Professor Ben Polak: So
last time we introduced a new
idea and the new idea was that
of best response.
And what was the idea?
The idea was to think of a
strategy that is the best you
can do, given your belief about
what the other people are doing:
what your opponents are doing,
what other players are doing.
And you could think of this
─ you could think of this
belief as the belief that
rationalizes that choice.
So if you have a boss you might
want to ─
and he or she is going to ask
you why you chose the action you
did.
If you took a best response to
some believe,
you can say I took this action
because I believed other people
are going to do this.
And since that was the best you
could do under that belief,
you'll hopefully keep your job.
I promised that today we would
look at the most important game
in the world.
And, as announced last time,
the most important game is the
penalty kick game.
So this is a game that occurs
in soccer and just to give an
idea of how important it is for
those people who are unfortunate
enough not to be soccer fans
here,
the last World Cup was decided
on penalty kicks.
In England's case,
England goes out of every
single World Cup and every
single European competition
because it loses on penalty
kicks,
usually to Germany,
it has to be said.
And more immediately,
this weekend,
as all of you are thinking the
most event in the world was
whatever was happening in
Congress to do with Iraq,
actually, the most important
event in the world was taking
place in England where my
favorite team,
Portsmouth, were playing Kaj,
the head TA's favorite team,
Liverpool.
And, about a third of a way
through that game there was a
penalty, and …
I'll let you know what happened
later.
So keep at the back of your
mind that the real world example
that matters here is Portsmouth
versus Liverpool this weekend.
(Kaj, the head TA is
Scandinavian so I've got no idea
why he's supporting Liverpool
anyway,
but I think maybe he spells it
like this or something like
this).
So what we're going to do is
we're going to look at some
numbers that are approximately
the probabilities of scoring
when you kick the penalty kick
in different directions.
But just make sure everyone
─ do I need to explain
what's going on here?
Is everyone familiar with this
situation?
There's one guy who's going to
run up and kick the ball.
The goal keeper is standing at
the goal.
And their aim is to kick it
into the goal.
That's probably enough.
You've all seen this right?
If you haven't seen this,
go see it.
I mean come on!
So things you should do in
life: read Shakespeare and see a
soccer game.
So the rough numbers for this
are as follows ─
and actually later on in the
class I'll give you some more
accurate numbers,
but these will do for now.
There are three ways,
the goal─
the attacker could kick the
ball.
He could kick the ball to the
left, the middle,
or the right.
And I shouldn't just say he
here of course,
I mean this is he or she but if
I get that wrong going on,
please forgive me for it.
The goalie can dive to the left
or the right.
In principle the goalie could
stay in the middle.
We'll come back and talk about
that later.
So this is the guy who is
shooting, he's called the
shooter and this is the goalie.
These are roughly ─
well, let me put up the payoffs
for this game and then I'll
explain them.
So you'll notice that I'm just
going to put in numbers here and
then the negative of the number
and the numbers are roughly like
this: (4,-4).
So the numbers are (4,
-4), (9, -9),
(6, -6), (6,
-6), (9, -9) and (4,
-4).
And the idea here is that the
number 4 represents 40% chance
of scoring if you shoot the ball
to the left of the goal and the
goal keeper dives to the left.
So the payoff here is something
like u_1(left) if the
goal keeper dives to the left is
equal to 4,
by which I mean there's a 40%
chance of scoring.
So the number for the--The
payoff for the shooter is his
probability of scoring and the
payoff for the goal keeper is
just the negative of that.
Let's keep things simple.
As I said before,
for now we'll ignore the
possibility that the goal keeper
could stay put.
So how should we start
analyzing this important game?
Well we start with the ideas we
learned already several weeks
ago now, or more than a week
ago, which is the idea of
dominant strategies.
Does either player here have a
dominated strategy?
Does either player have a
dominated strategy?
No, it's kind of clear that
they don't.
Let's just look at the shooter,
for example.
So you might think that maybe
middle dominates left,
but notice that middle has a
higher payoff against left than
shooting to the left.
It has as lower payoff if the
goalie dives to the right.
So, not surprisingly in this
game, it turns out,
that if the goalie dived to the
left you're best off shooting to
the right,
second best off shooting to the
middle, and worst off shooting
to the left.
That's if the goalie dives to
the left.
And if the goalie dives to the
right, you're best off shooting
to the left, second best off
shooting to the middle,
and worst off by shooting to
the right;
and that's kind of common sense.
Okay, so if we had stopped the
class after the first week where
all we learned to do was to
delete dominated strategies,
we'd be stuck.
We'd have nothing to say about
this game and as I said before,
this is the most important
game, so that would be bad news
for Game Theory.
But luckily,
we can do a little bit better
than that.
Before I do that,
let's just take a poll of the
class.
How many of you,
if you were playing for,
I guess it's going to be
America, which is a sad thing to
start with, never mind.
You guys are playing for
America and you're taking this
penalty kick and it's the last
kick in the World Cup,
how many of you,
show of hands,
how many of you would shoot to
the left?
How many of you would shoot to
the middle?
How many of you would shoot to
the right?
We've got kind of an even split
there, pretty much an even
split.
We're going to assume these are
the correct numbers and we're
going to see if that even split
is really a good idea or not.
So how should we go about
thinking about this?
What I suggest we do is we do
what we did last time and we
start to draw a picture to
figure out what my expected
payoff is,
depending on what I believe the
goalie is going to do.
So this is the same kind of
picture we drew last time.
So on the horizontal axis is my
belief, and my belief is
essentially the probability that
the goalie dives to the right.
Now as I did last time,
let me put in two axes to make
the picture a little easier to
draw.
So this is 0 and this is 1.
And you probably have lines in
your notes but I don't,
so let me just help myself a
bit.
So this is 2,4,
6,8, 10, so this is going to be
2,4, 6,8, and 10 and over here
2,4, 6,8, and 10,2,
4,6, 8, and 10.
This would be the basis of my
picture.
So it starts with a possibility
of shooting to the left.
Let's do this in red.
So I shoot to the left and the
goalie dives to the left,
my payoff is what?
It's 4.
If I shoot to the left and
there's no probability of the
goalie diving to the right,
which means that they dive to
the left, then my payoff is 4,
meaning I score 40% of the
time.
