WEBVTT

00:01.930 --> 00:06.350
Professor Ben Polak: So
last time we saw this,

00:06.353 --> 00:11.373
we saw an example of a mixed
strategy which was to play 1/3,

00:11.372 --> 00:15.372
1/3, 1/3 in our rock,
paper, scissors game.

00:15.370 --> 00:18.230
Today, we're going to be
formal, we're going to define

00:18.233 --> 00:20.993
mixed strategies and we're going
to talk about them,

00:20.988 --> 00:22.878
and it's going to take a while.

00:22.880 --> 00:29.240
So let's start with a formal
definition: a mixed strategy

00:29.244 --> 00:34.704
(and I'll develop notation as
I'm going along,

00:34.700 --> 00:40.020
so let me call it
P_i,

00:40.020 --> 00:49.290
i being the person who's
playing it) P_i is a

00:49.289 --> 00:56.669
randomization over i's pure
strategies.

00:56.670 --> 01:00.700
So in particular,
we're going to use the notation

01:00.696 --> 01:05.646
P_i (si) to be the
probability that Player i plays

01:05.645 --> 01:09.835
si given that he's mixing using
P_i.

01:09.840 --> 01:24.380
So P_i(si) is the
probability that P_i

01:24.378 --> 01:33.048
assigns to the pure strategy si.

01:33.050 --> 01:35.430
Let's immediately refer that
back to our example.

01:35.430 --> 01:39.800
So for example,
if I'm playing 1/3,1/3,1/3 in

01:39.799 --> 01:45.559
rock, paper, scissors then
P_i is 1/3,1/3,1/3 and

01:45.559 --> 01:51.219
P_i of rock--so
P_i(R)--is a 1/3.

01:51.220 --> 01:54.700
So without belaboring it,
that's all I'm doing here,

01:54.701 --> 01:56.751
is developing some notation.

01:56.750 --> 02:00.200
Let's immediately encounter two
things you might have questions

02:00.204 --> 02:02.984
about.
So the first is,

02:02.975 --> 02:10.385
that in principle
P_i(si) could be zero.

02:10.389 --> 02:12.689
Just because I'm playing a
mixed strategy,

02:12.686 --> 02:15.876
it doesn't mean I have to
involve all of my strategies.

02:15.879 --> 02:18.959
I could be playing a mixed
strategy on two of my strategies

02:18.955 --> 02:21.495
and leave the other one with
zero probability.

02:21.500 --> 02:24.340
So, for example,
again in rock,

02:24.343 --> 02:28.703
paper, scissors,
we could think of the strategy

02:28.703 --> 02:32.593
1/2,1/2,0.
In this strategy I assign--I

02:32.591 --> 02:37.261
play rock half the time,
I play paper half the time,

02:37.257 --> 02:39.907
but I never play scissors.

02:39.910 --> 02:42.530
So everyone understand that?

02:42.530 --> 02:46.750
And while we're here let's look
at the other extreme.

02:46.750 --> 02:52.100
The probability assigned by my
mixed strategy to a particular

02:52.096 --> 02:55.696
si could be one.
It could be that I assign all

02:55.702 --> 02:58.812
of the probability to a
particular strategy.

02:58.810 --> 03:01.640
What would we call a mixed
strategy that assigns

03:01.640 --> 03:04.410
probability 1 to one of the pure
strategies?

03:04.410 --> 03:06.340
What's a good name for that?

03:06.340 --> 03:08.780
That's a "pure strategy."

03:08.780 --> 03:12.480
So notice that we can think of
pure strategies as the special

03:12.479 --> 03:15.929
case of a mixed strategy that
assigns all the weight to a

03:15.931 --> 03:17.721
particular pure strategy.

03:17.720 --> 03:21.190
So, for example,
if Pi(R) was 1,

03:21.185 --> 03:26.995
that's equivalent to saying
that I'm playing the pure

03:26.998 --> 03:29.568
strategy rock,
i.e.

03:29.570 --> 03:33.820
a pure strategy.
So there's nothing here.

03:33.819 --> 03:37.649
I'm just being a little bit
nerdy about developing notation

03:37.645 --> 03:40.675
and making sure that everything
is in place,

03:40.680 --> 03:44.130
and just to point out again,
one consequence of this is

03:44.126 --> 03:47.566
we've now got our pure
strategies embedded in our mixed

03:47.572 --> 03:51.832
strategies.
When I've got a mixed strategy

03:51.830 --> 03:58.320
I really am including in those
all of the pure strategies.

03:58.320 --> 04:07.020
So let's proceed.

04:07.020 --> 04:19.630
I'm going to push that up a
little high, sorry.

04:19.629 --> 04:22.649
So now I want to think about
what are the payoffs that I get

04:22.651 --> 04:24.821
from mixed strategies,
and again, I'm going to go a

04:24.818 --> 04:26.828
little slowly because it's a
little tricky at first and we'll

04:26.829 --> 04:28.249
get used to this,
don't panic,

04:28.248 --> 04:31.738
we'll get used to this as we go
on and as you see them in

04:31.741 --> 04:34.051
homework assignments and in
class.

04:34.050 --> 04:48.430
So let's talk about the payoffs
from a mixed strategy.

04:48.430 --> 04:51.120
In particular,
what we're going to worry about

04:51.123 --> 04:52.503
are expected payoffs.

04:52.500 --> 04:59.020
So the expected payoff of the
mixed strategy P,

04:59.017 --> 05:05.387
let's be consistent and call it
P_i,

05:05.393 --> 05:11.773
the mixed strategy
P_i is what?

05:11.769 --> 05:26.669
It's the weighted average--it's
a weighted average or a weighted

05:26.674 --> 05:40.404
mixture if you like--of the
expected payoffs of each of the

05:40.396 --> 05:47.726
pure strategies in the mix.

05:47.730 --> 05:49.970
So this is a long way of saying
something again which I think is

05:49.973 --> 05:51.863
a little bit obvious,
but let me just say it again.

05:51.860 --> 05:55.070
The way in which we figure out
the expected payoff of a mixed

05:55.065 --> 05:57.945
strategy is, we take the
appropriately weighted average

05:57.951 --> 06:01.101
of the expected payoffs I would
get from the pure strategies

06:01.103 --> 06:02.603
over which I'm mixing.

06:02.600 --> 06:06.520
So to make that less abstract
let's immediately look at an

06:06.521 --> 06:09.841
example.
So here's an example we'll come

06:09.840 --> 06:13.410
back to several times,
but just once today,

06:13.405 --> 06:16.625
and this a game you've seen
before.

06:16.629 --> 06:21.809
Here is the game Battle of the
Sexes, in which Player A can

06:21.813 --> 06:25.213
choose--Player I can choose A
and B,

06:25.209 --> 06:30.599
and Player II can choose a and
b, and what I want to do is I

06:30.595 --> 06:35.975
want to figure out the payoff
from particular strategies.

06:35.980 --> 06:47.120
So suppose that P is being
played by Player I and P is

06:47.117 --> 06:51.737
let's say (1/5,4/5).

06:51.740 --> 06:52.900
So what do I mean by that?

06:52.899 --> 07:01.109
I mean that Player I is
assigning 1/5 to playing A and

07:01.107 --> 07:04.047
4/5 to playing B.

07:04.050 --> 07:07.010
And suppose that Q--so I am
going to use P and Q because

07:07.005 --> 07:10.335
it's convenient to do so rather
than calling them P_1

07:10.337 --> 07:11.517
and P_2.

07:11.519 --> 07:15.389
So suppose that Q is the
mixture that Player II is

07:19.346 --> 07:24.086
so she's putting a probability
1/2 on a and a probability 1/2

07:24.090 --> 07:26.270
on b.
Just to notice I switched

07:26.273 --> 07:29.323
notation on you a little bit,
for this example to keep life

07:29.319 --> 07:32.619
easy,
I'm going to use P to be row's

07:32.615 --> 07:36.445
mixtures and Q to be column's
mixtures.

07:36.449 --> 07:42.179
And the question I want to
answer is what is the expected

07:42.184 --> 07:44.954
payoff in this case of P?

07:44.950 --> 07:52.560
What is P's expected payoff?

07:52.560 --> 07:56.810
The way I'm going to do that
is, I'm first of all going to

07:56.810 --> 08:01.360
ask what is the expected payoff
of each of the pure strategies

08:01.358 --> 08:06.458
that P involves,
the pure strategies involved in

08:06.460 --> 08:10.570
P.
So to start off--so the first

08:10.566 --> 08:18.856
step is ask what is the expected
payoff for Player I of playing A

08:18.857 --> 08:26.107
against Q and what is the
expected payoff for Player I of

08:26.111 --> 08:29.351
playing B against Q?

08:29.350 --> 08:32.500
That will be our first question
and we'll come back and

08:32.498 --> 08:34.188
construct the payoff for P.

08:34.190 --> 08:36.910
So these are things we can do I
think.

08:36.909 --> 08:41.199
So the expected payoff of A
against Q is what?

08:41.200 --> 08:45.660
Well, half the time if you play
A you're going to find your

08:45.656 --> 08:49.346
opponent is playing a,
in which case you'll get 2,

08:49.349 --> 08:53.529
and half the time when you play
A you'll find your opponent is

08:53.529 --> 08:56.269
playing b in which case you'll
get 0.

08:56.270 --> 08:57.680
So let's just write that up.

08:57.679 --> 09:07.139
So I'm going to get 2 with
probability 1/2 plus 0 with

09:07.137 --> 09:11.457
probability 1/2.
Everyone happy with that?

09:11.460 --> 09:16.910
That gives me 1.
Please correct my math in this.

09:16.909 --> 09:19.469
It's very easy at the board to
make mistakes,

09:19.474 --> 09:21.344
but I think that one is right.

09:21.340 --> 09:23.120
Conversely, what if I played B?

09:23.120 --> 09:27.690
What's the expected payoff for
the row player of playing B

09:27.692 --> 09:30.262
against Q, where Q is 1/2,1/2?

09:30.259 --> 09:36.189
So half the time when I play B,
I'll meet a Player II playing a

09:36.186 --> 09:42.296
and I'll get 0 and half the time
I'll find Player II is playing b

09:42.303 --> 09:45.713
and I'll get 1.
So let's write that up.

09:45.710 --> 09:59.470
So I'll get 0 half the time and
I'll get 1 half the time for an

09:59.467 --> 10:04.227
average of 1/2.
That's the first thing I ask.

10:04.230 --> 10:08.600
And now to finish the job,
I now want to figure out what

10:08.596 --> 10:13.276
is the expected payoff for
Player I of using P against Q?

10:13.279 --> 10:16.819
That was the question I really
wanted to start off with.

10:16.820 --> 10:18.290
What's the way to think about
this?

