordinate onto a Nash

equilibrium and one way to think about

Nash equilibria is that they are self

enforcing agreements so it provid

everyone believes everyone else is going

to go along with this agreement then uh

everyone in fact will all right I want

to make another slightly more

philosophical remark associated with

this and it's to do with the idea of

leadership so leadership is kind of a

big word that you see written probably

too often these days in too many

newspaper articles and it probably comes

up in too many Yale classes and I don't

claim to know anything about leadership

all right but and I don't think game

theory is going to contribute anything

to understanding about leadership but

one thing we can do is tell you where

leadership may help all right and in a

coordination game where you where the

idea is to try and get people to

coordinate on a particular equilibria

rather than another equilibria or wor

still to be un coordinated entirely in

those kind of games leadership can help

tremendously a little bit of leadership

can help tremendously so these games

these coordination games are games where

there is a

scope for

leadership and just to see that uh in a

very simple example again we don't need

such a complicated example as the one we

looked at last time you could imagine a

game a really trivial coordination game

which look like

this all right 1 one Z

z0011 and clearly in this game what

matters is coordinating you either want

to coordinate on up left or you want to

coordinate on down right you don't want

to end up uh uh that's a d you don't

want to end

up uncoordinated on down left or ight

everyone see that so in this game if you

just played it it's quite likely going

to end up uncoordinated but if a little

bit of leadership can say okay let's

make sure this is where this is where we

coordinate or let's make sure this is

where we coordinate all right so this

matters a lot I don't want to overplay

the social importance of this but go

back a couple of years to what was

happening in the aftermath of of Katrina

and realize how important how how bad

things get when things fail to be

coordinated all right one other remark

before we leave this in the game we

played last time in the invest game uh

one feature of that game was that the

more you thought other people were going

to invest the more you wanted to invest

is that right the more you thought other

people going to invest the more likely

you were going to invest and if you go

back to the time before that we talked

about this partnership game and the

partnership game you were contributing

effort to a joint project it could have

been a law firm it could have been

working in a study group on a homework

assignment and that game if you remember

what the best responses looked

like they looked like

this where this was the effort of player

one and this was the effort of player

two this was the this was the best

response of player one and this was the

best response of player two this game

also has the feature that the more

effort the other person the more the

other person does the more you want to

do right the more effort your partner

provides into this project the more

effort you want to provide in this

project and I just want to introduce a

bit of jargon here these games in which

the more the other person does the more

I want to do these are called games of

strategic compliments right these are

games of strategic

compliments

strategic

complement all right so both the

investment game and the uh game with the

partnership the the the partnership firm

game are games of a treat complt we can

call the strategies strategic

complements and we'll come back to this

later on

today all right so before before we

leave coordination games I want to look

at another one a little bit more

complicated one perhaps that we

mentioned briefly last

time uh so we'll look at we'll look at

and play another game and we'll call

this game uh going to the

movies all right so I always regard one

purpose of this class to help hapless

Yale students in their dating strategies

uh that seems like a good thing to do do

uh and uh how many of you just we don't

need to show the camera on this but how

many of you are econ

majes there's a lot of people who

probably need a lot of help with their

dating strategies right right right okay

so the idea of this game is a couple is

going to meet up at the movies and uh

let's have a look at the movies

concerned here and we'll draw just three

possible

movies all right when the move we're

going to look at are

um the born

ultimatum and the uh movie called

Good

Shepherd and a movie called Snow

White and I'll explain the game in more

detail in a

second it used to be in the old days

before I had kids I could list off 15 Uh

current movies is and help you a bit

more with your dating strategy by giving

you instant movie reviews but now I have

kids I get to see precisely two movies a

year uh and the two movies I got to see

this year were uh the born ultimatum and

the Good Shepherd and now I think about

it both of us have Matt Damon in it and

my wife chose both so maybe I need to

work out some

more so just to give you some idea about

this m how many how many of you saw born

out ultimatum quite a lot of you this is

a movie it has pretty good action

virtually no plot basic lesson of this

movie is uh uh if if you take home

lesson it's that all spies are psychos

or something and then how many of you

saw Good Shepherd a lot of you saw that

as well that's actually pretty good it

has a lot of plot and no action so it's

the opp other way around and the basic

lesson of this movie is you probably

knew this already uh everyone at Yale

take place at Yale everyone at Yale is a

spy and by the way also a

psycho and the third movie is um is Snow

White which I haven't gone out to see

but uh my uh four-year-old daughter has

seen on 24 of the last 27 nights on

video uh and this movie um I don't know

if I'd recommend it that much uh it's

umap I'm being to PC but I'm not

convinced that for the Modern Woman

hanging around waiting for your prince

to come is really a good strategy um by

the way for those of you who are I don't

suppose you are but if you are doing

that strategy I'm going to get in

trouble never mind if you are doing that

strategy take it from a Brit most

princes are as dumb as toast it's not

worth waiting for all right so let's

let's put the payoffs in here so the

idea here is that these people are going

to Mo they're going to meet at the movie

they're going to go out to a movie and

they they they've decided to go to the

to the Criterion or the local movie

house and there are these three movies

showing and they're all excited about

going to this movie except being

economics majors and not very good at

dating they have forgotten to to tell

each other which movie they're actually

going to go to right they're going to

meet in the you know in there on the

back row probably and uh um uh but

they're not uh they're not telling us uh

which movie so it's a problem and let's

put in the payoffs and we'll put in the

payoffs that roughly I think would

correspond to my preferences and we'll

talk about it what these preferences

mean in a

second so here we go so -1 -1 0

-10 - 2 -2 so here here are the

preferences for these movies of player

one

and player two and you can see from

these preferences from these payoffs

that uh the the best thing for player

one is for both players both people to

meet and go to the born ultimatum this

is the action movie all right so so it's

not to be sexist let's assume this is

the this is the she and this is the he

