Professor Ben Polak:
So today I want to look at
two kinds of games and then
we'll change topic a bit.
The games I want to look at are
about ultimatums and bargaining.
And we'll start with ultimatums
and we'll move smoothly through,
and we'll see why that's an
easy transition in a while.
So the game we're going to play
involves two players,
1 and 2 and the game is this.
Player 1 is going to make a
take it or leave it offer to
Player 2 and this offer is going
to concern a pie and let's make
the pie worth $1 for now.
We'll probably play this for
real in a bit so we'll play it
for $1.
So it's a split of a pie,
we can think of the split as
offering S to Player 1 and
(1--S) to Player 2.
Player 2 has two choices:
2 can accept this offer,
and if 2 accepts the offer then
they get exactly the offer.
Player 1 gets S and Player 2
gets (1--S).
Alternatively,
Player 2 can reject,
and if Player 2 rejects the
offer then both players get 0,
both players gets nothing.
So a very simple game,
there's a dollar on the
table--I'll take it out in a
minute--so there's a dollar on
the table,
and our two players are going
to bargain for this,
but it isn't much of a bargain.
Player 1's basically going to
announce what the division's
going to be.
2 can either accept that
division or no one gets
anything.
Everyone understand the game?
So I thought we'd start off by
playing this game for real a
couple of times,
so why don't I come down and do
that.
So why don't we play with some
people in this row.
I've been playing with that row
all the time,
so Ale's going to help me find
somebody.
You found somebody okay,
so the person behind you?
Your name is?
You have to shout out,
this doesn't make a noise.
Equia, thank you.
So you'll be Player 1,
and Player 2 will be this
gentleman here whose name is?
Student: Noah.
Professor Ben Polak:
Noah, okay.
So Equia, do you know each
other?
Okay, so Equia can make any
offer she wants.
We'll play this for real money,
so there's a real dollar at
stake.
You can make any offer you
want--it can be in fractions of
pennies if you like--of the
amount of the dollar that you're
going to offer to Noah,
and the amount you're going to
keep yourself.
There you go,
shout it out so everyone can
hear.
Stand up, this is your moment
in the lights all right.
Will you stand up as well?
There we go.
Student: I'm going to
offer him a penny.
Professor Ben Polak:
You're going to offer him a
penny, and Noah are you going to
take a penny?
Student: No.
Professor Ben Polak:
Noah's not going to take a
penny.
So I didn't lose any money on
that: so no one made any money.
I didn't lose any money.
that seems pretty good.
Let's try a different couple.
Let's move around the room a
little bit.
So Ale why don't you take the
guy behind here who name is?
Shout out.
Student: Gary.
Professor Ben Polak:
Gary, all right so Gary why
don't you stand up,
and we'll let Gary play with
the gentleman in here,
what's your name in here?
Student: Anish.
Professor Ben Polak:
Anish, all right stand up,
do you know each other?
So why don't we let Anish be
Player 1 this time.
You understand the rule of the
game?
So make your offer.
Student: $.30.
I'm offering him $.30.
Professor Ben Polak:
You're offering him $.30?
He's saying no as well.
Okay, so let's raise the stakes
a bit.
Let's make it $10,
since we're not getting much
acceptance here.
Let me try a third couple,
working my way back,
so how about you,
what's your name?
Student: Courtney.
Professor Ben Polak:
So why don't you stand up
as well.
We'll let Courtney be Player 1.
Actually why don't you sit down
so I can do it from here,
and you are?
Student: Danny.
Professor Ben Polak:
Danny, all right so
Courtney's going to be Player 1
and Danny's going to be Player
2.
These weren't our dating couple
from earlier right?
We're safe on that?
Okay good, so Courtney what are
you going to offer?
Student: $5.
Professor Ben Polak:
$5, which is half of the
$10.
Student: Accept.
Professor Ben Polak:
All right accept,
all right.
So it turned out in this game
that a lot of people were
rejecting offers.
Let's have a look at it on the
board a second.
Let's think about it a second
and we'll come back,
we're not done with this
couple.
We're not going to need new
couples, we're going to come
back to you.
So in this game--it's pretty
simple to analyze this game by
backward induction.
By backward induction,
we're going to start with the
receiver of the offer and the
receiver of the offer is
choosing between the offer made
to them which in our notation is
(1-S).
And in our three examples that
was a penny, then it was $.30
and then it was half of it,
so $.50 just to be consistent.
We actually saw two of those
offers rejected,
but according to backward
induction,
assuming that people are trying
to maximize their dollar payoffs
here, what should we see the
receiver do?
They should accept even the
somewhat insulting offer of a
penny that was made:
even that somewhat insulting
offer should be accepted by
Noah.
So Noah didn't accept the offer
of a penny and let's come back
over there.
So Noah didn't accept the offer
a penny, and I forget who it was
but our second player didn't
accept an offer of $.30,
but we've just argued that they
should have been accepted.
In fact, when Equia made the
offer of one penny to Noah,
I think she assumed that Noah
would accept it,
is this right?
Why did you think Noah would
accept it?
Student: Because I felt
he would be better off with a
penny than nothing.
Professor Ben Polak:
He'd be better off with a
penny than nothing,
right.
Then we saw an offer.
The offer came up a bit.
Who was my second offerer?
You were my second offerer,
you offered a bit more.
Why did you offer more?
Student: I felt $.30 was
a pretty fair share.
It's a lot better than nothing.
Professor Ben Polak:
Thirty cents is better than
nothing, although not
necessarily fair,
but I guess it's better than
nothing.
Where was my rejector of the
second offer?
Who rejected the second offer?
Why did you reject $.30?
Student: Just a pride
thing.
Professor Ben Polak:
It's a pride thing.
Pretty soon we converged onto
$.50, which notice is no where
near backward induction.
So the third offer,
which is Courtney,
why did you offer half of it?
Student: Because half is
better for me than nothing.
Professor Ben Polak:
Half is better for you than
nothing, and you figured he's
going to reject otherwise,
and in fact he didn't reject.
Why didn't you reject?
Student: Because $5 is
better than nothing.
Professor Ben Polak:
$5 is better than nothing.
But the $5 is better than
nothing argument would have
argued against making any
rejection in this game.
Is that right?
So here's a game where backward
induction is giving a very clear
prediction.
The clear prediction is,
first of all,
the second player will accept
whatever's given to them;
and second, given that,
the first player should offer
them essentially nothing,
should offer them just a penny.