If I shoot to the left and the
goalie dived to the right,
then I score 90% of the time,
so my payoff is .9.
By the way why is it 90% of the
time and not 100% of the time?
I could miss;
okay, I could miss.
That happens rather often it
turns out, well 10% of the time.
So we know this is going to be
a straight line in between,
so let's put this line in.
So what's this?
It's the expected payoff to
Player I of shooting to the left
as it depends on the probability
that the goal keeper dives to
the right.
And conversely,
we can put in …
well let's do them in order.
So middle: so if I shoot to the
middle and the goal keeper dives
to the left,
then my payoff is .6,
is 6, or I score .6 of the
time, and if I shoot to the
middle and the goalie dives to
the right I still score 60% of
the time,
so once again it's a straight
line in between.
So this line represents the
expected payoff of shooting to
the middle as a function of the
probability that the goal keeper
dives to the right.
Finally ─
let's do it in green ─
let's look at the payoffs,
expected payoffs,
if I shoot to the right.
So if I shoot to the right and
the goalie dives to the left,
then I score with probability
.9, or my payoff is 9.
Conversely, if I shoot to the
right and he or she dives to the
right, then I score 40% of the
time, so here's my payoff .4.
And here's my green line
representing my expected payoff
as the shooter,
from shooting to the right,
as a function of the
probability that the goalie
dives to the right.
Did everyone understand how I
constructed this picture?
Easier picture than the one we
constructed last time.
So what does everyone notice
from this picture?
What's the first thing that
jumps out at you from this
picture?
Assuming these numbers are
true, what jumps out at you from
this picture?
Can we get some mikes up here?
So Ale, can we get this guy?
Stand up first, the guy in red.
What's your name?
Don't hold the mike;
just shout.
Student: There's no
point at which the 6,
at which it shooting in the
middle gets a higher payoff.
Professor Ben Polak:
Exactly, exactly.
So the thing that I hope jumps
out at you from this picture is
(no great guesses about figuring
out this is a ½),
so if the probability that the
goalie's going to jump to the
right is less than a ½,
then the best you can do is
represented by this green line,
which is shoot to the right.
So the goalie is going to shoot
to the right with the
probability less than a
½,
sorry he's going to dive to the
right with the probability less
than a ½,
you should shoot to the right.
Conversely, if you think the
goalie's going to shoot to the
right with probability more than
a ½,
then the best you can do is
represented by the pink line,
and that's shooting to the
left,
or if you think the goalie's
going to dive to the right with
the probability more than a
½,
the best you can do,
your best response is to shoot
to the left.
And there is no belief you
could possibly hold given these
numbers in this game that could
ever rationalize shooting the
ball to the middle.
Is that right?
So no: to say it another way,
middle is not a best response
to any belief I can hold about
the goal keeper,
to any belief.
So there's a lesson here,
and it's pretty much (just to
make the lesson resonate again):
imagine there you are in the
World Cup,
you're playing for England,
you have to justify your
actions not only to your
teammates and your manager,
and your boss,
but to about 60 million rather
angry fans.
What's the lesson here?
I'm hoping it was going to be
obvious, what's the lesson here?
The lesson is,
do not shoot to the middle.
Let me qualify that lesson
slightly, unless you're German.
Germans can do whatever they
like.
Now, it turns out that about a
third of the game between my
team Portsmouth and Kaj's team
Liverpool, this weekend there
was a penalty.
Portsmouth had a penalty and
the guy who was going to take
the penalty came up to kick the
penalty and he kicked it to the
middle and it was saved.
So just confirming these
actions, not only did that spoil
my weekend but it also spoiled
my opportunity to make fun of
Kaj all week,
so it was really a big deal.
So this weekend a penalty was
missed exactly by somebody
ignoring this rule.
There's a more general lesson
here, and the more general
lesson is, of course:
do not choose a strategy that
is never a best response to
anything you could believe.
The more general lesson,
do not choose a strategy that
is never a best response to any
belief.
Notice here,
just to underline something
which came up at the end last
time,
that doesn't just mean beliefs
of the form, the goalie's going
to dive left or the goalie's
going to dive right.
It means all probabilities in
between.
So we're allowing you to,
for example,
to hold the belief that it's
equally likely that the goalie
dives left or dives right.
But if there's no belief that
could possibly justify it,
don't do it.
And underlining what arises in
this game, notice that in this
game we're able to eliminate one
of the strategies,
in this case the strategy of
shooting to the middle,
even though nothing was
dominated.
So when we looked at domination
and deleted dominated
strategies, we got nowhere here.
Here, at least,
we got somewhere,
we got rid of the idea of
shooting to the middle.
Now if you can just persuade
the English and Portsmouth
soccer players of this lesson,
I'd be very happy.
So before we leave it,
I've been making a point in
this class of coming back to
reality from time to time,
so this is a very simple model
of the soccer game in reality.
Let's just try,
any of you on the Yale Soccer
Team?
No?
Have any of you played soccer
for your college?
One or two.
Have you ever played soccer?
How many of you have ever
played soccer?
Okay, good I was getting
worried there for a second.
So, one thing we said last time
was when we put up a model and
try and draw lessons from it,
we should just take a step back
and say, what's missing here?
So let's try and get some kind
of ─ I'll come off the
stage to make it easier for
Jude.
What's missing here?
What's missing in this model of
this piece of soccer,
this game within a game?
What's missing here?
Why is this not necessarily a
hundred percent accurate model?
I'll need some mikes up here.
Can you?
You have to really shout
because you're miles from the
mike there.
Student: You might be
better kicking to the left or to
the right depending on whether
you're right handed or left
handed.
Professor Ben Polak:
Good, so one thing that's
clearly missing here is I'm
ignoring that in fact right
footed players find it easier to
shoot to their left,
which is actually the goalie's
right.
So right footed players find it
easier to shoot to the left as
facing, to shoot across the
goal.
Does everyone confirm that's
true?
Yeah, anyone ever tried to this?
It's a little easier to hit the
ball hard to the opposite side
from the side which is your foot
and that's the same principle in
baseball.
It's a little bit easier to
pull the ball hard then it is to
hit the ball to the opposite
field.
Yes?
Student: Players don't
make their decision before,
and then stick with it
necessarily.
Professor Ben Polak: All
right, so players are making
decisions as they're running up.
