Speaker 1: In this video, we're going to talk about making a decision when outcomes are uncertain. And specifically, in this case we're looking at decision trees. Decision trees are an interesting tool that get used conceptually quite a bit. We're going to draw them visually, sometimes they get used in spreadsheets or there are software, but to make sure you understand the the process we're going to do a simple decision tree here as an example. So a decision tree requires that you know the uncertainty, so that you can quantify the uncertainty. Oops. Sorry. And that there are discrete outcomes. So A happens with this probability, B happens with this probability. There are ways that you can do it with less discrete, but we're going to look at it in the case of discrete probabilities. So let's begin then with an example. You've got a company is deciding whether to build a large facility or a small one. If they build a large facility and they have a favorable market, which has a probability of 50%, it's not always 50%, but in this circumstance, that is the number we're given, they will have a net profit of $200,000. If they build a large facility and they have an unfavorable market, in this case there are only two states of nature, there can be multiple, but in this case it's the other 50%, they will have a net profit of negative $180,000. If they build the small facility and have favorable, net profit will be $100,000. If the market is unfavorable, they will have a net profit of negative $20,000. And now they're trying to make a decision as to whether they should build the large facility or the small facility or frankly do nothing at all. So let's take a look. In this circumstance, we draw what is called a decision tree. And a decision tree starts with, well it doesn't always start with a decision, but here we're going to start with a decision and we draw a decision as a square. And a square means we have a choice. So in this case, we can build a large facility or we can build a small facility. So, here we have a choice that we're going to make. Then, we have an uncertain state of nature and in that case we draw a circle. And it's important to know the distinction between when you have a decision and when you have something that you have no control over, there are, there is uncertainties. And we always draw the decision tree from left to right and then analyze the decision tree from right to left. So in this case, we then have favorable markets. Favorable market and we would make 200,000 with a probability of 0.5 and we could have an unfavorable negative 180,000, probability of 0.5. And again, if we build a small, we would have favorable or we'd have an outcome of 100,000, probability of 0.5. Unfavorable which is negative 20,000, 0.5. And then there is the do nothing option where we have no costs and no revenues and it's zero. So now, we have what we know as a decision tree. We have a decision here and we have uncertainty. We could have more decisions here. We could have more uncertainty here and we'll talk about that in a second, but let's look at this simple one first. So the first thing we do is when we have uncertainty, we have to come up with an expected value. And an expected value is equal to the value that you would get with the outcome times the probability of that outcome. So we have 200,000. Times 0.5 plus negative 180,000 times 0.5. So that is the expected value of this uncertain node. And so we don't know what's going to happen, but the expected value 200,000 times 0.5 plus negative 180,000 times 0.5 gives us a value of $10,000 that we put on this node. That does not mean. That we will make $10,000 if we have a large facility. That means our expected value if we make the build a large facility is $10,000. In truth, we will either make $200,000 or we will lose $180,000. And we'll talk about that in a second. We do the exact same thing down here where we do the expected value. And we have 100,000 times 0.5. Plus negative 20,000 times 0.5. And that gives us an expected value of $40,000. So again, we don't make a $40,000 if we build a small facility. That's the expected value of that outcome. So if we build a small facility, the market is favorable. We make $100,000. If we build a small facility and it's unfavorable, we lose $20,000. So the expected value there is $40,000. The expected value of doing nothing is 0. So if our objective is to maximize the expected value, which is the most common objective in this circumstance, we would decide not to make that decision. We would decide not to make that decision. We would decide not to make this decision. And it is build a small facility. Expected value is equal to $40,000. There are other criterion. We could do maxi max. And maxi max means you maximize the maximum. In that case, you would look that we're not worried about the uncertainty in that circumstance. We're just saying where would we have the potential, not the certainty, the potential to make the most money. And that would be to build the largest facility because that has $200,000. If we wanted to minimize the maximum loss. Okay. Minimax, which would minimize the maximum loss. We would look here. It's $180,000. Here it's $20,000 and here it's zero. If we were completely risk averse, we would do nothing. Generally though, with decision trees, we look at the maximize expected value. And