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Decision Trees

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Influence Diagrams

Dr. Ayham Jaaron

Introduction

• decision trees and influence diagrams can be extremely useful in helping people to gain an understanding of the structure of the problems which confront them.

• Decision trees are models, and as such are simplifications of the real problem.

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Constructing a decision tree

• Two symbols are used in decision trees:

• Square: is used to represent a decision

• Circle: is used to represent a chance

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Example

• An engineer who works for a company which produces equipment for the food processing industry has been asked to consider the development of a new type of processor and to make a recommendation to the company’s board. Two alternative power sources could be used for the processor, namely gas and electricity, but for technical reasons each power source would require a fundamentally different design. Resource constraints mean that the company will only be able to pursue one of the designs, and because the processor would be more advanced than others which have been developed it is by no means certain that either design would be a success. The engineer estimates that there is a 75% chance that the electricity-powered design would be successful and only a 60% chance that the gas-powered design would be a success.

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Constructing a decision tree:An initial tree...Food processor problem

Example...continued

• After considering this tree the engineer realizes that if either design failed then the company would still consider modifying the design, though this would involve more investment and would still not guarantee success. He estimates that the probability that the electrical design could be successfully modified is only 30%, though the gas design would have an 80% chance of being modified successfully. This leads to the new tree which is shown in Figure. Note that the decision problem is now perceived to have two stages.

• At stage one a decision has to be made between the designs or not developing the problem at all.

• At stage two a decision may have to be made on whether the design should be modified.

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A new decision tree for the food-processor problem

Determining the optimal policy

• We will now show how the decision tree can be used to identify the optimal policy.

• The technique for determining the optimal policy in a decision tree is known as the rollback method.

• It can be seen that the rollback method allows a complex decision problem to be analyzed as a series of smaller decision problems.

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Rolling back the tree

Decision Trees and Utilities

• In the previous section we made the assumption that the decision maker was neutral to risk (he used the basics of EMV method).

• Let us now suppose that the engineer is concerned that his career prospects will be difficult if the development of the processor leads to a great loss of money for the company.

• He is therefore risk averse, and he should use the utility functions method for the monetary sums involved in this problem as shown in the next Figure.

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Decision trees and utility: The engineer’s utility function

Utility values from figure...

• $6 m= 0.94

• -$7 m= 0.65

• -$3 m= 0.8

• $10 m= 0.97

• -$12 m= 0

• $0 m= 0.85

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Applying rollback to utilities

Problem (1): Page 164

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Problem (1): page 164..continued

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Problem (1)...continued.

C) There is some debate in the company about the probability that was estimated by the research director. Assuming that all other elements of the problem remain the same, determine how low this probability would have to be before the option of not developing the product should be chosen.

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Problem...continued

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Decision trees involving continuousprobability distributions

• In the decision problem we considered above there were only two possible outcomes for each course of action, namely success and failure.

• in some problems the number of possible outcomes may be very large or even infinite.

• Example: the possible percentage market share a company might achieve after an advertising campaign.

• Variables like these could be represented by continuous probability distributions.

• But how can we incorporate such distributions in our decision tree? One way is to use Discrete probability distribution as approximation.

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The ExtendedPearson-Tukey (EP-T) approximation

• One method to make such type of approximation is the extended Pearson-Tukey approximation.

• Uses the median, 5th percentile, and 95th percentile to approximate the continuous chance node, and their probabilities are 0.63, 0.185, and 0.185, respectively

• It requires three estimates to be made by the decision maker:

(i) The value in the distribution which has a 95% chance of being exceeded. This value is allocated a probability of 0.185.

(ii) The value in the distribution which has a 50% chance of being exceeded. This value is allocated a probability of 0.63.

(iii) The value in the distribution which has only a 5% chance of being exceeded. This value is also allocated a probability of 0.185.

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Example

• let us suppose that a marketing manager has to decide whether to launch a new product and wishes to represent on a decision tree the possible sales levels which will be achieved in the first year if the product is launched. Suppose that she estimates that there is a 95% chance that first-year sales will exceed 10 000 units, a 50% chance that they will exceed 15 000 units and a 5% chance they will exceed 25 000 units.

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The extended Pearson-Tukey approximation method

Eliciting decision structure: One representation of the calculator problem

• Imagine that you are a businessman and you are considering making electronic calculators. Your factory can be equipped to manufacture them and you recognize that other companies have profited from producing them. However, equipping the factory for production will be very expensive and you have seen the price of calculators dropping steadily. What should you do?

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Eliciting decision structure: One representation of the calculator problem

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Towards a correct representation of the calculator problem?

Eliciting decision structure: One representation of the calculator problem

• Actually, you have probably guessed, there is no obviously right or wrong representation of any problem that is in any way related to real life.

• It is really a matter of the decision analyst’s judgment as to whether the elicited tree is a fair representation of the decision maker’s decision problem.

• The next figure presents a description of the typical phases in a decision analysis of a problem that the decision maker wishes to resolve with help of the practitioner of decision analysis – the decision analyst.

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Phases of a decision analysis

Eliciting decision tree: influence diagrams.

• What methods have been developed to help elicit decision tree representations from decision makers? One major method, much favoured by some decision analysts, is that of influence diagrams which are designed to summarize the dependencies that are seen to exist among events and acts within a decision. Such dependencies may be mediated by the flow of time, as we saw in our examples of decision trees

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Definitions used in influence diagrams: 1

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Definitions used in influence diagrams: 2

Turning influence diagrams into decision trees

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Example on influence diagrams

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Tree derived from influence diagram