I think that's okay here, right?
We can think of this as the
decision happening at the
instant at which you kicked.
So you're right that you could
have made your decision back in
the locker room,
or you could have made the
decision at half time,
but ultimately what matters
─ let's hope that goes
away.
We sure that it's not off my
mike?
Just in case I'll move my mike
a bit lower.
So I'm going to shout louder
because my mike is now lower.
It doesn't really matter
exactly when the decision is
made.
At the end of the day,
the goalie doesn't know the
decision of the shooter and the
shooter doesn't know the
decision of the goalie.
So it's as if that decision is
made instantaneously as the
shooter is running up.
What else?
Yeah, can we get this.
Tae can you get this guy here?
Stand up.
Shout out.
Student: The goalie
might stay in the middle.
Professor Ben Polak: The
goalie might stay in the middle.
That's a good point,
of course, I've abstracted from
that here, and in fact,
we'll come back,
I think, I'll try and put that
onto a problem set,
but I think you're right,
it is an issue here.
Anything else?
Well let me put up some real
numbers and we'll see about how
much the correspondence to what
we've got here.
So I gave you some numbers I
made up actually a long time
ago, but since I've been using
this game in class,
somebody went out and checked.
And it turns out that ignoring
middle for a second,
ignoring middle ─
so these are real numbers,
and these numbers come from a
paper in the AER by Chiappori
and some co-authors and for
everyone at Yale,
I'll make that paper available
to you through JSTORE or through
the Yale Library,
so you can go look at it if
you'd like.
What they worked out was the
following table.
And again, we need to be a
little bit careful here.
So I'm going to put the left
and right in inverted commas
because actually what they did
was they corrected for people's
natural direction and not
natural direction.
So the idea here is shooting to
the left if you're right footed
is the natural direction,
so left here means the natural
direction.
Of course, if you're left
footed it goes the other way,
but they've corrected for that.
It turns out that the
probabilities of scoring here
are as follows,
63.6,94.4,89.3,
and 43.7.
So things are not--I haven't
given you the numbers for the
middle but--So you can see that
whoever it was who said,
you're slightly better off,
you score with slightly higher
probabilities when you kick to
your natural side is exactly
right.
The thing is still not
dominated and we could still
have done exactly the same
analysis,
and actually you can see I'm
not very far off in the numbers
I made up, but things are not
perfectly symmetric.
I forget who it was who said
that, but that does turn out to
be true.
Certainly the goalie staying
put is an issue,
as I said we'll deal with that
in the problem set,
but there's another issue here.
Let me just raise one more
issue.
One more issue is,
you have another decision when
you run up to hit this,
hit the penalty other than just
left and right.
Someone whose played the game,
what's the other decision you
really face?
Can I get the woman here?
What's the other decision you
face?
Student: You could kick
up to the top corner.
Professor Ben Polak:
Okay, you can kick up and down,
that's true.
Okay, that's true actually,
that's true.
But I meant something else,
that's right,
but I meant something else.
What else is there?
Try this guy here.
Student: Spin.
Professor Ben Polak:
Well that's getting subtle here.
It's a much more basic thing,
what's a more basic thing?
What's a more basic thing here?
Take it, yeah right in front of
you.
Student: Speed.
Professor Ben Polak:
Speed, right.
So another decision you face is
do you just try and kick this
ball as hard as you can or do
you try and place it?
That's probably as important a
decision as placing it,
as deciding which direction to
hit, and it turns out to matter.
So, for example,
if you're the kind of person,
(which is I have to say all I
ever was),
if you're the kind of person
who can kick the ball fairly
hard but not very accurately,
then it actually might change
these numbers.
If you can kick the ball very
hard, but not very accurately,
then if you try and shoot to
the left or right,
you're slightly more likely to
miss.
On the other hand,
as you shoot to the middle,
since you're kicking the ball
hard, you're slightly more
likely to score.
Now, if this all seems like
arcane and irrelevant detail,
let's just see why this matters
in the picture,
and then we'll leave soccer,
at least for today.
So if you're the kind of person
who can kick the ball hard but
not accurately,
then it's going to lower the
probabilities of scoring as you
kick towards the right because
you might miss,
and it's going to lower the
probability of scoring as you
hit towards the left because
you're likely to miss,
and it might actually raise the
probability of your scoring as
you hit towards the middle,
because you hit the ball so
hard it's really pretty hard for
the goal keeper to stop it.
Here it goes in the middle,
and if you look carefully
there, I didn't really make it
clear enough,
you can see,
suddenly a strategy that looked
crazy shooting to the middle,
that suddenly started to seem
okay.
It turns out,
if you look at those dotted
lines, there's an area in the
middle,
the area between here and here,
this little area here,
you actually might be just fine
shooting to the middle.
So in reality,
we need to take into account a
little bit more,
and in particular,
we need to take into account
the abilities of players to hit
the ball accurately and/or hard.
And if those people,
if you're interested in that
─ and I realize at this
point I probably lost the
interest of most Americans in
the room,
but for the non-Americans in
the room, the people who are
interested in the real world
─ as I said before,
I'll put that article online
and that goes through all the
gory detail of this.
I should just say that the data
I just gave you is real data but
it's actually mixed ability
data.
This data comes half from the
Italian league,
which is pretty good and half
from the French league,
which sucks.
So who knows how much we should
trust it.
Okay, so that was our example
for the day and our first brush
with reality for the day.
Let's clean the board and do
some work.
Do a bit more formal stuff here.
So here we have an example but
I want to go back to the
generality and to a bit of
formalism.
By the way, I should tell you
that the game ended nil-nil or
0-0.
It's a moral victory for me I
think.
So I want to be formal about
these things I've been
mentioning informally.
And in particular,
I want to be formal about the
definition of best response.
I'm going to put down two
different definitions of best
response, one of which
corresponds to best response to
somebody else playing a
particular strategy like left
and right,
and the other is just going to
correspond to the more general
idea of a best response to a
belief.
It'll allow us to use our
notation and just be a little
bit more nerdy.
So Player i's strategy,
Ŝ_i (there's
going to be a hat to single it
out) is a best response (always
abbreviated BR),
to the strategy S_- i
of the other players if ─
and here's our real excuse to
use our notation ─
if the payoff from Player i
from choosing Ŝ_i
against S_- i is
weakly bigger than her payoff
from choosing some other
strategy,
S_i',
against S_- i and
this better hold for all
S_i' available to
Player i.
