4 Decision trees

Sometimes decisions can be complex and require a number of stages to arrive at a final outcome. Such a final outcome may be dependent on earlier, intermediate decisions. Alternatively, the final decision may be dependent on a series of uncertain, intermediate outcomes. Dealing with these types of decisions may appear, on the face of it, quite difficult. However, the technique of decision trees that you are going to explore in this section will help to simplify this process.

The best way to illustrate the technique is by a worked example in Activity 5. Before doing so, it is important to point out the meaning of two symbols that will be used in the decision trees.

Where a branch appears on your tree, this point will be called a node. A node may appear for one of two reasons. The first is that a decision is required. In other words, the node represents a series of choices. This type of node will be called a decision node and a square will be used to denote it. The second type of node is a chance node. Here, there is a range of possible events or outcomes of varying probabilities. Such nodes are denoted with a circle.

Activity 5 Introduction to decision trees

Timing: Allow approximately 30 minutes to complete this activity

In Videos 3 and 4, you will be introduced to the powerful technique of decision trees. This technique allows you to incorporate probabilities into a range of potential outcomes, which may themselves be conditional on other outcomes.

You may wish to watch the videos a few times and make notes in the text boxes to ensure that you understand the concept of decision trees, as well as to answer the questions.

Part 1

A company (MKOU) is assessing two outsourcing bids, A and B. Company A is more expensive but is reckoned to have a higher probability of delivering a high quality good than B. This is important as the higher the quality the more MKOU can charge and the less it will need to refund to dissatisfied customers. The data may be summarised as shown in Table 7.

Table 7 Possible financial benefits of using companies A and B

Company	Probability of acceptable service level	Net financial benefit if acceptable £M	Net financial cost if not acceptable £M
A	80%	120	-30
B	55%	160	-10

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Transcript: Video 3 A worked example on decision trees

NARRATOR:

In this video, we're going to introduce decision trees. A decision tree is a really useful tool for visualising and solving problems where there are various choices involved and we don't know which choices to make. So we're going to bring in probabilities with this.

Decision trees consist of nodes. These are where the branches of the decision tree split. And there are two types of nodes. A square node is where a decision is being taken, something that is our choice. And the other type of node we show is a circle, where there is some kind of chance element. This is where we apply probabilities, where things happen outside our control.

A company, MKOU, is assessing two outsourcing bids: A and B. Company A is more expensive but is reckoned to have a higher probability of delivering a high-quality good than B. This is important, as the higher the quality, the more MKOU can charge and the less it will need to refund dissatisfied customers.

So we need to assess two bids to make a decision between A or B. Let's set up a branch for A and a branch for B. There's a chance on each of them - will they deliver a good service or not? So let's look at A's first. It's a chance. So it's a circle node. Will the service of A be acceptable? And that's a 'Yes' or a 'No' branch. There's an 80% probability, or 0.8, of it being 'Yes'. So, therefore, there must be 0.2 of it being 'No'.

You'll recall from the beginning of this section that all probability must total 1. So if it's 'Yes', it has a final value of 120 million pounds, whereas if it's a 'No', it's minus 30 million pounds. And we're going to have to give refunds, and there's going to be costs incurred.

Coming down to B, is it acceptable? 'Yes' or 'No'? There is a 0.55 probability, or 55% chance of it being 'Yes'. So the 'No' must be 0.45 because probabilities have to equal 1. And the values here are 160 million pounds and minus 10 million pounds.

And what we do now is we just work out which of these two nodes - 1 or 2 - have the highest expected value. The expected value of 1 is going to be 0.8 times 120 plus 0.2 times minus 30, which is 90 million pounds. And the value of node 2 is going to be 0.55 times 160 plus 0.45 times minus 10. And that works out to be 83.5 million pounds. 90 million pounds goes here, and 83.5 million pounds goes here.

In this fairly simple decision tree, the root with the highest expected value is A. We would expect, on average, A to get 90 million pounds, whereas B only 83.5 million pounds. Therefore, on this basis, we would go with A.

Notice that although this is a simple decision tree, we can already see the technique. When we're setting up the model, we work from left to right, going through putting down what each of the various branches will do. And then we work backwards to get the expected value of each node.

