Hi, welcome back. The previous lecture we
talked about probabilities. And we did so
because we're gonna use probabilities in
this lecture to talk about decision making
under uncertainty. And to do that, we're
gonna introduce a new technique, a model
known as decision tree model. Now this
decision tree model is really gonna be
useful in terms of making decisions when
there's lots of contingencies. When
there's probabilistic events, when we
don't know the future state of the world.
So big reason we wanna learn this model is
just to better thinkers, to make better
choices, make better decisions, rather
than just sort of throw up our hands and
say, I can't figure out what to do. I
think I'm gonna choose this. Now there's
gonna be two other reasons as well. One is
gonna be, we're gonna use them to infer.
Odd things about the world, about other
people's choices. So we, we're going to
see someone choice and from that. We can
get some understanding of how that person
thinks about the world, so we can again
use it to explain what's going on. And
then a third reason, for fun, is we use
these decision trees to actually, maybe
learn a little bit about ourselves,
[inaudible] fun example at the end. So
let's get started. What is a decision
tree? Decision tree's pretty
straightforward. What you do is you think,
I've got some choice I can make. Maybe I
can, you know, buy something or not. And,
you know, if I don't buy it maybe my
benefit is zero, and if I buy it. Maybe my
benefit is plus five. Well, if that's the
case, then I should buy it, right, because
it's got a positive value. So decision
[inaudible] gets [inaudible] draw branches
representing our choices, and we choose
the branch that has the highest payoff.
Well we wanna do this, though when the
choices are a little bit harder and
there's all sorts of contingencies and
probabilities. So here's a example.
Imagine the following scenario. You're
planning a trip to a city and you've got a
ticket to go to the museum, lets say from
one to two. And suppose the museum is
quite a ways from the train station. So
you look at train ticket prices and you
see you can buy a ticket for the three
o'clock train. For only $200. But the four
o'clock train is $400. You're trying
[inaudible] boy, should I buy that? You
know, should I try and save money by
buying that ticket for the three o'clock
train before it sells out? Now there's a
40 percent chance you're not gonna make.
The train. So now you gotta think, oh my
gosh, should I but the ticket or not? Give
there's a 40 percent chance I'm not gonna
make it. And if I don't make it, then I'm
gonna have to buy two tickets. I'm gonna
basically throw away the $200. Well, how
do we make that sorta choice? Well, it's
not very hard. What we can do is we can
draw a decision tree. Now, the way to draw
these trees, is, we're gonna put a square
box to represent our decision [inaudible]
a choice. Do I buy or do I not buy? Well,
it's not quite that simple, right?
Because, if I buy, there's some
possibility, a 60 percent chance, and
let's put a six here, that I make the
train. And a 40 percent chance that I'm
late. So now I've gotta decide, okay, what
do I do? Because that's a little bit more
complicated. Well. To finish this off, to
use this tree, so I can make a good
decision, all I've got to do is put the
values of each choice. So if I don't buy
the ticket. Then I'm gonna be out $400.00,
forget about the expensive ticket. If I do
buy the ticket and I make my train, that's
great because what happens is I'm only out
$200.00. But if I buy my ticket and I'm
late, then I'm out $600.00. And, so now
I've got all the information that I need.
I've got all my payoffs at the end of each
branch. I've got the probability of each
branch, 60 percent of 40%, and I just have
to figure out what's the best choice for
me. So, let's make this all. So let's make
this all nice and clean. So here's all my
data, and I've just got to decide what's
the better choice. It's clear if I don't
buy ticket, I'm out $400. What if I buy
it? Well, there's a 60 percent chance that
I'm out $200 plus. A 40 percent chance
that I'm out $600. Well, if I add this up,
that's 120, 60 times 200, plus 240. And
120+240 is 360. So what I get is if I buy
the ticket, right, I'm out $360. And if I
don't buy the ticket, I'm out $400. So
it's a fairly easy choice, right? Buy for
360, right, or don't buy. For 400 and
here, since I want to have the lower cost,
what I'm going to do is buy. That was a
fairly simple example. Let's do one that's
more complicated. Suppose you think about
applying for a scholarship and there's,
it's worth $5,000. That seems like a
pretty good deal, and they limit this
scholarship to 200 applicants so you go
into the you know, the office, and you
realize, you know, you can be one of the
first 200. Now for this scholarship
[inaudible], you have to write a two page
essay, and after you write a two page
essay explaining why you deserve this
scholarship, they're gonna pick ten
finalists and those finalists are gonna
have to write ten page essays. So now
you've got this choice. I could. You know,
you can basically get $5000. That's a lot
of money, but you've got to write these
two essays. The two pager and then if you
make it as a finalist, a ten pager. And
there's some probability of making it as a
finalist and some probability of winning.