So in previous definitions,
we've seen the qualifier,
for all, be on the other
player's strategy.
Here, the qualifier for all is
on my strategy.
So strategy Ŝ_i
is a best response to the
strategy S_- i of the
other players if my payoff from
choosing Ŝ_i
against S_- i,
is weakly bigger than that from
choosing S_i' against
S_- i,
and this better hold for all
possible other strategies i
could choose.
There's another way of writing
that, that's kind of useful,
or equivalently,
Ŝ_i solves the
following.
It maximizes my payoff against
S_- i.
So you're all used to,
I'm hoping everyone is used to
seeing the term max.
As I solve the maximization
problem, how do I maximize my
payoff given that other people
are choosing S_- i?
Again, for the math phobics in
the room, don't panic,
this is just writing down in
words what we've already seen a
couple of times already today,
well today and last time.
Let's generalize this
definition a little bit,
since we want it to allow for
more general beliefs.
So just rewriting,
Player i's strategy,
same thing, Ŝ_i
is a best response.
But now let's be careful,
best response to the belief P
about the other player's
choices,
if ─ and it is going to
look remarkably similar except
now I'm going to have
expectation ─
if the expected payoff to
Player i from choosing
Ŝ_i,
given that she holds this
belief P, is bigger than her
expected payoff from choosing
any other strategy,
given she holds this belief P;
and this better hold for all
S_i' that she could
choose.
So very similar idea,
but the only thing is,
I'm slightly abusing notation
here by saying that my payoff
depends on my strategy and a
belief,
but what I really mean is my
expected payoff.
This is the expectation given
this belief.
Once again, we can write it the
other way, or Ŝ_i
solves max when I choose
S_i,
to maximize my expected payoff
this time from choosing
S_i against
S_-i.
What do I mean by expected
payoff?
Just in our example,
so just to make clear what that
expectation means,
so for example,
the expected payoff to Player i
in the game above from choosing
left given she holds the belief
P is equal to the probability
that the goal keeper dived to
the left,
times Player i's payoff from
choosing left against left,
plus the probability that the
goal keeper dived to the right,
times Player i's expected
payoff from choosing left
against right.
Okay, so expectation with
respect to P just means exactly
what you expect it to mean.
So this is a little bit of
math, a little formality,
but is everyone okay with that?
I haven't done anything here.
All I've done is write down
slightly boringly and nerdily,
exactly what we already saw in
a couple of occasions.
Student: [inaudible]
Professor Ben Polak:
Thank you.
So that right now is going to
seem a little bit like a sudden
blast of notation,
so let's just remind ourselves
what we really care about is the
idea, it's not the notation,
and let's spend the next half
hour applying these ideas to an
application.
So this application is not as
important as soccer,
but it's a bit more Economicsy,
so I can justify it under the
Economics title of the class.
So clearing off my soccer game.
So imagine--What we're going to
look at is a game involving a
partnership.
So Partnership Game.
And I believe this game is
covered in some detail in the
Watson textbook,
or something very close to it
is, if you're having trouble.
The idea is this.
There are two individuals who
are going to supply an input to
a joint project.
So that could be a firm,
it could be a law firm,
for example,
and they're going to share
equally in the profits .
So one example would be a firm
that they both earn,
sorry, they both own,
and another example would be
two of you working as a study
group on my homework assignment.
So they're going to share
equally in the profits of this
firm, or this joint project,
but you're going to supply
efforts individually.
So let's just be a bit more
formal.
So the players are going to be
the two agents and they own this
firm let's call it.
They own this firm jointly and
they split the profits,
so they share 50% of the
profits each.
So it's a profit- sharing
partnership.
Each agent is going to choose
her effort level to put into
this firm.
So, it could be that you're
deciding, as a lawyer,
how many hours you're going to
spend on the job.
So for most of you these
decisions will be a question of
whether you spend 20 hours a day
at the firm or 21 hours a day at
the firm,
something like that.
For most of you on your
homework assignments,
I'm hoping it's a little less
than that, but not much less
than that.
So the strategy choices,
we're not going to do it in
hours, let's just normalize and
regard these choices as living
in 0 to 4,
and you can choose any number
of hours between 0 and 4.
Just to mention as we go past
it, a novelty here.
Every game we've seen in the
class so far has had a discrete
number of strategies.
Even the game,
when you chose numbers,
you chose numbers 1,2,
3,4, 5, up to a 100,
there were 100 strategies.
Here there's a continuum of
strategies.
You could choose any real
number in the interval [0,4].
So you have a continuum of
possible choices.
That's not going to bother us
but let's point out it's there.
So there's a continuum of
strategies.
In principle,
you could bill your clients for
fractions of a second or
fractions of a minute.
Let's wonder what the firm
profit is given by.
So this partnership,
this law firm,
its profit is given by the
following expression:
4 times the effort of Player I,
plus the effort of Player II,
plus a parameter I'll call B,
times the product of their
efforts.
This is their profit.
And I won't tell you what B is
for now, but let's just--I mean
I won't tell you exactly what it
is--but I'll explain it.
We'll assume that B lives
between 0 and a 1/4 and it's
known, I just want to be able to
vary it later.
So what's the idea here?
The idea is Player I directly
contributes profits to the firm
by working, as does Player II.
But they also contribute
through this interaction term.
How do we think of that
interaction term?
How do we think of that term B
S_1 S_2?
When you're working on your
homework assignments,
if your product,
the thing you hand in was just
S_1 + S_2,
then you might think what?
You might think there's no
point working in a study group
at all.
If the product is just the sum
or multiple of a sum of the
inputs, there wouldn't be much
of a point working in a team at
all.
It's the fact you're getting
this extra benefit from working
with someone else that makes it
worth while working as a team to
start with.
Is that right?
So we can think of this term
has to do with complementarity,
or synergy, a very unpopular
word these days but still:
synergy.
So we're going to assume that
when you work together there are
some synergies.
Some of you are good at some
parts of the homework,
some of them are good at other
parts of the homework.
And so in this law firm,
one of these guys is an expert
on intellectual property and the
other one on fraud or something.
So I've got the agents.
I've got the strategy set.