so she would like to go and see Matt

Damon beat people up and uh coordinate

there that's her her her favorite thing

uh and she gets nothing if they fail to

coordinate

uh and uh so on and so forth uh her

second choice is to manage to coordinate

at Matt Damon being a Yale spy uh all

right and uh then she really doesn't

want to go and see Snow White and the

worst thing of all actually this is for

both players the worst thing of all is

to coordinate at Snow White because then

you've got to talk about it afterwards

at

coffee all right and similar to the

other person same exactly except that uh

he would rather uh meet at uh the Good

Shepherd uh you know since he's a a Yale

guy he probably think of himself as the

Matt Damon character there perhaps not

but and uh you know failing that he'd

like to coordinate at the born ultimatum

but again the disaster is to fail to

coordinate and go to see different

movies all right so these are the the

preferences what can we what can we tell

immediately if if you're stuck in this

situation what can you tell immediately

about this about this game what what's

an easy First Step let's get the the guy

in red

here Snow Snow White is dominated by

both players good so so for both players

for both players Snow White is a

dominated strategy so both players

having taken this class I told you it

help you with your dating but right so

both players will realize that they

should at least not go to Snow White is

that right right great animation but not

not perhaps a good date movie so uh it's

gone all right everyone see that so that

leaves us just with these two options uh

the two Matt Damon movies born ultimatum

and good Shephard and here the remaining

payoffs and let's let's see how we play

this out so so let's let's pick on some

people uh Al you want to pick on

somebody I get C on as well all

right all right so okay okay so we

picked on somebody there uh when do you

person who Ali picked on you want to

stand

up okay and your name is Nina say again

loudly Nina Ana Ana just shout out to

everyone here is Ana and K did you pick

on a

guy Nina Nina Nina okay Nina I'm sorry

Nina and Kai get get your guy to stand

out there and your name is David David

so do you know each other no no you're

okay so you're about to go on a date so

this is

okay all right all right so Nina's

preferences n's preferences are player

one's preferences here so she uh her

best thing now is to

successfully meets David at the born

ultimatum and David's preferences are uh

player two's preferences so his favorite

thing is to successfully meet at the

Good Shepherd and why don't you write

down uh a second uh on the corner of

your notepad what it is you're going to

choose to do with my ta's watching is

you really can't

cheat all right both written down all

right we have a good look a good look at

them so we just embarrass them a bit we

looked at them already I guess okay okay

so uh nah shout out to the crowd what it

is you where it is you've chosen to go

born ultimatum born ultimatum and and

David where did you choose to go I also

put down forign ultimate they're going

to coordinate very good

so all

right I I don't know what this I don't

know what this tells us about David and

Nina but don't don't don't let them off

the hook quite yet so now um that was

good we managed to coordinate although I

think we can all see that we might not

have done right we all see we might not

have done all right let's um let's try

some Communication in this game we said

that coordination games are helped by

communication all right so let's imagine

we're playing the game again once again

you're going to the movies once again

these the first time didn't happen so

you didn't for some reason it was

canceled that night someone had a bad

cold or something and you are about to

go off on Friday night to see these

movies and uh let's allow some

communication ahead of time so so David

uh uh uh you're about to go you you

realize you've got this coordination

problem so you phone Nina up and you can

say whatever you like so what would you

say to

Nina uh looks like you want to go to

Born

ultimatum uh I'd rather i' I'd rather go

and if you're going to be stubborn I'd

rather go seeing born ultimatum with

you then uh then not go on a date at all

so so here's another clue stubborn is

not a good first date

word all right but okay so Nina what

would you say in response to that I'll

go to Born ultimatum all right all right

all right so write down what you're

going to do this

time all right anina what did you do

forign ultimatum David forign ultimatum

so they're still managing to coordinate

but you can okay so so thank thank

thanks for this couple let give them a

round of applause

they we may come back and pick on you

later on in the course but we'll we'll

leave it for now all right so in this

case the communication worked all right

but am I right in thinking that the

communication isn't isn't such an

instant solution as it was in the game

we saw last time this game's a little

bit harder to get communication to work

now why I mean is that right first of

all so so what's what's the problem here

why is this a slightly more difficult

game let's keep the mics handy a second

uh why is this a more difficult game to

to attain coordination in um yeah that's

the okay can you neither of the best

responses is in Nash

equilibrium well I know hang on a second

so let's slow down a second that's a

good that's a good one let's let's start

with that what are the nashy what are

the nashy quili in this game what are

the nashy qu in this game uh somebody

say pick on somebody just pick on

anybody okay yeah what are the n equilib

in this game uh both players doing born

ultimatum and both players doing good

shepher all right so so in fact if we if

we check the nor the Nash equilibria in

this game the Nash equilibria you can

check that they are in fact responses

are uh both people doing the born

ultimatum or both

people doing the Good Shepherd and the

reason for that is if the other person

is going to go to born alatum I want to

do that and if the other person is going

to go to Good Shepherd I want to do that

so these are both National equilibria

that isn't too much the problem that was

true last time as well there were two

well-coordinated national equilibria

although one of them was better than the

other what's the tricky thing here

what's what's the extra trick here

somebody uh

yeah can uh is there a camera is do I

have a mic on this

side uh

sorry there was a hand up there yeah y

yep yep go go go go there you go stand

up

great each player prefers a different

Nash equilibrium all right so so unlike

the other game the the two examples of

coordination games we saw so far were

really pure coordination problems is

that right they were pure coordination

problems there was no

uh Conflict at all in in the case of the

investor game last week every player

preferred one equilibria to the other

right in the game I just put up this a

trivial example it didn't really

matter which equilibrium we played here

we just wanted to play an equilibrium is

that right is that right but

here there's a potential source of

conflict here both people would rather

be at an equilibrium than to be Mal

coordinated or uncoordinated but player

one wants to go to Born ultimatum and

player two wants to go to good sheeper

and actually I thought Nina's strategy

there was pretty good Nina's version of

communication was I'm going to Born

altimate right right that that basically

fixed things right but you could imagine

that communication could break down here

there could be a negotiation going on