So backward induction predicts
that the offer will be
essentially, let's say $.99 and
$.01, or even virtually $1 and
nothing.
But in fact we don't get that,
we get a lot of rejection of
these low offers,
and often we get offers made
much, much higher in the
vicinity of half.
Now why?
Why are we seeing a failure of
backward induction in this game?
I think this is not necessarily
you guys.
It's a very reliable result in
experimental data.
So why do we see so many people
in this ultimatum game both
offer more, and,
more importantly,
reject less than small amounts.
So let's talk about it,
so one person said it's a pride
thing.
Let's try the other aisle here,
so what's the smallest offer
you would have accepted.
What's your name first of all?
Student: Jeff. A cent.
Professor Ben Polak:
So there are some backward
induction players in the room.
Who would have rejected a cent?
Who would have rejected $.10?
We should be going down at
least, who would have rejected
$.30?
Few people rejected $.30,
not many actually.
How many people would have
rejected $.50?
One person even would have
rejected $.50,
but essentially no one.
So what's happening here?
Why do people think people are
rejecting what is essentially
money from my pocket,
there's nothing going on here,
I'm just giving you money.
Why are they rejecting being
given money?
Student: Overall the
stakes are really low,
so if you have any value on
sort of like pride,
what people said,
you know it's not worth a penny
or $.10.
Professor Ben Polak:
All right,
so it may be pride going on
here, so certainly one thing is
about pride.
It turns out that people do
this even in quite high stakes
games, but you're right,
certainly that trade off is
going to start to bite.
What else is going on?
So I agree, pride is part of
what's going on.
What else is going on here?
Let's try and get some
conversation going.
Somebody in here,
if I can get the mike in,
shout out your name and really
shout.
Student: Peter.
Professor Ben Polak:
Go on.
Student: Change is
cumbersome.
Professor Ben Polak:
Change is cumbersome,
you didn't want the change,
okay fair enough,
but if the stakes go up that
would get rid of that.
Student: Maybe people
are tying their own outcomes to
the other player's outcomes as
well.
Professor Ben Polak:
Right, so maybe people have
different payoffs here.
Maybe people are comparing
their payoff to the other
person's payoff that certainly
seems like a plausible thing to
be going on here.
You might feel less happy about
getting $.20 knowing the other
person's getting $.80 for
example.
Student: You want to try
to teach them a lesson to get
them to offer more.
Professor Ben Polak:
Right, you might be trying
to set a sort of moral standard
here.
So there's some notion of
indignation or even teaching
people that they really should
offer people more.
What else could be going on
here?
Student: If people know
I'm not going to accept less
than $.50 then they should give
me $.50 by backward induction.
Professor Ben Polak:
Right, so part of what's
going on here--actually this
game was a one shot game,
we just played it once.
We could have played it--in
fact they often do play this
this way in the lab--you could
have played it without anybody
knowing who the other player
was.
But particularly in this
setting where everyone can see
everyone else--even in the lab
where people actually can't see
people,
but they might imagine that the
game is really repeated--you
could imagine people trying to
establish a reputation.
Is that right?
So there's lots of these
reasons, these sort of moral
indignation reasons or teaching
a lesson reasons,
pride reasons.
There's also this basic reason
that people might be thinking,
I should play this game as I
would a similar situation in
life where I might want to be
establishing reputation.
So there's a certain amount of
confusion going on in the game,
and there's also a certain
amount of a lot of things make
sense.
Now notice that once we've
established that people are
going to reject small offers in
this game,
once we've established people
are going to reject small
offers, it makes perfectly good
sense to offer a lot more than
nothing.
So it's not that surprising
that once we've established the
idea that people are going to
reject small offers,
we're going to see people
making offers that are
reasonably large,
although not usually larger
than $.50.
Why is $.50 so focal here?
Why is $.50 so focal?
It's not a trick question.
I'm just asking you why do
people think $.50 is so focal?
I think it's typical that
people end up offering around
$.50, why?
It sounds fair, it seems fair.
There's some notion of fairness.
It's not clear by the way,
what ethical principal is
involved here.
It's not clear that if you're
walking along the streets and
you happen to find a dollar at
your feet that you should pick
it up and anyone else you'd
happen to see at that moment you
should give $.50 too.
That's essentially what the
situation is,
and you just chanced upon this
dollar that I just gave you.
It isn't clear that there's any
particularly great moral claim
to give it to someone else,
but I think people read this as
a situation about splitting a
cake in an environment of
distributional justice.
They view it as a larger
picture.
Is that right?
So it turns out there's a large
literature on this.
There's a large experimental
literature on the ultimatum
game.
And there's an even larger
literature on an even simpler
game in which I give people a
dollar,
I say you can give whatever
share you want to the other
person, and they don't even get
a chance to accept or reject.
That's called a dictator game.
In the dictator game,
literally, you're just simply
given a dollar and you can give
whatever share you want to the
other person.
It turns out that even in the
dictator game,
people give quite a lot of
money and that suggests that
there really is some notion of
fairness or some notion of
distributional justice going on
in people's heads here,
rightly or wrongly.
So one thing this should tell
us is, even in extremely simple
games, we should be a little bit
careful about reading backward
induction into what's going to
happen in the real world.
Part of this is because,
as we mentioned the very first
day of the class,
people care about other things
than just the obvious payoffs,
and part of it is about more
complicated things like
reputation and so on.
All right, having said that,
let's nevertheless for the
purpose of today,
act as if we are going to do
backward induction,
and let's embed this into a
slightly more complicated game.
So the more complicated game is
as follows.
So we're going to have a two
period bargaining game.
In this two period,
bargaining game,
the beginning of the game is
exactly the same.
So there's a dollar on the
table and Player 1 makes an
offer to 2.
And once again we can call this
offer S and 1 - S,
just the same as before.
And, just as before,
Player 2 can accept the offer,
and if they accept the offer
then this is indeed what they
get.
But now if 2 rejects,
which is 2's other
alternative--if Player 2 rejects
the offer then we flip the
roles.
We play the game again but we
flip the roles.
So we go to stage two,
everything we've said up here
is stage one.
And down here we go to stage
two, and in stage two,
Player 2 gets to make an offer
to 1.
And once again we can call
this--we better be careful.
Let's put ones here just to
indicate we're in the first
round and twos to indicate we're
in the second.