I know something about the
profits of the firm.
I need to tell you about their
payoffs.
So the payoffs:
the payoff for Player I is
going to depend,
of course, on her choice and on
the choice of her partner,
and it's going to equal a
½--because they're
splitting the profits--so a
½ of the profits.
So a ½
of 4 times S_1 plus
S_2 plus B S_1
S_2.
She gets half of those profits
but it also costs her
S_1 squared.
So S_1 squared is her
effort costs,
it's her input costs.
This is the effort cost.
Similarly, Player
II--everything's symmetric
here--Player II's payoff is the
same thing.
This term is the same except
we're going to subtract off
Player II's efforts squared:
S_2 squared.
So you get the profits of the
firm minus the disutility of
having missed all that sleep.
There's a guy in about the
fifth row there who's missed too
much sleep, so somebody just
nudge him.
That's it, good.
We won't put him on camera just
nudge him.
That's it, good.
There you go.
Next time we'll use the camera
for that.
So now we have everything we
want to analyze this firm and to
analyze how things are going to
work,
either when you're working on
your homework assignments or in
the law partnership.
Again, just to make this
relevant to you,
I mean this is very stylized of
course,
but a huge number of businesses
out there are partnerships and
do have this kind of profit
sharing rule and do have
synergies.
So this is a relevant issue in
a lot of businesses.
Now we're going to analyze
this--no secret here--we're
going to analyze this using the
idea of best response.
That's not a surprise to any of
you since that's where we
started the day.
So, in particular,
I want to figure out what is
Player I's best response to each
possible choice of Player II?
What is Player I's best
response for each possible
choice S_2 of Player
II?
How should I go about doing
that?
How should I do that?
So here what we did before was
we drew these graphs with
probabilities,
with beliefs of Player I and
the problem here is,
previously we had a nice simple
graph to draw because there were
just two strategies for Player
II.
Player II was a goalie,
he could dive to the left or
the right.
Problem is here,
that Player II has a continuum
of strategies and trying to draw
all possible probabilities over
an infinite number of objects on
the board is more than my
drawing can do.
Too hard.
So we need some other technique.
How are we going to find out
Player I's best response?
Somebody?
Wave your hands in the air,
way back in the corner.
Can I, can we,
let's get the mike.
Stand up but wait for the mike
to come to you.
How are we going to do it?
How are we going to figure out
what Player I's best response
is?
Shout loudly.
Student: [inaudible]
Professor Ben Polak:
Good, okay.
That's certainly the first step.
We've got that,
actually we've got that.
So here's Player I's payoff as
a function of what Player II
chooses and what Player I
chooses, so we have that
already.
We have Player I's payoff as a
function of the two efforts and
now I want to find out what is
Player I's best efforts given a
particular choice of
S_2?
Yeah.
Student: Take a
derivative of S_1.
Professor Ben Polak:
Good, take a derivative and--
Student: Set it equal to
zero.
Professor Ben Polak:
Okay, good.
So we're going to use calculus.
We're going to use calculus of
one variable.
We're just changing one
variable, S_1.
How many of you--we won't let
the camera see you--how many of
you have not--I'm not going to
show hands at all.
If you have not seen the
calculus that I'm about to use
on the board,
or more likely,
if you've forgotten it since
high school, don't panic.
There is a chapter in the back
of the book, I think it's
chapter 25, that goes over this,
it refreshes your memory of
such calculus.
And if you haven't,
if you've never seen it before,
if you haven't taken,
for example,
the equivalent of Math 112,
come and see us.
We'll probably try and line up
a quick calculus lesson,
a special section for those
people.
So if what I'm going to do now
is scary, come and see us and
we'll deal with it.
All right, what I'm going to
do, we want to take a derivative
of this thing.
What we're going to do is,
we're asking the question,
what is the maximum,
choosing S_1,
of this profit.
Can I multiply the ½
by the 4 just to save myself
some time?
So the profit is 2
S_1 plus S_2
plus B S_1
S_2 minus
S_1 squared.
We're asking the question,
taking S_2 as given,
what S_1 maximizes
this expression and as the
gentleman at the back said,
I'm going to differentiate and
then I'm going to set the thing
equal to 0.
So I'm almost bound to get this
wrong on the board.
So can you all watch me like a
hawk a second?
So if we differentiate this
object, I'm going to find a
first order condition in a
second.
All right, so we differentiate.
I'm going to have 2 still,
and then this S_1 is
going to become a 1,
and this S_1 here is
going to become a plus B
S_2,
everyone happy with that?
This S_1 squared is
going to become a minus
2S_1.
That was just differentiating.
Everyone happy with the way I
differentiated?
Is this coming back from high
school?
The cogs are spinning now?
To make this a first order
condition, I'm going to say "at
the best response," put a hat
over the 1.
At the best response this is
equal to 0.
Yeah Tae, can you get the guy
again.
Student: Wouldn't that
be 2, oh sorry,
never mind.
Professor Ben Polak:
Okay, you're right to shout out
because I'm very--I mean doing
it on the board I'm very likely
to make mistakes,
but okay.
So I differentiated this
object, this is my first
derivative and I set it equal to
0.
Now in a second I'm going to
work with that,
but I want to make sure I'm
going to find a maximum and not
a minimum,
so how do I make sure I'm
finding a maximum and not a
minimum?
I take a look at the second
derivative, which is the second
order condition.
So I'm going to differentiate
this object again with respect
to S_1,.
Pretend the hat isn't there a
second.
And none of this has an
S_1 in it,
so that all goes away.
And I'm going to get minus 2,
which came from here:
minus 2 and that is in fact
negative, which is what I wanted
to know.
To find a maximum I want the
second derivative to be
negative.
So here it is,
I've got my first order
condition.
It tells me that the best
response to S_2 is the
Ŝ_1 that solves
this equation,
that solves this first order
condition.
We can just rewrite that,
if I divide through by 2 and
rearrange, it's going to tell me
that Ŝ_1,
or if you like,
Ŝ_1 is equal to
1 plus B S_2.
So this thing is equal to
Player I's best response given
S_2.
Now I could go through again
and do exactly the same thing
for Player II,
but I'm not going to do that
because everything's symmetric.
So everyone happy with that?
So I could at the same--I could
do the same kind of analysis but
we know I'll get the same
answer.