all right so think about uh I mean I

don't I don't want to push this too hard

but think about other games where people

are communicating to fail to avoid a

failure of of coordination there's a

strike neg negotiation going on as we

speak in fact there's a critical moment

this morning uh up in Detroit between

General Motors and the United Auto

Workers and I don't want to say that

strikes are only about miscoordination

but clearly uh everyone's better off if

they come to some some agreement rather

than strike there's at least some

agreement that's going to be better for

everybody than a strike and yet because

there because there are conflicting

interests there basically in that case

conflicting interest about health and

pension payments it could well be that

you end up uh failing to

coordinate right all right so this game

has a name and actually somebody

mentioned it last time anyone who's the

person who mentioned this last time

somebody somebody mentioned this what

what was the what's this game called

Shout It Out Battle of the Sexes that's

that's good this game is called the

Battle of the

Sexes and we'll we'll see it in various

forms over the course of the semester

it's actually a very interesting game it

games like it they are coordination

games but different people disagree

about where you'd like to

coordinate all right so so much for

talking about coordination

games and helping you with a dating

strategy um this being a game theory

class there's at least one game we have

to discuss uh and we'll come to it now

so we're going to spend most of the rest

of today uh talking about uh corod Dooly

and before I do that let me just check

how how many of you have seen corod

Dooly before just raise your hand just

just for me right so maybe about half of

you so those of you who have don't worry

I mean this will be a bit review but

we'll see it more Through The Eyes of

Game Theory this time and for those of

you who haven't don't worry we're going

to go through it okay okay so this is a

classic game perhaps it's the one of the

most famous games and therefore worth

studying in the

class now as a purely Game Theory

exercise one reason for studying Coro

duopoly is that so far we've discussed

how to find Nash equilibrium when there

are few players each of whom has few

strategies and we've discussed how to

find Nash equilibrium where there are

many players each of whom has few

strategies and this is a game where

there are few players maybe two but they

have many strategies right they actually

have a Contin the strategies all right

and for those people who uh aren't

familiar with this and are worried about

the economics of it in particular it's

gone over in considerable detail in

chapter six of the textbook in chapter

six of the

textbook of the data

book all right so let's just motivate

this a little bit so there's two reasons

why this game is interesting I say what

it is so this is a game in which there

are two firms who are competing in the

same market and we'll give a bit more

detail in a second and one reason this

game is interesting from the point of

view of Economics is this game lies

between the two extreme cases that you

learn about in your intereconomics class

one extreme case is perfect competition

and the Other Extreme case is Monopoly

so this is this is really the first

attempt way back in the 19th century to

study a market that's somewhere in the

middle where in fact most markets are

right there are two firms and we're

interested in two things here we're

interested in what's going to happen in

these markets and then from a welfare

point of view from a policy point of

view we're interested about whether this

is good for consumers or good for uh

producers or what right how how this

relates uh to profits and consumer

surplus all right so with that in mind

let's just set the game

up so the players in this game are two

firms two firms and the strategies in

this game for the firms and this is

going to turn out to be important the

strategies are the quantities that they

produce of an identical

product right so they are the

quantities they produce or each of them

produces of an identical

product so far as far as the consumers

are concerned these two products are

perfect substitutes you could think of

these as two companies producing bottled

water and now I'm going to get hundreds

of letters saying not all bottle waters

are the same especially from Italians

and French but never mind let's let's

pretend that they are all right so just

to emphasiz that the strategies are

quantities rather than using S let me

use Q today to be the strategies so so

Qi and and uh Q minus I or q1 and Q2

will be the strategies

all right I need to give a little bit

more structure on the payoff before I

get to the payoffs so in particular I

need to tell you what is the cost of

production and the cost of production in

this game is simply going to be C *

Q all right so if I produce one unit it

cost me c if I produce two units it cost

me two c if I produce 100 units it cost

me 100 C and if I produce 735 units I

cost me

735c all right so for those people who

took intro economics which is most of

you this is a game well this is a

setting in which we have constant

marginal costs constant marginal cost

the constant marginal cost is

C all right and I'll need to tell you

about how prices are determined in this

market so prices are determined as

follows uh prices depend on some

parameters I'm just going to call A and

B uh and let let me write the equation

and then we'll see what it looks like so

basically the idea

is that the more these firms produce so

the more is the total quantity produced

q1 plus Q2 the lower is the price in the

marketplace for this product all right

and let's just draw a picture of that

we'll come back to this in a

minute let's draw a picture which we we

can keep for

later actually let's save myself some

time and bring this

down all right so this this

uh this equation I've just written you

can imagine q1 plus Q2 total quantity on

the horizontal axis and you could

imagine price on the vertical axis and

basically what we're saying is that

prices depend on total quantities as

follows where the slope of this line

slope of this line is minus

B all right now for those of you who

took economics 115 or equivalent what is

the name of this line I've just

drawn okay that that was way fewer

responses than I wanted all right let's

try again what's the name of this

downward sloping hint hint line that

I've just drawn it's a demand curve

thank you it's a demand curve all right

this is the demand it tells me look at

that way around and look at the uh look

at how prices correspond to quantities

it tells me the quantity demanded at any

given

price all right we'll come back to that

in a

second but meanwhile let's just finish

up what we're doing here and put in

payoffs so payoffs for these firms are

going to depend

on on uh on profits payoffs the

firms

aim to

maximize

profit and profit is going to be given

by

P * Q minus sorry it's just do let's do

this for firm one all right let's let's

be careful here let's do this for firm

one we do firm one firm two in a second

so the payoff for firm one as it depends

on the quantity that she produces and

the quantity that the other firm

produces is going to be

prices times the quantity that firm one

produces minus

costs times the quantity that she

produces all right all right so this is

this term here is

revenues

this term

here is revenues and this term here is

total costs right revenues minus costs

makes profits and again for those of you

who are less familiar with economics I'm