10:18.289 --> 10:22.879
Well P is 1/5 of the
time--according to P,

10:22.883 --> 10:29.163
1/5 of the time Player I is
playing A and 4/5 of the time

10:29.156 --> 10:33.746
Player I is playing B,
is that right?

10:33.750 --> 10:38.060
So to work out the expected
payoff what we're going to do is

10:38.062 --> 10:40.842
we're going to take 1/5 of the
time,

10:40.840 --> 10:45.250
and at which case he's playing
A and he'll get the expected

10:45.253 --> 10:49.213
payoff he would have got from
playing A against Q,

10:49.210 --> 10:56.970
and 4/5 of the time he's going
to be playing B in which case

10:56.971 --> 11:04.471
he'll get the expected payoff
from playing B against Q.

11:04.470 --> 11:07.840
Now just plugging in some
numbers to that from above,

11:07.838 --> 11:11.598
so we've got 1/5 of the time
he's doing the expected payoff

11:11.595 --> 11:15.865
from A against Q and that's this
number we worked out already.

11:15.870 --> 11:25.540
So this number here can come
down here, 1.

11:25.539 --> 11:32.359
And 4/5 of the time he's
playing B against Q,

11:32.358 --> 11:38.708
in which case his expected
payoff was 1/2,

11:38.711 --> 11:43.361
so this 1/2 comes in here.

11:43.360 --> 11:46.400
Everyone okay so far,
how I constructed it so far?

11:46.399 --> 11:48.389
Is this podium in the way of
you guys, are you okay?

11:48.390 --> 11:52.460
Let me push it slightly.

11:52.460 --> 11:54.920
So the total here is what?

12:01.360 --> 12:07.630
4/5 of 1/2 is 2/5,
so I've got a total of 3/5.

12:07.630 --> 12:11.960
So the total here is 3/5.

12:11.960 --> 12:15.850
Everyone understand how I did
that?

12:15.850 --> 12:19.380
Now while it's here let's
notice something.

12:19.379 --> 12:23.899
When I played P,
some of the time I played A and

12:23.902 --> 12:26.792
some of the time I played B.

12:26.789 --> 12:29.829
And when I ended up playing A,
I got A's expected payoff.

12:29.830 --> 12:32.350
And when I played B,
I got B's expected payoff.

12:32.350 --> 12:38.370
So the number I ended up with
3/5 must lie between the payoff

12:38.365 --> 12:44.575
I would have got from A which is
1, and the payoff I would have

12:44.582 --> 12:47.392
got from B which is 1/2.

12:47.390 --> 12:54.440
Is that right?
So 3/5 lies between 1/2 and 1.

12:54.440 --> 12:57.460
Everyone okay with that?

12:57.460 --> 13:02.890
Now that's a simple but very
general and very useful idea it

13:02.886 --> 13:06.076
turns out.
The idea here is that the

13:06.084 --> 13:11.284
payoff I'm going to get must lie
between the expected payoffs I

13:11.277 --> 13:14.877
would have got from the pure
strategies.

13:14.880 --> 13:15.630
Let me say it again.

13:15.629 --> 13:20.509
In general, when I play a mixed
strategy the expected payoff I

13:20.506 --> 13:25.296
get, is a weighted average of
the expected payoffs of each of

13:25.302 --> 13:31.302
the pure strategies in the mix,
and weighted averages always

13:31.300 --> 13:37.360
lie inside the payoffs that are
involved in the mix.

13:37.360 --> 13:42.870
So let me try and push that
simple idea a little harder.

13:42.870 --> 13:47.250
Suppose I was going to take the
average height in the

13:47.249 --> 13:50.449
class--average height in this
class.

13:50.450 --> 13:52.280
So let me just,
rather than use the class,

13:52.278 --> 13:53.838
let me just use some T.A.'s
here.

13:53.840 --> 13:59.700
So let me get these three
T.A.'s to stand up a second.

13:59.700 --> 14:03.930
Suppose I want to figure out
the average height of these

14:03.934 --> 14:06.194
three T.A.'s.
So stand up close together so I

14:06.188 --> 14:07.568
can at least see what's going on
here.

14:07.570 --> 14:10.760
So I think, from where I'm
standing, I've got that Ale is

14:10.758 --> 14:13.888
the tallest and Myrto is the
smallest, is that right?

14:13.889 --> 14:18.079
So I don't know instantaneously
what this average would be,

14:18.084 --> 14:22.644
but I claim that any weighted
average of their three heights,

14:22.639 --> 14:26.169
is going to give me a number
that's somewhere between the

14:26.167 --> 14:29.607
smallest height of the three,
which is Myrto's height,

14:29.611 --> 14:33.341
and the tallest height of the
three, which is Ale's height,

14:33.340 --> 14:35.800
is that right?
Is that correct?

14:35.800 --> 14:36.960
So that's a pretty general idea.

14:36.960 --> 14:40.500
Thanks guys I'll come back to
you in a second.

14:40.500 --> 14:42.320
Let's think about this
somewhere else,

14:42.320 --> 14:44.780
let's think about the batting
average of a team.

14:44.779 --> 14:48.759
The team batting average in
baseball, let's use the Yankees,

14:48.756 --> 14:51.286
for example.
We know that the team batting

14:51.292 --> 14:54.592
average, the average batting
average of the Yankee's--I don't

14:54.586 --> 14:56.976
know what it is,
I didn't look it up this

14:56.982 --> 15:00.312
morning--but I know it lies
somewhere between the player who

15:00.308 --> 15:03.858
has the highest batting average
which I'm guessing is Jeter,

15:03.860 --> 15:05.350
I'm guessing,
and the lowest,

15:05.345 --> 15:08.525
the person on the team who has
the lowest batting average,

15:08.529 --> 15:10.599
who is probably one of the
pitchers who played,

15:10.604 --> 15:13.314
who batted a few times in one
of those inter-league games.

15:13.309 --> 15:16.829
(It would have been better if
I'd used the Mets but I feel I

15:16.830 --> 15:20.470
should take pity on Mets fans
this week and not mention them.)

15:20.471 --> 15:23.851
So this is a very simple idea,
it's deceptively simple.

15:23.850 --> 15:26.040
It says averages,
weighted averages,

15:26.039 --> 15:29.789
lie between the highest thing
over which you're averaging and

15:29.794 --> 15:32.864
the lowest thing over which
you're averaging.

15:32.860 --> 15:35.400
Everyone okay with that idea?

15:35.399 --> 15:39.359
Now this very simple idea is
going to have an enormous

15:39.359 --> 15:43.169
consequence, and here's the
enormous consequence.

15:43.170 --> 15:49.500
Simple idea, big consequence.

15:49.500 --> 15:52.830
So there's going to be a lesson
that follows from this

15:52.826 --> 15:55.836
incredibly simple idea and this
is the lesson.

15:55.840 --> 16:05.590
If a mixed strategy is a best
response, so if a mixed strategy

16:05.593 --> 16:11.513
is the best thing you can be
doing,

16:11.509 --> 16:19.289
then each of the pure
strategies in the mix--I'm being

16:19.286 --> 16:26.766
a little bit loose here but I
mean assigned positive

16:26.768 --> 16:34.288
probability in the mix,
for those people who are nerdy

16:34.290 --> 16:41.290
enough to worry about it--each
of the pure strategies in the

16:41.293 --> 16:46.163
mix must themselves be best
responses.

16:46.159 --> 17:00.029
So, in particular,
each must yield the same

17:00.033 --> 17:07.913
expected payoff.
So here's a big conclusion that

17:07.912 --> 17:11.652
follows from that incredibly
simple idea about averages lying

17:11.652 --> 17:14.522
between the highest one and the
lowest one.

17:14.519 --> 17:19.359
Let's draw ourselves from that
lesson to this big conclusion.

17:19.360 --> 17:20.320
What is the conclusion?

17:20.319 --> 17:23.419
The conclusion is if a mixed
strategy is a best response,

17:23.417 --> 17:26.567
if the best thing I can do is
to play a mixed strategy,

17:26.569 --> 17:29.449
then each of the pure
strategies which I'm playing in

17:29.451 --> 17:32.781
that mix, which I'm assigning
positive probability to in that

17:32.776 --> 17:36.186
mix,
must themselves be best

17:36.189 --> 17:38.389
responses.
In particular,

17:38.391 --> 17:41.741
each of them therefore must
yield the same expected payoff.

17:41.740 --> 17:43.210
So let's go back to our example.

17:43.210 --> 17:48.030
Can I steal my three T.A.'s
again?

17:48.029 --> 17:49.689
Suppose the game,
suppose the thing I'm involved

17:49.686 --> 17:51.126
in--I should have made this
easier before,

17:51.131 --> 17:52.331
let me come down a little bit.

17:52.329 --> 17:54.679
I'll stand above here,
this is good.

17:54.680 --> 17:58.550
So suppose the game I'm
involved in, the payoff in the

17:58.547 --> 18:02.997
game is, a game in which I have
to choose the tallest group of

18:02.998 --> 18:06.228
my T.A.'s.
So my payoff is going to be the

18:06.228 --> 18:10.428
average height of whichever
subgroup of my T.A.'s I pick and

18:10.432 --> 18:12.572
these are my three choices.

18:12.569 --> 18:16.099
So if I pick more than one of
them I'm going to get a weighted

18:16.102 --> 18:18.132
average, that's a mixed
strategy.

18:18.130 --> 18:23.850
My aim here is to maximize the
height of whatever subgroup I

18:23.851 --> 18:25.391
pick.
So in this game,

18:25.386 --> 18:27.796
here's my three pure
strategies: my three pure

18:27.796 --> 18:30.486
strategies are to pick Myrto;
Ale;

18:30.490 --> 18:33.350
or Jake.
Those are my three pure

18:33.353 --> 18:34.823
strategies.
And my mixture,

18:34.821 --> 18:37.381
I could mix these two,
I could mix these two,

18:37.377 --> 18:38.827
I could mix all three.

18:38.829 --> 18:42.659
But remember my payoff here is
to get the group,

18:42.662 --> 18:45.192
the average as high as I can.

18:45.190 --> 18:52.460
So how am I going to get the
average as high as I can?

18:52.460 --> 18:55.940
I get the average as high I as
I can, I'm going to kick out

18:55.935 --> 18:59.225
Myrto for a start because
Myrto's just bringing down the

18:59.231 --> 19:00.851
average, is that right?

19:00.849 --> 19:04.689
Average height I should say,
there's nothing--and actually I

19:04.690 --> 19:08.010
think I'm going to kick out Jake
as well I think,

19:08.009 --> 19:11.669
I'm probably going to kick out
Jake as well because that way I

19:11.670 --> 19:14.010
just have Ale.
So if it was the case that I

19:14.006 --> 19:16.656
was picking both of them,
it would have to be they were

19:16.662 --> 19:19.172
equally tall but since they're
not equally tall,

19:19.170 --> 19:21.570
I should just pick the best one.