So similarly,
I would find that
Ŝ_2 equals 1 plus
B S_1 and this is the
best response of Player II,
as it depends on Player I's
choice of effort S_1.
Okay, now I found out what
Player I's best response is to
Player II, and what Player II's
best response is to Player I for
each possible choice of Player
II up here,
and for each possible choice of
Player I down there.
Now, let's see if we can get a
bit further.
And to get a bit further,
let's draw a picture.
What I'm going to do is I want
to draw the two functions we
just found and see what they
look like.
This is all in your notes
already, so I can get rid of it.
What I can do with is some more
chalk.
Excuse me.
So what I'm going to do is,
let's draw a picture that has
S_1 on the horizontal
axis and S_2 on the
vertical axis.
And there are different choices
here 1,2, 3, and 4 for Player I,
and here's the 45º
line.
If I'm careful I should get
this right 1,2,
3, and 4 are the possible
choices for Player I.
Now before I draw it I better
decide what B is going to be.
Okay, so I'm going to draw for
the case--I'm going to draw the
best response of Player I and
I'm going to draw the best
response for Player II in a
minute,
for the case B equals 1/4.
So we said B was somewhere
between 0 and 1/4,
let's draw the case for B
equals 1/4.
So the expression I want to
draw first of all is the best
response of Player I as a
function of S_2 and we
agreed that that was given by 1
plus 1/4 now,
1 plus 1/4 S_2.
So for each possible choice of
S_2,
I'm going to draw Player I's
best response and we'll do it in
red.
So if Player II chooses 0,
what is Player I's best
response?
Somebody shout out.
Student: 1.
Professor Ben Polak: 1,
okay.
So 1 plus 1/4 of 0 is 1,
so if Player II chooses 0,
Player I's best response is to
choose 1.
What if Player II chooses 4?
If Player II chooses 4,
what would be Player I's best
response?
So it'll be 1 plus 1/4 times
4,1/4 times 4 is 1,
so 1 plus 1 is 2,
so Player II's best response in
that case will be 2.
So if Player I chooses 4,
Player II should choose,
I'm sorry, Player II chooses 4,
Player I should choose 2,
and this is a straight line in
between.
So the line I've just drawn is
the best response for Player I
as it depends on Player II's
choice.
Everyone happy with how I drew
that?
I'm assuming you're taking it
on faith that it is a straight
line in between,
but it is.
The way we read this graph,
is you give me an
S_2,
I read across to the pink line
and drop down,
and that tells me the best
response for Player I.
Now we can do the same for
Player II, we can draw Player
II's best response as it depends
on the choices of Player I,
but rather than go through any
math, I already know what that
line's going to look like.
What does that line look like?
Somebody raise your hand.
Somebody?
What will Player II's best
response look like as a function
of Player I's choice in the same
picture?
Someone we haven't had before,
we've had all these guys
before, someone else.
Yeah, there's a guy in the
middle, can we get in to him?
Yeah, maybe it's easier from
that side.
Shout out loud so the mike can
hear you.
Student: It should be a
reflection across the 45°
line.
Professor Ben Polak:
Right, exactly.
So if I drew the equivalent
line for Player II,
which is Player II's best
response for each choice of
Player I,
we're simply flipping the
identities of the players,
which means we'll be reflecting
everything in that 45º
line.
So it'll go from 1 here to 2
here, and it'll look like this.
So this the best response for
Player II for every possible
choice of Player I,
and just to make sure we
understand it,
what this blue line tells me is
you give me an S_1,
an effort level of Player I,
I read up to the blue line and
go across and that tells me
Player II's best response.
Okay, now we're making some
progress.
What do we notice?
Remember we said that one of
the lessons of today's class,
the second lesson.
The first lesson was don't
shoot towards the middle of the
goal and the second more general
lesson was what?
It was don't ever play a
strategy that is not a best
response to everything.
I admit I'm cheating a little
bit here because I'm ignoring
beliefs, but trust me that's
okay in this game.
So are there strategies here
that are never a best response
to anything?
Put another way,
what strategies of Player I's
are ever a best response?
Anybody?
Well, let's have a look.
If Player II chooses 0 then
Player I's best response is 1,
and that's as low as he ever
goes.
So these strategies down here
less than 1 are never a best
response for Player I.
If Player II chooses 4,
then the synergy leads Player I
to raise his best response all
the way up to 2,
but these strategies up here
above 2 are never a best
response for Player I.
Is that right?
So the strategies below 1 and
above 2 are never a best
response for Player I.
Similarly, for Player II,
the lowest Player I could ever
do, is choose 0,
in which case Player II would
want to choose 1,
so the strategies below 1 are
never a best response for Player
II.
And the strategies above 2 were
never a best response for Player
II.
So let's actually--you might
want to be a little bit gentle
in your own notebook--but on my
board let's get rid of all these
strategies that are never a best
response.
So all of these strategies for
Player I are gone,
and all of these strategies for
Player I are gone.
You might want to not scribble
quite so much on your own
notebook, but still.
And all these strategies for
Player II are gone,
and all these strategies for
Player II are gone,
and what's left?
A lot of scribble is left.
What's left?
So I claim if you look
carefully there's a little box
in here that's still alive.
I've deleted all the strategies
that were best--that are never
best responses for Player I and
all the strategies that are
never best responses for Player
II,
and what I've got left is that
little box.
But I can't see that little
box, so what I'm going to do is
I'm going to redraw that little
box.
So let's redraw it.
So it goes from 1 to 2 this
time.
I'm just going to blow up that
box.
So this now is 1,1 and up here
is 2,2 and let's put in numbers
of quarters, so this will be,
what will it be?
It'll be 5/4,6/4 and 7/4,
and over here it'll be 5/4,
and 6/4, and 7/4.
And let's just draw how those,
that pink and blue line look in
that box.
This is just a picture of that
little box, so it's going to
turn out it goes from--the pink
line goes from here to here and
the blue line goes from here to
here.
We can work it out at home and
check it carefully,
but this isn't that incorrect.
So what I've done is I've
redrawn the picture we just had
and blown it up.
And have any of you seen that
picture before?
Anyone here seen that picture
before?
That's the picture we just had,
except I've changed the numbers
a bit.
