Hi. In our last lecture in this unit what we're going to do is we're going to look at something called the value of information. Now, in the previous lecture, we looked at how to make decisions under uncertainty. And, by definition, right, you don't know what's going to happen. That's what uncertainty means. Well you can ask a question in those models, what if I did know the answer, how much would the information be worth to you? And here's one of the real benefits of having this formal model, this formal structure. So we've got this formal structure, we can actually figure it out. We can actually solve for what the value of that information is. So, let me give an example. Think of the game, Roulette. So, Roulette has a wheel, and in the United States, a Roulette wheel has 38 different numbers you can bet on. Like the numbers one through 36 plus to other spots. So there's 38 things you can bet on. And if you put [inaudible], you know, if this is a classical probability thing. What if I guess a particular number. The odds of me winning would be one out of 38. So, what I could do is, I could say, what's the value of information? What if somebody knew what was gonna happen? There's two different questions we could ask. The first question we could ask is, what if the person would tell me, what if I said, Look, I always bet on number seventeen, 17's my lucky number. What are the odds that seventeen wins? Someone says, well, you know, I can tell you that before the wheel goes. What's, is it worth it for you. What's the value of that information. Well, first off. You, I thought you're gonna lose that roulette. Right. And so if you're gonna lose. You wouldn't wanna bet anything. So, if you didn't have information. You've got nothing. So you're pay off is zero. But what's the information worth to you. Suppose if they tell you this. You can win $100. Well then you think well boy, that's information's great, that's worth a $100 to me. That's not quite right, because it's only worth a $100 if he tells you that your number's gonna come up. So if he says it's not gonna be seventeen well then you don't bet, but if he tells you it is gonna be seventeen, you do bet, and you win, and she'll only gonna win, 38th of the time. So the value of that information would be a hundred, divided by 38. Now alternatively, suppose the person said, I can tell you the winning number. And supposedly, the most you're allowed to win is $100 per round. But this person's, look, I can tell you the winning number. What they tell you, they can tell you the winning number, you're gonna win 100 bucks. So if you're gonna win 100 bucks, then the value of that information is 100 bucks. So divide the winning number, it could be $100. The value of knowing whether your number wins, would be $100 divided by 38. 'Kay, so that's the idea. But now we want to apply this in a context that's a little more complicated, where we got decision trees and that sort of thing. So let's do an example. Let's suppose you're thinking about buyin' a car. And you could buy the car now but you're worried that there's going to be a cash back program. So you heard some rumors that in a month there's going to be a cash back program. And this cash back program's getting $1,000 cash back. And based on, let's suppose you've done a frequency analysis of the number of years in the past they've had a cash back program and you figure there's a 40 percent chance they're going to have a cash back program. So what you could do is you could rent a car for $500 right now and then wait and hope there's a cash back option. So, what we want to do is figure out, what would it be worth to you, if someone could tell you. Suppose you knew someone at the auto company, and they said well, I could tell you whether they're having a cash back program or not, but it's going to cost you. Right? So, what would that information be worth? How much would you pay to know if there is going to be a cash back program? That's what we want to figure out. Okay. First off, I want to say. That's not an easy question. [inaudible] so I said what's it worth to you, I don't know, $50, $100, $200, $300. Who knows, right. That's what we wanna try and figure out. So, to figure out the value of information we just have to do three things. First, we're gonna calculate the value without the information. So we're gonna say, suppose I just have to make my choice, what's my optimal choice, how do I come out? Second, I calculate the value if I had the information. So, if I knew what was going to happen what would I do, what would my net value be? And then third, I just take the difference. I take the value with the information minus the value without the information, the value without the information and that tells me sort of, what the information was worth. Totally straight forward, easy to do, provided again we use this decision tree model. Without the decision tree model, it's going to be pretty difficult. So, here's my choice. Do I rent, or do I buy? Right, now if I buy then you know, I'm just out nothing. I just use my net values, we just have this baseline value. If I rent, then there can be the cash back program. Right? Where there cannot be the cash back program. Okay, so lets make this more formal alright? So if I buy I get nothing, if I rent then and there's no cash back program and I basically wasted $500 renting for the month, but if there's a cash back program, I net 500. Why 500? Because whenever there's a $1000 right cash back minus the $500 to rent, so that's 500. And what was the probability of the cash back program was 40%, and was the probability there not being a cash back program is 60%. So, now I've gotta figure out a case, should I rent or not? Look, I've got 40 percent times 500 and 60%, make this little point there, times -500. So, if I multiply that out, .4 times 500, which is 200. +.6 X -500, which is -300, I get -100, so what I get is, if I were to rent I'm out a $100. So if I base it on my tree it's pretty straightforward, If I buy I get zero and if I rent I'm out $100 so what I should do, is I should buy. So without the information, right think about what I do, without the information I should just buy and I'm at this sort of at this net zero case. So that's what I've done, I've calculated without that information. Now what I wanna do is calculate the value with the information. Suppose I knew what was gonna happen. So now I can just draw another tree, and this tree's going to look different than the first tree because here's what's going to happen. First, this chance node is gonna be revealed to me. Somebody is gonna tell me is there a cache back program or not. And 40 percent of the time they're going to tell me yes there is, 60 percent of the time they're gonna tell me no there's not. Let's think about what's gonna happen. If they tell me there's a cash back program. Then I'm gonna rent. And get $500. If they tell me there's not gonna be a cash back program, then I'll buy. And I'm just in the same situation as I was before. So, if I had the information first. Then I've got easy choices. Once it's my choice, because look. Once information's revealed there's no longer uncertainty. So, with the information. I'm no longer making a choice under uncertainty. So clearly if there's a cash back program I rent, and I'm up $500, and if there's not a cash back program I buy, and I'm base back to back to this baseline case of zero. So what's the value if I knew this information first. Well 40 percent of the time I get $500, and 60 percent of the time I get nothing, so it's worth $200. So with the information my expected value's $200. So let's go back. Calculate the value without information, zero. Calculate the value with the information, $200. Calculate the difference, $200. Right, so you can make this more formal. With the information, $200, without the information, nothing, so therefore the value of that information is $200. In this case, right, this was a fairly simple one but we could go back to both of the two examples we did in the last lecture. One about buying the train ticket and one about writing the essay and we could say suppose we knew what was gonna happen, suppose we knew if we're gonna make the train and suppose we knew if we're gonna win that scholarship. We could ask what's the value of that information. How much would it be worth to us to know. What was gonna happen, and again you just do the same thing. We just follow that same technique. Solve it under uncertainty with the information, solve it as if you knew the answer. And then just take the difference between those two values. So what have we learned here? We've learned that we could take a model developed for one purpose, right? This decision tree model. We developed that to figure out how to make decisions under uncertainty. And we can repurpose it. We can use it to figure out the value of information. We can figure out, how much would it be worth for us to know what the future's gonna unfold, how the future's gonna unfold. And it's really straightforward to do, right? It was a three step process. First, we figure out what we do with that information. Then we figure out what we do with the information. And then we can see how much better off we are with the information. And that tells us the value of the information. And that would be a, in many cases it'd be a very hard thing to do in our head, but with this simple tree model it's really straight-forward and easy to do. So again we see the power of models to help us make better choices. Thanks.