19:21.569 --> 19:24.689
Let's go back to my Yankee's
example, if I want to pick a

19:24.693 --> 19:27.723
sub-team of the Yankee's,
I'm allowed to pick any number

19:27.717 --> 19:30.597
of people, to have the highest
average, batting average,

19:30.600 --> 19:32.050
in that sub-team.

19:32.049 --> 19:36.089
The way to do it is to find the
Yankee who has the highest

19:36.089 --> 19:38.639
batting average and just pick
him.

19:38.640 --> 19:40.050
Let's do one more example.

19:40.049 --> 19:43.209
Let me use the front row of
students here,

19:43.211 --> 19:45.991
so here's my,
can I get this front of

19:45.988 --> 19:48.608
students to stand up a second?

19:48.610 --> 19:51.020
This is a part of the row.

19:51.019 --> 19:57.679
And suppose my aim in life is
to construct the highest average

19:57.675 --> 19:59.575
GPA.
I'm not going to embarrass

19:59.583 --> 20:01.803
these guys and ask them what
their GPA's are.

20:01.799 --> 20:05.849
So my aim in life here is to
pick some sub-group of these

20:05.853 --> 20:07.593
one, two, three,
four,

20:07.589 --> 20:10.599
five, six, seven,
eight students,

20:10.599 --> 20:16.059
such that the average GPA of
that sub-group is as high as I

20:16.055 --> 20:19.145
can make it.
So what will I do here?

20:19.150 --> 20:24.310
So this being Yale I'll just
find the people who have the 4.0

20:24.308 --> 20:26.628
GPA's and just pick them.

20:26.630 --> 20:29.330
Is that right?
You might think well why not

20:29.330 --> 20:31.260
include somebody who has a 3.9
GPA?

20:31.260 --> 20:32.530
That's pretty good.

20:32.530 --> 20:35.480
So why not?
Because if there's anybody in

20:35.482 --> 20:39.292
this group who has a 4.0 GPA,
I'd do better just to pick that

20:39.294 --> 20:41.824
person.
The 3.9 person would just be

20:41.822 --> 20:43.582
pulling down the average.

20:43.579 --> 20:47.879
Now suppose there's nobody with
a 4.0 GPA and suppose it's the

20:47.884 --> 20:52.124
case that three of these people,
let's say these three people

20:52.119 --> 20:55.519
have a 3.9 GPA.
So these three have 3.9 GPA,

20:55.517 --> 20:58.677
imagine that,
and these other people they've

20:58.677 --> 21:01.687
got horrible grades like B+
somewhere.

21:01.690 --> 21:04.510
These are our future law school
students and these are the

21:04.511 --> 21:07.381
people--who knows what they're
going to end up doing--being

21:07.381 --> 21:08.521
President probably.

21:08.519 --> 21:11.889
So to construct the group with
the highest average GPA,

21:11.891 --> 21:13.391
what am I going to do?

21:13.390 --> 21:16.420
Well first I'll throw out all
these guys with low GPA's,

21:16.421 --> 21:19.561
so they can all sit down and
I'll look at these last three

21:19.562 --> 21:23.232
and these last three,
if they're all in the group

21:23.226 --> 21:26.186
they better all have the same
GPA.

21:26.190 --> 21:29.150
Why on earth?
If I'm trying to maximize the

21:29.154 --> 21:32.874
average of my group,
if any of them had a lower GPA

21:32.868 --> 21:36.198
I should kick them out,
and if one of them has a higher

21:36.196 --> 21:38.396
GPA than the other two,
I should kick out both the

21:38.404 --> 21:40.334
other two.
So if I'm including all three

21:40.327 --> 21:42.997
of them, in my constructing of
the average all of them must

21:43.001 --> 21:45.551
have the same GPA,
which I'm going to assume is

21:45.554 --> 21:48.444
3.9, to assume you can still
make into law school.

21:48.440 --> 21:49.640
Everyone understand that?

21:49.640 --> 21:54.860
Yeah?
Okay, thanks guys.

21:54.859 --> 21:58.679
So that's the way I want to
think about this.

21:58.680 --> 22:03.930
So the idea here is if I'm
using a mixed strategy as a best

22:03.928 --> 22:09.088
response, it must be the case
that everything on which I'm

22:09.087 --> 22:11.437
mixing is itself best.

22:11.440 --> 22:14.480
And the reason is,
if it wasn't,

22:14.477 --> 22:20.647
kick out the thing that isn't
best and my average will go up.

22:20.650 --> 22:24.840
So that leads us to the next
idea, but before I do just for

22:24.844 --> 22:27.524
formality, let me add a
definition.

22:27.519 --> 22:33.769
The definition is this,
a mixed strategy profile--what

22:33.770 --> 22:40.610
I'm going to do now is I'm going
to define Nash Equilibrium

22:40.610 --> 22:42.940
again,
just so we have it in our notes

22:42.941 --> 22:44.861
somewhere.
So a mixed strategy

22:44.855 --> 22:49.445
profile--there should be a
hyphen there--(P_1*,

22:49.450 --> 22:56.020
P_2*,
…all the way up to

22:56.021 --> 23:02.591
P_N*),
is a mixed strategy Nash

23:02.592 --> 23:12.282
Equilibrium if for each Player
i--so for each Player i--that

23:12.284 --> 23:20.014
player's mixed strategy
P_i* is a best

23:20.005 --> 23:29.035
response for Player i to the
strategies everyone else is

23:29.040 --> 23:34.790
picking P _-i*.

23:34.789 --> 23:41.849
So I'm exploiting,
by now, a well developed

23:41.846 --> 23:47.386
notation for player strategies.

23:47.390 --> 23:50.550
So this definition of Nash
Equilibrium, it's exactly the

23:50.547 --> 23:54.107
same as the definition of Nash
Equilibrium we've been using now

23:54.106 --> 23:57.466
for several weeks,
except everywhere where before

23:57.466 --> 24:00.116
we saw a pure strategy,
which was an S,

24:00.123 --> 24:02.293
I have replaced it with a P.

24:02.289 --> 24:05.519
So the same definition except
I'm using mixed strategies

24:05.524 --> 24:07.234
instead of pure strategies.

24:07.230 --> 24:11.870
But an implication of our
lesson is what?

24:11.869 --> 24:16.199
It's that if P_i* is
part of a Nash Equilibrium--so

24:16.201 --> 24:20.461
if Pi* is a best response to
what everyone else is doing,

24:20.460 --> 24:24.370
P_-i* --,
then each of the pure

24:24.366 --> 24:29.536
strategies involved in
P_i* must itself be a

24:29.542 --> 24:35.322
best response.
So an implication of the lesson

24:35.321 --> 24:40.251
is, the lesson implies the
following.

24:40.250 --> 24:45.600
If P_i* of a
particular strategy is positive,

24:45.600 --> 24:50.260
so in other words,
I'm using this strategy in my

24:50.257 --> 24:57.167
mix,
then that strategy is also a

24:57.168 --> 25:07.218
best response to what everyone
else is doing.

25:07.220 --> 25:15.030
Okay, so from a math point of
view this is the big idea of the

25:15.026 --> 25:18.236
day, this board.
If you're having trouble

25:18.243 --> 25:20.303
reading this at the back,
trust me I've written that up

25:20.298 --> 25:22.618
on the handout that will appear
magically on the computer,

25:22.620 --> 25:24.500
at the end of class.

25:24.500 --> 25:27.150
At the moment you're staring at
this, it's all a bit new,

25:27.151 --> 25:30.051
and as well as being new,
you're saying,

25:30.049 --> 25:34.789
okay but so what,
why do I care about this

25:34.792 --> 25:37.802
seemingly mundane fact?

25:37.799 --> 25:41.729
The reason we're going to turn
out to care about this seemingly

25:41.729 --> 25:46.119
mundane fact,
is that this fact is going to

25:46.122 --> 25:51.942
make it remarkably easy to find
Nash Equilibria.

25:51.940 --> 25:55.820
This fact, this lesson,
this idea that if I'm playing a

25:55.823 --> 25:58.343
pure strategy as part of the
mix,

25:58.339 --> 26:02.859
it must itself be a best
response, that's going to be the

26:02.864 --> 26:07.234
trick we're going to use in
finding mixed strategy Nash

26:07.228 --> 26:10.638
Equilibria.
The only way I can illustrate

26:10.642 --> 26:14.742
that to you is to do it,
so I'm going to spend the rest

26:14.743 --> 26:16.873
of today just doing that.

26:16.869 --> 26:20.309
I'm going to look at a game and
we're going to go through this

26:20.314 --> 26:22.454
game.
We'll discuss it a little bit

26:22.445 --> 26:25.495
because it's a fun game,
and we're going to find the

26:25.496 --> 26:28.006
mixed-strategy equilibria of
this game.

26:28.010 --> 26:29.970
Everyone know where we're going?

26:29.970 --> 26:33.530
I want to make sure before I go
on, are people looking very sort

26:33.526 --> 26:35.046
of deer in the headlamps?

26:35.049 --> 26:39.629
That was a lot of formality to
get through in a short period of

26:39.632 --> 26:40.822
time.
Does anyone want to ask a

26:40.818 --> 26:41.458
question at this point?

26:41.460 --> 26:44.920
Are you okay?
Okay to go on?

26:44.920 --> 26:47.650
So just remember that the
conclusion here comes from this

26:47.652 --> 26:48.582
very simple idea.

26:48.579 --> 26:50.959
The simple idea is,
the payoff to a weighted

26:50.957 --> 26:54.277
average must lie between the
best and worst thing involved in

26:54.275 --> 26:57.145
the average,
and therefore if I'm including

26:57.154 --> 26:59.954
things in there as part of a
best response,

26:59.947 --> 27:01.607
they must all be good.

27:01.609 --> 27:04.799
That's the simple idea,
this is the dramatic

27:04.796 --> 27:06.996
conclusion.
So the only way to prove this

27:06.998 --> 27:09.728
to you and the only way to prove
to you that this is useful is to

27:09.730 --> 27:10.670
go ahead and do it.

27:10.670 --> 27:17.590
So what I'm going to do is I'm
going to clean these boards and

27:17.585 --> 27:22.115
I'm going to start showing an
example.

27:22.119 --> 27:24.439
Again don't panic,
I think a lot of people at this

27:24.439 --> 27:26.569
part of the class have a
tendency to panic,

27:26.569 --> 27:29.749
because it's a new idea,
it seems like a lot of math

27:29.749 --> 27:31.689
around.
None of it's very hard math,

27:31.685 --> 27:33.105
it's all kind of arithmetic.