Once I deleted all the
strategies that were never a
best response and just focused
on that little box of strategies
that survived,
the picture looked exactly the
same as it did before,
albeit it blown up and the
numbers changed.
So what have we done so far?
We said players should never
play a strategy that's never a
best response to anything,
so we threw those away.
Now what's left?
What should we do now?
So some of the strategies that
we didn't throw away were best
responses to things,
but the things they were best
responses to have now been
thrown away.
Is that right?
This should be something
familiar from when we were
deleting dominated strategies.
The strategies I'm about to
throw away now,
they're not--it isn't that
they're not best responses,
they are best responses to
something.
But the things they were best
responses to,
we know are not going to be
played, because they themselves
were not best responses to
anything.
So what strategies do I have in
mind?
What strategies am I about to
throw away?
Well, for example,
for Player I we know now that
Player II is never going to
choose any strategy below 1,
and so the lowest Player II
will ever choose is 1,
and it turns out that the
lowest Player II would ever do
in response to anything is 1 and
above,
never leads Player I to choose
a strategy less than 5/4.
The highest Player II ever
chooses is 2,
and the highest response that
Player I ever makes to any
strategy 2 or less is 6/4,
so all these things bigger than
6/4 can go.
Let's be careful here.
These strategies I'm about to
delete, it isn't that they're
never best responses,
they were best responses to
things, but the things they were
best responses to,
are things that are never going
to be played,
so they're irrelevant.
So we're throwing away all of
the strategies less than 5/4 for
Player I and bigger than 6/4 for
Player I,
(which is 1½
for Player I) and similarly for
Player II.
And if I did this--and again,
don't scribble too much in your
notes--but if we just make it
clear what's going on here,
I'm actually going to delete
these strategies since they're
never going to be played--I end
up with a little box again.
So everyone see what I did?
I started with a game.
I found out what Player I's
best response was for every
possible choice of Player II,
and I found out what Player
II's best response was for every
possible strategy of Player I.
I threw away all strategies
that were never a best response,
then I looked at the strategies
that were left.
I said those strategies that
were a best response to things
that have now been thrown away,
but not best response
otherwise, I can throw those
away too.
And when I threw those away,
I was left once again with a
little box, and I could do it
again, and again,
and again.
If I go on doing this exercise
again, and again,
and again what am I going to
end up with?
Shout it out,
what am I going to end up with?
The intersection, right?
If I keep on constructing these
boxes within boxes,
so the next box would be a
little box in here.
I'm not going to draw it,
but it's something like this.
But if you keep on doing boxes
within boxes,
I'm going to converge in on
that intersection.
So if we know people are not
going to play a best
response--that's never a best
response,
and we know if we people are
not going to play something
which is never a best response,
and we know people are not
going to play which is not a
best response,
which is not a best etc.,
etc., etc.
We're going to converge in,
in this game,
to just one strategy for each
player, which is where they
intersect.
So what we're going to converge
in on to, is the S_1*,
let's call it in this case,
is equal to 1 plus B
S_2* and that S2* is
equal to 1 plus B
S_1*.
Actually, we can do it a little
better than that,
since we know the game is
symmetric,
we know that S_1* is
actually equal to
S_2*.
So taking advantage of the fact
that we know S_1 is
equal to S_2 (because
we're lying on the 45º
line),
I can simplify things by making
S_1* equal to
S_2*.
So now I've got--actually,
that looks like three
equations, it's really just two
equations, because one of them
implies the other.
And I can solve them,
and if I solve them out I'm
going to get something like (let
me just be careful) I'm going to
get something like:
1 minus B S_1* is
equal to 1,
or S_1* equals
S_2* is equal to 1
over (1--B).
And again, anytime I'm doing
algebra on the board,
someone should check me at
home, so just have a quick look
at that.
Is that right?
I think that's right.
My algebra, which is often
wrong, suggests that the
solution is S_1*
equals S_2* equals 1
over (1--B).
But what I'm doing,
it's just math,
there's nothing interesting
going on.
I'm just trying to solve out
for the equation of this point.
So what did we learn here?
We learned that in this game
deleting strategies that are
never best response,
and then deleting strategies
that are never best response to
anything that is a best response
and so on and so forth,
yielded a single strategy for
each player.
Just one strategy for each
player and that strategy was
given by this equation.
So if we were management
consultants working for Mckinsey
or something,
and we were brought in to
advise you on your homework
assignments,
or this law partnership on
their work practices,
we would come down with a
prediction that this is how much
work you're going to get.
Question, is this amount of
work a good amount of work or a
bad amount of work?
Here you are,
you're working for Mckinsey,
you've been hired by Joe Smith
and Ann Blogs to figure out
their strategy,
working on a problem in a team
on working on my homework
assignments.
You figured out how much work
they're going to contribute.
Is this a good amount of work?
Are they contributing too much,
too little?
Because the answer is,
depending on,
compared to what?
So let me rephrase,
are these people,
are these pair of partners in
the firm,
or two students working on
their homework assignments,
are they working more or less
than an efficient level?
Let's have a poll,
who thinks more?
Turn the camera out into the
audience, let's have a look.
Who thinks more?
Who thinks they're working just
right?
Who thinks less?
A lot of abstentions here.
I think they're working too
little here compared to what's
efficient.
I'll get you to solve it out on
a homework assignment,
so you can actually prove that.
You can prove that,
in fact, if you were writing a
contract, if there was a social
planner you'd work more.
But let's try and get to grips
why.
Why is it that when we see
these law partners,
or medical partners,
or whatever it happens to be,
or students together on a
homework assignment,
why is it we tend to get
inefficiently little effort when
we start figuring out the
strategy and working through the
game?
I'm conceding the answer.
I'm telling you they're going
to work too little.
Why do they end up not working
hard enough?
Any takers?
Can we get a mike in here, yeah.
Student: Because if they
work any harder than that,
then the other person is just
going to slack off instead.
Professor Ben Polak: All
right, so there's something
about that, there's something.
On the other hand,
this isn't really,
I mean the intuition you're
giving me is kind of a
Prisoner's Dilemma intuition.
Saying I'm going to let the
other guy work and I'll shirk.
But there's something,
I think there's something in
that, but there's a little bit
more going on here,
what more is going on?
I think that's a good first
step.
There is something of that.
Yeah.
Student: If there are
two people working together,
there's about half as much work
for each person to do.
