WEBVTT

00:02.260 --> 00:05.210
Professor Ben Polak:
So today I want to look at

00:05.214 --> 00:08.284
two kinds of games and then
we'll change topic a bit.

00:08.280 --> 00:13.900
The games I want to look at are
about ultimatums and bargaining.

00:13.900 --> 00:17.180
And we'll start with ultimatums
and we'll move smoothly through,

00:17.176 --> 00:20.136
and we'll see why that's an
easy transition in a while.

00:20.140 --> 00:29.530
So the game we're going to play
involves two players,

00:29.525 --> 00:35.115
1 and 2 and the game is this.

00:35.120 --> 00:43.550
Player 1 is going to make a
take it or leave it offer to

00:43.553 --> 00:53.213
Player 2 and this offer is going
to concern a pie and let's make

00:53.213 --> 00:57.663
the pie worth $1 for now.

00:57.660 --> 01:01.360
We'll probably play this for
real in a bit so we'll play it

01:01.356 --> 01:04.276
for $1.
So it's a split of a pie,

01:04.279 --> 01:09.349
we can think of the split as
offering S to Player 1 and

01:09.352 --> 01:11.422
(1--S) to Player 2.

01:11.420 --> 01:20.110
Player 2 has two choices:
2 can accept this offer,

01:20.110 --> 01:30.930
and if 2 accepts the offer then
they get exactly the offer.

01:30.930 --> 01:35.390
Player 1 gets S and Player 2
gets (1--S).

01:35.390 --> 01:40.620
Alternatively,
Player 2 can reject,

01:40.622 --> 01:49.392
and if Player 2 rejects the
offer then both players get 0,

01:49.393 --> 01:54.013
both players gets nothing.

01:54.010 --> 01:56.440
So a very simple game,
there's a dollar on the

01:56.440 --> 01:59.570
table--I'll take it out in a
minute--so there's a dollar on

01:59.573 --> 02:01.963
the table,
and our two players are going

02:01.960 --> 02:04.940
to bargain for this,
but it isn't much of a bargain.

02:04.939 --> 02:07.479
Player 1's basically going to
announce what the division's

02:07.484 --> 02:10.024
going to be.
2 can either accept that

02:10.017 --> 02:12.717
division or no one gets
anything.

02:12.720 --> 02:15.210
Everyone understand the game?

02:15.210 --> 02:19.060
So I thought we'd start off by
playing this game for real a

02:19.061 --> 02:22.181
couple of times,
so why don't I come down and do

02:22.182 --> 02:22.782
that.

02:22.780 --> 02:26.170


02:26.169 --> 02:29.579
So why don't we play with some
people in this row.

02:29.580 --> 02:32.030
I've been playing with that row
all the time,

02:32.025 --> 02:34.355
so Ale's going to help me find
somebody.

02:34.360 --> 02:37.960
You found somebody okay,
so the person behind you?

02:37.960 --> 02:41.870
Your name is?
You have to shout out,

02:41.870 --> 02:42.950
this doesn't make a noise.

02:42.950 --> 02:46.100
Equia, thank you.

02:46.099 --> 02:50.529
So you'll be Player 1,
and Player 2 will be this

02:50.534 --> 02:53.464
gentleman here whose name is?

02:53.460 --> 02:53.990
Student: Noah.

02:53.990 --> 02:55.000
Professor Ben Polak:
Noah, okay.

02:55.000 --> 02:57.340
So Equia, do you know each
other?

02:57.340 --> 03:02.310
Okay, so Equia can make any
offer she wants.

03:02.310 --> 03:05.400
We'll play this for real money,
so there's a real dollar at

03:05.403 --> 03:07.103
stake.
You can make any offer you

03:07.102 --> 03:09.942
want--it can be in fractions of
pennies if you like--of the

03:09.941 --> 03:12.831
amount of the dollar that you're
going to offer to Noah,

03:12.830 --> 03:17.070
and the amount you're going to
keep yourself.

03:17.069 --> 03:19.859
There you go,
shout it out so everyone can

03:19.863 --> 03:22.933
hear.
Stand up, this is your moment

03:22.928 --> 03:25.158
in the lights all right.

03:25.160 --> 03:28.590
Will you stand up as well?

03:28.590 --> 03:30.760
There we go.
Student: I'm going to

03:30.759 --> 03:31.579
offer him a penny.

03:31.580 --> 03:32.920
Professor Ben Polak:
You're going to offer him a

03:32.919 --> 03:34.109
penny, and Noah are you going to
take a penny?

03:34.110 --> 03:34.470
Student: No.

03:34.466 --> 03:35.456
Professor Ben Polak:
Noah's not going to take a

03:35.461 --> 03:37.441
penny.
So I didn't lose any money on

03:37.441 --> 03:39.441
that: so no one made any money.

03:39.440 --> 03:40.410
I didn't lose any money.

03:40.410 --> 03:41.560
that seems pretty good.

03:41.560 --> 03:42.450
Let's try a different couple.

03:42.449 --> 03:45.809
Let's move around the room a
little bit.

03:45.810 --> 03:50.780
So Ale why don't you take the
guy behind here who name is?

03:50.780 --> 03:52.130
Shout out.
Student: Gary.

03:52.129 --> 03:55.219
Professor Ben Polak:
Gary, all right so Gary why

03:55.221 --> 03:57.911
don't you stand up,
and we'll let Gary play with

03:57.911 --> 03:59.971
the gentleman in here,
what's your name in here?

03:59.970 --> 04:01.030
Student: Anish.

04:01.030 --> 04:02.700
Professor Ben Polak:
Anish, all right stand up,

04:02.701 --> 04:03.491
do you know each other?

04:03.490 --> 04:07.250
So why don't we let Anish be
Player 1 this time.

04:07.250 --> 04:10.440
You understand the rule of the
game?

04:10.440 --> 04:11.330
So make your offer.

04:11.330 --> 04:13.200
Student: $.30.

04:13.196 --> 04:15.236
I'm offering him $.30.

04:15.240 --> 04:17.730
Professor Ben Polak:
You're offering him $.30?

04:17.730 --> 04:18.810
He's saying no as well.

04:18.810 --> 04:23.360
Okay, so let's raise the stakes
a bit.

04:23.360 --> 04:26.080
Let's make it $10,
since we're not getting much

04:26.076 --> 04:28.636
acceptance here.
Let me try a third couple,

04:28.640 --> 04:30.780
working my way back,
so how about you,

04:30.775 --> 04:31.925
what's your name?

04:31.930 --> 04:32.900
Student: Courtney.

04:32.899 --> 04:34.029
Professor Ben Polak:
So why don't you stand up

04:34.032 --> 04:36.292
as well.
We'll let Courtney be Player 1.

04:36.290 --> 04:38.890
Actually why don't you sit down
so I can do it from here,

04:38.889 --> 04:40.149
and you are?
Student: Danny.

04:40.149 --> 04:41.439
Professor Ben Polak:
Danny, all right so

04:41.440 --> 04:43.150
Courtney's going to be Player 1
and Danny's going to be Player

04:43.151 --> 04:44.521
2.
These weren't our dating couple

04:44.523 --> 04:45.303
from earlier right?

04:45.300 --> 04:46.000
We're safe on that?

04:46.000 --> 04:48.600
Okay good, so Courtney what are
you going to offer?

04:48.600 --> 04:49.670
Student: $5.

04:49.670 --> 04:51.460
Professor Ben Polak:
$5, which is half of the

04:51.459 --> 04:54.379
$10.
Student: Accept.

04:54.379 --> 04:55.949
Professor Ben Polak:
All right accept,

04:55.951 --> 04:58.081
all right.
So it turned out in this game

04:58.080 --> 05:00.660
that a lot of people were
rejecting offers.

05:00.660 --> 05:02.600
Let's have a look at it on the
board a second.

05:02.600 --> 05:04.960
Let's think about it a second
and we'll come back,

05:04.955 --> 05:06.585
we're not done with this
couple.

05:06.589 --> 05:08.569
We're not going to need new
couples, we're going to come

05:08.566 --> 05:11.996
back to you.
So in this game--it's pretty

05:12.004 --> 05:17.354
simple to analyze this game by
backward induction.

05:17.350 --> 05:21.340
By backward induction,
we're going to start with the

05:21.338 --> 05:25.478
receiver of the offer and the
receiver of the offer is

05:25.484 --> 05:30.414
choosing between the offer made
to them which in our notation is

05:30.411 --> 05:33.281
(1-S).
And in our three examples that

05:33.283 --> 05:36.893
was a penny, then it was $.30
and then it was half of it,

05:36.893 --> 05:39.023
so $.50 just to be consistent.

05:39.019 --> 05:41.829
We actually saw two of those
offers rejected,

05:41.826 --> 05:44.246
but according to backward
induction,

05:44.250 --> 05:49.220
assuming that people are trying
to maximize their dollar payoffs

05:49.224 --> 05:52.624
here, what should we see the
receiver do?

05:52.620 --> 05:56.520
They should accept even the
somewhat insulting offer of a

05:56.516 --> 05:59.856
penny that was made:
even that somewhat insulting

05:59.855 --> 06:02.425
offer should be accepted by
Noah.

06:02.430 --> 06:06.070
So Noah didn't accept the offer
of a penny and let's come back

06:06.065 --> 06:08.595
over there.
So Noah didn't accept the offer

06:08.602 --> 06:12.012
a penny, and I forget who it was
but our second player didn't

06:12.014 --> 06:14.854
accept an offer of $.30,
but we've just argued that they

06:14.854 --> 06:15.994
should have been accepted.

06:15.990 --> 06:19.480
In fact, when Equia made the
offer of one penny to Noah,

06:19.476 --> 06:22.326
I think she assumed that Noah
would accept it,

06:22.328 --> 06:24.728
is this right?
Why did you think Noah would

06:24.733 --> 06:26.583
accept it?
Student: Because I felt

06:26.579 --> 06:28.829
he would be better off with a
penny than nothing.

06:28.829 --> 06:30.119
Professor Ben Polak:
He'd be better off with a

06:30.123 --> 06:30.823
penny than nothing,
right.

06:30.820 --> 06:31.840
Then we saw an offer.

06:31.840 --> 06:32.770
The offer came up a bit.

06:32.770 --> 06:34.180
Who was my second offerer?

06:34.180 --> 06:36.760
You were my second offerer,
you offered a bit more.

06:36.760 --> 06:37.880
Why did you offer more?

06:37.879 --> 06:40.109
Student: I felt $.30 was
a pretty fair share.

06:40.110 --> 06:40.880
It's a lot better than nothing.

06:40.879 --> 06:42.889
Professor Ben Polak:
Thirty cents is better than

06:42.885 --> 06:44.295
nothing, although not
necessarily fair,

06:44.297 --> 06:45.817
but I guess it's better than
nothing.

06:45.819 --> 06:48.479
Where was my rejector of the
second offer?

06:48.480 --> 06:50.050
Who rejected the second offer?

06:50.050 --> 06:51.290
Why did you reject $.30?

06:51.290 --> 06:52.490
Student: Just a pride
thing.

06:52.490 --> 06:55.210
Professor Ben Polak:
It's a pride thing.

06:55.209 --> 06:59.549
Pretty soon we converged onto
$.50, which notice is no where

06:59.548 --> 07:01.458
near backward induction.

07:01.459 --> 07:04.299
So the third offer,
which is Courtney,

07:04.300 --> 07:06.680
why did you offer half of it?

07:06.680 --> 07:09.280
Student: Because half is
better for me than nothing.

07:09.279 --> 07:10.929
Professor Ben Polak:
Half is better for you than

07:10.926 --> 07:12.596
nothing, and you figured he's
going to reject otherwise,

07:12.603 --> 07:13.613
and in fact he didn't reject.

07:13.610 --> 07:14.190
Why didn't you reject?

07:14.189 --> 07:15.389
Student: Because $5 is
better than nothing.

07:15.394 --> 07:16.774
Professor Ben Polak:
$5 is better than nothing.

07:16.769 --> 07:19.239
But the $5 is better than
nothing argument would have

07:19.243 --> 07:21.673
argued against making any
rejection in this game.

07:21.670 --> 07:24.250
Is that right?
So here's a game where backward

07:24.253 --> 07:27.223
induction is giving a very clear
prediction.

07:27.220 --> 07:29.720
The clear prediction is,
first of all,

07:29.719 --> 07:33.569
the second player will accept
whatever's given to them;

07:33.569 --> 07:37.359
and second, given that,
the first player should offer

07:37.359 --> 07:41.659
them essentially nothing,
should offer them just a penny.

