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Hi, welcome back. The previous lecture we
talked about probabilities. And we did so

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because we're gonna use probabilities in
this lecture to talk about decision making

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under uncertainty. And to do that, we're
gonna introduce a new technique, a model

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known as decision tree model. Now this
decision tree model is really gonna be

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useful in terms of making decisions when
there's lots of contingencies. When

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there's probabilistic events, when we
don't know the future state of the world.

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So big reason we wanna learn this model is
just to better thinkers, to make better

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choices, make better decisions, rather
than just sort of throw up our hands and

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say, I can't figure out what to do. I
think I'm gonna choose this. Now there's

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gonna be two other reasons as well. One is
gonna be, we're gonna use them to infer.

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Odd things about the world, about other
people's choices. So we, we're going to

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see someone choice and from that. We can
get some understanding of how that person

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thinks about the world, so we can again
use it to explain what's going on. And

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then a third reason, for fun, is we use
these decision trees to actually, maybe

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learn a little bit about ourselves,
[inaudible] fun example at the end. So

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let's get started. What is a decision
tree? Decision tree's pretty

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straightforward. What you do is you think,
I've got some choice I can make. Maybe I

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can, you know, buy something or not. And,
you know, if I don't buy it maybe my

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benefit is zero, and if I buy it. Maybe my
benefit is plus five. Well, if that's the

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case, then I should buy it, right, because
it's got a positive value. So decision

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[inaudible] gets [inaudible] draw branches
representing our choices, and we choose

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the branch that has the highest payoff.
Well we wanna do this, though when the

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choices are a little bit harder and
there's all sorts of contingencies and

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probabilities. So here's a example.
Imagine the following scenario. You're

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planning a trip to a city and you've got a
ticket to go to the museum, lets say from

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one to two. And suppose the museum is
quite a ways from the train station. So

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you look at train ticket prices and you
see you can buy a ticket for the three

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o'clock train. For only $200. But the four
o'clock train is $400. You're trying

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[inaudible] boy, should I buy that? You
know, should I try and save money by

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buying that ticket for the three o'clock
train before it sells out? Now there's a

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40 percent chance you're not gonna make.
The train. So now you gotta think, oh my

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gosh, should I but the ticket or not? Give
there's a 40 percent chance I'm not gonna

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make it. And if I don't make it, then I'm
gonna have to buy two tickets. I'm gonna

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basically throw away the $200. Well, how
do we make that sorta choice? Well, it's

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not very hard. What we can do is we can
draw a decision tree. Now, the way to draw

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these trees, is, we're gonna put a square
box to represent our decision [inaudible]

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a choice. Do I buy or do I not buy? Well,
it's not quite that simple, right?

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Because, if I buy, there's some
possibility, a 60 percent chance, and

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let's put a six here, that I make the
train. And a 40 percent chance that I'm

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late. So now I've gotta decide, okay, what
do I do? Because that's a little bit more

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complicated. Well. To finish this off, to
use this tree, so I can make a good

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decision, all I've got to do is put the
values of each choice. So if I don't buy

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the ticket. Then I'm gonna be out $400.00,
forget about the expensive ticket. If I do

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buy the ticket and I make my train, that's
great because what happens is I'm only out

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$200.00. But if I buy my ticket and I'm
late, then I'm out $600.00. And, so now

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I've got all the information that I need.
I've got all my payoffs at the end of each

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branch. I've got the probability of each
branch, 60 percent of 40%, and I just have

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to figure out what's the best choice for
me. So, let's make this all. So let's make

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this all nice and clean. So here's all my
data, and I've just got to decide what's

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the better choice. It's clear if I don't
buy ticket, I'm out $400. What if I buy

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it? Well, there's a 60 percent chance that
I'm out $200 plus. A 40 percent chance

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that I'm out $600. Well, if I add this up,
that's 120, 60 times 200, plus 240. And

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120+240 is 360. So what I get is if I buy
the ticket, right, I'm out $360. And if I

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don't buy the ticket, I'm out $400. So
it's a fairly easy choice, right? Buy for

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360, right, or don't buy. For 400 and
here, since I want to have the lower cost,

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what I'm going to do is buy. That was a
fairly simple example. Let's do one that's

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more complicated. Suppose you think about
applying for a scholarship and there's,

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it's worth $5,000. That seems like a
pretty good deal, and they limit this

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scholarship to 200 applicants so you go
into the you know, the office, and you

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realize, you know, you can be one of the
first 200. Now for this scholarship

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[inaudible], you have to write a two page
essay, and after you write a two page

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essay explaining why you deserve this
scholarship, they're gonna pick ten

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finalists and those finalists are gonna
have to write ten page essays. So now

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you've got this choice. I could. You know,
you can basically get $5000. That's a lot

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of money, but you've got to write these
two essays. The two pager and then if you

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make it as a finalist, a ten pager. And
there's some probability of making it as a

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finalist and some probability of winning.
So you look at this, and you think, how do

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I make this choice? Well, again, what do
we need to know? We need to know the

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probability of events happening. So the
probability of making it to be a finalist

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in the probability of winning, and that's
pretty straightforward. And we need to

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know the payoff. So we know that payoff
from the scholarship, but we need to know

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the cost. Of these assets. So to use the
decision tree the first step you're going

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to make is to figure out the cost. So
let's suppose you figure, well, what's the

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cost to me of writing a two page essay.
And if you're a, well, maybe twenty bucks.