27:33.109 --> 27:38.709
It's just this idea of not
panicking.

27:38.710 --> 27:45.370
So the example I want to look
at is going to be from tennis,

27:45.374 --> 27:50.914
and I'm going to consider a
game within a game,

27:50.910 --> 28:01.330
played by two tennis players,
and let's call them Venus and

28:01.327 --> 28:05.587
Serena Williams.
So a couple of years ago we

28:05.591 --> 28:07.731
used to use Venus and Serena
Williams for this example,

28:07.730 --> 28:10.500
and then for a while I worried,
that you wouldn't even remember

28:10.498 --> 28:12.148
who Venus and Serena Williams
were,

28:12.150 --> 28:16.700
and so we picked any two random
Russians, but now we're back.

28:16.700 --> 28:19.880
Seems like we're back to
picking Venus and Serena.

28:19.880 --> 28:25.420
So the game within the game is
this, suppose that they're

28:25.416 --> 28:32.036
playing and Serena is at the net
and the ball is on Venus' court,

28:32.039 --> 28:37.829
and Venus has reached the ball
and Venus has to decide whether

28:37.826 --> 28:43.416
to try to hit a passing shot
past Serena on Serena's left or

28:43.423 --> 28:45.513
on Serena's right.

28:45.509 --> 28:49.029
Notice I'm going to exclude the
possibility of throwing up a lob

28:49.026 --> 28:51.256
for now, just to make this
manageable.

28:51.259 --> 28:55.889
So basically the choice facing
Venus is should she try to pass

28:55.887 --> 29:00.557
Serena to Serena's left,
which is Serena's backhand side

29:00.556 --> 29:04.636
or to Serena's right,
which is Serena's forehand

29:04.644 --> 29:06.764
side.
People are familiar enough with

29:06.759 --> 29:08.819
tennis to understand what I'm
talking about?

29:08.819 --> 29:10.229
So we're going to assume this
is Wimbledon,

29:10.226 --> 29:12.266
otherwise no one would be at
the net to start with I guess.

29:12.270 --> 29:14.490
So this is at Wimbledon.

29:14.490 --> 29:19.600
Let's try and put up some
payoffs here.

29:19.599 --> 29:20.679
So these are going to be the
payoffs.

29:20.680 --> 29:25.620
I think that this example is
originally due to Dixit,

29:25.619 --> 29:28.089
but it's not a big deal.

29:28.089 --> 29:34.179
I think this example is due to
Dixit and Skeath.

29:34.180 --> 29:38.350
So here's some numbers and I'll
explain the numbers in a minute.

29:38.349 --> 29:52.879
So this is 50,50,
80,20, 90,10 and 20,80.

29:52.880 --> 29:54.610
So what are these numbers?

29:54.609 --> 29:58.459
So first of all let me just
explain what the strategies are,

29:58.460 --> 30:02.380
so I'm assuming the row player
is Venus and the column player

30:02.376 --> 30:06.076
is Serena.
I'm assuming that if Venus

30:06.077 --> 30:11.437
chooses L that means she
attempts to pass Serena to

30:11.437 --> 30:15.017
Serena's left,
we'll orient things from

30:15.019 --> 30:18.609
Serena's point of view,
and if she hits right that

30:18.613 --> 30:22.943
means she's attempting to pass
Serena on Serena's right.

30:22.940 --> 30:26.900
If Serena chooses L that means
she cheats slightly towards her

30:26.897 --> 30:30.397
left: not cheats in the sense of
breaking the rules,

30:30.400 --> 30:33.080
but cheats in terms of where
she's standing or leaning.

30:33.079 --> 30:36.039
And if she chooses right that
means she cheats slightly

30:36.035 --> 30:37.125
towards her right.

30:37.130 --> 30:40.570
So this is cheating towards her
backhand and this is cheating

30:40.568 --> 30:43.378
towards her forehand,
assuming she's right handed,

30:43.377 --> 30:44.807
which she in fact is.

30:44.809 --> 30:46.939
Okay, what do these numbers
mean?

30:46.940 --> 30:49.140
So let's start with the easy
ones.

30:49.140 --> 30:54.950
So if Venus chooses left and
Serena chooses right,

30:54.947 --> 30:58.737
then Serena has guessed wrong.

30:58.740 --> 31:04.690
Is that correct?
In which case Venus wins the

31:04.691 --> 31:10.531
points 80% of the time and
Serena wins it 20% of the time.

31:10.529 --> 31:15.889
Conversely, if Venus chooses
right and Serena chooses left,

31:15.888 --> 31:21.428
then again, Serena has guessed
wrong and this time Venus wins

31:21.432 --> 31:26.882
the points 90% of the time and
Serena wins the points 10% of

31:26.884 --> 31:30.164
the time.
This should be a familiar idea

31:30.164 --> 31:34.024
by now, but why is it the case
these nineties and eighties are

31:34.016 --> 31:36.796
not a 100%?
Why is it the case that if

31:36.795 --> 31:40.945
Serena guesses wrong Venus
doesn't win 100% of the time?

31:40.950 --> 31:42.670
Anybody?
Perhaps we can get a show of

31:42.668 --> 31:44.068
hands, get some mikes up.

31:44.070 --> 31:45.640
Why isn't it 100% here?

31:45.640 --> 31:48.210
Somebody?
Patrick?

31:48.210 --> 31:48.990
Wait for the mike.

31:48.990 --> 31:50.860
Student: Sometimes she
hits it out of bounds when she

31:50.864 --> 31:52.024
serves.
Professor Ben Polak:

31:52.022 --> 31:53.992
Right, this isn't even a serve,
this is a passing shot but the

31:53.993 --> 31:55.693
same is true.
So sometimes you're

31:55.688 --> 31:59.098
successfully going to hit it
past Serena but the ball is

31:59.097 --> 32:00.397
going to sail out.

32:00.400 --> 32:05.680
So that happens 10% of the time
here and 20% of the time here.

32:05.680 --> 32:11.280
Look at the other two boxes,
if Venus hits to Serena's left

32:11.279 --> 32:15.589
and Serena guesses left,
then we're going to assume that

32:15.590 --> 32:18.610
Serena's going to reach the ball
and make a volley,

32:18.609 --> 32:22.229
but her volley only manages to
go in--go over the net and go

32:22.232 --> 32:25.112
in--half the time,
so the payoffs are (50,50).

32:25.109 --> 32:29.179
Half the time Venus wins the
point and half the time Serena

32:29.177 --> 32:32.157
wins the point.
Conversely, if Venus hits the

32:32.159 --> 32:35.489
ball to Serena's right and
Serena guesses correctly and

32:35.490 --> 32:38.020
chooses right,
then we're in this box.

32:38.019 --> 32:41.439
Once again, Serena has guessed
correctly and she's going to

32:41.435 --> 32:45.025
successfully reach the volley
and this time she gets it in 80%

32:45.027 --> 32:49.107
of the time,
so Venus wins the point 20% of

32:49.105 --> 32:53.805
the time and Serena wins it 80%
of the time.

32:53.809 --> 32:57.979
So just to finish up the
description of the game here,

32:57.976 --> 33:02.606
notice that we're assuming that
Serena is a little better at

33:02.614 --> 33:07.414
volleying to her right than she
is volleying to her left.

33:07.410 --> 33:11.300
So this is her forehand volley
and we're going to assume that

33:11.295 --> 33:14.075
that's stronger than her
backhand volley.

33:14.079 --> 33:18.369
Conversely, we're assuming that
Venus' passing shot is a little

33:18.365 --> 33:22.165
better when she shoots it to
Serena's left than when she

33:22.167 --> 33:24.377
shoots it to Serena's right.

33:24.380 --> 33:28.250
This is her cross court passing
shot and this is her down the

33:28.250 --> 33:29.540
line passing shot.

33:29.539 --> 33:32.089
So none of that fine detail
matters a great deal,

33:32.086 --> 33:35.156
but just if you're interested
that's where the numbers come

33:35.162 --> 33:36.722
from.
I'm not claiming this is true

33:36.722 --> 33:38.292
data by the way,
I made up these numbers.

33:38.289 --> 33:40.709
Actually I think Dixit made up
these numbers,

33:40.707 --> 33:42.517
I forget where I got them from.

33:42.519 --> 33:45.679
So okay, everyone understand
the game?

33:45.680 --> 33:48.730
So now imagine,
either imagine you are Venus or

33:48.730 --> 33:51.710
Serena, or imagine perhaps more
realistically,

33:51.713 --> 33:54.833
that you've become Venus or
Serena's coach.

33:54.829 --> 33:57.739
Do I have any members of the
tennis team here?

33:57.740 --> 34:00.410
No.
Well imagine you've become

34:00.414 --> 34:03.784
their coach, so you take this
class and then you apply to

34:03.782 --> 34:06.492
replace their father as being
their coach.

34:06.490 --> 34:09.590
That's a tough assignment I
would think.

34:09.590 --> 34:12.800
So an obvious question is,
you're coaching Venus before

34:12.804 --> 34:16.024
Wimbledon, you know this
situation's going to arise and

34:16.018 --> 34:19.648
you might want to coach Venus on
what should she do here?

34:19.650 --> 34:24.620
Should she try and pass Serena
down the line or she should try

34:24.621 --> 34:29.431
and hit the cross court volley,
cross court passing shot?

34:29.429 --> 34:33.079
Notice that this is a question
of should you,

34:33.082 --> 34:38.062
Venus, play to your strength
which is the cross court passing

34:38.062 --> 34:42.672
shot,
or should you play to Serena's

34:42.665 --> 34:49.985
weakness, which would be to hit
it to Serena's backhand.

34:49.989 --> 34:53.079
Playing to your strength is to
choose right and playing to

34:53.079 --> 34:55.139
Serena's weakness is to choose
left.

34:55.139 --> 34:59.209
Conversely, for Serena,
should you lean towards your

34:59.209 --> 35:03.679
strength, which I guess is
leaning to the right or should

35:03.677 --> 35:06.627
you lean towards Venus'
weakness,

35:06.630 --> 35:09.560
which I guess is leaning left?

35:09.559 --> 35:12.139
When you look at coaching
manuals on this stuff,

35:12.144 --> 35:15.344
or you listen to the terrible
guys who commentate on tennis

35:15.335 --> 35:18.575
for ESPN--oh no I'm getting in
trouble again--very nice guys

35:18.579 --> 35:20.779
who commentate on tennis for
ESPN,

35:20.780 --> 35:24.640
they say just incredibly dumb
things at this point.

35:24.639 --> 35:27.409
They say things like,
you should always play to your

35:27.411 --> 35:30.781
strengths and don't worry about
the other person's weakness.