So they make an offer is S2 and
(1-S2) where S2 is that that
goes to Player 1 and (1-S2) is
that that goes to Player 2.
So 2 gets to make an offer to
Player 1 and now 1 can accept or
reject.
If she accepts then she gets
her share from here.
The offer is accepted,
and if she rejects then we get
nothing.
So this game is exactly the
same as playing the previous
game, except we flip roles,
but we're going to add one
catch.
The catch is this,
in the first round the money on
the table is $1,
but if we end up going into the
second round,
so the first offer is not
accepted and we go into the
second round,
then part of the money is lost.
In particular,
we'll assume that the money on
the table is δ.
If you think of δ
as being just some number less
than 1--so if you want a
concrete example think of this
as being $.90.
So the idea here is if you get
into the second round,
time has past,
it's costly,
and so money in the second
round is--I think of it as money
in the second round as being
worth less or could actually
think of this cake being eaten
up,
some of it's thrown away,
some of it's wasted.
Everyone understand the game?
So this is very similar to the
previous game,
but we've got this second stage
coming in, and we've got
discounting.
So this is the idea of
discounting.
How many of you have heard the
term discounting before?
You probably saw it in a
finance class or a macro class
where we think about there being
a value of time.
Money today is worth more than
money tomorrow,
partly because you could put
the money today into the bank
and it could earn interest,
partly because you're simply
impatient to get that money and
go and have lunch,
particularly on the day in
which the clocks changed.
Okay, so let's try this game
again and let's just play it for
real, so let's come down again.
Everyone understand the game?
Basically the same rules except
we're just flipping around and
with the possibility that the
cake may shrink.
Let's see what people have
learned, so who were our first
pair?
Our first pair were Equia and
Noah all right.
So Equia, what are you going to
offer?
You're Player 1 here but if
your offer is rejected Noah's
going to get to make an offer to
you.
All right so what are you going
to offer this time?
Student: $25.99,
so $.25, $.46.
Professor Ben Polak:
$.46 okay,
so he gets $.46 if he accepts
the offer, is that right?
Student: Yes.
Professor Ben Polak:
$.46 if he accepts the
offer.
Student: I accept that.
Professor Ben Polak:
He accepts that,
okay that was easy.
So Equia got $.54 and Noah got
$.46.
Who was our second pair?
So that was,
I've forgotten,
Anish right and?
Student: Gary.
Professor Ben Polak:
Gary.
So Anish what are you going to
offer this time?
Student: I'll offer
$.43.
Professor Ben Polak:
$.43, you're going to push
the envelope a little bit.
Student: All right.
Professor Ben Polak:
All right,
that one got accepted as well.
Okay, so people are converging
here.
What about our third offerer,
receiver it was Courtney and?
Student: Danny.
Professor Ben Polak:
Danny.
So Courtney?
Student: $.30.
Professor Ben Polak:
$.30, so it's $.30 for him.
Student: I'll accept.
Professor Ben Polak:
Three acceptances,
all right.
Let's find out something here,
so I was hoping to get into the
second round.
Okay so you accepted,
that's fine--no chicken sounds
around the room.
So, it's Danny right?
Student: Yeah.
Professor Ben Polak:
So Danny had you
rejected--you acceptedbut had
you accepted what would you have
offered in the second round?
Student: $.45.
Professor Ben Polak:
$.45, all right,
and would you have accepted
that in the second round if $.30
hadn't been accepted?
Courtney: Yes.
Professor Ben Polak:
Okay, so you might have
done better it turns out.
Let's go back through to the
other rejections,
to the other acceptances.
So my second couple you offered
$.43, is that right?
You said yes to $.43.
Had you rejected $.43 what
would you have offered back in
return?
Student: $.43.
Professor Ben Polak:
$.43, the same thing back.
Would you have accepted it?
Student: He gets $.47?
Professor Ben Polak: He
would have got $.47 in that
case.
Student: Yeah.
Professor Ben Polak: You
would have accepted that,
okay.
Equia went first and she
offered $.45 to Noah.
And Noah had,
in fact, you rejected,
what would you have offered
back?
Student: I would have
also done $.45.
Professor Ben Polak:
Same thing back and would
you have accepted it?
Student: Yes.
Professor Ben Polak:
Okay.
So we can see here that the
decision to accept or reject
partly depends on what you think
the other side is going to do in
the second round,
is that right?
So here you are,
if you're in the middle of this
game.
If you're Player 2 you've
received an offer.
None of these offers sounded
crazy.
$.30 was the lowest one,
but none of them sounded crazy.
And you're trying to decide
whether you should accept or
reject this offer.
And one thing you should have
in mind is what would I offer if
I reject.
And will that offer that I then
offer in the next round be
accepted or rejected,
is that right?
So if we just work backwards we
can see what you should offer in
the first round should be just
enough to make sure it's
accepted knowing that the person
who's receiving the offer in the
first round is going to think
about the offer they're going to
make in the second round,
and they're going to think
about whether you're going to
accept or reject in the second
round, is that right?
So that sounds like a bit of a
mouthful but that mouthful of
reasoning is exactly backward
induction.
It's exactly backward induction.
It's saying:
to figure out what I should do
in the first round or what I
should offer in the first round,
I need to figure out whether
Player 2 is going to accept or
reject.
And to figure out whether he or
she is going to accept or
reject, and I have to put myself
in his or her shoes,
and figure out what he or she
would offer if she did reject,
and what he or she thinks I
would do if I got that second
round offer.
Is that right?
All right, so let's try and
analyze this as if backward
induction was going to work
here, as if we didn't have to
worry about things like pride
here.
So this is the game we're
actually playing,
so let's keep that one and
actually analyze it on the
board.
I want to walk us from a
largely mundane game of take it
or leave it offers to a more
complicated game in which there
can be several rounds of offers.
But we're going to go slowly so
we'll start just with two
rounds.
So first of all let's just look
at the stage one game,
and let's keep in track what
the offer is and what the
receiver.
This is the offerer and this is
the receiver.
In the one stage game,
the game only has one stage,
then we know from backward
induction what the results
should be.
It isn't what we'd find in the
lab, it isn't what we find in
the classroom,
but we know what we should get.
The offerer should offer to
keep everything essentially or
maybe $.99 but let's call it $1
and the receiver gets nothing.
So again I'm approximating a
little bit because it could be
$.99 and a penny but who cares.