07:41.660 --> 07:48.880
So backward induction predicts
that the offer will be

07:48.881 --> 07:57.351
essentially, let's say $.99 and
$.01, or even virtually $1 and

07:57.353 --> 08:00.693
nothing.
But in fact we don't get that,

08:00.693 --> 08:03.203
we get a lot of rejection of
these low offers,

08:03.202 --> 08:06.102
and often we get offers made
much, much higher in the

08:06.103 --> 08:07.333
vicinity of half.

08:07.329 --> 08:10.619
Now why?
Why are we seeing a failure of

08:10.618 --> 08:13.558
backward induction in this game?

08:13.560 --> 08:15.200
I think this is not necessarily
you guys.

08:15.199 --> 08:18.819
It's a very reliable result in
experimental data.

08:18.819 --> 08:23.339
So why do we see so many people
in this ultimatum game both

08:23.340 --> 08:25.910
offer more, and,
more importantly,

08:25.912 --> 08:28.642
reject less than small amounts.

08:28.639 --> 08:32.149
So let's talk about it,
so one person said it's a pride

08:32.149 --> 08:35.609
thing.
Let's try the other aisle here,

08:35.605 --> 08:40.715
so what's the smallest offer
you would have accepted.

08:40.720 --> 08:42.190
What's your name first of all?

08:42.190 --> 08:47.860
Student: Jeff. A cent.

08:47.860 --> 08:49.640
Professor Ben Polak:
So there are some backward

08:49.643 --> 08:50.723
induction players in the room.

08:50.720 --> 08:55.880
Who would have rejected a cent?

08:55.880 --> 08:58.470
Who would have rejected $.10?

08:58.470 --> 09:02.650
We should be going down at
least, who would have rejected

09:02.647 --> 09:04.607
$.30?
Few people rejected $.30,

09:04.607 --> 09:05.697
not many actually.

09:05.700 --> 09:09.280
How many people would have
rejected $.50?

09:09.279 --> 09:11.239
One person even would have
rejected $.50,

09:11.236 --> 09:12.456
but essentially no one.

09:12.460 --> 09:14.840
So what's happening here?

09:14.840 --> 09:17.750
Why do people think people are
rejecting what is essentially

09:17.747 --> 09:20.257
money from my pocket,
there's nothing going on here,

09:20.261 --> 09:21.741
I'm just giving you money.

09:21.740 --> 09:24.280
Why are they rejecting being
given money?

09:24.279 --> 09:26.749
Student: Overall the
stakes are really low,

09:26.745 --> 09:29.205
so if you have any value on
sort of like pride,

09:29.210 --> 09:32.580
what people said,
you know it's not worth a penny

09:32.578 --> 09:34.348
or $.10.
Professor Ben Polak:

09:34.351 --> 09:36.261
All right,
so it may be pride going on

09:36.260 --> 09:38.450
here, so certainly one thing is
about pride.

09:38.450 --> 09:40.900
It turns out that people do
this even in quite high stakes

09:40.898 --> 09:43.088
games, but you're right,
certainly that trade off is

09:43.090 --> 09:44.250
going to start to bite.

09:44.250 --> 09:45.110
What else is going on?

09:45.113 --> 09:46.923
So I agree, pride is part of
what's going on.

09:46.920 --> 09:48.800
What else is going on here?

09:48.799 --> 09:50.759
Let's try and get some
conversation going.

09:50.759 --> 09:52.369
Somebody in here,
if I can get the mike in,

09:52.372 --> 09:53.872
shout out your name and really
shout.

09:53.870 --> 09:54.850
Student: Peter.

09:54.850 --> 09:56.410
Professor Ben Polak:
Go on.

09:56.409 --> 09:58.019
Student: Change is
cumbersome.

09:58.019 --> 10:00.539
Professor Ben Polak:
Change is cumbersome,

10:00.544 --> 10:02.864
you didn't want the change,
okay fair enough,

10:02.857 --> 10:05.747
but if the stakes go up that
would get rid of that.

10:05.750 --> 10:08.570
Student: Maybe people
are tying their own outcomes to

10:08.566 --> 10:10.376
the other player's outcomes as
well.

10:10.379 --> 10:11.879
Professor Ben Polak:
Right, so maybe people have

10:11.883 --> 10:12.583
different payoffs here.

10:12.580 --> 10:15.000
Maybe people are comparing
their payoff to the other

10:15.003 --> 10:17.903
person's payoff that certainly
seems like a plausible thing to

10:17.902 --> 10:18.902
be going on here.

10:18.899 --> 10:23.129
You might feel less happy about
getting $.20 knowing the other

10:23.132 --> 10:25.632
person's getting $.80 for
example.

10:25.629 --> 10:27.819
Student: You want to try
to teach them a lesson to get

10:27.815 --> 10:28.575
them to offer more.

10:28.580 --> 10:30.090
Professor Ben Polak:
Right, you might be trying

10:30.089 --> 10:31.199
to set a sort of moral standard
here.

10:31.200 --> 10:33.820
So there's some notion of
indignation or even teaching

10:33.824 --> 10:36.354
people that they really should
offer people more.

10:36.350 --> 10:39.540
What else could be going on
here?

10:39.539 --> 10:44.199
Student: If people know
I'm not going to accept less

10:44.199 --> 10:49.339
than $.50 then they should give
me $.50 by backward induction.

10:49.340 --> 10:51.080
Professor Ben Polak:
Right, so part of what's

10:51.076 --> 10:52.876
going on here--actually this
game was a one shot game,

10:52.880 --> 10:53.800
we just played it once.

10:53.799 --> 10:56.589
We could have played it--in
fact they often do play this

10:56.586 --> 10:59.626
this way in the lab--you could
have played it without anybody

10:59.625 --> 11:01.495
knowing who the other player
was.

11:01.500 --> 11:04.620
But particularly in this
setting where everyone can see

11:04.620 --> 11:08.140
everyone else--even in the lab
where people actually can't see

11:08.144 --> 11:10.174
people,
but they might imagine that the

11:10.165 --> 11:12.755
game is really repeated--you
could imagine people trying to

11:12.757 --> 11:13.917
establish a reputation.

11:13.920 --> 11:16.050
Is that right?
So there's lots of these

11:16.047 --> 11:18.617
reasons, these sort of moral
indignation reasons or teaching

11:18.622 --> 11:20.152
a lesson reasons,
pride reasons.

11:20.149 --> 11:23.319
There's also this basic reason
that people might be thinking,

11:23.318 --> 11:26.278
I should play this game as I
would a similar situation in

11:26.276 --> 11:29.336
life where I might want to be
establishing reputation.

11:29.340 --> 11:32.550
So there's a certain amount of
confusion going on in the game,

11:32.553 --> 11:35.503
and there's also a certain
amount of a lot of things make

11:35.503 --> 11:37.923
sense.
Now notice that once we've

11:37.917 --> 11:42.127
established that people are
going to reject small offers in

11:42.126 --> 11:44.606
this game,
once we've established people

11:44.609 --> 11:47.579
are going to reject small
offers, it makes perfectly good

11:47.581 --> 11:49.811
sense to offer a lot more than
nothing.

11:49.809 --> 11:52.149
So it's not that surprising
that once we've established the

11:52.145 --> 11:54.235
idea that people are going to
reject small offers,

11:54.240 --> 11:57.010
we're going to see people
making offers that are

11:57.006 --> 11:59.596
reasonably large,
although not usually larger

11:59.596 --> 12:02.496
than $.50.
Why is $.50 so focal here?

12:02.500 --> 12:08.130
Why is $.50 so focal?

12:08.130 --> 12:09.090
It's not a trick question.

12:09.090 --> 12:12.140
I'm just asking you why do
people think $.50 is so focal?

12:12.139 --> 12:15.689
I think it's typical that
people end up offering around

12:15.690 --> 12:18.520
$.50, why?
It sounds fair, it seems fair.

12:18.520 --> 12:20.250
There's some notion of fairness.

12:20.250 --> 12:23.280
It's not clear by the way,
what ethical principal is

12:23.279 --> 12:25.839
involved here.
It's not clear that if you're

12:25.842 --> 12:29.122
walking along the streets and
you happen to find a dollar at

12:29.121 --> 12:32.291
your feet that you should pick
it up and anyone else you'd

12:32.290 --> 12:35.570
happen to see at that moment you
should give $.50 too.

12:35.570 --> 12:37.270
That's essentially what the
situation is,

12:37.269 --> 12:39.859
and you just chanced upon this
dollar that I just gave you.

12:39.860 --> 12:42.220
It isn't clear that there's any
particularly great moral claim

12:42.218 --> 12:45.228
to give it to someone else,
but I think people read this as

12:45.225 --> 12:48.505
a situation about splitting a
cake in an environment of

12:48.508 --> 12:50.088
distributional justice.

12:50.090 --> 12:51.620
They view it as a larger
picture.

12:51.620 --> 12:54.420
Is that right?
So it turns out there's a large

12:54.419 --> 12:55.289
literature on this.

12:55.289 --> 12:58.199
There's a large experimental
literature on the ultimatum

12:58.199 --> 12:59.979
game.
And there's an even larger

12:59.983 --> 13:03.143
literature on an even simpler
game in which I give people a

13:03.136 --> 13:04.886
dollar,
I say you can give whatever

13:04.893 --> 13:07.503
share you want to the other
person, and they don't even get

13:07.501 --> 13:08.941
a chance to accept or reject.

13:08.940 --> 13:10.410
That's called a dictator game.

13:10.409 --> 13:13.379
In the dictator game,
literally, you're just simply

13:13.378 --> 13:16.998
given a dollar and you can give
whatever share you want to the

13:17.000 --> 13:19.330
other person.
It turns out that even in the

13:19.327 --> 13:21.137
dictator game,
people give quite a lot of

13:21.144 --> 13:23.784
money and that suggests that
there really is some notion of

13:23.778 --> 13:26.368
fairness or some notion of
distributional justice going on

13:26.367 --> 13:29.457
in people's heads here,
rightly or wrongly.

13:29.460 --> 13:33.930
So one thing this should tell
us is, even in extremely simple

13:33.927 --> 13:38.537
games, we should be a little bit
careful about reading backward

13:38.542 --> 13:43.012
induction into what's going to
happen in the real world.

13:43.009 --> 13:44.789
Part of this is because,
as we mentioned the very first

13:44.785 --> 13:46.325
day of the class,
people care about other things

13:46.331 --> 13:49.451
than just the obvious payoffs,
and part of it is about more

13:49.447 --> 13:52.727
complicated things like
reputation and so on.

13:52.730 --> 13:56.100
All right, having said that,
let's nevertheless for the

13:56.098 --> 13:58.908
purpose of today,
act as if we are going to do

13:58.905 --> 14:01.905
backward induction,
and let's embed this into a

14:01.914 --> 14:03.784
slightly more complicated game.

14:03.779 --> 14:06.289
So the more complicated game is
as follows.

14:06.290 --> 14:15.790


14:15.789 --> 14:24.259
So we're going to have a two
period bargaining game.

14:24.259 --> 14:25.989
In this two period,
bargaining game,

14:25.988 --> 14:28.358
the beginning of the game is
exactly the same.

14:28.360 --> 14:36.140
So there's a dollar on the
table and Player 1 makes an

14:36.141 --> 14:40.201
offer to 2.
And once again we can call this

14:40.203 --> 14:43.263
offer S and 1 - S,
just the same as before.

14:43.259 --> 14:49.019
And, just as before,
Player 2 can accept the offer,

14:49.016 --> 14:55.576
and if they accept the offer
then this is indeed what they

14:55.578 --> 14:58.698
get.
But now if 2 rejects,

14:58.700 --> 15:03.940
which is 2's other
alternative--if Player 2 rejects

15:03.938 --> 15:07.708
the offer then we flip the
roles.

15:07.710 --> 15:12.010
We play the game again but we
flip the roles.

15:12.009 --> 15:15.059
So we go to stage two,
everything we've said up here

15:15.062 --> 15:15.962
is stage one.

15:15.960 --> 15:19.600


15:19.600 --> 15:29.950
And down here we go to stage
two, and in stage two,

15:29.952 --> 15:37.822
Player 2 gets to make an offer
to 1.

15:37.820 --> 15:43.410
And once again we can call
this--we better be careful.

15:43.409 --> 15:46.989
Let's put ones here just to
indicate we're in the first

15:46.985 --> 15:50.225
round and twos to indicate we're
in the second.

15:50.230 --> 15:55.560
So they make an offer is S2 and
(1-S2) where S2 is that that

15:55.564 --> 16:00.994
goes to Player 1 and (1-S2) is
that that goes to Player 2.

16:00.990 --> 16:06.180
So 2 gets to make an offer to
Player 1 and now 1 can accept or

16:06.183 --> 16:10.373
reject.
If she accepts then she gets

16:10.372 --> 16:12.942
her share from here.