35:30.780 --> 35:35.350
I think it won't take much time
today to figure out that's not

35:35.346 --> 35:38.196
great advice.
But can people at least see

35:38.200 --> 35:41.210
that this is a difficult
problem, this is not an

35:41.213 --> 35:44.293
immediately obvious problem,
is that correct?

35:44.289 --> 35:48.309
One reason it's not immediately
obvious is not only is no

35:48.312 --> 35:52.822
strategy dominated here,
but there is no pure strategy

35:52.817 --> 35:57.907
Nash Equilibrium in this game,
in this little sub game.

35:57.909 --> 36:01.499
There is no pure strategy Nash
Equilibrium--and notice that I

36:01.496 --> 36:03.046
added the qualifier now.

36:03.050 --> 36:05.300
Previously I would just have
said Nash Equilibrium,

36:05.297 --> 36:07.767
but now that we have mixed
strategies in the picture,

36:07.769 --> 36:10.719
I'm going to talk about pure
strategy Nash Equilibria to be

36:10.717 --> 36:13.357
those that are the only
involving pure strategies.

36:13.360 --> 36:17.020
Okay, so why is there no pure
strategy Nash Equilibrium?

36:17.020 --> 36:18.240
Well let's have a look.

36:18.239 --> 36:21.949
So if Venus--If Serena thought
that Venus was going to choose

36:21.946 --> 36:25.146
left then her best response,
not surprisingly,

36:25.150 --> 36:29.920
is to lean left and if Serena
thought that Venus was going to

36:29.918 --> 36:33.868
choose right,
then her best response is to

36:33.868 --> 36:37.978
cheat to the right,
so 50 is bigger than 20,

36:37.981 --> 36:40.661
and 80 is bigger than 10.

36:40.659 --> 36:44.389
And conversely,
if Venus thought that Serena

36:44.387 --> 36:49.757
was cheating a bit to the left
then her best response is to hit

36:49.762 --> 36:54.792
it to Serena's right,
and if Venus thought Serena was

36:54.788 --> 37:00.408
leaning to the right then Venus'
best response is to hit it to

37:00.412 --> 37:03.112
Serena's left.
So I think that's not at all

37:03.109 --> 37:05.489
surprising when you think about
it, not at all surprising,

37:05.489 --> 37:07.919
you're going to get this little
cycle like this,

37:07.916 --> 37:10.856
but we can see immediately that
these best responses never

37:10.858 --> 37:23.298
coincide,
so there is no pure strategy

37:23.296 --> 37:30.556
equilibrium.
So that leaves us a bit stuck

37:30.556 --> 37:34.266
except I guess you know what the
next question's going to be,

37:34.274 --> 37:37.314
and I shouldn't leave it in too
much suspense.

37:37.309 --> 37:40.239
The next question's going to
be, okay there's not pure

37:40.244 --> 37:43.404
strategy Nash Equilibrium,
but we've just introduced a new

37:43.401 --> 37:44.731
idea which was what?

37:44.730 --> 37:47.990
It was Nash Equilibrium in
mixed strategies.

37:47.989 --> 37:51.879
Maybe there's going to be a
mixed strategy Nash Equilibrium.

37:51.880 --> 37:53.980
In fact, there is,
there is going to be one.

37:53.980 --> 38:00.620
So our exercise now is,
let's find a mixed strategy

38:00.624 --> 38:05.814
Nash Equilibrium,
and before we find it,

38:05.807 --> 38:12.317
let's just interpret what it's
going to mean.

38:12.320 --> 38:16.350
A mixed strategy Nash
Equilibrium in this game,

38:16.354 --> 38:21.354
is going to be a mix for Venus
between hitting the ball to

38:21.354 --> 38:24.604
Serena's left and Serena's
right,

38:24.599 --> 38:30.029
and a mix for Serena between
leaning left and leaning right,

38:30.029 --> 38:35.869
such that each person's mix,
each person's randomization is

38:35.865 --> 38:41.295
a best response to the other
person's randomization.

38:41.300 --> 38:44.320
Since these players are sisters
and have played each other many,

38:44.317 --> 38:46.207
many times,
not just in competition but

38:46.210 --> 38:48.660
probably in practice,
it seems like a reasonable idea

38:48.658 --> 38:51.198
that they might have arrived in
playing each other,

38:51.199 --> 38:54.949
at a mixed strategy Nash
Equilibrium.

38:54.949 --> 39:01.509
That's what we're going to try
and do, now how are we going to

39:01.513 --> 39:04.693
do that?
So what we're going to do is

39:04.694 --> 39:08.454
we're going to exploit the trick
that we have here,

39:08.452 --> 39:12.492
the lesson here.
The lesson we have here says if

39:12.488 --> 39:16.948
players are playing a mixed
strategy as part of a Nash

39:16.947 --> 39:20.667
Equilibrium,
each of the pure strategies

39:20.666 --> 39:25.126
involved in the mix,
each of their pure strategies

39:25.126 --> 39:28.216
must itself be a best response.

39:28.220 --> 39:33.820
We're going to use that idea.

39:33.820 --> 39:37.830
So let's try and do that.

39:37.829 --> 39:40.169
So I'm hoping that by doing
this, I'm going to illustrate to

39:40.174 --> 39:43.904
you immediately,
that this idea is actually

39:43.900 --> 39:50.620
useful, at least useful if you
end up coaching the Williams

39:50.616 --> 39:53.236
sisters.
Alright, I want to keep this so

39:53.244 --> 39:54.254
you can still read it.

39:54.250 --> 39:55.120
Ill bring it down a bit.

39:55.120 --> 39:59.100
Can people still read it?

39:59.099 --> 40:04.089
Okay, so what I want to do is,
I want to find a mixture for

40:04.092 --> 40:08.742
Serena and a mixture for Venus
that are equilibrium.

40:08.739 --> 40:11.279
Having put it up there let me
bring it down again.

40:11.280 --> 40:13.010
This was not so intelligent of
me.

40:13.010 --> 40:16.350
I actually want to bring in
some notation,

40:16.348 --> 40:19.848
so as before,
let's assume that Serena's mix

40:19.849 --> 40:23.609
is,
let's use Q and (1-Q) to be

40:23.614 --> 40:30.364
Serena's mix and let's use P and
(1-P) to be Venus' mix.

40:30.360 --> 40:40.830
Let's establish that notation.

40:40.829 --> 40:45.129
So here's the trick,
So this is the slightly magic

40:45.131 --> 40:48.121
bit of the class,
so pay attention,

40:48.117 --> 40:51.977
I'm about to pull a rabbit out
of a hat.

40:51.980 --> 41:03.080
Trick, what should I do first,
to find Serena's Nash

41:03.077 --> 41:10.037
Equilibrium mix,
so that's (Q,

41:10.039 --> 41:15.449
(1-Q)), what I'm going to do is
I'm going to look at

41:15.452 --> 41:18.002
Venus' payoffs.

41:18.000 --> 41:26.690
So to find Serena's Nash
Equilibrium mix the trick is to

41:26.687 --> 41:35.687
look at Venus' payoffs,
that's going to be my magic

41:35.690 --> 41:39.600
trick.
Let's try and see why.

41:39.599 --> 41:50.239
So let's look at Venus'
payoffs, Venus' payoffs against

41:50.241 --> 41:53.451
Q.
So if Serena is choosing (Q,

41:53.445 --> 41:56.115
1-Q), what are Venus' payoffs?

41:56.119 --> 42:05.369
So if she chooses left then her
payoff is 50 with probability

42:05.374 --> 42:11.544
Q--and I'm going to use the
pointer here,

42:11.544 --> 42:18.644
and hope that the camera can
see this too.

42:18.639 --> 42:33.859
She gets 50 with probability Q
and she gets 80 with probability

42:33.857 --> 42:43.127
1-Q.
If she chooses right then she

42:43.127 --> 42:59.527
gets 90 with probability Q and
she gets 20 with probability of

42:59.527 --> 43:10.967
1-Q.
I meant to point to that.

43:10.970 --> 43:14.240
So what?
So what is this:

43:14.237 --> 43:18.277
we're looking for a mixed
strategy Nash Equilibrium,

43:18.276 --> 43:22.406
so in particular,
not only Serena is mixing but

43:22.409 --> 43:27.859
in this case what we're claiming
is, Venus is mixing as well.

43:27.860 --> 43:34.870
So if Venus is mixing as well,
that means that Venus is using

43:34.871 --> 43:41.181
the strategy left with some
probability P and using the

43:41.181 --> 43:46.441
strategy right with some
probability 1-P.

43:46.440 --> 43:53.090
Since Venus sometimes chooses
left and sometimes chooses right

43:53.088 --> 43:59.088
as her best response to Q,
her best response to Serena,

43:59.090 --> 44:05.940
what must be true of the payoff
to left and the payoff to right?

44:05.940 --> 44:10.860
Let's go through it again,
so we're going to assume that

44:10.859 --> 44:13.719
Venus is mixing.
So sometimes she chooses left

44:13.720 --> 44:16.000
and sometimes she chooses right
and she's going to be,

44:16.004 --> 44:18.254
she's in a Nash Equilibrium,
so she's choosing a best

44:18.246 --> 44:21.136
response.
So whatever that mix P,

44:21.136 --> 44:24.116
1-P is, it's a best response.

44:24.119 --> 44:28.539
Since she's playing a best
response of P and that sometimes

44:28.544 --> 44:32.594
involves choosing left and
sometimes involves choosing

44:32.587 --> 44:36.287
right,
it must be the case that what?

44:36.289 --> 44:43.529
It must be the case that both
left itself and right itself are

44:43.531 --> 44:47.331
both themselves best response.

44:47.329 --> 44:51.329
If she's mixing between them,
it must be that both choosing

44:51.333 --> 44:55.133
left or choosing right are
themselves best responses.

44:55.130 --> 44:57.620
If they weren't she should just
drop them out of the mix,

44:57.620 --> 44:59.310
that would raise her average
payoff.

44:59.309 --> 45:04.569
Right, just like we dropped out
the short T.A.'s to get a high

45:04.570 --> 45:09.660
height and we dropped out the
failing Yale students to get a

45:09.658 --> 45:19.798
high GPA.
So if Venus is mixing in this

45:19.799 --> 45:38.459
Nash Equilibrium then the payoff
to left and to right must be

45:38.460 --> 45:42.860
equal,
they must both be best

45:42.862 --> 45:46.942
responses, both left and right
must be a best response,

45:46.940 --> 45:50.810
so in particular,
the expected payoffs must be

45:50.805 --> 45:54.395
the same.
Is that right, is that correct?

45:54.400 --> 45:56.930
So what does that allow me to
do?

45:56.929 --> 46:04.659
It allows me to put an equals
sign in here.