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Maybe it's worth $20.00 of my time to
write a two page essay. And what about the

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ten page essay. Well the ten page essay,
you could say, well maybe that's only

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$40.00. That means it's only 40. I've
already written the two pages. I've

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outlined my ideas and I'd be sorta excited
about having to be a finalist in, and it's

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not that having to expand on my ideas, so
let's just assume, $40.00. So, now I've

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got everything I need. I've got my
benefits and my costs and all my

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probabilities, so you just have to. Draw
the tree, right? Well, that's right. First

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step is draw the tree but once I draw the
tree I've got to write down all those

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payoffs and probabilities. So I've got to
make sure I've got everything right. And

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once I've got everything right, I can
solve it backwards just like I did before.

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I can figure out the value of each branch,
right, and then figure out what choices I

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should make. So let's draw the tree. The
question is do I write the essay or not.

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If I don't write the essay I get nothing,
and If I write the essay, well, now it's a

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little more complicated because what can
happen. Well, there's going to be some

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random note here where I could be
selected. Or not. And then if I'm

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selected, I can decide whether I want to
write essay two. Were not, but it probably

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will. And then there's going to be some
random thing, whether I win. Yeah, that

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will be great. Or, whether I lose. >> And
that won't be so great. But no, what I

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want to do is, not have these smiling and
happy faces, the happy faces, and the sad

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faces. Actually, want to like put in the
numbers. So let's do it. This isn't too

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hard. Again, if I don't that's zero. The,
what's the probability I've selected?

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Well, 200 applicants, ten make it to be
finalists, so we can assume this is five%,

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right?.05, and there's a 95 percent chance
I lose. Now SA two, I can either do it or

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don't do it. And then, if I win, here's
another change node right here. What are

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the odds of me winning? The odds of me
winning here are ten%, one out of ten. And

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there's a 90 percent chance they lose. So
those are all my probabilities. Now I

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gotta figure out my payoffs. Well, even if
I don't like [inaudible], my [inaudible]

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zero. If I write the first essay and lose,
I'm out $twenty, so that's minus twenty.

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If I'm selected but then don't write the
second essay which is sort of a crazy

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thing to do and also have $twenty, if I do
write the second essay and lose, I'm out

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$60 because I wrote two essays, one for
$twenty, one for $40. But if I win, right,

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then I get $4,940. I get the $5,000 minus
the $60. For, the cost of writing the

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essay. So let's clean this up a little
bit. So here's the total analysis, right?

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Here's the beautiful game tree with all my
probabilities. What I've gotta do is I've

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gotta figure out what's my payoff, right?
What's the payoff in doing these things?

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So let's just work our way backwards. So
let's start right here. If I win, there's

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a ten percent chance I win. That's 49/40,
so I can take point one. Times 4940, plus

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point nine. Times minus 60. Well, what is
that?.4 times 4940 is 494. Right,

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and.9-60=54. So what I get is 440. So what
I can do is I can put 440 right here, I

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can basically wipe out all this stuff over
here and put a 440 there. Now so if I look

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at this question that do I write an essay
too, it seems really obvious, right. If I

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write the essay, my expected winnings are
440. If I don't write the essay, my

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expected winnings are minus twenty. So
again, let's clean this up. So if I write

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the essay, 440, if I don't write it, it's
minus twenty. It seems pretty clear,

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right, that I should write the essay. So
now, it just comes down to this. If I

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write essay one, there's some chance I'm
gonna get selected. If I'm selected, my

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expected winning is 440. If I'm not
selected, I'm gonna end up losing twenty.

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So what's this worth? Well. 440, I'm going
to get that ten to five percent of the

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time, but 95 percent of the time, right.
I'm gonna lose twenty. So I've gotta add

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these two things up. Well, 440 times five
percent is 22. And minus twenty times 95

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is minus nineteen. So if I add those two
things up, I get three. So what that means

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is I can replace this whole branch, in
working backwards, with a three. So now if

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I look at my decision, should I write the
essay or not? If I don't write the essay,

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I get nothing. And if I write the essay,
my expected value's $three. So, what

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should I do? Well, I should probably write
the essay, 'cause it's got a positive

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expected value. And the interesting thing
here is, if there'd been 300 applicants,

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or 400 applicants, right? Maybe I don't
want to write the essay. So, what the tree

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does, what this decision tree analysis
does, is it helps us figure out, was it

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really a good thing to do? So that's how
you use decision trees to make decisions.

