[MUSIC] Hello, and welcome to week six, our final week of this MOOC course. So what are we going to achieve in these last few sessions. Well, I'd like to show you an eclectic mix of various quantitative methods designed to solve various real world problems. And where it possible, I'd like to draw on much of the probability and statistical concepts we've seen in previous weeks. So we begin with the topic called decision tree analysis. And arguably, this sort of brings us full circle with our initial look in week one at decision making under uncertainty. Remember, we need to make decisions in the present for which we don't know exactly what's going to happen in the future. So we have these unknown or uncertain future outcomes. So how do we look at modelling this kind of situation? Well, decision trees help us along the way. So I'd like to consider some examples of managerial decision making under uncertainty. Imagine you are a factory producing some product and you need various suppliers for your raw materials. Well, how many suppliers should you have? And where should you source these materials from? Clearly, some uncertainties about the reliability maybe of the supply chains. Or maybe your R&D, your research and development department, is trying to choose between several different new prototypes, but they've only got the resources to perhaps launch one or two to the market. So which one are you going to launch? Well, clearly, the one which will be most popular, but you cannot know with certainty today which one that would be. Maybe in human resources and your hiring someone for a position. To hire or not to hire? Well, when you're deciding whether or not to give a job offer to someone, you'd really like to know how good they will be at that particular job. Of course, there is uncertainty when making this decision. Sure, you can look at their CV, their resume, to see what they've achieved in the past. Which could be some indicator, but not a perfect predictor of what their future performance might be. So here we're going to consider an example of decision analysis. Where by we will keep it very simplified and have a single decision maker. We're going to make an assumption, we'll assume the decision maker is rational individual. Now, perhaps there is a big question mark about whether people truly are rational. But for the sake of simplicity, remember back in week one when we were thinking about designing models as a deliberate simplification of reality. The assumption of rationality will make our life somewhat simpler for the time being. So here we will just consider that single decision maker, but start to appreciate we could easily extend this problem to two or more decision makers, where there's actually going to be some strategic decision making. And this looks into the realm of game theory, which some of you may wish to study later on. For example, when you're playing a game of chess, you're deciding which piece to move. But if you're a good chess player, you're not deciding on your next move, you're trying to anticipate the next move of your opponent. If you're a really good chess player, you'll then start to think not just those two moves ahead but further moves ahead as well. Of course you cannot know exactly what your opponent is going to do, there's some uncertainty there but your decision making will be based on your expectations of what your opponent may do. So, game theory, a fascinating subject. Some of you may have seen the film, A Beautiful Mind, looking at the life of John Nash really the father of game theory. And I strongly recommend this as perhaps another field of study for you. But returning to our single decision maker, assumed to be rational, and hence as there's no other decision maker involved, we're looking at non-strategic decisions. So let's take a nice example. Let's consider you are an ice cream manufacturer. Now, I love ice cream, chocolate is my favorite but I'm quite passion to vanilla too. So, let's imagine we are this ice cream manufacturer. And in our very simplified world, let's assume our sales can either be high or low. And hence, the profits, the amount of money we'll make will depend on whether sales are high or low. We'll just choose some arbitrary numbers just to get this example up and running. Let's suppose if sales are high, which are clearly very good for us, suppose we make profits of 300,000 pounds. However, if sales are low, let's suppose we actually make a loss of 100,000 pounds. So remember, just think of a loss as a negative profit. So we could write it as -100,000 pounds in sort of profit terms, but that minus sign indicates it's negative and hence represents a loss. Now we have uncertainty and therefore we'll need to draw on probabilities to make this problem work. So let's assume that there is a 60% chance of high sales, and by extension a 40% chance of low sales. So the probability of high sales 0.6, the probability of low sales 0.4. Now you may wish to relate this back to our look at the Bernoulli distribution some time ago, a situation where there were only two possible outcomes, success and failure. So here we might say high sales represents a success because we make money, whereas low sales represents a failure, we lose money. Now how can we make this into a problem of decision making under uncertainty. Well, let's suppose, as an ice cream manufacturer, we have an option, if we so choose, to do some advertising. Now advertising doesn't guarantee success. So if we decide to pay out for advertising, let's model it as not guaranteeing a high level of sales, rather it acts to increase the probability of this good outcome. Now if you think about marketing agencies around the world, one hopes as a client if you pay them money to do some advertising campaign for you, one hopes it works either to increase demand, that demand generation form of advertising. Or maybe your goal is just to perhaps raise your brand awareness. But you hope that the advertising campaign would be successful. Now one likes to think, most of the time, advertising campaigns are successful but clearly not every advertising campaign did reach those particular