Let's just call it a $1 or
nothing if it's a one stage
game.
So now let's consider a two
stage game.
In the two stage game the
person who's making the offer in
the first stage needs to look
forward,
anticipate what would happen if
her offer was rejected by Player
2 and Player 2 went forward into
the second stage.
Is that right?
So in the two stage game,
in the first stage of the two
stage game, the person making
the offer wants to anticipate
what the receiver would offer in
the second round were the
receiver to reject her offer.
But we can do that by backward
induction.
We know that in the second
round if the receiver rejects
the offer, then the second round
of the two stage game is what?
It's a one stage game,
and we've just argued,
at least if we believe in
backward induction,
in that case,
Player 2 who is then the
offerer, will offer $1 and
Player 1,
who is now the receiver,
will accept it and get nothing.
So Player 1 in the first round
of the two stage game wants to
make an offer that's just enough
to get Player 2 to accept it
now.
So let's think about this.
So if Player 1 offers 2
something more than what?
Tomorrow Player 2 can get $1
but that's $1 tomorrow.
So $1 tomorrow is worth how
much today if we're discounting?
It's just worth δ right.
It's just worth δ.
So if Player 1 offers Player 2
more than δ
x $1, which is what Player 2
can get tomorrow,
then 2 will accept.
If Player 1 offers 2 less than
δ x $1--because you can get
a $1 tomorrow but that's only
worth $δ--a $1 tomorrow is
worth just $δ
today--then 2 will reject.
So the offer has to be exactly
enough to get accepted,
which is exactly $δ.
So Player 2 knows that she can
get $1 tomorrow so you need to
offer her at least $δ
today to make it as good for
her as getting $1 tomorrow.
So we know the receiver must be
offered at least $δ
tomorrow, which means the
offerer is going to keep $[1 -
δ].
So in the first round of the
two stage game,
Player 1 should offer $[1 -
δ]
for herself and $δ
for Player 2 and Player 2
should accept that because
$δ dollars today is as good
as $1 tomorrow.
Now, another way to see that is
in a picture,
so let's just draw a picture.
Let's put the payoff of Player
1 here and the payoff of Player
2 on this axis.
And we're going to assume that
they're just going to maximize
dollars where there's no pride
in here.
And if we just look at the one
stage game, we're simply looking
at this line.
The offers in the one stage
game: it could be that Player 1
gets everything herself and
gives nothing to Player 2,
it could be that Player 2 keeps
everything, ends up getting
everything and Player 1 gets
nothing,
and it could be any combination
in between.
We argued by backward
induction--although not in
reality--in backward induction,
in the one stage game,
Player 1 makes an offer to
Player 2 which is kind of an
insulting offer.
Player 1 says I get everything
and you get a $.01.
So this is the one stage game.
In the two stage,
if things are settled in the
first stage this line represents
the possible divisions between
Player 1 and Player 2.
But if we end up going into the
second stage,
the pie is shrunk.
The pie is shrunk,
instead of going from $1 to $1,
it goes from $δ1 to
$δ1.
Or if you like,
if δ is .9 it goes from
$.9 to $.9.
So let's draw that line in.
So if we head into the second
stage, we'll end up being here,
and this goes from $δ
here to $δ
here where these dollars are
being evaluated at time one.
All right so the pie has shrunk.
If we get into the second stage
then, by backward induction,
Player 2 is in an ultimatum
game,
Player 2 will be making the
offer and Player 2 says:
whatever cake is left I'm going
to take all of it and you're
just going to get a $.01.
So if we get into the second
stage then Player 2 will make
this offer to Player 1.
Player 2 will say I'm going to
keep the whole of the pie,
which in first period dollars
is worth $δ.
So I'm going to end up with a
payoff of δ
and you're going to end up with
a payoff of essentially nothing.
Player 2 knows that they can
therefore get at least
$δ--or $δ
in current day dollars
worth--from rejecting your
offer.
Since they can get at least
$δ current day offers from
rejecting your offer,
the lowest offer you can make
to them is an offer that gives
them at least $δ.
So the offer you're going to
make is this offer:
this is the two stage offer.
It happens in the first stage.
Player 1 makes an offer that
gives Player 2 what Player 2
could get in the second round,
so gives Player 2 $δ
and keeps $[1 - δ]
for herself.
Everyone understand the picture?
So this picture is just
corresponding to this table.
The thing people tend to get
confused about here,
I think, is they get confused
between current dollars and
discounted dollars,
so we're going to do all the
analysis here in terms of the
first period dollars,
dollars tomorrow are going to
be worth δ.
There's a hand up,
can I get a shouting out?
Yeah?
Student: [Inaudible]
Professor Ben Polak:
Yes, sorry.
So this is the outcome if it
was a one stage game and this is
the outcome if it was a two
stage game.
The offer is made and accepted.
Let's roll it forward,
let's look at a three stage
game.
Let's keep this picture handy
and think about a three stage
game.
So the beginning of the game is
the same.
We're going to look at three
stage bargaining,
and the rules in three stage
bargaining are pretty much the
same as in two stage bargaining,
but now there's two possible
flips.
In three stage bargaining,
in the first period,
in period one,
1 makes the offer and if it's
accepted the game is over.
In period two,
if we reject,
then we go to period two when 2
makes the offer and if it's
rejected now,
this time by Player 1,
then we go to period three
where once again 1 makes the
offer.
So you can see where we're
heading, we're heading towards
an alternate offer bargaining
model.
I'm going to make an offer,
Jake's going to either accept
or reject, then he'll make an
offer and we'll flip to and fro.
There's a question,
let me try and get a mike out
to the question.
Yeah?
Student: I have a
question about the two player
game, if δ
is the best that Player 2 can
get tomorrow then why wouldn't 1
offer Player 2 δ
discounted by δ
today?
Professor Ben Polak:
Good.
Right so I think I was
confusing about it,
so let me make it clear.
So tomorrow Player 2 can get
everything, everything that
there is.
So whatever pie is left
tomorrow Player 2 can get all of
it.
So call that pie tomorrow 1 and
evaluate it in period one
dollars as being worth $δ.
Does that make sense?
Okay, so I think I misspoke on
that, so let me say it again.
So every period there's this
pie and every period,
if it was the last period of
the game,
the person making the offer is
going to get the whole pie,
but if I view that pie tomorrow
from today,
a pie of $1 tomorrow is only
worth $δ
today.