46:04.659 --> 46:07.349
Since left is a best response
and right is a best response,

46:07.351 --> 46:09.811
since they're both best
responses, they must yield the

46:09.810 --> 46:10.970
same expected payoff.

46:10.969 --> 46:13.429
Here's their expected payoffs,
they must be the same.

46:13.429 --> 46:18.089
Now, I've got one equation and
one unknown, and now I'm down to

46:18.088 --> 46:20.388
algebra.
So let me do the algebra.

46:20.389 --> 46:24.279
I claim this expression is
equal to that expression,

46:24.283 --> 46:29.023
so simplifying a bit I'm going
to get--you should just watch to

46:29.016 --> 46:33.516
make sure I don't get this
wrong--I'm going to get 40Q,

46:33.519 --> 46:41.479
so this implies 40Q is equal to
60(1-Q).

46:41.480 --> 46:49.080
So I took this 50 onto this
side and this 20 onto that side,

46:49.080 --> 46:56.810
so I have 40Q is equal to 60(1-
Q) and that implies that Q is

46:56.808 --> 47:02.258
equal to .6.
So those last two steps were

47:02.256 --> 47:05.156
just algebra.
So what was the trick here?

47:05.159 --> 47:11.679
The trick was I found Q,
which is how Serena is mixing

47:11.684 --> 47:18.064
by looking at Venus' payoffs,
knowing that Venus is mixing

47:18.056 --> 47:23.256
and hence I can set Venus'
payoffs equal to one another.

47:23.260 --> 47:26.520
Say that again,
I found the way in which Serena

47:26.524 --> 47:29.864
is mixing by knowing that if
Venus is mixing,

47:29.860 --> 47:37.950
her expected payoffs must be
equal and I solved out for

47:37.948 --> 47:43.488
Serena's mix,
this is Serena's mix.

47:43.490 --> 47:47.780
Let's do it again.

47:47.780 --> 47:49.720
Here I'm wishing I had another
board.

47:49.719 --> 47:55.119
I don't want to lose those
numbers entirely,

47:55.119 --> 48:00.519
so I'm going to try and squeeze
in a bit.

48:00.520 --> 48:01.320
I know what I can do.

48:01.320 --> 48:04.210
Let's get rid of this one
entirely.

48:04.210 --> 48:08.950
There we go, that works.

48:08.949 --> 48:11.809
Let's get rid of this one
entirely.

48:11.810 --> 48:15.140
I can still see my numbers.

48:15.140 --> 48:17.130
Let's do the converse.

48:17.130 --> 48:22.080
Let's do the trick again,
this time what I'm going to do

48:22.079 --> 48:26.489
is I'm going to figure out how
Venus is mixing.

48:26.489 --> 48:31.279
I know how Serena is mixing
now, so now I'm going to work

48:31.284 --> 48:33.514
out how Venus is mixing.

48:33.510 --> 48:41.640
Now, to figure out how Serena
was mixing, I used Venus'

48:41.635 --> 48:44.885
payoffs.
So to find out how Venus is

48:44.892 --> 48:46.842
mixing what am I going to do?

48:46.840 --> 48:50.580
I'm going to use Serena's
payoffs.

48:50.579 --> 48:58.659
So to find Venus' mix,
which is P, 1-P,

48:58.662 --> 49:09.722
--let's be careful it's her
Nash Equilibrium mix--use

49:09.723 --> 49:14.193
Serena's payoffs.

49:14.190 --> 49:22.770
Here we go, so if Serena
chooses, this is S's payoffs,

49:22.765 --> 49:31.175
if Serena chooses L then her
payoffs will be what?

49:31.179 --> 49:34.369
So again, just watch to make
sure I don't get this wrong and

49:34.371 --> 49:37.401
I'll point to the things to try
and help myself a bit.

49:37.400 --> 49:44.640
So with probability P she'll
get 50.

49:44.639 --> 49:59.479
So 50 with probability P,
and with probability 1-P she'll

49:59.475 --> 50:07.345
get 10.
And if she chooses to lean to

50:07.354 --> 50:18.914
the right, to lean towards her
forehand, then with probability

50:18.911 --> 50:29.901
P she'll get 20 and with
probability 1-P she'll get 80.

50:29.900 --> 50:38.460
We know that Serena is mixing,
so since Serena is mixing what

50:38.456 --> 50:43.586
must be true of these two
payoffs?

50:43.590 --> 50:44.830
What must be true of the two
payoffs?

50:44.829 --> 50:48.909
The payoff to l and the payoff
to r, what must be true about

50:48.914 --> 50:53.004
them since Serena is using a
mixture of these two strategies

50:52.998 --> 50:54.658
in Nash Equilibrium?

50:54.659 --> 50:58.969
It must be the case that both l
is a best response and r is a

50:58.972 --> 51:02.062
best response,
in which case the payoff must

51:02.062 --> 51:05.442
be, someone shout it out,
equal, thank you.

51:05.440 --> 51:10.780
They must be equal,
these must be equal.

51:10.780 --> 51:14.810
They must be equal since Serena
is indifferent between choosing

51:14.809 --> 51:17.799
left or right and hence is
mixing over them.

51:17.800 --> 51:21.130
So again, using the fact that
they're equal reduces this to

51:21.130 --> 51:23.830
algebra, and again,
I'll probably get this wrong

51:23.829 --> 51:30.189
but let me try.
So I claim, let's take 20 away

51:30.191 --> 51:37.221
from here, I've got 30P equals
70(1-P).

51:37.219 --> 51:39.199
I hope that's right,
that looks right.

51:39.199 --> 51:41.149
Again, this is just algebra at
this point.

51:41.150 --> 51:46.770
So I took 20 away from here and
10 away from there,

51:46.766 --> 51:50.806
and this implies that P equals
.7.

51:50.809 --> 51:56.149
So I claim I have now found the
mixed strategy Nash Equilibrium.

51:56.150 --> 52:01.120
Here it is.
The Nash Equilibrium is as

52:01.117 --> 52:05.087
follows.
Let's be careful,

52:05.088 --> 52:08.708
this is Venus' mix.

52:08.710 --> 52:18.790
So if Venus is mixing .7,
.3, .7 on left and .3 on right,

52:18.789 --> 52:28.329
and Serena is mixing .6,
.4, so this is Venus' mix and

52:28.329 --> 52:32.289
this Serena's mix.

52:32.289 --> 52:41.109
Venus is shooting to the left
of Serena with probability of .7

52:41.106 --> 52:49.196
and Serena is leaning that way
with probability of .6.

52:49.199 --> 52:54.879
So we were able to find this
Nash Equilibrium by using the

52:54.876 --> 52:58.826
trick before.
Now let's just reinforce this a

52:58.830 --> 53:01.370
little bit by talking about it.

53:01.369 --> 53:07.559
So suppose it were the case
that Serena, instead of leaning

53:07.558 --> 53:14.278
to the left .6 of the time leant
to the left more than .6 of the

53:14.279 --> 53:19.239
time.
So suppose you're Venus' coach,

53:19.235 --> 53:26.185
and suppose you know that
Serena leans to the left more

53:26.194 --> 53:32.644
than .6 of the time,
what would you advise Venus to

53:32.638 --> 53:34.528
do?
Let me try it again.

53:34.530 --> 53:38.750
So suppose your Venus' coach
and suppose you've observed the

53:38.747 --> 53:43.177
fact that Serena leans to the
left more than .6 of the time,

53:43.179 --> 53:47.869
what would you advise Venus to
do?

53:47.870 --> 53:50.240
Pass to the right, shout out.

53:50.239 --> 53:50.649
Student: Pass to the
right.

53:50.650 --> 53:51.980
Professor Ben Polak:
Pass to the right,

53:51.983 --> 53:55.113
exactly.
So if Serena cheats to the left

53:55.107 --> 53:59.937
more than .6 of the time,
then Venus' best response is

53:59.941 --> 54:02.861
always to shoot to the right.

54:02.860 --> 54:06.020
That maximizes her chance of
winning the point.

54:06.019 --> 54:15.299
Conversely, if Serena leans to
the left less than .6 of the

54:15.300 --> 54:20.740
time, then Venus should do what?

54:20.740 --> 54:24.150
Shoot to the left all the time.

54:24.150 --> 54:28.440
So if Serena doesn't choose
exactly this mix,

54:28.435 --> 54:33.885
then Venus' best response is
actually a pure strategy.

54:33.889 --> 54:36.969
Say it again,
if Serena leans to the left too

54:36.966 --> 54:40.526
often, more than .6,
then Venus should just go right

54:40.533 --> 54:43.823
and if Serena leans to the left
too little,

54:43.820 --> 54:46.610
then Venus should always go
left.

54:46.610 --> 54:49.770
We can do exactly the same the
other way around.

54:49.769 --> 54:53.659
If Venus shoots to the right,
so that's her cross hand

54:53.656 --> 54:56.586
passing shot more than .7 of the
time,

54:56.590 --> 55:04.310
and you're Serena's coach,
what should you tell Serena to

55:04.310 --> 55:08.400
do?
Go that way all the time.

55:08.400 --> 55:12.460
So if Venus is hitting it to
Serena's left more than .7 of

55:12.462 --> 55:16.242
the time, Serena should just
always go to her left,

55:16.239 --> 55:20.319
and if Venus is hitting to the
left less than .7 of the time,

55:20.317 --> 55:23.237
so to the right more than .3 of
the time,

55:23.239 --> 55:28.179
then Serena should always go to
the right.

55:28.179 --> 55:31.389
So that's how this kind of
comes back into the sort of the

55:31.392 --> 55:33.142
coaching manuals if you like.

55:33.140 --> 55:35.450
Okay, so how am I doing so far?

55:35.449 --> 55:39.359
Have I lost everyone yet or are
people still with me?

55:39.360 --> 55:42.160
How many of you play tennis,
ever?

55:42.159 --> 55:47.699
So all your tennis is going to
dramatically improve after

55:47.696 --> 55:50.806
today, right?
So now let's make life more

55:50.813 --> 55:53.663
interesting.
Let's go back to the start.

55:53.659 --> 55:56.599
We've figured out this is an
equilibrium, this is how Venus

55:56.598 --> 55:58.668
and Serena play,
Venus and Serena know each

55:58.673 --> 56:00.873
other perfectly well,
they know that they mix this

56:00.866 --> 56:02.446
way,
they're going to best respond

56:02.448 --> 56:04.458
to it, this is going to be where
they end up.

56:04.460 --> 56:09.130
But in the meantime,
Serena hires a new coach and

56:09.125 --> 56:15.025
Serena's new coach is just very,
very good at teaching Serena

56:15.028 --> 56:18.828
how to play at the net,
and in particular,

56:18.834 --> 56:21.994
how to hit the backhand volley.