So you look at this, and you think, how do
I make this choice? Well, again, what do
we need to know? We need to know the
probability of events happening. So the
probability of making it to be a finalist
in the probability of winning, and that's
pretty straightforward. And we need to
know the payoff. So we know that payoff
from the scholarship, but we need to know
the cost. Of these assets. So to use the
decision tree the first step you're going
to make is to figure out the cost. So
let's suppose you figure, well, what's the
cost to me of writing a two page essay.
And if you're a, well, maybe twenty bucks.
Maybe it's worth $20.00 of my time to
write a two page essay. And what about the
ten page essay. Well the ten page essay,
you could say, well maybe that's only
$40.00. That means it's only 40. I've
already written the two pages. I've
outlined my ideas and I'd be sorta excited
about having to be a finalist in, and it's
not that having to expand on my ideas, so
let's just assume, $40.00. So, now I've
got everything I need. I've got my
benefits and my costs and all my
probabilities, so you just have to. Draw
the tree, right? Well, that's right. First
step is draw the tree but once I draw the
tree I've got to write down all those
payoffs and probabilities. So I've got to
make sure I've got everything right. And
once I've got everything right, I can
solve it backwards just like I did before.
I can figure out the value of each branch,
right, and then figure out what choices I
should make. So let's draw the tree. The
question is do I write the essay or not.
If I don't write the essay I get nothing,
and If I write the essay, well, now it's a
little more complicated because what can
happen. Well, there's going to be some
random note here where I could be
selected. Or not. And then if I'm
selected, I can decide whether I want to
write essay two. Were not, but it probably
will. And then there's going to be some
random thing, whether I win. Yeah, that
will be great. Or, whether I lose. >> And
that won't be so great. But no, what I
want to do is, not have these smiling and
happy faces, the happy faces, and the sad
faces. Actually, want to like put in the
numbers. So let's do it. This isn't too
hard. Again, if I don't that's zero. The,
what's the probability I've selected?
Well, 200 applicants, ten make it to be
finalists, so we can assume this is five%,
right?.05, and there's a 95 percent chance
I lose. Now SA two, I can either do it or
don't do it. And then, if I win, here's
another change node right here. What are
the odds of me winning? The odds of me
winning here are ten%, one out of ten. And
there's a 90 percent chance they lose. So
those are all my probabilities. Now I
gotta figure out my payoffs. Well, even if
I don't like [inaudible], my [inaudible]
zero. If I write the first essay and lose,
I'm out $twenty, so that's minus twenty.
If I'm selected but then don't write the
second essay which is sort of a crazy
thing to do and also have $twenty, if I do
write the second essay and lose, I'm out
$60 because I wrote two essays, one for
$twenty, one for $40. But if I win, right,
then I get $4,940. I get the $5,000 minus
the $60. For, the cost of writing the
essay. So let's clean this up a little
bit. So here's the total analysis, right?
Here's the beautiful game tree with all my
probabilities. What I've gotta do is I've
gotta figure out what's my payoff, right?
What's the payoff in doing these things?
So let's just work our way backwards. So
let's start right here. If I win, there's
a ten percent chance I win. That's 49/40,
so I can take point one. Times 4940, plus
point nine. Times minus 60. Well, what is
that?.4 times 4940 is 494. Right,
and.9-60=54. So what I get is 440. So what
I can do is I can put 440 right here, I
can basically wipe out all this stuff over
here and put a 440 there. Now so if I look
at this question that do I write an essay
too, it seems really obvious, right. If I
write the essay, my expected winnings are
440. If I don't write the essay, my
expected winnings are minus twenty. So
again, let's clean this up. So if I write
the essay, 440, if I don't write it, it's
minus twenty. It seems pretty clear,
right, that I should write the essay. So
now, it just comes down to this. If I
write essay one, there's some chance I'm
gonna get selected. If I'm selected, my
expected winning is 440. If I'm not
selected, I'm gonna end up losing twenty.
So what's this worth? Well. 440, I'm going
to get that ten to five percent of the
time, but 95 percent of the time, right.
I'm gonna lose twenty. So I've gotta add
these two things up. Well, 440 times five
percent is 22. And minus twenty times 95
is minus nineteen. So if I add those two
things up, I get three. So what that means
is I can replace this whole branch, in
working backwards, with a three. So now if
I look at my decision, should I write the
essay or not? If I don't write the essay,
I get nothing. And if I write the essay,
my expected value's $three. So, what
should I do? Well, I should probably write
the essay, 'cause it's got a positive
expected value. And the interesting thing
here is, if there'd been 300 applicants,
or 400 applicants, right? Maybe I don't
want to write the essay. So, what the tree
does, what this decision tree analysis
does, is it helps us figure out, was it
really a good thing to do? So that's how
you use decision trees to make decisions.