A pie of $1 the day after
tomorrow is only worth $δ
tomorrow and $δ²
today and so on.
So that's the way in which
we're going to do discounting
here.
Good, all right.
So in this game,
if 1 makes an offer,
if it's accepted it's over.
If it's accepted then we're
done.
And if this offer's accepted
then we're done.
And if this offer's accepted
then we're done.
And in the third round,
if it's rejected then both
players get nothing.
Once again we're going to
assume that the players are
discounting.
So what does it mean to say
they're discounting?
It means that dollars in period
one are worth dollars,
dollars in period two are
discounted by δ,
and dollars in period three are
discounted by δ
x δ, or if you like by
δ².
Just to put this into real
notions of money,
if you think of δ
as being 90%,
then $1 in period one is worth
$1, a $1 in period two viewed
from period one is worth $.90,
and a $1 in period three viewed
from period one is worth $.81.
Okay, so what do we think is
going to happen here?
Well, once again we can do
backward induction.
Here we are in our picture.
Let's look at the three-stage
game.
Once again, when we analyze,
as always when we analyze these
games using backward induction,
we want to start at the end.
If we start at the end,
we know that the last stage,
that's the third stage of the
three stage game looks like
what?
It looks like a one stage game.
In the one stage game we know
the offerer will get everything.
Say it again,
so the last stage of the three
stage game, we know the person
who makes the offer who this
time will be Player 1 will get
everything.
However, that everything is
only worth δ
in period two dollars and it's
only worth δ²
in period one dollars.
So in period one,
in the first period of this
game, we know that if their
offer is rejected we know what's
going to happen.
Say it again,
in the first period of this
three stage game,
if the offer is rejected then
we'll go into a two stage game,
and we already know what
happens in a two stage game.
In a two stage game,
the person who gets to make an
offer gets $[1 - δ]
and the person who receives the
offer gets $δ.
So we know in the first stage
of the game that the person who
receives the offer always has
the outside option of saying,
no I reject.
And we know that that person
tomorrow will get $[1 - δ].
But $[1 - δ]
tomorrow is worth how much
today?
It's worth $δ[1 - δ].
Tomorrow they're going to get
$[1 - δ],
so today that's worth
$δ[1--δ].
So the offer I have to make in
the first round to make sure
that the other person accepts it
has to be just better in
discounted dollars,
than what they're going to get
tomorrow.
They're going to get $[1 -
δ]
tomorrow, so I have to give
them $δ[1--δ]
today, which means I keep for
myself $[1 - δ[1--δ]].
And if you don't like the
algebra let's look at the
picture.
In the picture,
in the one stage game,
this is the offer.
In the two stage game,
we know if we get to the second
round, Player 2 gets everything
so we have to give him that much
today.
And if we get into the third
round, now we're looking at
δ² here and
δ² here.
We know that if we get into the
third round the person who makes
the offer in the third round
will get everything,
so we can actually work our way
along.
In the third round,
the person who makes the third
offer will get everything,
so in the second round you'd
have to give them that much,
so in the first round you'd
have to give Player 2 that much.
Say it again,
in period three,
the person making the offer can
get everything.
So in period two,
they must be getting δ
times that, so in period one
you have to give them at least
this much.
And this here is the offer
you'd make in the three stage
game.
So in the picture we're just
doing a little zigzag;
on the chart we're also always
working across the diagonal.
So we've done the one stage
game: the one stage game,
the person making the offer
gets everything.
In the two stage game,
the person making the offer
offers just enough to get the
offer accepted which is $δ
because that's what $1 is worth
tomorrow.
In the three stage game,
the person making the offer
makes just enough to get the
offer accepted,
which is δ
times what the receiver would
get tomorrow.
What they get tomorrow is $[1 -
δ], so they get
$δ[1--δ]
today.
How about the four stage game?
Let's see if we can do that.
So if we go to the four stage
game now, in the four stage game
if the person receiving the
offer rejects the offer,
then tomorrow they can get $[1
- δ[1--δ]].
So I need to offer them enough
now in current dollars so they
will prefer that to getting
$[1--δ[1 - δ]]
tomorrow,
so how much must I offer them?
I have to offer them at least
δ times that much,
so I have to offer them
$[δ x [1 - δ
x [1--δ]]].
Again, I'll keep the rest for
myself so I'll get $[1 -
δ[1--δ[1--δ]]].
And so the principle is always
give people just enough today so
they'll accept the offer,
and just enough today is
whatever they get tomorrow
discounted by δ.
So actually this backward
induction isn't so bad.
What makes it a little bit
easier is you don't actually,
when you go through an extra
stage of this you don't actually
have to go all the way to the
beginning,
you could actually start where
you were last time and just
discount once more by δ.
Let's see if we can see any
kind of pattern emerging in this
algebra, so let's just multiply
out these brackets.
In the four stage game,
this thing is actually equal
to--just multiplying
through--it's 1 - δ
+ δ²
- δ³--I hope it is
anyway.
That's what this is.
And this thing is equal to
δ - δ²
+ δ³,
just multiplying out the
brackets.
Does anyone see a pattern
emerging here in these offers?
We had offers of 1 and then
1--δ.
We could also multiply out this
one.
It might be helpful to do so:
This is 1 - δ
+ δ².
Anyone see a pattern what these
offers look like?
They kind of alternate.
So let's have a look,
rather than do every stage.
Should I do one more stage to
see if we can see a pattern
emerging or should I jump
straight to ten stages and see
what happens?
Go straight to ten people say?
Let's do one more,
nah, let's jump through to ten.
So imagine that this game
actually had ten rounds,
so this is a ten stage game,
and let's just continue our
chart down here.
So here's--need a bit more
space here--ten stages:
ten stage game.
I'm going to continue my chart
and my chart says in the ten
stage game what am I going to
get?
So the offer is going to
be--it's going to be the same
pattern--1 - δ
+ δ²
- δ³…
+ δ^(8) - δ^(9),
everyone see that?
So what I'm doing is I'm
continuing the pattern from
above.
So if I had ten stages,
I always start with a 1,
the positive and negative terms
just alternate,
and I have as many terms as 1
minus the stage I'm in.
So in the four stage game,
I ended up with δ³,
so in the ten stage game I'll
end up at δ^(9).
Everyone happy with that?
