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.