16:12.940 --> 16:22.220
The offer is accepted,
and if she rejects then we get

16:22.224 --> 16:25.564
nothing.
So this game is exactly the

16:25.561 --> 16:28.721
same as playing the previous
game, except we flip roles,

16:28.721 --> 16:30.791
but we're going to add one
catch.

16:30.789 --> 16:35.699
The catch is this,
in the first round the money on

16:35.703 --> 16:40.423
the table is $1,
but if we end up going into the

16:40.415 --> 16:43.925
second round,
so the first offer is not

16:43.930 --> 16:46.830
accepted and we go into the
second round,

16:46.827 --> 16:49.287
then part of the money is lost.

16:49.289 --> 16:53.319
In particular,
we'll assume that the money on

16:53.316 --> 16:55.326
the table is δ.

16:55.330 --> 16:59.590
If you think of δ
as being just some number less

16:59.586 --> 17:04.006
than 1--so if you want a
concrete example think of this

17:04.006 --> 17:08.276
as being $.90.
So the idea here is if you get

17:08.279 --> 17:11.639
into the second round,
time has past,

17:11.640 --> 17:14.220
it's costly,
and so money in the second

17:14.220 --> 17:16.940
round is--I think of it as money
in the second round as being

17:16.943 --> 17:19.583
worth less or could actually
think of this cake being eaten

17:19.576 --> 17:21.566
up,
some of it's thrown away,

17:21.570 --> 17:22.940
some of it's wasted.

17:22.940 --> 17:25.230
Everyone understand the game?

17:25.230 --> 17:28.260
So this is very similar to the
previous game,

17:28.263 --> 17:32.053
but we've got this second stage
coming in, and we've got

17:32.054 --> 17:33.644
discounting.

17:33.640 --> 17:38.840
So this is the idea of
discounting.

17:38.839 --> 17:42.549
How many of you have heard the
term discounting before?

17:42.549 --> 17:46.859
You probably saw it in a
finance class or a macro class

17:46.859 --> 17:50.929
where we think about there being
a value of time.

17:50.930 --> 17:53.230
Money today is worth more than
money tomorrow,

17:53.234 --> 17:56.154
partly because you could put
the money today into the bank

17:56.152 --> 17:59.262
and it could earn interest,
partly because you're simply

17:59.264 --> 18:02.044
impatient to get that money and
go and have lunch,

18:02.039 --> 18:04.259
particularly on the day in
which the clocks changed.

18:04.259 --> 18:09.469
Okay, so let's try this game
again and let's just play it for

18:09.472 --> 18:12.342
real, so let's come down again.

18:12.340 --> 18:13.430
Everyone understand the game?

18:13.430 --> 18:16.560
Basically the same rules except
we're just flipping around and

18:16.563 --> 18:19.033
with the possibility that the
cake may shrink.

18:19.029 --> 18:21.309
Let's see what people have
learned, so who were our first

18:21.305 --> 18:24.585
pair?
Our first pair were Equia and

18:24.586 --> 18:28.566
Noah all right.
So Equia, what are you going to

18:28.573 --> 18:30.523
offer?
You're Player 1 here but if

18:30.523 --> 18:33.743
your offer is rejected Noah's
going to get to make an offer to

18:33.740 --> 18:35.150
you.
All right so what are you going

18:35.149 --> 18:35.819
to offer this time?

18:35.819 --> 18:42.709
Student: $25.99,
so $.25, $.46.

18:42.710 --> 18:44.010
Professor Ben Polak:
$.46 okay,

18:44.012 --> 18:45.912
so he gets $.46 if he accepts
the offer, is that right?

18:45.913 --> 18:46.623
Student: Yes.

18:46.617 --> 18:48.337
Professor Ben Polak:
$.46 if he accepts the

18:48.342 --> 18:49.352
offer.
Student: I accept that.

18:49.346 --> 18:50.126
Professor Ben Polak:
He accepts that,

18:50.129 --> 18:50.529
okay that was easy.

18:50.529 --> 18:58.919
So Equia got $.54 and Noah got
$.46.

18:58.920 --> 19:00.500
Who was our second pair?

19:00.500 --> 19:02.690
So that was,
I've forgotten,

19:02.692 --> 19:05.942
Anish right and?
Student: Gary.

19:05.940 --> 19:07.170
Professor Ben Polak:
Gary.

19:07.172 --> 19:09.062
So Anish what are you going to
offer this time?

19:09.059 --> 19:09.939
Student: I'll offer
$.43.

19:09.943 --> 19:11.453
Professor Ben Polak:
$.43, you're going to push

19:11.454 --> 19:12.314
the envelope a little bit.

19:12.309 --> 19:12.889
Student: All right.

19:12.889 --> 19:13.709
Professor Ben Polak:
All right,

19:13.714 --> 19:14.454
that one got accepted as well.

19:14.450 --> 19:17.010
Okay, so people are converging
here.

19:17.009 --> 19:22.389
What about our third offerer,
receiver it was Courtney and?

19:22.390 --> 19:23.000
Student: Danny.

19:22.996 --> 19:23.906
Professor Ben Polak:
Danny.

19:23.906 --> 19:26.046
So Courtney?
Student: $.30.

19:26.054 --> 19:30.684
Professor Ben Polak:
$.30, so it's $.30 for him.

19:30.680 --> 19:31.640
Student: I'll accept.

19:31.640 --> 19:33.180
Professor Ben Polak:
Three acceptances,

19:33.184 --> 19:35.154
all right.
Let's find out something here,

19:35.153 --> 19:37.493
so I was hoping to get into the
second round.

19:37.490 --> 19:42.640
Okay so you accepted,
that's fine--no chicken sounds

19:42.641 --> 19:45.861
around the room.
So, it's Danny right?

19:45.861 --> 19:47.331
Student: Yeah.

19:47.329 --> 19:49.369
Professor Ben Polak:
So Danny had you

19:49.369 --> 19:52.259
rejected--you acceptedbut had
you accepted what would you have

19:52.262 --> 19:53.782
offered in the second round?

19:53.779 --> 19:54.949
Student: $.45.

19:54.948 --> 19:57.338
Professor Ben Polak:
$.45, all right,

19:57.340 --> 20:00.620
and would you have accepted
that in the second round if $.30

20:00.623 --> 20:02.073
hadn't been accepted?

20:02.069 --> 20:02.869
Courtney: Yes.

20:02.873 --> 20:04.793
Professor Ben Polak:
Okay, so you might have

20:04.788 --> 20:05.858
done better it turns out.

20:05.859 --> 20:08.289
Let's go back through to the
other rejections,

20:08.291 --> 20:09.751
to the other acceptances.

20:09.750 --> 20:13.810
So my second couple you offered
$.43, is that right?

20:13.810 --> 20:15.610
You said yes to $.43.

20:15.609 --> 20:18.719
Had you rejected $.43 what
would you have offered back in

20:18.720 --> 20:20.080
return?
Student: $.43.

20:20.084 --> 20:22.354
Professor Ben Polak:
$.43, the same thing back.

20:22.350 --> 20:24.540
Would you have accepted it?

20:24.539 --> 20:25.299
Student: He gets $.47?

20:25.302 --> 20:26.802
Professor Ben Polak: He
would have got $.47 in that

20:26.799 --> 20:27.809
case.
Student: Yeah.

20:27.810 --> 20:29.950
Professor Ben Polak: You
would have accepted that,

20:29.945 --> 20:32.295
okay.
Equia went first and she

20:32.295 --> 20:34.165
offered $.45 to Noah.

20:34.170 --> 20:35.230
And Noah had,
in fact, you rejected,

20:35.234 --> 20:36.304
what would you have offered
back?

20:36.299 --> 20:37.529
Student: I would have
also done $.45.

20:37.529 --> 20:39.019
Professor Ben Polak:
Same thing back and would

20:39.015 --> 20:39.725
you have accepted it?

20:39.730 --> 20:40.160
Student: Yes.

20:40.159 --> 20:40.889
Professor Ben Polak:
Okay.

20:40.890 --> 20:43.700
So we can see here that the
decision to accept or reject

20:43.702 --> 20:46.982
partly depends on what you think
the other side is going to do in

20:46.976 --> 20:48.946
the second round,
is that right?

20:48.950 --> 20:51.980
So here you are,
if you're in the middle of this

20:51.975 --> 20:54.025
game.
If you're Player 2 you've

20:54.029 --> 20:55.199
received an offer.

20:55.200 --> 20:57.400
None of these offers sounded
crazy.

20:57.400 --> 21:00.410
$.30 was the lowest one,
but none of them sounded crazy.

21:00.410 --> 21:02.960
And you're trying to decide
whether you should accept or

21:02.961 --> 21:03.891
reject this offer.

21:03.890 --> 21:08.020
And one thing you should have
in mind is what would I offer if

21:08.015 --> 21:11.125
I reject.
And will that offer that I then

21:11.125 --> 21:14.705
offer in the next round be
accepted or rejected,

21:14.712 --> 21:17.662
is that right?
So if we just work backwards we

21:17.660 --> 21:20.960
can see what you should offer in
the first round should be just

21:20.960 --> 21:23.940
enough to make sure it's
accepted knowing that the person

21:23.942 --> 21:27.192
who's receiving the offer in the
first round is going to think

21:27.189 --> 21:30.649
about the offer they're going to
make in the second round,

21:30.650 --> 21:32.920
and they're going to think
about whether you're going to

21:32.915 --> 21:35.135
accept or reject in the second
round, is that right?

21:35.140 --> 21:38.760
So that sounds like a bit of a
mouthful but that mouthful of

21:38.761 --> 21:41.341
reasoning is exactly backward
induction.

21:41.340 --> 21:43.130
It's exactly backward induction.

21:43.130 --> 21:45.430
It's saying:
to figure out what I should do

21:45.425 --> 21:48.865
in the first round or what I
should offer in the first round,

21:48.869 --> 21:51.579
I need to figure out whether
Player 2 is going to accept or

21:51.579 --> 21:53.349
reject.
And to figure out whether he or

21:53.348 --> 21:55.828
she is going to accept or
reject, and I have to put myself

21:55.829 --> 21:58.329
in his or her shoes,
and figure out what he or she

21:58.328 --> 22:01.268
would offer if she did reject,
and what he or she thinks I

22:01.274 --> 22:03.604
would do if I got that second
round offer.

22:03.599 --> 22:06.389
Is that right?
All right, so let's try and

22:06.387 --> 22:10.057
analyze this as if backward
induction was going to work

22:10.058 --> 22:13.998
here, as if we didn't have to
worry about things like pride

22:14.002 --> 22:17.302
here.
So this is the game we're

22:17.298 --> 22:21.678
actually playing,
so let's keep that one and

22:21.680 --> 22:25.350
actually analyze it on the
board.

22:25.350 --> 22:29.700


22:29.700 --> 22:33.100
I want to walk us from a
largely mundane game of take it

22:33.102 --> 22:36.752
or leave it offers to a more
complicated game in which there

22:36.752 --> 22:38.982
can be several rounds of offers.

22:38.980 --> 22:43.120
But we're going to go slowly so
we'll start just with two

22:43.115 --> 22:43.775
rounds.

22:43.780 --> 22:48.400


22:48.400 --> 22:53.820
So first of all let's just look
at the stage one game,

22:53.820 --> 22:59.240
and let's keep in track what
the offer is and what the

22:59.240 --> 23:04.480
receiver.
This is the offerer and this is

23:04.482 --> 23:08.062
the receiver.
In the one stage game,

23:08.057 --> 23:12.487
the game only has one stage,
then we know from backward

23:12.493 --> 23:15.783
induction what the results
should be.

23:15.779 --> 23:18.119
It isn't what we'd find in the
lab, it isn't what we find in

23:18.117 --> 23:20.017
the classroom,
but we know what we should get.

23:20.019 --> 23:23.839
The offerer should offer to
keep everything essentially or

23:23.840 --> 23:28.130
maybe $.99 but let's call it $1
and the receiver gets nothing.

23:28.130 --> 23:30.530
So again I'm approximating a
little bit because it could be

23:30.526 --> 23:31.886
$.99 and a penny but who cares.

23:31.890 --> 23:35.630
Let's just call it a $1 or
nothing if it's a one stage

23:35.626 --> 23:38.576
game.
So now let's consider a two

23:38.577 --> 23:41.637
stage game.
In the two stage game the

23:41.636 --> 23:46.426
person who's making the offer in
the first stage needs to look

23:46.434 --> 23:49.934
forward,
anticipate what would happen if

23:49.933 --> 23:55.373
her offer was rejected by Player
2 and Player 2 went forward into

23:55.370 --> 23:57.070
the second stage.