So the four [error:
ten]
stage offer is this slightly
ugly thing, 1 - δ
+ δ²
- δ³…
+ δ^(4) - δ^(5),
etc., + δ^(8) - δ^(9).
That's a pretty ugly thing,
but fortunately some point in
high school you learned how to
sum that thing.
Do you remember what this is?
What do you call objects like
this in high school?
Anyone remember?
Objects like 1,
δ, δ²,
δ³,
δ^(4) what are they
called?
They're called geometric series
right, they're called geometric
series.
Anyone remember how to sum them?
We know that S is equal to
this, this is what the offer is,
if there is ten rounds.
We know the offer is accepted.
So the way to remember how to
sum it, the trick for summing it
is to multiple both sides of
this equation by the common
ratio,
so multiply both sides by
δ.
So if I multiply this side by
δ, I'm going to multiply
the other side by δ.
And this 1 will become a
δ, this δ
will become a δ²,
this δ²
will become a δ³,
this δ³
will become a
δ^(4)….
There will be a -δ^(8)
coming from the seventh term.
There will be a +δ^(9)
coming from the δ^(8) term,
and there will be a
-δ^(10).
Everyone okay with that?
I just multiplied everything
through by δ
and I just shifted along one
for convenience.
What do I do now,
anyone remember?
Add the two lines together.
So by summing this side I get
[1 + δ]
S^(10).
On the other side,
what's kind of convenient is
everything cancels.
The 1 comes through, I get a 1.
These two terms cancel and
these two terms cancel,
and these two terms cancel,
and so on and so on,
all the way up to the end where
I get -δ^(10).
Everyone okay.
All the other terms have
cancelled out.
 So now just sorting out my
algebra a bit--I'm going to take
it on the other side--I'm going
to have that the offer you
make--this is what you're going
to get to keep--so the amount I
claim I should keep in the first
round is [1 - δ^(10])/[1 +
δ].
Just be a bit careful with the
notation here because it may be
a little bit confusing.
The 10 here doesn't mean to the
tenth power, it's just the offer
in the tenth round [error:
ten-round game],
whereas the 10 here really does
mean in the tenth power.
So if we play this game for ten
rounds, the offer you'd make
would be [1 - δ^(10)]/[1 +
δ]
which means the amount you're
offering to the other side,
which is 1 - S would be [δ
+ δ^(10)]/[1 + δ].
So to summarize where we are:
we started off by considering a
very simple game,
a one stage take it or leave it
offer.
We know that,
in that take it or leave it
offer, Player 1 is going to
claim everything for herself and
offer nothing to Player 2.
Then we considered a two stage
game which was the same as the
one stage game except that if
Player 2 rejects the offer,
he--let's call player 2,
"he"--he gets to make an offer
to Player 1 in the second
period.
We know that in the second
period of that two stage game
Player 2 can keep everything for
himself.
Everything for himself tomorrow
is worth δ
today, so you have to offer him
at least δ
today and keep 1 - δ
for yourself.
Then we looked at a three stage
game.
In this three stage game,
if Player 2 rejects in the
first round, Player 2 can make
you an offer in the second
round,
but now if you're Player 1 and
you reject in the second round
you get to make an offer to
Player 2 in the third round.
We argued that in the second
round of this game,
if Player 2 rejects you in the
first round and makes an offer
in the second round,
it'll be in a two stage game
and they'll be able to keep 1 -
δ of the pie for
themselves.
So you have to offer them at
least δ x [1--δ]
today for them to accept the
offer, keeping the rest for
yourself.
Then we looked at a four stage
game.
In the four stage game if
Player 2 rejects your offer,
he can make you an offer,
but if you reject the offer you
can make him an offer,
but if he rejects that offer he
can make you an offer.
And once again we asked how
much do I have to offer Player 2
now for him to accept the offer
now?
He knows that if he rejects the
offer, he can get this amount
1--δ x [1--δ]
tomorrow.
So I have to offer him δ
times that today,
and once again I keep the same
for myself.
That's just a summary of what
we did.
And then what we did was we
cheated.
We jumped to the tenth stage
[error: ten stages],
just noticing that a pattern
had emerged,
and we found that in the tenth
stage [error:
ten-stage game],
this is the offer you'd make
just according to the same
pattern.
And it was this horrible thing,
and then we used a little bit
of high school math to simply
this thing.
And it turns out this is the
amount you keep for yourself and
this is the offer you'll make to
Player 2.
In each case I've accepted--Did
I make a math mistake?
Thank you.
Let's put a superscript in
here.
good.
So what do we observe here?
So the first thing to observe
is in the one stage game if we
believe backward induction you
certainly want to be the person
making the offer.
In the one stage game,
in the ultimatum game,
there's a huge first-mover
advantage.
In the two stage game it's not
clear if you want to make the
offer, it depends on how large
δ is,
but if δ
is a big number like .9 you'd
rather be the person receiving
the offer.
In the three stage game,
it looks like you'd probably
rather make the offer,
but it's not so clear.
So where does it go to as we go
down the path,
as it goes down towards the ten
stage game?
It looks like in the ten stage
game you'd probably still prefer
to make the offer than not,
but they're certainly much
closer together than they were
before.
Some of that initial bargaining
power has been washed out by the
fact that there are ten stages.
So let's try and push this just
a little bit harder.
Instead of looking at the tenth
stage offer, what if we look at
the infinite stage offer.
So in principle we look at the
infinite stage of this game.
So I can make you an offer,
you can say no and make me an
offer, and then I can reject and
make you an offer,
and then you can reject and
make me an offer,
and so on and so forth.
So we look at this term.
If in principle and you can
make an infinite number of
offers--so, what's this term
going to look like if I can make
an infinite number of offers?
So I claim it's going to look
like this [1 - δ^(∞)]
/ [1 + δ]
and over here,
at least it's going to converge
towards this.
We'll be a bit more formal,
and over here we'll have
[δ + δ^(∞)]
/ [1 + δ].
However now I get a little bit
simpler.
What is δ^(∞)?
It's 0 right,
so .9 x .9 x .9 x .9 x .9 x.
9 x .9 x .9 x .9 x …
is 0.
So this last term disappears as
does this one,
and we just get 1 / [1 +
δ]
and δ / [1 + δ].
So if we make alternating offer
bargaining--a bargaining game
where in each round I make you
an offer,
you can accept it or you can
reject and make me an offer,
and we imagine there's no bound
to this game,
it just goes on arbitrarily
long--then our prediction is
that Player 1,
the person who makes the first
offer will get 1 / [1 + δ]
of the initial pie and Player 2
will get δ
/ [1 + δ]
of the initial pie.