hoping this is not going to be too hard

but you can read up on it otherwise all

right what I'm going to do here is I'm

going to substitute in for

prices all right so I've got this

expression for prices

here here it is and I want to plug

it into that P there okay all right so

that's I'm just going to rewrite it now

with that with that P expanded out all

right so I'm going to get and I'm going

to multiply out at the same time so I'm

going to have a

q1 minus

b q1

2 minus B

q1

Q2 minus

C q1 right so all I'm doing here nothing

nothing particularly exciting I'm simply

plugging this expression

in for p and then I multiplied the whole

thing out because I because it was all

multiplied by q1 and I should warn you

again that I'm very likely to make

mistakes when I'm doing this kind of

thing so please catch me if I do all

right so here I have a new expression

for firm

ones profit and I could do the same for

firm two but I'm not going to because

it's that would get boring okay all

right so this is this is just firm one's

profit okay and with this in hand what I

want to do is and firm two have a

similar expression what I want to do is

I want to figure out the Nash

equilibrium of this of this

game right now I will need the other

board I want to figure out the Nash

equilibrium

of this game both firms are producing

quantities both firms are trying to

maximize profit and I want to find out

the Nash equilibrium how do I go about

that exercise what's the what do I need

to do to find each firm's Nash

equilibrium how how do we always find

National equilibrium when we're not

guessing what do we

do anybody somebody way over there who's

going be way off camera but perap we get

a mic in there we get him sorry he's way

in the corner there come come come this

way along the a bit closer to the

aisle there you go well we would just

delete the uh we would iter iteratively

delete uh dominated okay we could do

that we could do that we could try it

deleting dominated strategies uh and see

if that and see that process converged I

had something else in mind here

something else more more straightforward

what what else so the guy in

red um identify the best responses of

each player as a function of the others

and find out where intersect okay say

that L more that was correct let say it

more loudly shout um identify the fun

the best responses of each player as a

function of the others and find out

where they intersect okay good so what

we're going to do is we're going to

figure out player One's best response

quantity to each possible choice of

player two and then we're going to flip

it around and figure out player two's

best response quantity to each possible

choice of player one and then we're

going to see where they where they where

those where those coincide where they

cross right so just like we've done

several times before all right right so

what I need to do then is I need to

figure out what is player One's best

response for each possible Choice Q2 of

player

two all right so how do I do

that how do I do that I want to find the

best response how do I do

that everyone's looking like this is but

I bet you all know

really anybody me to call call somebody

no

somebody yeah yeah sorry athletic day

for you somebody right at the back

there sorry stand up and

shout um find the best response function

by differentiating this and maximizing

the function for every Q2 okay good good

so what we're going to do is all right

good we're g to we're trying to maximize

the best response is going to maximize

these uh these profits all right so in

particular what we're going to do to

find out what what quantity q1 maximizes

this profits for each choice of of Q2

we're going to differentiate this with

respect to q1 and then what and then set

it equal to zero okay so again let me

just take a look out there how many of

you remember this we did this once

before I did this class how many of you

remember this from from your uh high

school calculus days or from

112 quite quite a number of you actually

remember okay good so what we're going

to do is we're going to differentiate

this thing to find a first order

condition

we're going to

differentiate with respect to q1 right

the thing we're trying our control

variable we're trying to to maximize the

thing with respect to and set it equal

to zero all right so what I'm going to

get so This this term here aq1 is going

to become an A and this term here minus

B q^2 is going to become

a 2B

q1 and this term here minus bq1 Q2 is

going to become a minus B Q2 and the

last term minus cq1 is going to become a

minus C all right everyone happy with

that what I did was I differentiated

this uh fairly simple function with

respect to q1 and since I want to find a

maximum what I'm going to do is I'm

going to set this thing equal to zero

all right at my maximum I put a hat over

it to indicate this is the a Max at my

maximum I'm going to set this thing

equal to zero now just one since we're

being nerdy in this class despite our

attempts to go dating we're being nerdy

let's actually uh uh be a little bit

careful this was a first order condition

or first order necessary condition I

actually need to check the second order

condition so how do I check the second

order

condition I I differentiate again right

I differentiate a second time and check

the sign so the second second order

condition I differentiate this

expression again with respect to q1 the

only place q1 appears here is here so

when I differentiate again I'm just

going to get minus 2B and sure enough

that's negative which is what I wanted

to know to just just to check that when

I'm finding this thing I'm finding a

maximum and not a minimum all right good

good okay so this is our key

expression and I'm going to use this

expression I'm going to solve it for q1

so the best response for player one as a

function of what player two chooses Q2

is just equal to the q1 Hat in this

expression and if I solve that out

carefully I will no doubt make a mistake

but let's try it uh I'm going to have a

minus C over 2

B

minus

Q2

over 2 two all right so try that at home

but I think I did that right so I took

the I took this to the other

side and I divided through by 2B and the

B here

canceled all right everyone just stare

at that a second make sure I get it

right I'm very very capable of getting

the things

wrong all right good so what I have here

is an equation that tells me player

One's best response for each possible

choice of player

two all right and I could do the same

thing for player two to find player

two's best response for every possible

choice of player one on the other hand

we know it you know the math is going to

be the same so let's

cheat I know that player two's best

response for every possible choice of

player one which if we' done it would be

Q2 hat is going to equal a minus C over

2B minus q1 over two right so I'm just

I'm just flipping around the twos and

the

ones so at this point I found player

One's best response as a function of Q2

I found player two's best response as a

function of q1 and the way I did it just

to remember the way I did it was I

applied a little bit of 112 Andor high

school calculus right essentially single

variable

calculus all right now before we could

just solve at this but it's not let's

draw these let's draw these okay so what

I'm going to do is I'm going to go back

to my

picture and draw these functions I'll

just remind ourselves what these

functions

are it's a minus C over 2B minus Q2 over

two and equivalently for uh player two

all right that's the thing I is hidden

by the board now what I want to do is I