23:57.070 --> 23:59.880
Is that right?
So in the two stage game,

23:59.876 --> 24:04.576
in the first stage of the two
stage game, the person making

24:04.579 --> 24:09.529
the offer wants to anticipate
what the receiver would offer in

24:09.525 --> 24:14.385
the second round were the
receiver to reject her offer.

24:14.390 --> 24:18.050
But we can do that by backward
induction.

24:18.049 --> 24:22.559
We know that in the second
round if the receiver rejects

24:22.564 --> 24:27.904
the offer, then the second round
of the two stage game is what?

24:27.900 --> 24:30.560
It's a one stage game,
and we've just argued,

24:30.559 --> 24:33.399
at least if we believe in
backward induction,

24:33.400 --> 24:36.860
in that case,
Player 2 who is then the

24:36.863 --> 24:40.423
offerer, will offer $1 and
Player 1,

24:40.420 --> 24:44.490
who is now the receiver,
will accept it and get nothing.

24:44.490 --> 24:49.650
So Player 1 in the first round
of the two stage game wants to

24:49.650 --> 24:54.810
make an offer that's just enough
to get Player 2 to accept it

24:54.811 --> 24:59.981
now.
So let's think about this.

24:59.980 --> 25:08.520
So if Player 1 offers 2
something more than what?

25:08.519 --> 25:13.229
Tomorrow Player 2 can get $1
but that's $1 tomorrow.

25:13.230 --> 25:16.610
So $1 tomorrow is worth how
much today if we're discounting?

25:16.610 --> 25:18.640
It's just worth δ right.

25:18.640 --> 25:19.580
It's just worth δ.

25:19.579 --> 25:23.989
So if Player 1 offers Player 2
more than δ

25:23.994 --> 25:28.314
x $1, which is what Player 2
can get tomorrow,

25:28.312 --> 25:30.522
then 2 will accept.

25:30.520 --> 25:34.350


25:34.349 --> 25:40.049
If Player 1 offers 2 less than
δ x $1--because you can get

25:40.053 --> 25:45.573
a $1 tomorrow but that's only
worth $δ--a $1 tomorrow is

25:45.572 --> 25:50.082
worth just $δ
today--then 2 will reject.

25:50.079 --> 25:55.159
So the offer has to be exactly
enough to get accepted,

25:55.161 --> 25:57.751
which is exactly $δ.

25:57.750 --> 26:05.380
So Player 2 knows that she can
get $1 tomorrow so you need to

26:05.380 --> 26:12.250
offer her at least $δ
today to make it as good for

26:12.247 --> 26:16.187
her as getting $1 tomorrow.

26:16.190 --> 26:21.840
So we know the receiver must be
offered at least $δ

26:21.843 --> 26:27.503
tomorrow, which means the
offerer is going to keep $[1 -

26:27.496 --> 26:31.976
δ].
So in the first round of the

26:31.981 --> 26:36.861
two stage game,
Player 1 should offer $[1 -

26:36.858 --> 26:40.338
δ]
for herself and $δ

26:40.341 --> 26:46.261
for Player 2 and Player 2
should accept that because

26:46.263 --> 26:52.653
$δ dollars today is as good
as $1 tomorrow.

26:52.650 --> 26:55.080
Now, another way to see that is
in a picture,

26:55.084 --> 26:56.804
so let's just draw a picture.

26:56.800 --> 27:00.520


27:00.519 --> 27:05.879
Let's put the payoff of Player
1 here and the payoff of Player

27:05.884 --> 27:08.464
2 on this axis.
And we're going to assume that

27:08.455 --> 27:10.605
they're just going to maximize
dollars where there's no pride

27:10.605 --> 27:14.375
in here.
And if we just look at the one

27:14.377 --> 27:19.667
stage game, we're simply looking
at this line.

27:19.670 --> 27:23.390
The offers in the one stage
game: it could be that Player 1

27:23.389 --> 27:26.979
gets everything herself and
gives nothing to Player 2,

27:26.980 --> 27:29.860
it could be that Player 2 keeps
everything, ends up getting

27:29.862 --> 27:31.802
everything and Player 1 gets
nothing,

27:31.799 --> 27:34.889
and it could be any combination
in between.

27:34.890 --> 27:38.550
We argued by backward
induction--although not in

27:38.550 --> 27:42.690
reality--in backward induction,
in the one stage game,

27:42.690 --> 27:46.540
Player 1 makes an offer to
Player 2 which is kind of an

27:46.544 --> 27:50.524
insulting offer.
Player 1 says I get everything

27:50.521 --> 27:52.271
and you get a $.01.

27:52.270 --> 27:53.610
So this is the one stage game.

27:53.610 --> 27:59.330


27:59.329 --> 28:03.469
In the two stage,
if things are settled in the

28:03.467 --> 28:09.167
first stage this line represents
the possible divisions between

28:09.169 --> 28:11.559
Player 1 and Player 2.

28:11.559 --> 28:14.649
But if we end up going into the
second stage,

28:14.646 --> 28:16.046
the pie is shrunk.

28:16.049 --> 28:19.939
The pie is shrunk,
instead of going from $1 to $1,

28:19.941 --> 28:22.801
it goes from $δ1 to
$δ1.

28:22.799 --> 28:25.369
Or if you like,
if δ is .9 it goes from

28:25.372 --> 28:28.212
$.9 to $.9.
So let's draw that line in.

28:28.210 --> 28:33.920
So if we head into the second
stage, we'll end up being here,

28:33.917 --> 28:37.817
and this goes from $δ
here to $δ

28:37.817 --> 28:43.617
here where these dollars are
being evaluated at time one.

28:43.620 --> 28:45.430
All right so the pie has shrunk.

28:45.430 --> 28:50.190
If we get into the second stage
then, by backward induction,

28:50.193 --> 28:53.023
Player 2 is in an ultimatum
game,

28:53.019 --> 28:55.569
Player 2 will be making the
offer and Player 2 says:

28:55.566 --> 28:58.506
whatever cake is left I'm going
to take all of it and you're

28:58.512 --> 28:59.962
just going to get a $.01.

28:59.960 --> 29:06.960
So if we get into the second
stage then Player 2 will make

29:06.959 --> 29:10.029
this offer to Player 1.

29:10.029 --> 29:14.719
Player 2 will say I'm going to
keep the whole of the pie,

29:14.718 --> 29:18.818
which in first period dollars
is worth $δ.

29:18.819 --> 29:21.239
So I'm going to end up with a
payoff of δ

29:21.236 --> 29:24.776
and you're going to end up with
a payoff of essentially nothing.

29:24.779 --> 29:29.819
Player 2 knows that they can
therefore get at least

29:29.815 --> 29:33.935
$δ--or $δ
in current day dollars

29:33.943 --> 29:37.673
worth--from rejecting your
offer.

29:37.670 --> 29:42.210
Since they can get at least
$δ current day offers from

29:42.209 --> 29:46.069
rejecting your offer,
the lowest offer you can make

29:46.074 --> 29:49.924
to them is an offer that gives
them at least $δ.

29:49.920 --> 29:54.310
So the offer you're going to
make is this offer:

29:54.314 --> 29:57.124
this is the two stage offer.

29:57.120 --> 29:59.810
It happens in the first stage.

29:59.809 --> 30:03.329
Player 1 makes an offer that
gives Player 2 what Player 2

30:03.329 --> 30:07.529
could get in the second round,
so gives Player 2 $δ

30:07.525 --> 30:10.755
and keeps $[1 - δ]
for herself.

30:10.760 --> 30:12.060
Everyone understand the picture?

30:12.059 --> 30:15.279
So this picture is just
corresponding to this table.

30:15.279 --> 30:18.329
The thing people tend to get
confused about here,

30:18.326 --> 30:21.876
I think, is they get confused
between current dollars and

30:21.880 --> 30:25.200
discounted dollars,
so we're going to do all the

30:25.198 --> 30:28.648
analysis here in terms of the
first period dollars,

30:28.650 --> 30:31.090
dollars tomorrow are going to
be worth δ.

30:31.089 --> 30:33.619
There's a hand up,
can I get a shouting out?

30:33.619 --> 30:36.399
Yeah?
Student: [Inaudible]

30:36.403 --> 30:39.973
Professor Ben Polak:
Yes, sorry.

30:39.970 --> 30:47.850
So this is the outcome if it
was a one stage game and this is

30:47.853 --> 30:53.243
the outcome if it was a two
stage game.

30:53.240 --> 30:54.900
The offer is made and accepted.

30:54.900 --> 30:59.790
Let's roll it forward,
let's look at a three stage

30:59.791 --> 31:02.441
game.
Let's keep this picture handy

31:02.435 --> 31:04.845
and think about a three stage
game.

31:04.849 --> 31:08.889
So the beginning of the game is
the same.

31:08.890 --> 31:12.050
We're going to look at three
stage bargaining,

31:12.052 --> 31:16.132
and the rules in three stage
bargaining are pretty much the

31:16.129 --> 31:21.559
same as in two stage bargaining,
but now there's two possible

31:21.564 --> 31:26.074
flips.
In three stage bargaining,

31:26.069 --> 31:31.009
in the first period,
in period one,

31:31.012 --> 31:39.592
1 makes the offer and if it's
accepted the game is over.

31:39.589 --> 31:44.369
In period two,
if we reject,

31:44.365 --> 31:54.615
then we go to period two when 2
makes the offer and if it's

31:54.623 --> 31:59.653
rejected now,
this time by Player 1,

31:59.646 --> 32:04.596
then we go to period three
where once again 1 makes the

32:04.595 --> 32:07.245
offer.
So you can see where we're

32:07.251 --> 32:11.411
heading, we're heading towards
an alternate offer bargaining

32:11.406 --> 32:12.826
model.
I'm going to make an offer,

32:12.825 --> 32:14.505
Jake's going to either accept
or reject, then he'll make an

32:14.512 --> 32:15.532
offer and we'll flip to and fro.

32:15.529 --> 32:18.739
There's a question,
let me try and get a mike out

32:18.744 --> 32:19.954
to the question.

32:19.950 --> 32:23.520


32:23.519 --> 32:25.819
Yeah?
Student: I have a

32:25.818 --> 32:29.308
question about the two player
game, if δ

32:29.305 --> 32:34.135
is the best that Player 2 can
get tomorrow then why wouldn't 1

32:34.140 --> 32:37.390
offer Player 2 δ
discounted by δ

32:37.390 --> 32:39.810
today?
Professor Ben Polak:

32:39.809 --> 32:41.569
Good.
Right so I think I was

32:41.574 --> 32:44.254
confusing about it,
so let me make it clear.

32:44.250 --> 32:47.800
So tomorrow Player 2 can get
everything, everything that

32:47.799 --> 32:50.459
there is.
So whatever pie is left

32:50.457 --> 32:53.587
tomorrow Player 2 can get all of
it.

32:53.589 --> 32:59.319
So call that pie tomorrow 1 and
evaluate it in period one

32:59.315 --> 33:02.685
dollars as being worth $δ.

33:02.690 --> 33:05.120
Does that make sense?

33:05.119 --> 33:08.219
Okay, so I think I misspoke on
that, so let me say it again.

33:08.220 --> 33:11.730
So every period there's this
pie and every period,

33:11.732 --> 33:14.602
if it was the last period of
the game,

33:14.599 --> 33:18.419
the person making the offer is
going to get the whole pie,

33:18.422 --> 33:21.442
but if I view that pie tomorrow
from today,

33:21.440 --> 33:25.870
a pie of $1 tomorrow is only
worth $δ

33:25.874 --> 33:28.734
today.
A pie of $1 the day after

33:28.734 --> 33:33.204
tomorrow is only worth $δ
tomorrow and $δ²

33:33.203 --> 33:35.963
today and so on.
So that's the way in which

33:35.957 --> 33:37.667
we're going to do discounting
here.

33:37.670 --> 33:42.110
Good, all right.
So in this game,

33:42.114 --> 33:45.714
if 1 makes an offer,
if it's accepted it's over.

33:45.710 --> 33:50.780
If it's accepted then we're
done.

33:50.779 --> 33:52.359
And if this offer's accepted
then we're done.

33:52.359 --> 33:53.919
And if this offer's accepted
then we're done.

33:53.920 --> 33:59.490
And in the third round,
if it's rejected then both

33:59.489 --> 34:01.989
players get nothing.

34:01.990 --> 34:06.250
Once again we're going to
assume that the players are

34:06.252 --> 34:08.532
discounting.
So what does it mean to say

34:08.530 --> 34:09.410
they're discounting?