Let's try and get a handle
about what those numbers are.
So if you imagine these offers
can be made fairly rapidly,
for example,
I can make offer today,
you can make an offer back to
me in half an hour's time,
and then I can make an offer
back to you in half an hour's
time,
then it's reasonable to assume
that the pie is not shrinking
very fast.
The discount factor is not a
big deal here.
So these offers can be made in
rapid succession,
but we might think that δ
itself is approximately 1:
the pie isn't shrinking very
fast.
If δ is approximately 1,
and if we take δ
to 1 here--the time isn't that
valuable given how rapidly we
can make offers to and fro--then
what does this make this equal?
In the case where δ
is equal to 1 what do we get?
We get 1/2, which means this
will also be 1/2.
So we learned something from
this which is kind of
surprising.
If you do alternating
offers--the sort of standard,
very natural game of
bargaining--sort of the kind of
bargaining you might do in the
bazaar,
in a market,
or the kind of bargaining you
might imagine going on in
negotiations between baseball
players or their agents and
teams,
or general managers of
teams--in which offers just go
to and fro and they go to and
fro fairly rapidly,
and in principle they could
make lots and lots of offers.
In principle--what this moral
tells us is, in principle,
we're going to end up with each
side splitting whatever the pie
was equally.
Very, very different from the
ultimatum game where all the
bargaining power was on the
person who made the first offer.
So what are the lessons here?
What can we conclude from this?
We've looked at alternating
offer bargaining,
and we've concluded,
under special conditions,
we've concluded that you get an
even split.
You get an even share,
an even split,
a fifty-fifty split if three
things are true.
The first thing is there's
potentially infinitely many
offers: potentially can bargain
forever.
And if discounting is not a big
deal.
What does discounting really
not being a big deal means?
It means those offers can be
made in rapid succession.
So no discounting,
or if you like,
rapid offers.
If you have to wait a year
between every offer then that
discount factor would be a big
deal.
But I actually made a third
assumption, and I made it
without telling you.
What was the third assumption I
made?
I snuck the third assumption
past you without telling you.
What was that third assumption?
Let me get the mike here.
So I claim I snuck a third
assumption.
There's somebody,
let me start over here.
What was the third assumption?
Student: I don't know if
this is what you're looking for
but they know how big the pie
is.
Professor Ben Polak:
They know how big the pie is,
that's true,
that's a big deal actually.
That's true,
but there's something else
going on here.
What is it?
Student: We assume that
both players were rational.
Professor Ben Polak:
We assumed that.
That's true,
but we've kind of been assuming
that throughout backward
induction.
You're right none of this
backward induction would apply
so cleanly if we didn't assume
that.
What else do I assume?
It's hidden actually,
I snuck it in.
We assume the discount factor
is a constant,
that's true,
but not just constant but
something else.
You're on the right lines.
They're the same.
I've assumed here,
implicitly, I've assumed that
both people are equally
impatient,
they have the same discount
factor, δ_1 =
δ_2.
Why does that matter?
Well let's just think about it
intuitively a second.
Suppose that one of these
players is very,
very impatient.
They need the money now.
If it's cake,
they need the cake now.
They're very impatient,
and the other person is very
patient.
They can wait forever to get
this bargain to come across.
Who do you think is going to do
better, the patient player or
the impatient player?
The patient player is going to
do better.
The way we ended up,
the way we did all this
analysis is we assumed that
those discount factors were the
same.
We assumed that each person was
discounting time at the same
rate, perhaps because they were
facing the same bank with the
same interest rate.
But in practice,
often one side is going to be
in a hurry to get the dispute
resolved, and the other side can
sit around forever.
In that world,
the side who could sit around
forever is going to do much
better.
Now, we're going to look at
relaxing this assumption and
this assumption in particular
we're going to relax it on a
homework exercise.
So in your homework exercise
you're going to try redoing part
of this analysis--good practice
anyway--but doing it in a
setting where the discount
factors are different.
So one thing we learned here
was: yes, you get an even split,
but it depends on these three
assumptions.
It's kind of important because
when you think about bargaining,
I think a lot of people simply
assume intuitively that whatever
the bargain is about,
people will eventually split in
the middle.
When you're bargaining about a
house, or the price you're going
to pay for a house,
all these things,
you kind of implicitly have
this assumption you're going to
end up splitting the difference.
What I'm arguing here is you
will split the difference in
this natural bargaining game,
but only under very special
assumptions, and in particular,
the assumption of patience is
critical.
There's another remarkable
thing here though,
it's also hidden.
So not only did we end up with
an even split,
but something else remarkable
happened in this bargaining
game.
What was the other thing?
Somewhat amazingly,
a very unrealistic thing that
occurred in this bargaining
game?
See if you can spot it.
So one thing was an assumption
I made and the other is actually
a prediction.
Yeah.
Student: The first offer
will be the offer that's
accepted.
Professor Ben Polak:
Good.
Did everyone see that?
So in this bargaining game,
I set it up as alternating
offer bargaining,
so the image you had in your
mind was of haggling.
I made an offer,
you guys thought about this
offer.
Should I take this offer or not.
Maybe I won't take this offer.
You make an offer back to me,
and we kind of haggled to and
fro.
But actually,
in the equilibrium of the game,
none of that happened.
That all happened in our mind.
We thought about what offer we
would make, and we thought about
what offer you would make back
to me if I made you this offer
and you rejected it and so on.
We did this backward induction
exercise but it was all in our
heads.
In this game the actual
prediction is:
the very first offer is
accepted.
There's no haggling,
there's no bargaining.
Now that doesn't seem very
realistic, there's no haggling.
Backward induction suggests
that we should never see
bargaining, never see the actual
process of bargaining.
What you should see is an offer
is made and it's accepted.
Now what is it about the real
world that allows for haggling
to take place?
If this was a model of the real
world and we believed in
backward induction,
then we're done.
So why is it in fact in the
real world we see people make
offers to and fro?
What's different about the real
world than this model?
Let's talk about it a bit.
You must have all bargained for
something in the real world.
None of you have probably
bargained for a house yet,
but you might have bargained
for a car or something.
In the real world you make
offers go to and fro,
right?
What's going on?
Why are we getting offers in
the real world whereas we don't
in this game?
