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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â probabilities were. Okay, thanks a lot.

Â