34:09.409 --> 34:13.639
It means that dollars in period
one are worth dollars,

34:13.639 --> 34:17.549
dollars in period two are
discounted by δ,

34:17.550 --> 34:22.320
and dollars in period three are
discounted by δ

34:22.324 --> 34:26.354
x δ, or if you like by
δ².

34:26.349 --> 34:29.059
Just to put this into real
notions of money,

34:29.062 --> 34:31.462
if you think of δ
as being 90%,

34:31.460 --> 34:36.430
then $1 in period one is worth
$1, a $1 in period two viewed

34:36.432 --> 34:42.552
from period one is worth $.90,
and a $1 in period three viewed

34:42.545 --> 34:45.955
from period one is worth $.81.

34:45.960 --> 34:49.230
Okay, so what do we think is
going to happen here?

34:49.230 --> 34:51.650
Well, once again we can do
backward induction.

34:51.650 --> 34:59.500


34:59.500 --> 35:00.790
Here we are in our picture.

35:00.790 --> 35:05.000


35:05.000 --> 35:07.460
Let's look at the three-stage
game.

35:07.460 --> 35:11.090
Once again, when we analyze,
as always when we analyze these

35:11.092 --> 35:14.912
games using backward induction,
we want to start at the end.

35:14.909 --> 35:18.739
If we start at the end,
we know that the last stage,

35:18.735 --> 35:22.925
that's the third stage of the
three stage game looks like

35:22.934 --> 35:25.644
what?
It looks like a one stage game.

35:25.639 --> 35:30.539
In the one stage game we know
the offerer will get everything.

35:30.539 --> 35:33.919
Say it again,
so the last stage of the three

35:33.918 --> 35:38.478
stage game, we know the person
who makes the offer who this

35:38.475 --> 35:42.085
time will be Player 1 will get
everything.

35:42.090 --> 35:46.170
However, that everything is
only worth δ

35:46.171 --> 35:51.091
in period two dollars and it's
only worth δ²

35:51.088 --> 35:53.498
in period one dollars.

35:53.500 --> 35:57.410
So in period one,
in the first period of this

35:57.409 --> 36:02.649
game, we know that if their
offer is rejected we know what's

36:02.652 --> 36:05.222
going to happen.
Say it again,

36:05.215 --> 36:07.865
in the first period of this
three stage game,

36:07.873 --> 36:11.743
if the offer is rejected then
we'll go into a two stage game,

36:11.739 --> 36:16.569
and we already know what
happens in a two stage game.

36:16.570 --> 36:19.890
In a two stage game,
the person who gets to make an

36:19.890 --> 36:23.540
offer gets $[1 - δ]
and the person who receives the

36:23.542 --> 36:25.072
offer gets $δ.

36:25.070 --> 36:29.440
So we know in the first stage
of the game that the person who

36:29.440 --> 36:33.660
receives the offer always has
the outside option of saying,

36:33.664 --> 36:37.574
no I reject.
And we know that that person

36:37.565 --> 36:40.785
tomorrow will get $[1 - δ].

36:40.789 --> 36:46.019
But $[1 - δ]
tomorrow is worth how much

36:46.017 --> 36:51.007
today?
It's worth $δ[1 - δ].

36:51.010 --> 36:56.260
Tomorrow they're going to get
$[1 - δ],

36:56.255 --> 37:01.495
so today that's worth
$δ[1--δ].

37:01.500 --> 37:05.560
So the offer I have to make in
the first round to make sure

37:05.564 --> 37:09.494
that the other person accepts it
has to be just better in

37:09.488 --> 37:12.028
discounted dollars,
than what they're going to get

37:12.030 --> 37:14.500
tomorrow.
They're going to get $[1 -

37:14.495 --> 37:17.305
δ]
tomorrow, so I have to give

37:17.313 --> 37:21.623
them $δ[1--δ]
today, which means I keep for

37:21.623 --> 37:24.693
myself $[1 - δ[1--δ]].

37:24.690 --> 37:26.910
And if you don't like the
algebra let's look at the

37:26.914 --> 37:28.854
picture.
In the picture,

37:28.854 --> 37:33.044
in the one stage game,
this is the offer.

37:33.039 --> 37:35.559
In the two stage game,
we know if we get to the second

37:35.559 --> 37:38.549
round, Player 2 gets everything
so we have to give him that much

37:38.554 --> 37:41.924
today.
And if we get into the third

37:41.916 --> 37:46.866
round, now we're looking at
δ² here and

37:46.865 --> 37:49.025
δ² here.

37:49.030 --> 37:52.220


37:52.219 --> 37:55.929
We know that if we get into the
third round the person who makes

37:55.928 --> 37:58.928
the offer in the third round
will get everything,

37:58.929 --> 38:02.239
so we can actually work our way
along.

38:02.239 --> 38:05.299
In the third round,
the person who makes the third

38:05.300 --> 38:08.710
offer will get everything,
so in the second round you'd

38:08.711 --> 38:11.971
have to give them that much,
so in the first round you'd

38:11.973 --> 38:14.053
have to give Player 2 that much.

38:14.050 --> 38:16.960
Say it again,
in period three,

38:16.961 --> 38:21.881
the person making the offer can
get everything.

38:21.880 --> 38:24.910
So in period two,
they must be getting δ

38:24.910 --> 38:28.910
times that, so in period one
you have to give them at least

38:28.905 --> 38:32.495
this much.
And this here is the offer

38:32.503 --> 38:36.153
you'd make in the three stage
game.

38:36.150 --> 38:40.280


38:40.280 --> 38:44.030
So in the picture we're just
doing a little zigzag;

38:44.030 --> 38:49.600
on the chart we're also always
working across the diagonal.

38:49.600 --> 38:53.080


38:53.079 --> 38:55.209
So we've done the one stage
game: the one stage game,

38:55.207 --> 38:57.087
the person making the offer
gets everything.

38:57.090 --> 39:00.250
In the two stage game,
the person making the offer

39:00.246 --> 39:04.106
offers just enough to get the
offer accepted which is $δ

39:04.111 --> 39:07.011
because that's what $1 is worth
tomorrow.

39:07.010 --> 39:08.560
In the three stage game,
the person making the offer

39:08.561 --> 39:09.961
makes just enough to get the
offer accepted,

39:09.960 --> 39:12.470
which is δ
times what the receiver would

39:12.465 --> 39:15.415
get tomorrow.
What they get tomorrow is $[1 -

39:15.418 --> 39:17.988
δ], so they get
$δ[1--δ]

39:17.990 --> 39:20.530
today.
How about the four stage game?

39:20.530 --> 39:22.540
Let's see if we can do that.

39:22.539 --> 39:27.789
So if we go to the four stage
game now, in the four stage game

39:27.791 --> 39:32.441
if the person receiving the
offer rejects the offer,

39:32.440 --> 39:37.850
then tomorrow they can get $[1
- δ[1--δ]].

39:37.849 --> 39:43.089
So I need to offer them enough
now in current dollars so they

39:43.085 --> 39:47.535
will prefer that to getting
$[1--δ[1 - δ]]

39:47.535 --> 39:48.665
tomorrow,

39:48.670 --> 39:52.250


39:52.250 --> 39:53.600
so how much must I offer them?

39:53.599 --> 39:58.599
I have to offer them at least
δ times that much,

39:58.604 --> 40:02.944
so I have to offer them
$[δ x [1 - δ

40:02.935 --> 40:09.345
x [1--δ]]].
Again, I'll keep the rest for

40:09.346 --> 40:18.006
myself so I'll get $[1 -
δ[1--δ[1--δ]]].

40:18.010 --> 40:23.480
And so the principle is always
give people just enough today so

40:23.478 --> 40:27.908
they'll accept the offer,
and just enough today is

40:27.914 --> 40:32.114
whatever they get tomorrow
discounted by δ.

40:32.110 --> 40:37.850


40:37.849 --> 40:40.269
So actually this backward
induction isn't so bad.

40:40.269 --> 40:42.909
What makes it a little bit
easier is you don't actually,

40:42.905 --> 40:45.775
when you go through an extra
stage of this you don't actually

40:45.781 --> 40:47.891
have to go all the way to the
beginning,

40:47.889 --> 40:50.729
you could actually start where
you were last time and just

40:50.733 --> 40:52.283
discount once more by δ.

40:52.280 --> 40:55.790


40:55.789 --> 41:00.119
Let's see if we can see any
kind of pattern emerging in this

41:00.120 --> 41:04.010
algebra, so let's just multiply
out these brackets.

41:04.010 --> 41:10.440
In the four stage game,
this thing is actually equal

41:10.439 --> 41:15.989
to--just multiplying
through--it's 1 - δ

41:15.986 --> 41:21.276
+ δ²
- δ³--I hope it is

41:21.281 --> 41:24.491
anyway.
That's what this is.

41:24.489 --> 41:34.029
And this thing is equal to
δ - δ²

41:34.027 --> 41:41.937
+ δ³,
just multiplying out the

41:41.941 --> 41:46.931
brackets.
Does anyone see a pattern

41:46.928 --> 41:49.838
emerging here in these offers?

41:49.840 --> 41:53.120
We had offers of 1 and then
1--δ.

41:53.119 --> 41:54.479
We could also multiply out this
one.

41:54.480 --> 42:01.070
It might be helpful to do so:
This is 1 - δ

42:01.067 --> 42:06.927
+ δ².
Anyone see a pattern what these

42:06.929 --> 42:08.949
offers look like?

42:08.950 --> 42:10.150
They kind of alternate.

42:10.150 --> 42:22.240
So let's have a look,
rather than do every stage.

42:22.240 --> 42:30.800


42:30.800 --> 42:33.470
Should I do one more stage to
see if we can see a pattern

42:33.473 --> 42:36.103
emerging or should I jump
straight to ten stages and see

42:36.098 --> 42:38.668
what happens?
Go straight to ten people say?

42:38.670 --> 42:43.010
Let's do one more,
nah, let's jump through to ten.

42:43.010 --> 42:47.700
So imagine that this game
actually had ten rounds,

42:47.696 --> 42:52.956
so this is a ten stage game,
and let's just continue our

42:52.957 --> 42:59.277
chart down here.
So here's--need a bit more

42:59.282 --> 43:06.072
space here--ten stages:
ten stage game.

43:06.070 --> 43:11.640
I'm going to continue my chart
and my chart says in the ten

43:11.641 --> 43:15.101
stage game what am I going to
get?

43:15.099 --> 43:22.729
So the offer is going to
be--it's going to be the same

43:22.733 --> 43:27.493
pattern--1 - δ
+ δ²

43:27.486 --> 43:35.116
- δ³…
+ δ^(8) - δ^(9),

43:35.120 --> 43:36.330
everyone see that?

43:36.329 --> 43:40.399
So what I'm doing is I'm
continuing the pattern from

43:40.401 --> 43:42.571
above.
So if I had ten stages,

43:42.568 --> 43:46.218
I always start with a 1,
the positive and negative terms

43:46.224 --> 43:49.154
just alternate,
and I have as many terms as 1

43:49.148 --> 43:51.008
minus the stage I'm in.

43:51.010 --> 43:53.870
So in the four stage game,
I ended up with δ³,

43:53.866 --> 43:56.616
so in the ten stage game I'll
end up at δ^(9).

43:56.620 --> 44:02.010
Everyone happy with that?

44:02.010 --> 44:03.580
So the four [error:
ten]

44:03.579 --> 44:06.989
stage offer is this slightly
ugly thing, 1 - δ

44:06.990 --> 44:09.450
+ δ²
- δ³…

44:09.446 --> 44:16.006
+ δ^(4) - δ^(5),
etc., + δ^(8) - δ^(9).

44:16.010 --> 44:19.700
That's a pretty ugly thing,
but fortunately some point in

44:19.703 --> 44:22.873
high school you learned how to
sum that thing.

44:22.870 --> 44:25.600
Do you remember what this is?

44:25.599 --> 44:27.859
What do you call objects like
this in high school?

44:27.860 --> 44:31.130
Anyone remember?
Objects like 1,

44:31.125 --> 44:34.195
δ, δ²,
δ³,

44:34.197 --> 44:37.357
δ^(4) what are they
called?

44:37.360 --> 44:39.510
They're called geometric series
right, they're called geometric

44:39.508 --> 44:44.538
series.
Anyone remember how to sum them?

44:44.539 --> 44:51.489
We know that S is equal to
this, this is what the offer is,

44:51.486 --> 44:54.476
if there is ten rounds.

44:54.480 --> 44:55.940
We know the offer is accepted.

44:55.940 --> 45:00.640
So the way to remember how to
sum it, the trick for summing it

45:00.636 --> 45:04.866
is to multiple both sides of
this equation by the common

45:04.870 --> 45:09.390
ratio,
so multiply both sides by

45:09.389 --> 45:11.569
δ.
So if I multiply this side by

45:11.572 --> 45:13.472
δ, I'm going to multiply
the other side by δ.