want to draw a picture a little bit like

we did for the partnership

game so in the partnership game we put

efforts on these axes and now I'm going

to put quantities on these axes so this

is going to be the choice of player one

and this is going to

be the choice of player two and what I

want to do is I want to figure out what

what this looks like what uh for each Q2

that you give me or that player two

chooses I want to find out and draw what

is player One's best

response all right everyone happy what

I'm doing here all right so let's start

somewhere what we're going to do is

we'll refer back to this other picture

which is why I left it

here all right so in particular what

would be player One's best

response if player two didn't produce at

all how we find that without worrying

about the economics of a second how

would we find that just as a math

exercise so suppose player two doesn't

produce at

all right what is player One's best

response someone read it off for

me when we when we co call somebody you

can

just a minus C over 2B all right so so

shout that out a minus C over 2B good so

if I if I plug if I plug Q2 equal Z into

here this term disappears right and I

just get Aus C over 2B so that what

what's that telling me let me put the 45

degree line in here what's that telling

me it say if player two chooses not to

produce then player One's best

response is a minus

C over

2B now I claim that that quantity a

minus c/ 2B

actually has another name what's the

name of that

quantity what's the name of the quantity

a minus C over 2B once we once we escape

the algebra and just think about it in

terms of

economics well let's think about it a

bit let's go back to the previous

picture if player two is producing

nothing player two is producing nothing

then what is player one

effectively he's a monopolist or she's a

monopolist all right so we know how to

figure out Monopoly quantities from what

we did in 115 or equivalent courses let

me finish the picture here here's my

demand

curve here's my constant marginal cost

at

C all right I want to I want to use this

picture on the left now to figure out

what the Monopoly quantity is can anyone

tell me let's Co call somebody again

well raise your hands if you took 115 or

110 or

150 oh no no no no I know more of you

took it than that it's it's a prct for

the class right right raise your hands

if you didn't take any of those things

right all right so Al keep your hands up

call somebody who hasn't got that hand

up anybody anybody go

ahead passing the back

here do you want to tell me what what

the where's where's the Monopoly

quantity on this

picture try try the there the woman

woman one end yeah that's that where you

are the woman here yeah oh no I really

don't remember don't remember don't

remember who else who else doesn't

remember somebody must don't remember is

a good reply if you're if you're

auditioning for it to be the Attorney

General but it's not such a

good

anybody somebody somebody help me out

here where's where's the Monopoly

quantity on this

picture it's a good review exercise this

yeah it's where the marginal revenue is

equal to marginal cost ah it's where

marginal revenue equals marginal cost I

should have picked on one of the S

people by the way it's it's where

marginal revenue equals marginal cost

right the reason I should pick on the

when people is they their ambition in

life is to run a monopoly right say all

right trouble is I haven't drawn the

marginal revenue curve yet right I

haven't right you can't actually see the

marginal revenue here what does the

marginal revenue look like in this

picture don't we go to the same guy the

guy in brown where's where's the

marginal revenue look what the mar

Revenue look like in this picture on the

left it's one half the slope of the

price yeah oh it's I guess it's twice

the slope but that's far enough I know

what you mean all right so the marginal

revenue here let's put in a different

color the marginal

revenue look something like

this right and as I don't know what your

name is as the guy said the Monopoly

quantity the Monopoly

quantity is when marginal revenue equals

marginal cost all right and as as we

know this line here has twice the slope

of the original line of the demand curve

so the slope of this thing is minus 2

B all right now I'm going to claim that

this Monopoly quantity I claimed before

is going to co this our intuition says

if the other firm isn't producing my

best response must be to produce My

Monopoly quantity which I can see on

this picture and I also claim that the

math is telling me just a kind of nerdy

math is telling me that that quantity is

a minus C over 2B and I claim that

they're the same thing how can I see

they're the same thing well look here I

have a line of slope minus two how far

down does it have to go it has to go a

minus C down this pink line it has to

drop off from a all the way to C how far

along do I have to go to drop off a

minus C when I have slope minus 2B

answer a minus C over 2B right not even

calcul something you really all in high

school all right okay so this Monopoly

quantity is is

indeed a minus

C over 2B all right now I'm going to ask

again is is some of this being coming

out of the fog I mean you may not seen

it with the algebra before but some of

this sort of pictures have you've seen

before it's good review exercise for

some of you who are have taken 150 and

are about to take 150 or there

now all right so we found precisely one

point on this best response picture and

there's a lot of points to find and it's

20 past 12 so we to get going

all right so let's try and find another

Point all right it's also a different

question how much quantity would firm

two have to produce in order to induce

firm one not to produce at

all then again how much quantity would

firm two have to produce in order to

induce firm One's best response to be

zero all right uh Al help me out the

it's Katie right is that right the woman

in Green Let's get a mic to Katie oh

there's a mic Closer by never mind sorry

yeah a minus C over 2 all right so so

that's just solving out the algebra

we're saying we're saying what what Q

solves a minus C over 2B minus Q2 over 2

equals z right what

Q2 makes this equal to zero and Katie's

answer is solving out the algebra here

is that

Q2 that solves this must be a minus C

over over

B all right that's right where can I see

that one on the picture though let's go

back to the economics right so I'm

claiming that there some quantity up

here I claim this quantity is a minus C

over

B such that if firm 2 produces that much

quantity firm one will just produce

nothing right but that's just kind of

algebra and math why can I see it on

this

picture what quantity if produced by

firm two would cause firm one to shut

down on this

picture yeah it's going to make it in

here sorry

again that's where the marginal cost and

the demand thing uh demand lines

intersect all right so it's here right

it's here it's where it's where uh

marginal it's where the marginal cost

and the demand curve

intersect and what's the

intuition let's work it out suppose firm

two has produced all of this quantity up

to

here all right so firm two is already

producing all of this quantity so

already just from firm 2's production

the price has been driven driven all the

way down to

costs all right so if I I'm firm one if

I produce any more quantity what's going

to happen to the

price if I prod any quantity at all in

addition to what's out there already

what's going to happen to the price it's

going to be pushed

below

costs right so I'll be producing this