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Let's do something a little bit trickier
with them. Let's do something where we try

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and infer what other people think about
probabilities. So suppose you have a

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friend and they say look, I know about
this investment and it sounds a little

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risky to you and they say it's going to
pay $50,000. You know, but almost sure and

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you're gonna put $2,000 in. She says,
look, I'm in. I'm investing my 2,000

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bucks, this is a great deal you
[inaudible] invest. So you've gotta

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decide, you know, do you want to invest?
Well, the first thing [inaudible] what

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does she think the likelihood of this
thing really is? Well, what we can do, we

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can draw a tree and say, you know, I can
invest, or I can not invest. And there's

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some probability that this will succeed
and there's some probability that it's

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going to fail. And if it succeeds, she's
going to make $50,000 and so would I. And

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if it fails, I'll lose $2,000. So let's
try and figure out what our friend is

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thinking. So what our friend is thinking
is that 50P, right, minus 2x1-P is bigger

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than zero. So she's figuring the end of
this branch right here, before chance

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takes its move, is higher than zero. So if
I work this through, it says 50P. Minus

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two plus 2P is bigger than zero. So if I
bring the P's all to this side, we're

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gonna get 52P, has gotta be bigger than
two. So what she's assuming is P is bigger

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than two over 52. Or about, you know,
right around four%. So now I can look at

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this investment thing. Do I really think
there's a four percent chance it's going

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to pay off? Clearly my friend does,
because she's in, and I can decide whether

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or not to make the decision or not. I can
also infer and this is the key point, I

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can infer from her decision that she
thinks that even though this is risky,

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that there's way more than a four percent
chance that it's going to happen. Because

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otherwise she wouldn't put her money in
it. Okay, so decision trees, even if we

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don't know the probabilities, if we look
at someone else's actions, we can infer

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what they think the probabilities are. Now
one last thing that's sort of fun. We can

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use these trees to infer payoffs and
sometimes we can use them even to infer

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payoffs about ourselves, like how we think
about things. So here's the scenario, it's

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kind of a fun one. You've got a standby
ticket, right? Got some standby ticket to

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go visit your parents, you call the
airlines in the morning of the flight and

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it's like a one-third chance that you're
going to make the flight. Two-thirds

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chance you're probably not going to make
it. So you've got to decide do you go to

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the airport, right, or do you just stay on
campus and not go home for the weekend.

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Well, suppose you decide not to go. You
decide to stay at the airport. You can use

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a decision tree to find out exactly how
much you really wanted to see your

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parents. What do I mean by that? Well,
let's see. So here's the decision; you

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stay on campus and let's suppose, let's
make that a baseline payoff of zero. You

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can go to the airport, and there's a
one-third change you're gonna make the

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flight. And let's call this V, the value
of seeing your parents. Now, there's a

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two-thirds chance, right, right, and we'll
put in a little cost here. Minus C for,

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you know, a couple hours of your time to
take the taxi to the airport and back, or

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take the train to the airport and back.
Alternatively, you cannot make the flight,

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and there, the cost is just gonna be the
straight minus C. Well, since you chose to

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stay at home, what that means is this.
That means one-third. Times the value of

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seeing your parents, minus the cost of
going to the airport. Right? Plus.

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Two-thirds times minus C, the cost of
going to the airport, has got to be less

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than zero. What that means is, one-third
V, if I add up the Cs, minus C is less

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than zero. So, if I work all the way
through this, what this means is that V.

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Is less than 3C. So it means your value of
going to see your parents is less than

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tree times the cost of going to the
airport. What's that's telling you is,

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well maybe I didn't want to see my parents
very much. Now if you did go to the

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airport and try and fly standby that's
saying the opposite. That's saying V is

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bigger that 3C and it's saying that you
really did want to see your parents. Which

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is a great thing since I'm sure your
parents would love to see you. Okay we've

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done decision trees here, lots of fun.
What we've shown is when we've got these

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decisions to make, where there's lots of
probabilities and contingencies, these

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trees are really helpful. They're really
useful in helping us make these reasoned

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decision, now again. You don't have to
adhere to what the model tells you to do,

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but the model is again a crutch, an aide,
a guide to help you making better

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decisions. We also side to use these trees
to infer what other people are thinking

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about probabilities. Right, cuz when our
friend made that investment, we could

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infer that she thought that there was a
more, at least a four percent chance that

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thing was gonna pay off. And the last
thing you could do is after the fact you

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could think, I made this choice. What is
this choice saying about how I think about

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the world or how I think about my parents
depending on what you thought the

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probabilities were. Okay, thanks a lot.