45:13.469 --> 45:18.809
And this 1 will become a
δ, this δ

45:18.812 --> 45:24.282
will become a δ²,
this δ²

45:24.278 --> 45:29.778
will become a δ³,
this δ³

45:29.783 --> 45:33.093
will become a
δ^(4)….

45:33.090 --> 45:38.420
There will be a -δ^(8)
coming from the seventh term.

45:38.420 --> 45:44.180
There will be a +δ^(9)
coming from the δ^(8) term,

45:44.181 --> 45:47.601
and there will be a
-δ^(10).

45:47.600 --> 45:48.860
Everyone okay with that?

45:48.860 --> 45:51.100
I just multiplied everything
through by δ

45:51.095 --> 45:53.425
and I just shifted along one
for convenience.

45:53.429 --> 45:55.639
What do I do now,
anyone remember?

45:55.640 --> 45:58.800
Add the two lines together.

45:58.800 --> 46:10.120
So by summing this side I get
[1 + δ]

46:10.115 --> 46:13.545
S^(10).
On the other side,

46:13.545 --> 46:16.115
what's kind of convenient is
everything cancels.

46:16.120 --> 46:21.010
The 1 comes through, I get a 1.

46:21.010 --> 46:22.810
These two terms cancel and
these two terms cancel,

46:22.812 --> 46:26.522
and these two terms cancel,
and so on and so on,

46:26.518 --> 46:33.378
all the way up to the end where
I get -δ^(10).

46:33.380 --> 46:35.440
Everyone okay.
All the other terms have

46:35.441 --> 46:39.481
cancelled out.
So now just sorting out my

46:39.477 --> 46:45.917
algebra a bit--I'm going to take
it on the other side--I'm going

46:45.915 --> 46:51.735
to have that the offer you
make--this is what you're going

46:51.739 --> 46:58.179
to get to keep--so the amount I
claim I should keep in the first

46:58.177 --> 47:02.977
round is [1 - δ^(10])/[1 +
δ].

47:02.980 --> 47:05.040
Just be a bit careful with the
notation here because it may be

47:05.036 --> 47:05.876
a little bit confusing.

47:05.880 --> 47:09.250
The 10 here doesn't mean to the
tenth power, it's just the offer

47:09.245 --> 47:11.645
in the tenth round [error:
ten-round game],

47:11.650 --> 47:15.520
whereas the 10 here really does
mean in the tenth power.

47:15.519 --> 47:20.549
So if we play this game for ten
rounds, the offer you'd make

47:20.546 --> 47:23.776
would be [1 - δ^(10)]/[1 +
δ]

47:23.783 --> 47:28.983
which means the amount you're
offering to the other side,

47:28.980 --> 47:41.010
which is 1 - S would be [δ
+ δ^(10)]/[1 + δ].

47:41.010 --> 47:44.800


47:44.800 --> 47:47.770
So to summarize where we are:
we started off by considering a

47:47.774 --> 47:50.154
very simple game,
a one stage take it or leave it

47:50.154 --> 47:51.734
offer.
We know that,

47:51.725 --> 47:56.185
in that take it or leave it
offer, Player 1 is going to

47:56.191 --> 48:01.321
claim everything for herself and
offer nothing to Player 2.

48:01.320 --> 48:05.610
Then we considered a two stage
game which was the same as the

48:05.605 --> 48:09.815
one stage game except that if
Player 2 rejects the offer,

48:09.820 --> 48:14.790
he--let's call player 2,
"he"--he gets to make an offer

48:14.789 --> 48:18.009
to Player 1 in the second
period.

48:18.010 --> 48:21.280
We know that in the second
period of that two stage game

48:21.276 --> 48:23.826
Player 2 can keep everything for
himself.

48:23.829 --> 48:27.669
Everything for himself tomorrow
is worth δ

48:27.666 --> 48:31.496
today, so you have to offer him
at least δ

48:31.503 --> 48:35.093
today and keep 1 - δ
for yourself.

48:35.090 --> 48:37.260
Then we looked at a three stage
game.

48:37.260 --> 48:40.000
In this three stage game,
if Player 2 rejects in the

48:39.997 --> 48:42.997
first round, Player 2 can make
you an offer in the second

48:43.003 --> 48:45.023
round,
but now if you're Player 1 and

48:45.017 --> 48:47.817
you reject in the second round
you get to make an offer to

48:47.823 --> 48:49.353
Player 2 in the third round.

48:49.349 --> 48:52.529
We argued that in the second
round of this game,

48:52.532 --> 48:56.592
if Player 2 rejects you in the
first round and makes an offer

48:56.594 --> 49:00.854
in the second round,
it'll be in a two stage game

49:00.850 --> 49:05.730
and they'll be able to keep 1 -
δ of the pie for

49:05.734 --> 49:08.464
themselves.
So you have to offer them at

49:08.461 --> 49:11.181
least δ x [1--δ]
today for them to accept the

49:11.183 --> 49:13.203
offer, keeping the rest for
yourself.

49:13.199 --> 49:15.029
Then we looked at a four stage
game.

49:15.030 --> 49:17.950
In the four stage game if
Player 2 rejects your offer,

49:17.945 --> 49:20.595
he can make you an offer,
but if you reject the offer you

49:20.604 --> 49:22.614
can make him an offer,
but if he rejects that offer he

49:22.612 --> 49:23.562
can make you an offer.

49:23.559 --> 49:27.229
And once again we asked how
much do I have to offer Player 2

49:27.233 --> 49:29.603
now for him to accept the offer
now?

49:29.599 --> 49:35.369
He knows that if he rejects the
offer, he can get this amount

49:35.366 --> 49:38.726
1--δ x [1--δ]
tomorrow.

49:38.730 --> 49:42.800
So I have to offer him δ
times that today,

49:42.803 --> 49:46.703
and once again I keep the same
for myself.

49:46.699 --> 49:48.029
That's just a summary of what
we did.

49:48.030 --> 49:49.930
And then what we did was we
cheated.

49:49.929 --> 49:51.719
We jumped to the tenth stage
[error: ten stages],

49:51.723 --> 49:53.333
just noticing that a pattern
had emerged,

49:53.329 --> 49:55.369
and we found that in the tenth
stage [error:

49:55.367 --> 49:57.447
ten-stage game],
this is the offer you'd make

49:57.452 --> 49:59.302
just according to the same
pattern.

49:59.300 --> 50:02.460
And it was this horrible thing,
and then we used a little bit

50:02.463 --> 50:04.733
of high school math to simply
this thing.

50:04.730 --> 50:08.510
And it turns out this is the
amount you keep for yourself and

50:08.508 --> 50:11.278
this is the offer you'll make to
Player 2.

50:11.280 --> 50:15.560
In each case I've accepted--Did
I make a math mistake?

50:15.559 --> 50:18.289
Thank you.
Let's put a superscript in

50:18.289 --> 50:21.879
here.
good.

50:21.880 --> 50:24.440
So what do we observe here?

50:24.440 --> 50:27.950
So the first thing to observe
is in the one stage game if we

50:27.954 --> 50:31.594
believe backward induction you
certainly want to be the person

50:31.588 --> 50:32.838
making the offer.

50:32.840 --> 50:36.360
In the one stage game,
in the ultimatum game,

50:36.360 --> 50:39.480
there's a huge first-mover
advantage.

50:39.480 --> 50:42.250
In the two stage game it's not
clear if you want to make the

50:42.250 --> 50:44.270
offer, it depends on how large
δ is,

50:44.269 --> 50:47.869
but if δ
is a big number like .9 you'd

50:47.867 --> 50:51.547
rather be the person receiving
the offer.

50:51.550 --> 50:56.190
In the three stage game,
it looks like you'd probably

50:56.187 --> 51:00.377
rather make the offer,
but it's not so clear.

51:00.380 --> 51:03.300
So where does it go to as we go
down the path,

51:03.304 --> 51:06.234
as it goes down towards the ten
stage game?

51:06.230 --> 51:10.450
It looks like in the ten stage
game you'd probably still prefer

51:10.454 --> 51:13.174
to make the offer than not,
but they're certainly much

51:13.170 --> 51:14.280
closer together than they were
before.

51:14.280 --> 51:18.440
Some of that initial bargaining
power has been washed out by the

51:18.440 --> 51:20.620
fact that there are ten stages.

51:20.619 --> 51:26.249
So let's try and push this just
a little bit harder.

51:26.250 --> 51:30.350
Instead of looking at the tenth
stage offer, what if we look at

51:30.346 --> 51:32.126
the infinite stage offer.

51:32.130 --> 51:41.430
So in principle we look at the
infinite stage of this game.

51:41.429 --> 51:43.239
So I can make you an offer,
you can say no and make me an

51:43.243 --> 51:44.963
offer, and then I can reject and
make you an offer,

51:44.960 --> 51:47.860
and then you can reject and
make me an offer,

51:47.861 --> 51:49.511
and so on and so forth.

51:49.510 --> 51:51.300
So we look at this term.

51:51.300 --> 51:54.260
If in principle and you can
make an infinite number of

51:54.258 --> 51:57.608
offers--so, what's this term
going to look like if I can make

51:57.607 --> 51:59.447
an infinite number of offers?

51:59.449 --> 52:04.779
So I claim it's going to look
like this [1 - δ^(∞)]

52:04.780 --> 52:07.230
/ [1 + δ]
and over here,

52:07.227 --> 52:11.507
at least it's going to converge
towards this.

52:11.510 --> 52:17.480
We'll be a bit more formal,
and over here we'll have

52:17.481 --> 52:22.751
[δ + δ^(∞)]
/ [1 + δ].

52:22.750 --> 52:26.490
However now I get a little bit
simpler.

52:26.490 --> 52:30.700
What is δ^(∞)?

52:30.699 --> 52:34.499
It's 0 right,
so .9 x .9 x .9 x .9 x .9 x.

52:34.500 --> 52:37.270
9 x .9 x .9 x .9 x …
is 0.

52:37.269 --> 52:45.569
So this last term disappears as
does this one,

52:45.569 --> 52:51.289
and we just get 1 / [1 +
δ]

52:51.286 --> 52:56.816
and δ / [1 + δ].

52:56.820 --> 53:00.900
So if we make alternating offer
bargaining--a bargaining game

53:00.902 --> 53:03.762
where in each round I make you
an offer,

53:03.760 --> 53:06.900
you can accept it or you can
reject and make me an offer,

53:06.897 --> 53:09.527
and we imagine there's no bound
to this game,

53:09.530 --> 53:13.920
it just goes on arbitrarily
long--then our prediction is

53:13.919 --> 53:17.429
that Player 1,
the person who makes the first

53:17.431 --> 53:22.381
offer will get 1 / [1 + δ]
of the initial pie and Player 2

53:22.380 --> 53:24.690
will get δ
/ [1 + δ]

53:24.694 --> 53:26.854
of the initial pie.

53:26.849 --> 53:29.969
Let's try and get a handle
about what those numbers are.

53:29.969 --> 53:34.419
So if you imagine these offers
can be made fairly rapidly,

53:34.422 --> 53:37.392
for example,
I can make offer today,

53:37.389 --> 53:39.729
you can make an offer back to
me in half an hour's time,

53:39.730 --> 53:42.160
and then I can make an offer
back to you in half an hour's

53:42.156 --> 53:44.486
time,
then it's reasonable to assume

53:44.487 --> 53:47.147
that the pie is not shrinking
very fast.

53:47.150 --> 53:51.140
The discount factor is not a
big deal here.

53:51.139 --> 53:54.009
So these offers can be made in
rapid succession,

53:54.010 --> 53:57.430
but we might think that δ
itself is approximately 1:

53:57.429 --> 53:59.749
the pie isn't shrinking very
fast.

53:59.750 --> 54:04.530
If δ is approximately 1,
and if we take δ

54:04.526 --> 54:10.156
to 1 here--the time isn't that
valuable given how rapidly we

54:10.163 --> 54:16.663
can make offers to and fro--then
what does this make this equal?

54:16.659 --> 54:20.019
In the case where δ
is equal to 1 what do we get?

54:20.019 --> 54:29.739
We get 1/2, which means this
will also be 1/2.