Let's do something a little bit trickier
with them. Let's do something where we try
and infer what other people think about
probabilities. So suppose you have a
friend and they say look, I know about
this investment and it sounds a little
risky to you and they say it's going to
pay $50,000. You know, but almost sure and
you're gonna put $2,000 in. She says,
look, I'm in. I'm investing my 2,000
bucks, this is a great deal you
[inaudible] invest. So you've gotta
decide, you know, do you want to invest?
Well, the first thing [inaudible] what
does she think the likelihood of this
thing really is? Well, what we can do, we
can draw a tree and say, you know, I can
invest, or I can not invest. And there's
some probability that this will succeed
and there's some probability that it's
going to fail. And if it succeeds, she's
going to make $50,000 and so would I. And
if it fails, I'll lose $2,000. So let's
try and figure out what our friend is
thinking. So what our friend is thinking
is that 50P, right, minus 2x1-P is bigger
than zero. So she's figuring the end of
this branch right here, before chance
takes its move, is higher than zero. So if
I work this through, it says 50P. Minus
two plus 2P is bigger than zero. So if I
bring the P's all to this side, we're
gonna get 52P, has gotta be bigger than
two. So what she's assuming is P is bigger
than two over 52. Or about, you know,
right around four%. So now I can look at
this investment thing. Do I really think
there's a four percent chance it's going
to pay off? Clearly my friend does,
because she's in, and I can decide whether
or not to make the decision or not. I can
also infer and this is the key point, I
can infer from her decision that she
thinks that even though this is risky,
that there's way more than a four percent
chance that it's going to happen. Because
otherwise she wouldn't put her money in
it. Okay, so decision trees, even if we
don't know the probabilities, if we look
at someone else's actions, we can infer
what they think the probabilities are. Now
one last thing that's sort of fun. We can
use these trees to infer payoffs and
sometimes we can use them even to infer
payoffs about ourselves, like how we think
about things. So here's the scenario, it's
kind of a fun one. You've got a standby
ticket, right? Got some standby ticket to
go visit your parents, you call the
airlines in the morning of the flight and
it's like a one-third chance that you're
going to make the flight. Two-thirds
chance you're probably not going to make
it. So you've got to decide do you go to
the airport, right, or do you just stay on
campus and not go home for the weekend.
Well, suppose you decide not to go. You
decide to stay at the airport. You can use
a decision tree to find out exactly how
much you really wanted to see your
parents. What do I mean by that? Well,
let's see. So here's the decision; you
stay on campus and let's suppose, let's
make that a baseline payoff of zero. You
can go to the airport, and there's a
one-third change you're gonna make the
flight. And let's call this V, the value
of seeing your parents. Now, there's a
two-thirds chance, right, right, and we'll
put in a little cost here. Minus C for,
you know, a couple hours of your time to
take the taxi to the airport and back, or
take the train to the airport and back.
Alternatively, you cannot make the flight,
and there, the cost is just gonna be the
straight minus C. Well, since you chose to
stay at home, what that means is this.
That means one-third. Times the value of
seeing your parents, minus the cost of
going to the airport. Right? Plus.
Two-thirds times minus C, the cost of
going to the airport, has got to be less
than zero. What that means is, one-third
V, if I add up the Cs, minus C is less
than zero. So, if I work all the way
through this, what this means is that V.
Is less than 3C. So it means your value of
going to see your parents is less than
tree times the cost of going to the
airport. What's that's telling you is,
well maybe I didn't want to see my parents
very much. Now if you did go to the
airport and try and fly standby that's
saying the opposite. That's saying V is
bigger that 3C and it's saying that you
really did want to see your parents. Which
is a great thing since I'm sure your
parents would love to see you. Okay we've
done decision trees here, lots of fun.
What we've shown is when we've got these
decisions to make, where there's lots of
probabilities and contingencies, these
trees are really helpful. They're really
useful in helping us make these reasoned
decision, now again. You don't have to
adhere to what the model tells you to do,
but the model is again a crutch, an aide,
a guide to help you making better
decisions. We also side to use these trees
to infer what other people are thinking
about probabilities. Right, cuz when our
friend made that investment, we could
infer that she thought that there was a
more, at least a four percent chance that
thing was gonna pay off. And the last
thing you could do is after the fact you
could think, I made this choice. What is
this choice saying about how I think about
the world or how I think about my parents
depending on what you thought the
probabilities were. Okay, thanks a lot.