56:21.989 --> 56:25.359
So Serena's new coach,
let's say it's Tony Roche or

56:25.359 --> 56:29.539
somebody, it's just a brilliant
coach and Tony Roche is able to

56:29.537 --> 56:34.117
improve Serena's backhand volley
and that changes these payoffs.

56:34.119 --> 56:38.199
So you should rewrite the whole
matrix but I'm going to cheat.

56:38.199 --> 56:43.779
So the new game is exactly the
same as it was everywhere else,

56:43.775 --> 56:48.705
except for now when Serena gets
to the backhand volley,

56:48.710 --> 56:51.910
she gets in it 70% of the time.

56:51.909 --> 56:59.689
So there used to 50,50 in that
box and now it's 30,70.

56:59.690 --> 57:04.110
So the game has changed because
Serena has got better at hitting

57:04.107 --> 57:05.437
backhand volleys.

57:05.440 --> 57:13.700
We want to figure out how is
this going to affect play at

57:13.699 --> 57:17.199
Wimbledon?
Now it doesn't take much to

57:17.203 --> 57:21.123
check that there is still no
pure strategy Nash Equilibrium.

57:21.119 --> 57:24.459
It's still the case,
in fact even more so,

57:24.463 --> 57:29.363
that Serena's best response to
Venus choosing left is to lean

57:29.355 --> 57:33.485
to the left.
So it's still the case that the

57:33.490 --> 57:39.150
best responses do not coincide,
there is still no pure strategy

57:39.150 --> 57:41.940
equilibrium.
What we're going to do of

57:41.942 --> 57:45.242
course is we're going to find a
mixed strategy equilibrium,

57:45.242 --> 57:47.632
but before we do so,
let's think about this

57:47.632 --> 57:51.572
intuitively.
Let's see if we can intuit an

57:51.570 --> 57:53.780
answer.
I'm guessing we can't,

57:53.778 --> 57:56.428
but let's see if we can intuit
an answer.

57:56.429 --> 58:01.149
So Serena has improved her
backhand volley,

58:01.150 --> 58:07.670
and hence when she reaches it
she gets it in more often.

58:07.670 --> 58:10.140
So one effect,
you might think,

58:10.140 --> 58:14.750
is what we might want to call a
direct effect and I think

58:14.751 --> 58:17.141
there's two effects here.

58:17.139 --> 58:21.869
There are two effects,
one of these I'm going to call

58:21.867 --> 58:25.047
the direct effect,
and by effect,

58:25.050 --> 58:30.500
I mean in particular an effect
on how Serena should play the

58:30.503 --> 58:33.723
game.
So since Serena has improved

58:33.715 --> 58:38.105
her backhand volley,
when she reaches that volley

58:38.113 --> 58:42.473
she gets it in more often,
so one might say in that

58:42.471 --> 58:46.801
case--your Serena's coach--in
that case you should lean to the

58:46.795 --> 58:49.555
left more often than you did
before,

58:49.559 --> 58:52.019
because at least when you get
that backhand volley you're

58:52.022 --> 58:53.432
going to get it in more often.

58:53.429 --> 59:03.369
So the direct effect says
Serena should lean left more,

59:03.373 --> 59:09.453
in other words,
Q should go up.

59:09.450 --> 59:11.490
Is that right?
So Serena's now better at

59:11.485 --> 59:15.085
playing this backhand volley,
so she may as well favor it a

59:15.092 --> 59:17.272
bit more and hence Q will go up.

59:17.269 --> 59:21.899
So that's the direct effect,
but of course there's a "but"

59:21.899 --> 59:23.839
coming.
What's the but?

59:23.840 --> 59:25.730
Again, let's see my tennis
players here,

59:25.729 --> 59:27.569
raise your hands if you play
tennis.

59:27.570 --> 59:30.030
Suddenly nobody plays tennis,
come on raise your hands okay.

59:30.030 --> 59:31.260
What's the but here?

59:31.260 --> 59:37.670
We think Serena's backhand has
improved so she might be tempted

59:37.666 --> 59:42.316
to play towards her backhand a
bit more often,

59:42.316 --> 59:46.046
what's the but?
So I claim the but is this--you

59:46.053 --> 59:49.493
tell me if I'm wrong--the but is
that Venus (she's her sister

59:49.486 --> 59:52.676
after all,
right, so Venus knows that

59:52.676 --> 59:58.416
Serena's backhand has improved)
so Venus is going to hit it to

59:58.416 --> 1:00:02.176
Serena's left less often than
before.

1:00:02.180 --> 1:00:05.400
Is that right?
So since Serena's backhand has

1:00:05.396 --> 1:00:09.846
improved, Venus is going to hit
it to Serena's backhand less

1:00:09.853 --> 1:00:13.543
often than before,
and that might make Serena less

1:00:13.539 --> 1:00:17.399
inclined to cheat towards her
backhand because the ball is

1:00:17.398 --> 1:00:19.428
coming that way less often.

1:00:19.429 --> 1:00:24.459
So this is a indirect or a
strategic effect.

1:00:24.460 --> 1:00:32.430
The strategic effect is Venus
hits L less often,

1:00:32.430 --> 1:00:42.100
so Serena should reduce the
number of times that she leans

1:00:42.096 --> 1:00:52.946
to the left because the ball is
coming that way fewer times.

1:00:52.949 --> 1:00:57.239
Now notice that these two
effects go in opposite

1:00:57.242 --> 1:00:59.802
directions, is that right?

1:00:59.800 --> 1:01:02.470
One of them tends to argue that
Q would go up,

1:01:02.469 --> 1:01:06.029
that's the direct effect and
the other one is more subtle,

1:01:06.030 --> 1:01:09.720
it says we now think about not
just how my play has improved,

1:01:09.719 --> 1:01:13.589
but also how the other person's
going to respond to knowing that

1:01:13.592 --> 1:01:16.832
my play has improved,
that's the more subtle effect

1:01:16.832 --> 1:01:18.772
and that's going to push Q down.

1:01:18.769 --> 1:01:22.699
That's going to make it less
likely, that's an argument

1:01:22.697 --> 1:01:24.877
against leaning to the left.

1:01:24.880 --> 1:01:26.900
So imagine you're going to be
Serena's coach,

1:01:26.899 --> 1:01:29.239
which of these effects do you
think is going to win,

1:01:29.240 --> 1:01:30.250
let's have a poll.

1:01:30.250 --> 1:01:32.390
Which of these effects do you
think is going to win?

1:01:32.389 --> 1:01:34.359
The direct effect or the
indirect effect?

1:01:34.360 --> 1:01:35.910
The direct effect or the
strategic effect?

1:01:35.910 --> 1:01:38.200
Who thinks the direct effect?

1:01:38.199 --> 1:01:40.759
Who thinks Serena,
who'd advise Serena to play to

1:01:40.763 --> 1:01:43.333
her strength a bit more and lean
left a bit more,

1:01:43.327 --> 1:01:45.087
who thinks the direct effect?

1:01:45.090 --> 1:01:47.690
Raise your hands,
let's have a poll.

1:01:47.690 --> 1:01:51.710
Who thinks the indirect effect,
the effect of Serena hitting it

1:01:51.713 --> 1:01:54.183
that way less often is going to
win?

1:01:54.179 --> 1:01:56.729
Who's abstaining and basically
refusing to be a coach?

1:01:56.730 --> 1:01:58.850
Quite a number of you,
all right.

1:01:58.849 --> 1:02:10.789
Well we're going to find out by
re-solving for the Nash

1:02:10.785 --> 1:02:17.275
Equilibrium.
What we're going to do is redo

1:02:17.275 --> 1:02:23.155
the calculation we did before
starting with Serena.

1:02:23.159 --> 1:02:26.319
So to find Serena's mix,
to find Serena's new

1:02:26.317 --> 1:02:29.257
equilibrium mix,
what do we have to do?

1:02:29.260 --> 1:02:31.280
The question is,
in equilibrium,

1:02:31.282 --> 1:02:35.262
is Serena going to lean to the
left more (so Q is going go up)

1:02:35.261 --> 1:02:37.741
or less (so Q's going to do
down).

1:02:37.739 --> 1:02:41.009
So I need to find out what is
Serena's new equilibrium mix.

1:02:41.010 --> 1:02:42.600
What's the new Q?

1:02:42.599 --> 1:02:44.989
How do I go about finding
Serena's equilibrium Q,

1:02:44.993 --> 1:02:46.193
what's the trick here?

1:02:46.190 --> 1:02:51.470
Shout it out.
Use Venus' payoffs.

1:02:51.469 --> 1:03:08.699
So to find the new Q for
Serena, use Venus' payoffs.

1:03:08.700 --> 1:03:11.610
Now let's do that.

1:03:11.610 --> 1:03:18.170
So from Venus' point of view,
if she chooses left then her

1:03:18.167 --> 1:03:22.997
payoffs are now,
and again I should use the

1:03:22.999 --> 1:03:27.759
pointer,
30 with probability Q,

1:03:27.760 --> 1:03:34.670
this is the new Q and 80 with
probability 1-Q,

1:03:34.673 --> 1:03:42.973
30 with probability Q plus 80
with probability 1-Q.

1:03:42.969 --> 1:03:46.209
Again, this is the new Q,
I should really give it,

1:03:46.205 --> 1:03:48.775
put Q prime or something but I
won't.

1:03:48.780 --> 1:03:55.520
If she chooses right then her
payoff is what?

1:03:55.519 --> 1:04:09.349
It's going to be 90 with
probability Q and 20 with

1:04:09.349 --> 1:04:16.359
probability 1-Q.
What do we know about these two

1:04:16.361 --> 1:04:19.041
payoffs if Venus is mixing in
equilibrium?

1:04:19.039 --> 1:04:21.389
We know she's mixing in
equilibrium because we saw there

1:04:21.386 --> 1:04:22.876
was no pure strategy
equilibrium,

1:04:22.880 --> 1:04:27.240
so what we do know about these
two payoffs since Venus is using

1:04:27.238 --> 1:04:29.978
both these strategies in
equilibrium?

1:04:29.980 --> 1:04:30.930
They must be the same.

1:04:30.929 --> 1:04:32.989
Since she's using both these
strategies, these strategies

1:04:32.992 --> 1:04:33.842
must be equally good.

1:04:33.840 --> 1:04:38.790
They must both be best
responses so these two payoffs

1:04:38.788 --> 1:04:41.818
are equal.
Since they're equal all I have

1:04:41.821 --> 1:04:44.471
to do is solve out for Q,
so let's do it.

1:04:44.469 --> 1:04:57.229
So I'm going to get 90 minus 30
is 60Q, is equal to 80 minus 20

1:04:57.227 --> 1:05:04.427
which is 60(1-Q),
so Q equals .5.