54:29.739 --> 54:32.109
So we learned something from
this which is kind of

54:32.110 --> 54:33.890
surprising.
If you do alternating

54:33.889 --> 54:36.569
offers--the sort of standard,
very natural game of

54:36.572 --> 54:39.912
bargaining--sort of the kind of
bargaining you might do in the

54:39.912 --> 54:41.332
bazaar,
in a market,

54:41.329 --> 54:44.749
or the kind of bargaining you
might imagine going on in

54:44.748 --> 54:48.288
negotiations between baseball
players or their agents and

54:48.293 --> 54:50.173
teams,
or general managers of

54:50.166 --> 54:53.316
teams--in which offers just go
to and fro and they go to and

54:53.323 --> 54:55.803
fro fairly rapidly,
and in principle they could

54:55.802 --> 54:57.252
make lots and lots of offers.

54:57.250 --> 55:02.600
In principle--what this moral
tells us is, in principle,

55:02.596 --> 55:08.616
we're going to end up with each
side splitting whatever the pie

55:08.624 --> 55:11.984
was equally.
Very, very different from the

55:11.978 --> 55:15.388
ultimatum game where all the
bargaining power was on the

55:15.389 --> 55:17.559
person who made the first offer.

55:17.560 --> 55:21.230
So what are the lessons here?

55:21.230 --> 55:25.050


55:25.050 --> 55:27.340
What can we conclude from this?

55:27.340 --> 55:31.820


55:31.820 --> 55:38.600
We've looked at alternating
offer bargaining,

55:38.597 --> 55:45.527
and we've concluded,
under special conditions,

55:45.529 --> 55:52.769
we've concluded that you get an
even split.

55:52.769 --> 55:56.089
You get an even share,
an even split,

55:56.088 --> 56:00.418
a fifty-fifty split if three
things are true.

56:00.420 --> 56:06.840
The first thing is there's
potentially infinitely many

56:06.841 --> 56:11.931
offers: potentially can bargain
forever.

56:11.930 --> 56:15.930


56:15.929 --> 56:21.619
And if discounting is not a big
deal.

56:21.619 --> 56:23.659
What does discounting really
not being a big deal means?

56:23.659 --> 56:26.709
It means those offers can be
made in rapid succession.

56:26.710 --> 56:33.870
So no discounting,
or if you like,

56:33.866 --> 56:38.596
rapid offers.
If you have to wait a year

56:38.596 --> 56:41.946
between every offer then that
discount factor would be a big

56:41.945 --> 56:44.235
deal.
But I actually made a third

56:44.238 --> 56:47.438
assumption, and I made it
without telling you.

56:47.440 --> 56:49.280
What was the third assumption I
made?

56:49.280 --> 56:53.400
I snuck the third assumption
past you without telling you.

56:53.400 --> 56:57.220
What was that third assumption?

56:57.220 --> 57:01.540
Let me get the mike here.

57:01.539 --> 57:03.779
So I claim I snuck a third
assumption.

57:03.780 --> 57:05.900
There's somebody,
let me start over here.

57:05.900 --> 57:07.070
What was the third assumption?

57:07.070 --> 57:08.580
Student: I don't know if
this is what you're looking for

57:08.584 --> 57:09.444
but they know how big the pie
is.

57:09.440 --> 57:11.160
Professor Ben Polak:
They know how big the pie is,

57:11.159 --> 57:12.449
that's true,
that's a big deal actually.

57:12.449 --> 57:14.269
That's true,
but there's something else

57:14.272 --> 57:16.322
going on here.
What is it?

57:16.320 --> 57:18.510
Student: We assume that
both players were rational.

57:18.510 --> 57:19.340
Professor Ben Polak:
We assumed that.

57:19.340 --> 57:21.220
That's true,
but we've kind of been assuming

57:21.215 --> 57:22.825
that throughout backward
induction.

57:22.829 --> 57:25.819
You're right none of this
backward induction would apply

57:25.822 --> 57:27.892
so cleanly if we didn't assume
that.

57:27.890 --> 57:31.250
What else do I assume?

57:31.250 --> 57:33.720
It's hidden actually,
I snuck it in.

57:33.719 --> 57:36.429
We assume the discount factor
is a constant,

57:36.430 --> 57:38.760
that's true,
but not just constant but

57:38.762 --> 57:40.922
something else.
You're on the right lines.

57:40.920 --> 57:42.560
They're the same.

57:42.559 --> 57:46.969
I've assumed here,
implicitly, I've assumed that

57:46.965 --> 57:50.335
both people are equally
impatient,

57:50.340 --> 58:01.070
they have the same discount
factor, δ_1 =

58:01.065 --> 58:05.155
δ_2.

58:05.160 --> 58:06.570
Why does that matter?

58:06.570 --> 58:08.750
Well let's just think about it
intuitively a second.

58:08.750 --> 58:11.870
Suppose that one of these
players is very,

58:11.867 --> 58:15.097
very impatient.
They need the money now.

58:15.099 --> 58:18.289
If it's cake,
they need the cake now.

58:18.289 --> 58:22.089
They're very impatient,
and the other person is very

58:22.094 --> 58:24.604
patient.
They can wait forever to get

58:24.598 --> 58:26.418
this bargain to come across.

58:26.420 --> 58:29.810
Who do you think is going to do
better, the patient player or

58:29.810 --> 58:31.110
the impatient player?

58:31.110 --> 58:34.740
The patient player is going to
do better.

58:34.739 --> 58:37.389
The way we ended up,
the way we did all this

58:37.390 --> 58:40.970
analysis is we assumed that
those discount factors were the

58:40.965 --> 58:43.265
same.
We assumed that each person was

58:43.266 --> 58:46.586
discounting time at the same
rate, perhaps because they were

58:46.590 --> 58:49.520
facing the same bank with the
same interest rate.

58:49.519 --> 58:52.279
But in practice,
often one side is going to be

58:52.279 --> 58:56.019
in a hurry to get the dispute
resolved, and the other side can

58:56.019 --> 58:57.429
sit around forever.

58:57.429 --> 59:00.449
In that world,
the side who could sit around

59:00.445 --> 59:03.035
forever is going to do much
better.

59:03.039 --> 59:08.189
Now, we're going to look at
relaxing this assumption and

59:08.187 --> 59:13.517
this assumption in particular
we're going to relax it on a

59:13.521 --> 59:15.581
homework exercise.

59:15.579 --> 59:20.519
So in your homework exercise
you're going to try redoing part

59:20.522 --> 59:25.132
of this analysis--good practice
anyway--but doing it in a

59:25.134 --> 59:29.504
setting where the discount
factors are different.

59:29.500 --> 59:33.840
So one thing we learned here
was: yes, you get an even split,

59:33.838 --> 59:37.018
but it depends on these three
assumptions.

59:37.019 --> 59:40.579
It's kind of important because
when you think about bargaining,

59:40.577 --> 59:44.137
I think a lot of people simply
assume intuitively that whatever

59:44.135 --> 59:46.965
the bargain is about,
people will eventually split in

59:46.968 --> 59:48.318
the middle.
When you're bargaining about a

59:48.316 --> 59:49.736
house, or the price you're going
to pay for a house,

59:49.739 --> 59:51.809
all these things,
you kind of implicitly have

59:51.807 --> 59:54.907
this assumption you're going to
end up splitting the difference.

59:54.909 --> 59:57.199
What I'm arguing here is you
will split the difference in

59:57.201 --> 1:00:00.501
this natural bargaining game,
but only under very special

1:00:00.496 --> 1:00:04.996
assumptions, and in particular,
the assumption of patience is

1:00:04.999 --> 1:00:05.899
critical.

1:00:05.900 --> 1:00:10.270


1:00:10.269 --> 1:00:12.949
There's another remarkable
thing here though,

1:00:12.951 --> 1:00:14.111
it's also hidden.

1:00:14.110 --> 1:00:16.210
So not only did we end up with
an even split,

1:00:16.213 --> 1:00:18.893
but something else remarkable
happened in this bargaining

1:00:18.889 --> 1:00:20.669
game.
What was the other thing?

1:00:20.670 --> 1:00:23.680
Somewhat amazingly,
a very unrealistic thing that

1:00:23.683 --> 1:00:25.883
occurred in this bargaining
game?

1:00:25.880 --> 1:00:27.220
See if you can spot it.

1:00:27.219 --> 1:00:30.809
So one thing was an assumption
I made and the other is actually

1:00:30.813 --> 1:00:32.033
a prediction.
Yeah.

1:00:32.030 --> 1:00:34.060
Student: The first offer
will be the offer that's

1:00:34.063 --> 1:00:35.103
accepted.
Professor Ben Polak:

1:00:35.097 --> 1:00:36.577
Good.
Did everyone see that?

1:00:36.579 --> 1:00:39.919
So in this bargaining game,
I set it up as alternating

1:00:39.923 --> 1:00:42.763
offer bargaining,
so the image you had in your

1:00:42.762 --> 1:00:44.342
mind was of haggling.

1:00:44.340 --> 1:00:47.250
I made an offer,
you guys thought about this

1:00:47.248 --> 1:00:48.838
offer.
Should I take this offer or not.

1:00:48.840 --> 1:00:49.930
Maybe I won't take this offer.

1:00:49.929 --> 1:00:53.559
You make an offer back to me,
and we kind of haggled to and

1:00:53.564 --> 1:00:54.824
fro.
But actually,

1:00:54.821 --> 1:00:58.551
in the equilibrium of the game,
none of that happened.

1:00:58.550 --> 1:01:00.250
That all happened in our mind.

1:01:00.250 --> 1:01:03.000
We thought about what offer we
would make, and we thought about

1:01:02.996 --> 1:01:05.646
what offer you would make back
to me if I made you this offer

1:01:05.653 --> 1:01:07.163
and you rejected it and so on.

1:01:07.159 --> 1:01:10.589
We did this backward induction
exercise but it was all in our

1:01:10.592 --> 1:01:16.232
heads.
In this game the actual

1:01:16.231 --> 1:01:24.571
prediction is:
the very first offer is

1:01:24.566 --> 1:01:29.476
accepted.
There's no haggling,

1:01:29.483 --> 1:01:32.033
there's no bargaining.

1:01:32.030 --> 1:01:37.250
Now that doesn't seem very
realistic, there's no haggling.

1:01:37.250 --> 1:01:45.140


1:01:45.139 --> 1:01:48.799
Backward induction suggests
that we should never see

1:01:48.799 --> 1:01:52.889
bargaining, never see the actual
process of bargaining.

1:01:52.889 --> 1:01:57.169
What you should see is an offer
is made and it's accepted.

1:01:57.170 --> 1:02:02.110
Now what is it about the real
world that allows for haggling

1:02:02.110 --> 1:02:05.040
to take place?
If this was a model of the real

1:02:05.036 --> 1:02:07.236
world and we believed in
backward induction,

1:02:07.237 --> 1:02:10.127
then we're done.
So why is it in fact in the

1:02:10.131 --> 1:02:13.621
real world we see people make
offers to and fro?

1:02:13.619 --> 1:02:18.479
What's different about the real
world than this model?

1:02:18.480 --> 1:02:21.980
Let's talk about it a bit.

1:02:21.980 --> 1:02:25.510
You must have all bargained for
something in the real world.

1:02:25.510 --> 1:02:27.500
None of you have probably
bargained for a house yet,

1:02:27.501 --> 1:02:29.611
but you might have bargained
for a car or something.

1:02:29.610 --> 1:02:32.520
In the real world you make
offers go to and fro,

1:02:32.515 --> 1:02:33.695
right?
What's going on?

1:02:33.699 --> 1:02:37.139
Why are we getting offers in
the real world whereas we don't

1:02:37.144 --> 1:02:45.104
in this game?
What are we missing?

1:02:45.099 --> 1:02:46.599
Student: In the real
world you don't actually know

1:02:46.597 --> 1:02:47.797
what the other person's discount
factor is,

1:02:47.800 --> 1:02:50.400
therefore, you have uncertainly
as to what your highest possible

1:02:50.398 --> 1:02:52.568
offer could be.
Professor Ben Polak:

1:02:52.570 --> 1:02:54.490
Good.
So in the real world,

1:02:54.487 --> 1:02:58.127
unlike the model on the board,
not only are those discount

1:02:58.131 --> 1:03:01.711
factors different but you
probably don't know how patient

1:03:01.710 --> 1:03:04.140
or impatient the other side is.

1:03:04.139 --> 1:03:06.969
You can get some ideas about
how patient or impatient the

1:03:06.967 --> 1:03:09.437
other side is by looking at
their characteristics,

1:03:09.442 --> 1:03:10.712
for example.
For example,

1:03:10.713 --> 1:03:12.803
if you know that the person
you're bargaining with over

1:03:12.800 --> 1:03:14.930
their car--you're trying to buy
their car--and they're a

1:03:14.925 --> 1:03:16.775
graduate student who's just a
got a job in,

1:03:16.780 --> 1:03:18.470
I don't know,
Uzbekistan or something,

1:03:18.468 --> 1:03:21.068
and they aren't going to be
taking their car with them,

1:03:21.070 --> 1:03:24.180
and they're leaving next week,
you know they're in a hurry.