What are we missing?
Student: In the real
world you don't actually know
what the other person's discount
factor is,
therefore, you have uncertainly
as to what your highest possible
offer could be.
Professor Ben Polak:
Good.
So in the real world,
unlike the model on the board,
not only are those discount
factors different but you
probably don't know how patient
or impatient the other side is.
You can get some ideas about
how patient or impatient the
other side is by looking at
their characteristics,
for example.
For example,
if you know that the person
you're bargaining with over
their car--you're trying to buy
their car--and they're a
graduate student who's just a
got a job in,
I don't know,
Uzbekistan or something,
and they aren't going to be
taking their car with them,
and they're leaving next week,
you know they're in a hurry.
So there's times when you're
going to know something about
other people's discount factors,
how patient they are,
but lots of times you're not
going to know.
So one thing is you just don't
know what the discount factor
is.
By the way, what else might you
not know about the other side?
What else might you not know?
Student: How big the
surplus is they were splitting.
Professor Ben Polak:
Good.
You might not know this good
that you're selling.
We've been talking about the
big one pie which you're carving
up, and everyone knows the size
of the pie.
But in the real world,
I might not know this object
that's being sold,
I might not know how much this
object is worth to the other
side,
and he or she may not know how
much it's worth to me.
So that lack of information is
going to change the game
considerably.
In particular,
I might want to turn down some
offers in this game in order to
appear like a patient person.
Why I might want to turn down
some offers in the game in order
to appear like somebody who
doesn't really value this all
very much.
And in so doing,
I'm going to try and get you to
make me a better offer.
So what's going on in haggling
and bargaining,
according to this model,
what's missing in this model,
is the idea that you don't know
who it is you're bargaining
with.
You don't know how much they
value the objects in question.,
You don't know how impatient
they are to get away with the
cash.
So it's a big assumption here,
a very big assumption,
is that everything is known.
So both the size of the pie,
let's call it the value of the
pie, and the value of time is
assumed to be known,
but in the real world you
typically don't know the value
of the pie on the other side,
and you typically don't know
how much they value time.
So that produces a whole
literature on bargaining,
none of which we really have
time to do in this class,
which is a pity because
bargaining's kind of important.
So instead, I want to spend the
last five minutes just
introducing, is it really worth
introducing a new topic in the
last five minutes?
No, I think it is,
let me talk a little bit more
about bargaining rather than
that.
So what does this suggest if
we're going out in the real
world?
I'm actually taking this to
reality.
So one thing it suggests is
people for whom it's known that
they're going to be
impatient--people for whom it's
known that they desperately need
this deal to go through--are
going to do less well in
bargains.
We already know people may do
less well in bargaining because
they're less sophisticated
players,
but here it isn't that they're
less sophisticated--they can be
as sophisticated as you
like--but they're just going to
be in a hurry.
We already talked about the
graduate student whose leaving
for Uzbekistan,
but who else typically in
bargaining is going to need cash
now?
Who else is going to be in a
weaker position in their
bargaining, socially in our
society?
Student: When labor
management disputes labor.
Professor Ben Polak:
So that's a good question.
That's a good question in labor
management disputes--there's one
going on right now in
Hollywood--it's not clear there.
It could be,
it could be the management side
who's in a hurry because they
just need right now to get David
Letterman's script written.
That would tend to favor labor,
but it could be the labor side.
Why might it be the labor side?
Does everyone know this?
There's a writer's strike going
on in Hollywood right now,
so the people on the management
side who are in the weakest
position are the people who are
in the most hurry to get this
resolved,
and those are the guys with the
fewest scripts in the pipeline,
and that tends to be late night
TV shows.
So those guys really want this
thing settled fast.
They're in a weak bargaining
position.
On the other hand,
there may be a reason why
labor's in a weak bargaining
position.
Why might labor be in a weak
bargaining position relative to
management?
They have rents to pay.
They have immediate demands on
their cash.
The typical worker is typically
poorer than your typical
manager, not always but
typically, and they have to pay
the rent.
They have to feed their
children.
So there's a more general idea
there.
More generally in bargaining,
the people who are poorer,
typically--it isn't just poor
in terms of income,
it's poor in terms of
wealth--are going to be more
impatient to get things
resolved;
and that's going to put them in
a weaker bargaining position.
So in bilateral bargaining,
having low wealth and being
known to have low wealth puts
you in a weaker position.
And that means that typically
people who are poorer are going
to do less well,
although the late night TV show
may be an exception.
It makes you think a little bit
about whether adjusting up a
bargaining position makes things
equal for everybody.
Any other thoughts about who
has strength and who has
weakness in bargaining?
What other kind of stunts do we
see people do in bargaining?
What else is kind of missing
here when you think about a
particular--what we want to do
in this class is develop these
ideas and take them to the real
world,
so what other real world things
here are kind of missing?
Student: Usually people
will make their first offer a
lot higher than what they're
actually willing to accept.
Professor Ben Polak:
Right, so typically,
you're right,
typically bargaining isn't just
a series of random numbers,
typically people start out high
and then they concede towards
the middle, is that right?
So if I'm the buyer I start out
with a low price and come up,
and if you're the seller you
start off with a high price and
come down.
So again that seems to be about
establishing reputation and
trying to indicate how much I
want this good.
There's something worth saying
here which we haven't got time
to do in this class,
and I hope you all have time to
take the follow up class 156.
We can actually show formally
that in a setting in which
buyers and sellers are
bargaining,
and buyers and sellers do not
know how much the good is worth
to the buyer or how much it
costs to the seller,
typically you cannot expect to
get efficiency.
Let me say it again.
So it's kind of an important
economic fact that's missing
from 115 and unfortunately
missing from this class,
but if we go to a more real
world setting in which people's
values are not known,
not only are the offers not
accepted immediately and not
only is there some inequity in
that the poor tend to be more
impatient and do less well,
but also you get bad
inefficiency.
The inefficiency occurs
essentially because the sellers
want to seem like they're hard
and the buyers want to seem like
they're hard,
and you get a failure for deals
to be made.
So some deals are actually
going to be lost or take a long
time in coming,
you're going to get some
strikes before the deals occur
and that's all inefficient.
So bargaining,
not in this model,
but in the real world tends to
lead to inefficiency.
So I'll leave it there,
we'll have an earlyish lunch
since we're all starving because
of the clock change anyway.