goals either of demand generation or increasing brand awareness. So in our problem, let's assume that if we decide to undertake some advertising this will increase the probability of success. Here, a high level of sales. So without advertising, we model the probability of success of 0.6. Let's increase this to 0.9 in the event where we take some advertising. Of course, if we are increasing the probability of the good outcome, we must be decreasing the probability of the bad outcome. So, the risk now of having a low level of sales was originally not 0.4, but that would be revised down to 0.1. So our question is very straight forward, to advertise or not to advertise? This is the decision we will need to make. Well, we just need to make it slightly more realistic and we know that advertising typically cost some money to undertake. Let's assume the cost of this advertising campaign will be 100,000 pounds. And we have to pay this 100,000 pounds regardless of whether we end up with a high or low level of sales. Now, here we may think about sort of conventional forms of advertising maybe in the print media, maybe online advertising, or billboards posted around a city. Of course some advertising could be free if we think about word of mouth advertising. Remember, if you have a good or bad experience with some product, you're likely to tell your friends, family, associates about that. About whether to recommend buying the product, or to perhaps avoid it. But here we'll stick with conventional advertising which costs money. So with all of this, we now want to depict it in a decision tree. So on the screen is our decision tree for this problem. So how do we read this? Well, you read a decision tree from left to right and think of this as the time order in which events takes place. You will see on the far left that we have a square depicted. So in a decision tree, a square is going to represent a decision node. So when you're effectively standing at that square, you have a decision to make. Now here we've kept it very simplified and we have a binary decision, either to advertise and hence pay for the advertising or not to advertise. So, we see some branches stemming out from that decision node. These branches then lead to a couple of circles. Now these depict so-called chance nodes. So this is where we leave things down to chance, fate, destiny, call it what you will. But here is something about which we have no control. We do have control of whether we advertise or not, this is our conscious decision at the start of the game. But what happens there after regardless of our advertising decision is beyond our control. So this is the uncertainty being modeled. And note here how the square comes before the circles. As in, we need to make the decision at the start of the game before we know whether these sales turn out to be high or low. You also see the probabilities of high sales and low sales in those different states in the world where we've chosen to advertise or not advertise, depicted. And at the very end of these branches, one sees the payoffs that would accrue to us. So note in the arguably the best scenario, we didn't pay for advertising but we did get those high sales anyway. We make 300,000 pounds profit. Whereas if we had high sales but I had to pay advertising, then we would have to deduct the cost of the advertising from our payoff. And hence we would only make 200,000 pounds. But a profit, nonetheless. So the question, to advertise or not to advertise? How are we going to solve this problem? Well, here we can appeal to something called the expected monetary value. So remember, a few weeks ago we introduced the concept of an expectation, a probability-weighted average. And that's how we're going to solve this decision tree. Namely, we'll consider each possible routes, namely the advertise and not advertise routes. And work out the expected monetary value in each situation. So, all we do is do probabilities times payoffs and sum them over the respective branches along one route of our tree. So, for example, if we decided to advertise, we have a 0.9 probability of high sales times the payoff of 200,000 pounds + 0.1 probability of low sales with the loss of 200,000 pounds. The initial loss of 100,000 less the advertising costs. So as an expectation, this would then give us 160,000 pounds. Same kind of calculation, just different numbers with the non-advertising route, will give us an expected monetary value of 140,000 pounds. So in this very simplified world, our decision is straightforward. We will simply look down the two different branches and paths of our tree and compare the expected monetary values. And given the one route of advertising gives us the higher expected monetary value, this would represent our preferred course of action. And hence the answer of our question to advertise or not to advertise is indeed to advertise. So, that's a very simple form of decision tree analysis. Note that 160,000 pounds in expectation, we would never get in a single play of this game. Either sales are high and we make 200,000 pounds. Or sales are low and we've lost 200,000 pounds. But remember when we introduced expectations, you view these as long-run averages. Such that, if we play this game a large number of times, lets say, we play the same game every year for ten years and we chose to advertise each time. Then on average, nine years out of those ten, we would end up with high sales and make 200,000 pounds. But one year in every ten, on average, we would lose 200000 pounds. So if you now average your profits and losses over those ten years, you will be coming up with this 160,000 pound figure. So a nice example of applying the Bernoulli distribution, the binary set of outcomes, probabilities attached to it, and of course the role of expectations too. Now perhaps one major caveat which we've not dealt with here but we will consider later on in week six is the issue of risk. Namely, we've purely based this decision on expected values, expected monetary values but we have not taken into account the risks involved with the different courses of action. We'll see that in the next section. [MUSIC]