1:03:24.179 --> 1:03:26.299
So there's times when you're
going to know something about

1:03:26.297 --> 1:03:28.717
other people's discount factors,
how patient they are,

1:03:28.716 --> 1:03:31.136
but lots of times you're not
going to know.

1:03:31.139 --> 1:03:35.189
So one thing is you just don't
know what the discount factor

1:03:35.187 --> 1:03:37.487
is.
By the way, what else might you

1:03:37.489 --> 1:03:39.449
not know about the other side?

1:03:39.450 --> 1:03:43.720
What else might you not know?

1:03:43.719 --> 1:03:45.869
Student: How big the
surplus is they were splitting.

1:03:45.869 --> 1:03:47.079
Professor Ben Polak:
Good.

1:03:47.080 --> 1:03:49.010
You might not know this good
that you're selling.

1:03:49.010 --> 1:03:52.500
We've been talking about the
big one pie which you're carving

1:03:52.501 --> 1:03:55.121
up, and everyone knows the size
of the pie.

1:03:55.119 --> 1:03:58.419
But in the real world,
I might not know this object

1:03:58.421 --> 1:04:01.591
that's being sold,
I might not know how much this

1:04:01.590 --> 1:04:04.100
object is worth to the other
side,

1:04:04.099 --> 1:04:08.049
and he or she may not know how
much it's worth to me.

1:04:08.050 --> 1:04:12.990
So that lack of information is
going to change the game

1:04:12.988 --> 1:04:15.538
considerably.
In particular,

1:04:15.536 --> 1:04:20.656
I might want to turn down some
offers in this game in order to

1:04:20.661 --> 1:04:23.351
appear like a patient person.

1:04:23.349 --> 1:04:26.559
Why I might want to turn down
some offers in the game in order

1:04:26.558 --> 1:04:29.498
to appear like somebody who
doesn't really value this all

1:04:29.503 --> 1:04:31.403
very much.
And in so doing,

1:04:31.400 --> 1:04:35.460
I'm going to try and get you to
make me a better offer.

1:04:35.460 --> 1:04:37.960
So what's going on in haggling
and bargaining,

1:04:37.959 --> 1:04:41.069
according to this model,
what's missing in this model,

1:04:41.070 --> 1:04:43.700
is the idea that you don't know
who it is you're bargaining

1:04:43.702 --> 1:04:45.542
with.
You don't know how much they

1:04:45.536 --> 1:04:48.726
value the objects in question.,
You don't know how impatient

1:04:48.731 --> 1:04:50.791
they are to get away with the
cash.

1:04:50.789 --> 1:04:54.849
So it's a big assumption here,
a very big assumption,

1:04:54.854 --> 1:04:57.204
is that everything is known.

1:04:57.199 --> 1:05:11.899
So both the size of the pie,
let's call it the value of the

1:05:11.895 --> 1:05:25.065
pie, and the value of time is
assumed to be known,

1:05:25.070 --> 1:05:27.810
but in the real world you
typically don't know the value

1:05:27.807 --> 1:05:31.627
of the pie on the other side,
and you typically don't know

1:05:31.631 --> 1:05:33.751
how much they value time.

1:05:33.750 --> 1:05:35.620
So that produces a whole
literature on bargaining,

1:05:35.616 --> 1:05:37.746
none of which we really have
time to do in this class,

1:05:37.750 --> 1:05:39.820
which is a pity because
bargaining's kind of important.

1:05:39.820 --> 1:05:42.300
So instead, I want to spend the
last five minutes just

1:05:42.298 --> 1:05:45.148
introducing, is it really worth
introducing a new topic in the

1:05:45.151 --> 1:05:46.181
last five minutes?

1:05:46.179 --> 1:05:50.159
No, I think it is,
let me talk a little bit more

1:05:50.160 --> 1:05:53.210
about bargaining rather than
that.

1:05:53.210 --> 1:05:58.050
So what does this suggest if
we're going out in the real

1:05:58.046 --> 1:06:00.856
world?
I'm actually taking this to

1:06:00.857 --> 1:06:03.517
reality.
So one thing it suggests is

1:06:03.521 --> 1:06:07.151
people for whom it's known that
they're going to be

1:06:07.153 --> 1:06:11.733
impatient--people for whom it's
known that they desperately need

1:06:11.730 --> 1:06:15.510
this deal to go through--are
going to do less well in

1:06:15.507 --> 1:06:17.967
bargains.
We already know people may do

1:06:17.973 --> 1:06:20.543
less well in bargaining because
they're less sophisticated

1:06:20.535 --> 1:06:22.345
players,
but here it isn't that they're

1:06:22.351 --> 1:06:24.731
less sophisticated--they can be
as sophisticated as you

1:06:24.731 --> 1:06:26.891
like--but they're just going to
be in a hurry.

1:06:26.889 --> 1:06:30.929
We already talked about the
graduate student whose leaving

1:06:30.933 --> 1:06:33.773
for Uzbekistan,
but who else typically in

1:06:33.771 --> 1:06:36.681
bargaining is going to need cash
now?

1:06:36.679 --> 1:06:38.809
Who else is going to be in a
weaker position in their

1:06:38.811 --> 1:06:40.371
bargaining, socially in our
society?

1:06:40.370 --> 1:06:48.300


1:06:48.300 --> 1:06:50.430
Student: When labor
management disputes labor.

1:06:50.429 --> 1:06:51.349
Professor Ben Polak:
So that's a good question.

1:06:51.349 --> 1:06:53.969
That's a good question in labor
management disputes--there's one

1:06:53.965 --> 1:06:56.285
going on right now in
Hollywood--it's not clear there.

1:06:56.289 --> 1:06:59.859
It could be,
it could be the management side

1:06:59.863 --> 1:07:04.933
who's in a hurry because they
just need right now to get David

1:07:04.933 --> 1:07:07.513
Letterman's script written.

1:07:07.510 --> 1:07:10.080
That would tend to favor labor,
but it could be the labor side.

1:07:10.080 --> 1:07:11.700
Why might it be the labor side?

1:07:11.700 --> 1:07:12.710
Does everyone know this?

1:07:12.710 --> 1:07:15.340
There's a writer's strike going
on in Hollywood right now,

1:07:15.340 --> 1:07:18.020
so the people on the management
side who are in the weakest

1:07:18.018 --> 1:07:20.788
position are the people who are
in the most hurry to get this

1:07:20.787 --> 1:07:22.787
resolved,
and those are the guys with the

1:07:22.787 --> 1:07:25.397
fewest scripts in the pipeline,
and that tends to be late night

1:07:25.403 --> 1:07:27.123
TV shows.
So those guys really want this

1:07:27.121 --> 1:07:27.931
thing settled fast.

1:07:27.929 --> 1:07:30.349
They're in a weak bargaining
position.

1:07:30.349 --> 1:07:32.139
On the other hand,
there may be a reason why

1:07:32.142 --> 1:07:33.812
labor's in a weak bargaining
position.

1:07:33.809 --> 1:07:37.459
Why might labor be in a weak
bargaining position relative to

1:07:37.456 --> 1:07:38.256
management?

1:07:38.260 --> 1:07:41.840


1:07:41.840 --> 1:07:42.720
They have rents to pay.

1:07:42.719 --> 1:07:45.829
They have immediate demands on
their cash.

1:07:45.829 --> 1:07:49.189
The typical worker is typically
poorer than your typical

1:07:49.188 --> 1:07:52.488
manager, not always but
typically, and they have to pay

1:07:52.486 --> 1:07:55.586
the rent.
They have to feed their

1:07:55.586 --> 1:07:58.126
children.
So there's a more general idea

1:07:58.130 --> 1:08:00.300
there.
More generally in bargaining,

1:08:00.295 --> 1:08:03.715
the people who are poorer,
typically--it isn't just poor

1:08:03.720 --> 1:08:06.430
in terms of income,
it's poor in terms of

1:08:06.434 --> 1:08:09.704
wealth--are going to be more
impatient to get things

1:08:09.701 --> 1:08:12.271
resolved;
and that's going to put them in

1:08:12.274 --> 1:08:14.024
a weaker bargaining position.

1:08:14.019 --> 1:08:18.889
So in bilateral bargaining,
having low wealth and being

1:08:18.894 --> 1:08:24.044
known to have low wealth puts
you in a weaker position.

1:08:24.039 --> 1:08:27.519
And that means that typically
people who are poorer are going

1:08:27.516 --> 1:08:30.236
to do less well,
although the late night TV show

1:08:30.239 --> 1:08:31.629
may be an exception.

1:08:31.630 --> 1:08:34.910
It makes you think a little bit
about whether adjusting up a

1:08:34.906 --> 1:08:37.956
bargaining position makes things
equal for everybody.

1:08:37.960 --> 1:08:39.860
Any other thoughts about who
has strength and who has

1:08:39.857 --> 1:08:40.767
weakness in bargaining?

1:08:40.770 --> 1:08:45.860
What other kind of stunts do we
see people do in bargaining?

1:08:45.859 --> 1:08:47.689
What else is kind of missing
here when you think about a

1:08:47.688 --> 1:08:49.678
particular--what we want to do
in this class is develop these

1:08:49.683 --> 1:08:51.083
ideas and take them to the real
world,

1:08:51.079 --> 1:08:57.939
so what other real world things
here are kind of missing?

1:08:57.939 --> 1:09:00.549
Student: Usually people
will make their first offer a

1:09:00.547 --> 1:09:03.107
lot higher than what they're
actually willing to accept.

1:09:03.109 --> 1:09:05.639
Professor Ben Polak:
Right, so typically,

1:09:05.641 --> 1:09:08.011
you're right,
typically bargaining isn't just

1:09:08.010 --> 1:09:11.590
a series of random numbers,
typically people start out high

1:09:11.585 --> 1:09:15.085
and then they concede towards
the middle, is that right?

1:09:15.090 --> 1:09:18.230
So if I'm the buyer I start out
with a low price and come up,

1:09:18.234 --> 1:09:21.334
and if you're the seller you
start off with a high price and

1:09:21.326 --> 1:09:24.186
come down.
So again that seems to be about

1:09:24.191 --> 1:09:28.041
establishing reputation and
trying to indicate how much I

1:09:28.042 --> 1:09:30.842
want this good.
There's something worth saying

1:09:30.840 --> 1:09:33.440
here which we haven't got time
to do in this class,

1:09:33.441 --> 1:09:36.771
and I hope you all have time to
take the follow up class 156.

1:09:36.770 --> 1:09:40.370
We can actually show formally
that in a setting in which

1:09:40.366 --> 1:09:42.716
buyers and sellers are
bargaining,

1:09:42.720 --> 1:09:46.320
and buyers and sellers do not
know how much the good is worth

1:09:46.319 --> 1:09:49.319
to the buyer or how much it
costs to the seller,

1:09:49.319 --> 1:09:52.829
typically you cannot expect to
get efficiency.

1:09:52.830 --> 1:09:53.370
Let me say it again.

1:09:53.369 --> 1:09:56.349
So it's kind of an important
economic fact that's missing

1:09:56.354 --> 1:09:59.184
from 115 and unfortunately
missing from this class,

1:09:59.180 --> 1:10:02.090
but if we go to a more real
world setting in which people's

1:10:02.094 --> 1:10:04.544
values are not known,
not only are the offers not

1:10:04.544 --> 1:10:07.324
accepted immediately and not
only is there some inequity in

1:10:07.324 --> 1:10:10.204
that the poor tend to be more
impatient and do less well,

1:10:10.199 --> 1:10:13.579
but also you get bad
inefficiency.

1:10:13.579 --> 1:10:16.519
The inefficiency occurs
essentially because the sellers

1:10:16.523 --> 1:10:19.903
want to seem like they're hard
and the buyers want to seem like

1:10:19.903 --> 1:10:24.383
they're hard,
and you get a failure for deals

1:10:24.376 --> 1:10:26.996
to be made.
So some deals are actually

1:10:27.002 --> 1:10:29.222
going to be lost or take a long
time in coming,

1:10:29.217 --> 1:10:31.817
you're going to get some
strikes before the deals occur

1:10:31.818 --> 1:10:33.358
and that's all inefficient.

1:10:33.359 --> 1:10:35.569
So bargaining,
not in this model,

1:10:35.566 --> 1:10:39.286
but in the real world tends to
lead to inefficiency.

1:10:39.289 --> 1:10:41.419
So I'll leave it there,
we'll have an earlyish lunch

1:10:41.415 --> 1:10:43.995
since we're all starving because
of the clock change anyway.