in this case, that is to build a. That is to build a. That is to build a. That is to build a large, a small plant and with an expected value of $40,000. If we build a small plant, we may make $100,000 and the most we'll lose is $20,000. So let's look at sort of expanding that a little bit. What if they could do some work? For another company, if they build large and have unfavorable market. So they would have excess capacity and they could build for someone. They could choose to do that. Okay. If they have a large plant, work, income of $150,000. So in that circumstance, if they have a large plant, unfavorable, and I'm just extending what I drew before. They could make a decision, work for external, and then they would have a net loss of $30,000. Because they would generally have a loss of 180. We get 150 from the other company. So we'd end up losing 30,000. Don't do it. And it's negative 180. So in that circumstance, we would choose to do the work and we would then have an expected value here of negative 30. And then that would affect our losses by increasing the expected value of the loss. Okay. So we would then have the expected value of the large plant because now we would bring negative 30,000 here. We would have 200,000 here. And our expected value would be 200,000 times 0.5 plus negative plus negative 30,000 times 0.5. And that would mean that the expected value. Here. be eighty five thousand dollars so in this circumstance we would choose the large facility. So all we've done here is put another branch on the tree. So let's look at another outcome. What if unfavorable large and there was a 50% chance that they could do that work. So rather than being a certainty there's a 50% chance. So then we have large and we have favorable and unfavorable and then we have rather than a decision we have an uncertain outcome where we might be able to make some money or we might not. So now we can't make the choice here because it's a probability. So now we have an expected value here and this expected value is negative 105 and then we have to come up with an expected value. Here remember this was .5 and 200,000 and then the expected value would equal to 37,500 and that would still be smaller than the small plant. So in that case we would choose the small facility where students often go wrong here. Is there... Is there... they say, well, we're going to build a large plant and if we have unfavorable, we're going to choose to sell to the other company and have this. Well, if it's uncertain, we can't choose that. And often, even in this case, I will see students say, we're going to choose to have a large facility and then choose to have a favorable market. When you have uncertainty, you can't choose which of those uncertain outcomes you will have. So that's why it's important to understand the difference between a square, which is a decision, and a circle, which is an uncertain outcome. The last thing I want to cover here is a concept called the expected value of perfect information. And the expected value of perfect information is a expected value of perfect information. Expected value of perfect information means you know which uncertain outcome will happen. Now that sounds implausible, but say you could invest money in market research that would give you greater certainty as to as to whether you would have a favorable market or unfavorable market. So in this case, then we go back to and say, OK, now, rather than having this uncertainty, we would look at what happens. If we what happens if we know for sure what's going to happen. So in this case, if we know the market is going to be favorable, there's no difference in do nothing. If we built a small facility, it would be 100,000. If we built a large facility, it would be 200,000. So if we know for sure. OK. If we knew for sure that we're going to have favorable conditions, we would choose 200,000. Now what would happen if we knew for sure we were going to have unfavorable, well, unfavorable here we lose 180, unfavorable here we lose 20,000 and unfavorable here nothing happens. So in this circumstance, if we knew for sure we were going to have unfavorable market conditions, we would choose to do nothing. So then we can go favorable. Pick large 200,000 unfavorable, right? This is in the circumstance that we knew exactly what was going to happen. Pick do nothing. OK. So. OK. So. So we would do the expected value there, 200,000 times .5 plus 0.5 times 0, which was the outcome if we do nothing, is equal to 100,000. So if we have perfect information, we would have an expected value of 100,000. So that is the expected value of 100,000. OK. OK. I said, well, how much would you be willing to pay for perfect information? How much would you be willing to pay for certainty relative to knowing favorable versus unfavorable? You would then look at the difference between this and best choice with uncertainty. So that would be a hundred thousand minus 40,000, which was the best choice when we were uncertain. Equals $60,000. So you would be able to, you would be willing to spend as much as 60, sorry, $60,000. You'd be willing to spend as much as $60,000 to know for certain whether you would have a favorable or unfavorable outcomes. So that's very simple introduction to decision trees. I think that that, gives you a quick overview. And if you have any questions, please ask.