stuff this water at Cost C and only

getting P which is now going to be less

than C in return so I'll be losing money

on any any product that I produce

everyone see that I say again so firm

two if firm two produces all the way up

to

here then any product produced by firm

one is going to push prices even lower

in particular they'll be lower here than

costs and so you'll make losses on that

product all right what's the name going

back to economics 115 what's the name

for this quantity the quantity Where

demand and marginal cost equal

that's the that's the competitive

quantity all right so this is the

perfect competition quantity in a

perfectly competitive market that's

exactly where prices are going to end up

this is not a perfectly competitive

market but if it were that's where pric

is going to end

up all right so we have the Monopoly

quantity here and we have the

competitive quantity here and in between

what does this best response curve look

like

somebody it's a straight line it's a

straight line thank you all right so in

between this is just a straight

line all right so here is the best

response of firm one to each possible

choice of firm

two all right everyone's looking really

like this was hard but it can't be that

hard you've seen most of you seen this

before all right so don't if you haven't

seen it before don't panic all I did was

I did a little bit of calculus a little

bit of algebra and then I drew the thing

I just drew the thing right okay so this

is firm One's best response as a

function of Q2 what is firm 2's best

response as a function of q1 just just

just we get there just to remind you the

way we read this is you give me a

quantity of firm two I find firm One's

best response by going across to the

pink line and dropping down down so if

firm two produces this amount then firm

One's best response is this amount if

firm two produces this amount then firm

One's best response is this am all right

now conversely how do we find firm 2's

best response as a function of q1 what

does that look

like yeah tell you don't get the two

guys in front in front of

you um it's symetrical it's uh

symmetrical across the 4 4° all right

it's reflected across the 45° line so

let's reflect these two

points this will be here and this will

be be a bit more

careful missed by mile I need to try

again I missed by a mile which is which

is a problem in drawing my lines but

never

mind

all right so this will be the Monopoly

quantity for firm two and this is the

competitive quantity for FM

one all right so if I drawn that better

it would look as if this was half of

this distance right but I didn't draw it

well try and draw it better in your

notes all

right right so this

is Aus C over B B and this is a minus c/

2B so this distance and this distance

are meant to be the same and the way

I've drawn it this don't look the same

at

all all

right and the way we read this green

graph is you give me a choice of firm

one q1 I go up to the green line and go

across and this tells me the best

response for firm

two all right

so at this point how many of you have

seen this picture

before good many of you have seen it

before all right so uh I think we can

really call call somebody for this one

all right so uh uh somebody tell me and

pick some time to call when you call

call somebody anybody can you close

by there we go okay okay so here's the

tough question what's the Nash

equilibrium on this picture where the

green and the pink line intersect all

right where the green and the pink line

line intersect all right all right so

this must be this must be the Nash

equilibrium right that wasn't hard right

okay okay all right now okay so why

let's go back again why is this the N

equilibrium because at this point as in

the partnership game we saw a similar

thing as in the partnership game where

the best response is intersect is where

player one is playing a best response to

player two and player two is playing a

best to player one so this is the Nash

equilibrium in this Corno game all right

let's do a little bit of algebra just to

make sure we can find this

thing all right so what we're going to

do

is set these two things equal to each

other put in Stars

here so at at at the na equilibrium

quantities I shouldn't really do this

but never mind at the Nash equilibrium

quantities I'm going to have q1 star is

equal to a - c/ 2b Q2 star Etc and Q2

star is equal Aus C 2B q1 star/

two all right and I'm going to solve

this out by setting q1 equal to Q2 since

I know the game is

symetric I know we're on the 45 degree

line all right so all I have to do here

to do this algebraically is to solve out

these equations so let's try and do

that so putting q1 in here right I if I

substitute q1 in here what will it give

me it'll give me q1 Star is equal to a -

c/

2b minus q1 star over

two let's multiply both sides by two

I'll get 2 q1 Star is equal to Aus C

over

B minus q1 star take that to the other

side I have three q1 star is equal to a

minus

C over B and finally divide by three q1

star is equal to a minus

C over

3B all right so that Crossing Point

actually occurs at a minus C over 3B and

again I'm going to look my ta check I

didn't make a I didn't screw that up I

think I think I didn't screw that up all

right so this is sometimes called the

Coro

quantity now this game this game of

imperfect competition between two firms

competing in quantities was thought up

and and and uh studied by a French

Economist called Coro uh almost 100

years before Nash right so 100 years

really before the invention of Game

Theory right someone figured out this

answer for this

game

right okay so what do we know so far I

I'll leave the algebra there transfer it

up

here Q star q1 star = Q2 star equal a

minus

C over

3B all

right so so far we've we've been working

pretty hard and we haven't I guess

learned a lot we just kind of solve the

thing out now we get to draw some

lessons out of this thing all right so

everybody who's feeling a little bit

Shell Shocked from having been doing

algebra and calculus and drawing

pictures and feeling like they've been

cheated into taking a class that looks

far too much like economics calm down

we're going to actually talk about now

okay all

right so one thing to remark about this

this game one we've learned immediately

is that this game is different in a

significant way it's different from the

partnership game I mean obviously it's

different because it's about something

different but I mean in terms of the

game theory it's different it's

different from the partnership game it's

different from the investment game what

is it that's different about this game

from the part what's the obvious thing I

looking at this picture that makes it

different from the partnership

game in the partnership game the best

response lines sloped up is that right

the more I did of my strategy the more

the other player did as a best response

in the investment game the more likely I

was to invest the more likely you the

more you wanted to invest right but in

this game we have the exact opposite the

more player one

produces the the more player one

produces the less player two wants to

produce and the more player two produces

the less player one wants to produce

produce all right so this game is a game

not of strategic

complement but of strategic

substitutes and I want to be careful

here it's not it's not that these goods

are substitutes I mean clearly that's

also the case right if both these firms

are producing bottled water and it's

identical right then then the goods

themselves are substitute

that's not the point I'm making here