Professor Ben Polak:
That's true, but that would
suggest it doesn't matter if
they slack off.
What's going on here,
so go back to your Economics
115 or 150, if you took either
of those courses.
What's the problem here?
What's underlying the problem?
Let's get this guy in the pink
down here.
Student: They only
capture fifty percent of their
marginal benefit.
Professor Ben Polak:
That's the point.
Good, well your name is?
Student: Patrick.
Professor Ben Polak: So
Patrick is giving,
I think, it's the correct
answer here.
The problem here isn't really
about the amount of work.
It isn't even,
by the way, about the synergy.
You might think it's because of
this synergy that they don't
take into account correctly.
That isn't the problem here.
It turns out even without the
synergy this problem would be
there.
The problem is what Patrick
said.
The problem is that at the
margin, I, a worker in this
firm, be it a law partnership or
a homework solving group,
I put in, I bear the cost,
at the margin,
I bear the full cost at the
margin for any extra unit of
effort I put in,
but I only reap half the
benefits.
At the margin,
I'm reaping,
I'm bearing the cost for the
extra unit of effort I
contribute,
but I'm only reaping half of
the induced profits of the firm,
because of profit sharing.
That leads all of us to put in
too little effort.
What's the general term that
captures all such situations in
Economics?
It's an "externality."
It's an externality.
There's an externality here.
When I'm figuring out how much
effort to contribute to this
firm I don't take into account
that other half of profits that
goes to you.
So this isn't to do with the
synergy.
It isn't to do with something
complicated.
It's something you knew back in
115.
If you have profit sharing in a
firm or profit sharing in
homework assignment,
or any joint projects,
you have to worry about too
little effort being contributed
because there's an externality.
My efforts benefits you not
just me.
While we've got this on the
board, let's just think a little
bit more.
What would happen if we changed
the degree of the synergy?
What would happen if we lowered
B?
So B is the degree to which the
synergy across these workers,
if we lowered B,
what would happen to our
picture?
Let me redraw a picture
unscribbled.
We had a picture that looked
something like this.
This was S_1 and this
was S_2.
If we lowered the degree of
synergy, what would happen to
the effort level that we'd find
by this method?
What would happen?
What would happen to the
picture?
Anybody, again this is a 115
kind of exercise,
we're going to be moving lines
around.
Yeah, Henry isn't it?
So let's get a mike in to Henry.
Student: The lines will
get shallower and eventually
become horizontal and vertical
respectively.
Professor Ben Polak: All
right, good.
So the pink line is actually
going to get steeper,
but I know what you mean.
So the pink line is going to
move towards the vertical,
and the blue line is going to
move towards the horizontal,
and notice that the amount of
effort that we generate,
goes down dramatically,
goes down in this direction.
So if we lower the synergy
here, not only do I contribute
less effort, but you know I
contribute less effort,
and therefore you contribute
less effort and so on.
So we get this scissors effect
of looking at it this way.
We could draw other lessons
from this, but let me try and
move on a little bit.
We decided in this game to
solve it by looking at best
responses, deleting things that
were never a best response,
looking again,
deleting things that were never
a best response,
and so on and so forth,
and luckily,
in this particular game,
things converged and they
converged to the points where
the pink and the blue line
crossed.
What do we call that point?
What do we call the point where
the pink and the blue line
cross?
That's an important idea for
this class.
That's going to turn out to be
what's called a Nash
Equilibrium.
So we know what it's called.
How many of you have heard the
term Nash Equilibrium before?
How many of you saw the movie
about Nash?
We'll come back and talk about
that a bit next time.
So this is a Nash Equilibrium,
but okay we know what it is in
jargon, and we know,
we kind of knew that was going
to be an important point,
because most of you have taken
Economics courses before and you
know that whenever lines cross
in Economics it's important,
right?
But what does it mean here?
Why is it--what's going on at
that line?
What does it tell us that the
pink and the blue line cross?
What makes that point special?
What does it mean to say the
pink and the blue line cross?
Can I get the guy way back,
like three rows behind you in
purple?
Shout out again.
Student: It means that
neither player has an incentive
to deviate from that point.
Professor Ben Polak: All
right, well that's correct.
So let's try and read that
through.
So I don't know your name,
your name is?
Student: Allen.
Professor Ben Polak: So
Allen is saying if Player I is
choosing this strategy and
Player II is choosing her
corresponding strategy here,
neither player has an incentive
to deviate.
Another way of saying it is:
neither player wants to move
away.
So if Player I chooses
S_1*,
Player II will want to choose
S_2* since that's her
best response.
If Player II is playing
S_2*,
Player I will want to play
S_1* since that's his
best response.
Neither has any incentive to
move away.
So more succinctly,
Player I and Player II,
at this point where the lines
cross, Player I and Player II
are playing a best response to
each other.
The players are playing a best
response to each other.
So clearly in this game,
it's where the lines cross.
Let's go back to the game we
played with the numbers.
Everyone had to choose a number
and the winner was going to the
person who was closest to 2/3 of
the average.
(By the way,
the winner's never picked up
their winnings for that,
so they still can.) So in that
game, what's the Nash
Equilibrium in that game?
Everyone choosing 1.
How do we know that's the Nash
Equilibrium of that game?
How do we know that everyone
choosing 1 is the Nash
Equilibrium in the game where
you all chose numbers?
Well let's just use the
definition.
If everybody chose a 1,
the average in the class would
be 1,2/3 of that would be 2/3
and you can't go down below 1,
so everyone's best response
would be to choose 1.
Say it again:
if everyone chose 1,
then everyone's best response
would be to choose 1,
so that would be a Nash
Equilibrium.
Did people play Nash
Equilibrium when we played that
game?
No, they didn't.
Not initially at any rate,
but notice as we played the
game repeatedly,
what happens?
As we played the game
repeatedly, we noticed that play
seemed to converge down towards
1.
Is that right?
In this game,
when we analyzed the game
repeatedly, it seemed like our
analysis converged towards the
equilibrium.
Now that's not always going to
happen but it's kind of a nice
feature about Nash Equilibrium.
Sometimes play tends to
converge there.
Nash Equilibrium's going to be
a huge idea from now to the
mid-term exam and we're going to
pick it up and see more examples
on Wednesday.
 