End transcript: Video 3 A worked example on decision trees

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Video 3 A worked example on decision trees

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Part 2

A company is considering launching a new product. It can either launch immediately or in one year’s time. If it launches immediately there is a 0.75 chance of the launch being successful. If it is unsuccessful then the launch will be halted at a cost of £1M and relaunched in a year’s time. If the company launches immediately it may opt to also have a promotion, which has a 0.6 chance of success. If the promotion is successful the financial benefit is £10M, if not £2M. If the company does not do the promotion the benefit is £5M. If the company launches in a year’s time the benefit is £6M. What should the company do?

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Transcript: Video 4 A second worked example on decision trees

NARRATOR:

Let's look at a more complicated example. A company is considering launching a new product. It can either launch immediately or in 1 year's time. If it launches immediately, there's a 0.75 chance of the launch being successful. If it's unsuccessful, then the launch will be halted at a cost of 1 million pounds and re-launch in a year's time.

If the company launches immediately, it may also opt to have a promotion. If the promotion is successful, the financial benefit is 10 million pounds. If not, it's 2 million pounds. If the company doesn't do the promotion, the benefit is 5 million pounds. If the company launches in a year's time, the benefit is 6 million pounds.

So we're considering launching a product. It can launch immediately or in one year's time. We put our two branches down. At the end, we'll go with the one with the highest expected value. If the product launches immediately, there's a 0.75 chance of the launch being successful.

This is something outside our control, so we write successful with a question mark. Yes, it will be successful, is a 0.75 chance, and therefore, no, it won't be successful, has a 0.25 chance. If it's unsuccessful, the launch will be halted at a cost of 1 million pounds. So let's write that here. -1 million pounds and re-launched in a year's time. So it'll come down here to 1 year, and we'll join those lined up in a minute.

If the company launches immediately, it may also opt to have a promotion. Well, we'll only do the promotion if it's successful. So we had another choice. Do we launch a promo? Yes or no? If we do a promotion, will it be successful? Yes or no?

The promotion has a 0.6 probability of being successful and, therefore, 0.4 probability of not being successful. If the promotion is successful, the benefit is 10 million pounds after all costs. And if it's not successful, it's 2 million after all costs.

If the company doesn't do the promotion, the benefit is 5 million pounds. Here's this line. And if the company launches in a year's time, the benefit is 6 million pounds, so these two lines join up. So we now work from right to left. We'll label this node number 1.

The expected value here is 0.6 times 10 plus 0.4 times 2, which is 6.8 million pounds. We'll write that on node 1. On node 2, where we have a choice, 'Yes' is worth 6.8 in expected value terms, whereas 'No' is worth 5. So we'd pick 'Yes' and with an expected value of 6.8.

Coming now to node 3, the expected value at this point is 0.75 times 6.8 because that's the expected value at node 2 plus 0.25. Now will launch in a year's time, but now the expected value of coming down this route is -1 plus 6, which is 5. Node 3 is therefore worth 6.35 million pounds.

The value of this route is simply 6. Nothing else happened in between. So what we can conclude, based on our diagram, is that launching a new product now has an expected value of 6.35 million pounds, whereas launching a new product in a year's time has an expected value of 6. Therefore, on this basis, we would go with the 6.35 million pounds, so we would launch now.

If we were to launch now and it was successful, we then have another choice to make at node 2. And here, we would choose 'Yes', we would promote. So our strategy is the launch now, and then, if it's successful, to launch a promotion.

Just to recap. Decision nodes are shown as squares. Chance nodes are shown as circles. Terminal nodes are the ones where their values go right at the end, and we show them as rectangles. Probabilities coming out of a chance node must total 1 because they must cover every single eventuality.

We calculated expected values moving from right to left and then carried back the expected values which formed later decisions. We saw this in the example question, do we promote? The expected value of promoting was higher than not promoting, so that node no longer becomes a decision node because we made the decision now. And then, coming right back to the beginning, you can see which initial branch gives you the highest expected value.

End transcript: Video 4 A second worked example on decision trees

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Video 4 A second worked example on decision trees

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Discussion

To summarise, you can use decision trees to break down a decision into a series of events that involve the decision-maker making a sub-decision (‘decision node’) or there being a chance event outside of the decision-maker’s control (‘chance node’). (Note that ‘sub-decision’ means a decision taken after the first, main, decision.)

By allocating probabilities to the chance nodes you can evaluate the expected value from the various combinations of sub-decisions and chance events.

This then informs which initial decision and then subsequent sub-decisions should be taken.

Now that you have watched the videos on decision trees, you will consider potential decisions faced by businesses. In the next section you will see some more applied examples of how decision trees are used in making business decisions.