strategic substitutes is a strat is a

statement about the nature of the game

so strategic substitutes is telling me

that my strategy is a strategic

substitute of your strategy if the more

I do of my strategy the less you want to

do of yours and conversely the more you

want to do of your strategy the less I

want to do of

mine all

right

okay let's let's come back to the

question we started with which is to do

with profits and so on in society we

know that if these players play these

games if we at least if we believe in

natural equilibrium uh then they're

going to produce these quantities

they're going to produce

here all right but let's ask a different

question each of these firms is trying

to maximize profit all right but how

about the total profit in the

industry each of these firms we know

where the fact they're playing a best

response is Max imizing their profits

taking us given what the other firm's

doing but how about total industry

profits are they being maximized

here who thinks total industry profits

are being maximized who thinks total

industry profits are not being maximized

all right good good so shout out okay so

total profits are not being maximized

where on the picture let me get another

bit of cold calling in here where on the

picture maximize these total industry

profits

where on the picture maximizes total

industry

profits we had we had it before some

else yeah no no idea no

idea people are taking the fifth here

yeah the guy in there yeah is it at the

Monopoly at Monopoly exactly so clearly

if we produce the Monopoly quantity by

definition the Monopoly quantity

maximizes total industry profit so for

example if firm two shut down and firm

one

produced its Monopoly profits it

Monopoly quantity that would maximize

firm profits and conversely if firm one

shut down and firm two produced its

Monopoly quantity that would maximize

industry

profits right where else on this picture

maximize IND industry profit so so this

point maximize industry industry profits

and this point maximizes industry

profits where else maximizes industry

profits

anybody well how about all the points in

between here firm one produced nothing

and firm two produce the Monopoly

quantity and here firm two produce

nothing and firm one produce the

Monopoly quantity but you could also

just split the Monopoly quantity for

example half half at this point here all

right so if the firms wanted to make

more money one thing they could do is

they could sign an

agreement saying

why don't each of us produce not our

Coro quantity but produce half Monopoly

profit sorry half monop half the

Monopoly

quantity and that would produce this

much

each all right so firm one would be

producing half its Monopoly property

quantity and firm two would be producing

half its Monopoly

quantity all right so what's wrong with

that

agreement so here they what are they

they're uh two water companies so they

Poland Spring and uh and uh um Coca-Cola

I guess these days have signed this

agreement saying each one's going to

produce half Monopoly quantity and

what's going to go wrong with that

agreement yeah the guy in

red first of all it's illegal it's

illegal it's illegal right so so even

without there being an attorney general

somebody might be awake in the justice

department and and notice that that's

actually illegal all right it's illegal

to sign an agreement uh to to sign a

contract saying you're going to restrict

quantities it's illegal so they can't

sign that as a contract but hey they

could just agree to do it right so

Coca-Cola and Poland Spring could just

agree to produce half the quantity of

water each all right what's going to go

what what might go wrong in that

agreement let's let's just suppose that

that game is just being played once for

now all right and suppose that this is

Coca-Cola's quantity and this is Poland

Spring's quantity and suppose you're the

you're the manager of Poland Spring

right so water is coming out of the

ground in in Maine for you all right and

you you know that this Coca-Cola guy is

going to produce uh this quantity here

right so this is qm over two qm over two

all right so let's let's pick on our two

managers let's have two people who've

spoken before so the guy in the red

shirt's name is Steve Steve and I guess

in red he should be the co Coca-Cola guy

right all right and let's pick on our

our Jimmy Stewart from last week whose

name is Patrick right uh get can I get a

mic on on Patrick all right right so

Steve and Patrick up are respectively

the managers of Coca-Cola and uh and

Poland Spring and here he is sorry right

and Patrick Patrick believes that the

Coca-Cola manager is going to may we

just make this Coke and Pepsi it's two

identical they're near enough identical

right so these are Coke and Pepsi

confuse me less all right so Patrick our

manager of Pepsi believes that Coca-Cola

is going to produce this quantity and

he's agreed to produce this quantity

what is Patrick actually going to do

Patrick what are you going to do um so I

should cheat and and pick the quantity

on my best response line that's much

farther out so I'll produce more than

that quantity all right so Patrick being

Jimmy Stewart or not actually when he's

in the in the when he's playing the

manager of Pepsi is going to produce

more of this uh undrinkable liquid and

produce this quantity here all right so

what is this this is Patrick's best

response to the other guy producing the

Monopoly

quantity all right so Patrick's response

in Pepsi is to overproduce relative to

the Monopoly quantity actually

overproduce even relative to the Coro

quantity right and produce all the way

out

here okay so Patrick's producing

here but what about Mr Coca-Cola guy all

right now Mr Coca-Cola guy knows Patrick

pretty well you know they've been in the

same industry for a while and Mr Mr

cocaa guy's name is St Steven Stephen

presumably knows that Patrick is

actually going to

produce this quantity sorry this

quantity so Stephen what what should you

produce um increase my own production to

to match my my best response line good

so stepen anticipating that Patrick is

going to think that Steven's a sucker

but Patrick's going to cheat on him and

produce too much Steven's going to

produce a best response to that and he's

going to

produce this quantity here so what is

this this is player two's best

response to player One's best response

to player two

producing half Monopoly

output all right but Patrick knows

Steven pretty well he knows those guys

Who Running Coca-Cola are pretty smart

guys so Patrick knows that the guy who

runs Coca-Cola is going to anticipate

that the guy who runs Pepsi is going to

cheat on the guy who runs Coca-Cola and

hence the guy who runs Coca-Cola is

going to play a best response to

Patrick's cheating quantity so what

quantity is uh is Patrick G to produce

so I reduce my production then and I'll

I'll end up pretty close to the cor

equilibrium all right so you're going to

go to here so this point here is I don't

even write this this is the best

response of player one to the best

response of player two to the best

response of player one to player two

producing half Monopoly output under

lots of brackets here all right and can

anyone see where this process is going

see where this process is going right

it's going back it's going back to the

Nash equilibrium right each of them is

trying to play a best response what they

anticipate the other person's going to

do and in this game if they keep on

doing that it's going to drag them back

to N equ I want to be careful here this

won't be true of all games but in this

game playing best response to each other