1:05:04.429 --> 1:05:07.789
If I did the algebra too
quickly just trust me,

1:05:07.786 --> 1:05:09.606
I think I got it right.

1:05:09.610 --> 1:05:12.820
From here on in,
it was just algebra.

1:05:12.820 --> 1:05:15.100
So what have I found out?

1:05:15.100 --> 1:05:18.380
Did Q go up or go down?

1:05:18.380 --> 1:05:21.370
Well it used to be,
Q used to be what?

1:05:21.369 --> 1:05:24.819
.6 and now its .5,
so let me ask what I think is

1:05:24.820 --> 1:05:27.830
an easy question,
did it go up or down?

1:05:27.830 --> 1:05:32.970
It went down.
Q went down,

1:05:32.973 --> 1:05:37.793
the equilibrium Q went down.

1:05:37.789 --> 1:05:40.039
So which effect turned out to
be bigger?

1:05:40.039 --> 1:05:43.269
The direct effect of playing
more to your strength or the

1:05:43.267 --> 1:05:46.667
indirect effect of taking into
account that your opponent is

1:05:46.668 --> 1:05:49.318
going to play less often to your
strength.

1:05:49.320 --> 1:05:52.280
Which effect turned out to be
the bigger effect?

1:05:52.280 --> 1:05:54.160
The indirect effect,
the strategic effect.

1:05:54.159 --> 1:05:58.219
Of course I really did want the
strategic effect to be bigger

1:05:58.220 --> 1:06:02.080
because this is a course about
strategy, but the strategic

1:06:02.077 --> 1:06:04.037
effect actually won here.

1:06:04.039 --> 1:06:12.239
The strategic effect,
the indirect effect is bigger.

1:06:12.239 --> 1:06:15.499
That's good news for me because
it says the slightly dumb coach

1:06:15.502 --> 1:06:18.552
who didn't bother to take Game
Theory would have stopped at

1:06:18.554 --> 1:06:21.764
this direct effect and they'd
have told Serena to go the wrong

1:06:21.763 --> 1:06:23.683
way,
but the smart coach who takes

1:06:23.683 --> 1:06:26.473
my class, and therefore somehow
contributes to my salary,

1:06:26.469 --> 1:06:33.839
in an extraordinarily indirect
way, gets it right.

1:06:33.840 --> 1:06:38.370
Now we can also solve out for
Venus' new mix and we'll do it

1:06:38.374 --> 1:06:40.914
in a second.
But before I do it,

1:06:40.914 --> 1:06:45.234
let me just point out that we
actually, we really can now

1:06:45.230 --> 1:06:47.080
intuit Venus' effect.

1:06:47.079 --> 1:06:49.359
It may not be exact numbers but
we can intuit here.

1:06:49.360 --> 1:06:52.820
As I claim, I claim if we think
this through carefully,

1:06:52.817 --> 1:06:56.207
we know whether Venus is
shooting more to the left,

1:06:56.210 --> 1:06:59.190
than she was before,
or less to the left,

1:06:59.190 --> 1:07:00.830
than she was before.

1:07:00.829 --> 1:07:04.639
Notice that in the new
equilibrium Serena is going less

1:07:04.636 --> 1:07:08.576
often to her left even though
she's better at hitting the

1:07:08.583 --> 1:07:10.823
backhand,
she's better at hitting the

1:07:10.819 --> 1:07:12.089
ball when she gets there.

1:07:12.090 --> 1:07:17.240
So since Serena is leaning left
less often what must be true

1:07:17.235 --> 1:07:20.545
about Venus in this new
equilibrium?

1:07:20.550 --> 1:07:23.620
It must be the case that Venus
is hitting the ball to the left

1:07:23.616 --> 1:07:25.286
less often.
Does that make sense?

1:07:25.289 --> 1:07:32.149
We have enough information
already on the board to tell us

1:07:32.147 --> 1:07:36.957
that, nevertheless,
let's do the math.

1:07:36.960 --> 1:07:51.560
Let's go and retrieve a board
to do the math.

1:07:51.559 --> 1:08:02.489
Just to complete this,
let's figure out exactly what

1:08:02.490 --> 1:08:07.850
Venus does do.
So to figure out what Venus is

1:08:07.847 --> 1:08:09.837
going to do, what's our trick?

1:08:09.840 --> 1:08:11.690
I want to figure out how Venus
is going to mix.

1:08:11.690 --> 1:08:17.200
I'm going to find out Venus'
new P, how do I find out Venus'

1:08:17.196 --> 1:08:19.246
new equilibrium mix?

1:08:19.250 --> 1:08:22.270
I look at Serena's payoffs.

1:08:22.270 --> 1:08:25.400
So if Serena chooses left,
her payoff is,

1:08:25.400 --> 1:08:28.610
and I'll read it off quickly
this time,

1:08:28.609 --> 1:08:36.549
is 70P plus 10(1-P) and if
Serena chooses right her payoff

1:08:36.550 --> 1:08:44.070
is 20P plus 80(1-P) and I'm
praying that the T.A.'s are

1:08:44.072 --> 1:08:50.622
going to catch me if I make a
mistake here,

1:08:50.619 --> 1:08:56.579
and I know these have to be
equal because I know that in

1:08:56.579 --> 1:09:00.859
fact Venus is mixing--sorry,
I know that Serena is mixing,

1:09:00.859 --> 1:09:01.959
so I know these must be equal.

1:09:01.960 --> 1:09:12.370
So since they're equal I can
solve out and hope that I've got

1:09:12.366 --> 1:09:19.646
this right, so I've got 50P
equals 70(1-P),

1:09:19.650 --> 1:09:24.160
so P is equal to 7/12.

1:09:24.159 --> 1:09:26.369
So again, that's just algebra,
I rushed it a bit,

1:09:26.369 --> 1:09:27.289
it's just algebra.

1:09:27.290 --> 1:09:28.930
Same idea, just algebra.

1:09:28.930 --> 1:09:35.420
So 7/12 is indeed smaller than
what it used to be,

1:09:35.419 --> 1:09:43.099
because it used to be 7/10,
so that confirms our result.

1:09:43.100 --> 1:09:45.060
So the strategic effect
dominated.

1:09:45.060 --> 1:09:49.850
Venus shot to Serena's backhand
less often, and as a

1:09:49.848 --> 1:09:54.898
consequence, so much so,
that Serena actually found it

1:09:54.899 --> 1:10:00.299
worthwhile going more to the
right than she used to before.

1:10:00.300 --> 1:10:02.310
Now let's just talk this
through one more time.

1:10:02.310 --> 1:10:04.780
This was a comparative statics
exercise.

1:10:04.779 --> 1:10:08.019
We looked at a game,
we found an equilibrium,

1:10:08.019 --> 1:10:11.699
we changed something
fundamental about the game,

1:10:11.699 --> 1:10:14.499
and we looked again to look at
the new equilibrium,

1:10:14.502 --> 1:10:16.522
that's called comparative
statics.

1:10:16.520 --> 1:10:18.770
Let's talk through the
intuition.

1:10:18.770 --> 1:10:27.640
Before we made any changes
Venus was indifferent.

1:10:27.640 --> 1:10:31.410
She was indifferent between
shooting to the left and

1:10:31.407 --> 1:10:33.177
shooting to the right.

1:10:33.180 --> 1:10:38.800
Then we improved Serena's
ability to hit the volley to her

1:10:38.797 --> 1:10:42.737
left, we improved her backhand
volley.

1:10:42.739 --> 1:10:50.709
If we had not changed the way
Serena played then what would

1:10:50.707 --> 1:10:56.397
Venus have done?
So suppose in fact Serena's Q

1:10:56.395 --> 1:11:01.335
had not changed.
If Serena's Q had not changed,

1:11:01.342 --> 1:11:05.942
remembering that Venus was
indifferent before,

1:11:05.935 --> 1:11:10.115
how would Venus have changed
her play?

1:11:10.120 --> 1:11:13.090
Somebody?
If we started from the old Q

1:11:13.085 --> 1:11:17.085
and then we improved Serena's
ability to play the backhand

1:11:17.087 --> 1:11:21.437
volley, and if Q didn't change,
what would Venus have done?

1:11:21.439 --> 1:11:24.409
She'd never,
ever have shot to the left

1:11:24.405 --> 1:11:29.235
anymore, she'd only have shot to
the right which can't possibly

1:11:29.243 --> 1:11:30.963
be an equilibrium.

1:11:30.960 --> 1:11:34.940
So something about Serena's
play has to bring Venus back

1:11:34.943 --> 1:11:38.423
into equilibrium,
it brings Venus back into being

1:11:38.419 --> 1:11:40.809
indifferent, and what was it?

1:11:40.810 --> 1:11:47.590
It was Serena moving to the
left less often and moving to

1:11:47.593 --> 1:11:50.383
the right more often.

1:11:50.380 --> 1:11:52.870
To say it again,
if we didn't change Q,

1:11:52.871 --> 1:11:56.741
Venus would only go to the
right, so we need to reduce Q,

1:11:56.739 --> 1:12:00.959
have Serena go to the right,
to bring Venus back into

1:12:00.955 --> 1:12:03.665
equilibrium.
Conversely, if Venus hadn't

1:12:03.670 --> 1:12:06.600
changed her behavior,
if Venus had gone on shooting

1:12:06.595 --> 1:12:09.705
exactly the same as she was,
P and 1-P as before,

1:12:09.713 --> 1:12:13.493
then Serena would have only
gone to the left and that can't

1:12:13.491 --> 1:12:14.861
be an equilibrium.

1:12:14.859 --> 1:12:17.509
So it must be something about
Venus' play that brings Serena

1:12:17.512 --> 1:12:19.312
back into equilibrium,
and what is it?

1:12:19.310 --> 1:12:24.320
It's that Venus starts shooting
to the right more often.

1:12:24.319 --> 1:12:27.879
So just two reminders,
before you leave two reminders.

1:12:27.880 --> 1:12:28.610
Wait, wait, wait.

1:12:28.609 --> 1:12:32.479
First, in about five minutes
time a handout will magically

1:12:32.477 --> 1:12:36.817
appear on the website that goes
through these arguments again,

1:12:36.819 --> 1:12:40.779
all of them in two other games,
so you can have a look at the

1:12:40.779 --> 1:12:42.279
handout.
Second thing,

1:12:42.276 --> 1:12:45.486
a problem set has already
appeared by magic on that

1:12:45.492 --> 1:12:49.612
website that gives you lots of
examples like this to work on.

1:12:49.609 --> 1:12:52.419
Play tennis over the weekend
for practice and we'll see you

1:12:52.418 --> 1:12:52.998
on Monday.