figuring out that in fact the other guy

is going to going to cheat on me or the

other guys going anticipate that I'm

going to cheat on him or the other guys

going to anticipate that I'm going to

anticipate that he's going to cheat on

me etc etc will eventually drag us back

to the Coro quantity all right so it's

going to be pretty difficult for us to

sustain this joint Monopoly output this

collusive agreement we can't let the

courts enforce it where Coca-Cola and

Pepsi so we really don't want to bring

the mafia in although maybe we do I

don't know all right so basically we're

stuck with this verbal agreement and we

both have an incentive to cheat and

produce more whatever it is sugar water

right okay now in practice this is not

the only problem facing two firms who

are produc who are trying to produce the

Monopoly output right in practice when

firms try and have these uh agreements

which are not contracts to try and

produce joint Monopoly output what else

goes

wrong what else goes wrong so you could

imagine some firms trying to have an

agreement we know they can't sign a

legal agreement but let's let's take

that off the table so for example in

around 1900 in America or a little bit

earlier let's say let's say 1880 in

America it really isn't clear that it

was illegal to write agreements to say

We'll restrict quantities right so firms

did used to write that kind of

agreements but still something went

wrong what went wrong so we're getting a

little bit further away from the game

and more back into the real world now

what kind of things went wrong I I want

to get some news nearer M can can I get

the the woman here the the Yale

sh without a contract there's no means

of enforcement all right so so it might

be that again there's a problem with

enforcement although actually in 1880 in

America it's not clear that they

couldn't have enforced that contract all

right so one problem is UN

enforceability right but that might far

enough ago that might a problem what

else the guy next

next one

yeah if they're not at the competitive

outcut they could undercut price and

take the entire Market let's be careful

we'll come back to that next time but

right now we're competing in quantities

all right so so we're not we're not

fixing prices here we're not even naming

prices we're just producing Coca-Cola

and Pepsi and for want of a better word

spewing it out into the market right

what what else goes wrong

here well one thing that goes wrong is

that suppose COC Pepsi were able to

either to write a contract or whatever

to sustain joint Monopoly outputs at

this high price this price which is well

sorry producing positive profits in the

industry it is going to turn out

therefore that prices are going to be

above costs these firms are making a lot

of profits what do you think's going to

happen in this industry this industry

that's got a lot of profit around it

what's going to happen in this industry

yeah the guy in the blue shirt here I

like this guy

here another competitor could join the

market all right another firm is going

to come in and produce a similar product

so Dr Pepper or something is that same

is that the thir one someone else is

going to come in and produce Cola in

this industry and that's exactly that's

exactly what happened in the US at the

turn of the last century companies did

produce agreements to restrict for

example the production of paper and the

production of rubber and the production

of Steel and the production of iron and

the production of of uh of railway lines

actually in quite complicated things and

sure enough what you see very quickly

emerging is new entrance entering the

market to say hey these firms are

restricting their their quantities we

can get in there and make money all

right so new and a a competitive Fringe

of firms is going to enter and drive

prices down now I talked about that in

1900 America whereas that happened in

the 20th century uh more dramatically so

we know the early example in sort of

sort of golden age America but where

else do we see a competitive Fringe

enter when there was a collusive

agreement to keep prices keep quantities

down

aines so Airlines okay good so Airlines

is an example we've seen that where else

oil is probably the classic example

Airlines is a good example okay so in

oil we all know that in the in the late

60s early 70s OPEC was formed precisely

to restrict quantity among the major oil

producing countries and very quickly we

see companies take other countries take

advantage of this to start producing oil

profitably so who who are the

competitive Fringe against OPEC

anybody well the Brits were for a start

the Brits uh you know England started or

Scotland I should say Britain anyway

started producing oil in the North Sea

lots of countries uh uh in in Latin

America started looking and finding oil

and of course Russian oil started being

very very profitable all right so what

there so there are various reasons why

it's hard to maintain this collusive

agreement one is there's an incentive to

cheat and another is is that other firms

are going to enter but before we leave

this let's just pose the question we

started with suppose in fact we do end

up back at the Corno quantity so here we

are back at the Corno

quantity right how does this quantity

compare and how therefore the prices

compare and how the profits compare with

Monopoly prices on the one hand and

competitive prices on the other right so

we know we know the quantity that's

going to be produced we figured out here

it is we know that the quantity is going

to be

produced each

firm each firm is going to produce a

minus C over 3B and there are two such

firms so the total quantity produced in

the

market the total

quantity will be 2 A minus C over

3B all right how does that compare to

the Monopoly quantity well the Monopoly

quantity was a minus

C over

2B and the competitive

quantity was a minus C over B remind you

of those they're on our

picture here's Aus C over 2B and here's

a minus C over

B so this is the total quantity produced

in this equilibrium in this corash

equilibrium how does it compare to the

Monopoly

quantity and to the competitive

quantity

anybody little bit of just staring at

the board which which is the is bigger

so I claim that this total quantity

being produced is less than the

competitive

quantity but more in total than the

Monopoly prop

quantity all right I was about to get

the wrong way around right right right

the total quantity being produced is

less than would be produced under

perfect competition but more than would

produced under Monopoly and consequently

prices are going to go the other way

right and consequently prices are going

to be highest under Monopoly lowest

under competition and somewhere in

between in this cor no

situation all right so from the point of

view of of point of view of The

Producers this Corno equilibrium is

worse than properly but better than

perfect competition and from the point

of view of the rest of us the consumers

this Corno quantity is worse than

perfect competition but better than

Monopoly right the only rer on that

being if they're producing Coca-Cola and

you have any kind of concern for your

teeth maybe you're maybe you shouldn't

produce so much

anyway all right so so far what we've

done is we've looked at this

classic I admit not the most exciting

game in the world but classic

application of Game Theory to imperfect

competition next time I'm actually going

to take this a bit further but I'm going

to leave this game behind and look at

other ways in which we can study

imperfect competition using Game Theory