# Part 3 Probabilistic Decision Models

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Part 3 Probabilistic Decision Models
Chapter 11 Decision Theory

Learning Objectives After completing this chapter, you should be able to: Outline the characteristics of a decision theory approach to decision making. Describe and give examples of decisions under certainty, risk, and complete uncertainty. Make decisions using maximin, maximax, minimax regret, Hurwicz, equally likely, and expected value criteria and use Excel to solve problems involving these techniques. Use Excel to solve decision-making problems under risk using the expected value criterion. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Learning Objectives (cont’d)
After completing this chapter, you should be able to: Develop decision trees that consist of a combination of decision alternatives and events. Use TreePlan to develop decision trees with Excel. Determine if acquiring additional information in a decision problem will be worth the cost. Calculate revised probabilities manually and with Excel. Analyze the sensitivity of decisions to probability estimates. Describe how utilities can be used in lieu of monetary value in making decisions. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Decision Theory Decision theory problems are characterized by the following: A list of alternatives. A list of possible future states of nature. Payoffs associated with each alternative/state of nature combination. An assessment of the degree of certainty of possible future events. A decision criterion. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Suppose that the developer views the possibilities as
Example 11-1 Suppose that a real estate developer must decide on a plan for developing a certain piece of property. After careful consideration, the developer has ruled out “do nothing” and is left with the following list of acceptable alternatives: 1. Residential proposal. 2. Commercial proposal #1. 3. Commercial proposal #2. Suppose that the developer views the possibilities as 1. No shopping center. 2. Medium-sized shopping center. 3. Large shopping center. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 11–1 General Format of a Decision Table

Table 11–2 Payoff Table for Real Estate Developer

Decision Making under Certainty
Table 11–3 If It Is Known That No Shopping Center Will be Built, Only the First Column Payoffs Would Be Relevant Decision Making under Certainty Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Decision Making under Complete Uncertainty
Approaches to decision making under complete uncertainty: Maximin Maximax. Minimax regret. Hurwicz Equal likelihood Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 11–4 Maximin Solution for Real Estate Problem
The maximin strategy is a conservative one; it consists of identifying the worst (minimum) payoff for each alternative and then selecting the alternative that has the best (maximum) of the worst payoffs. In effect, the decision maker is setting a floor for the potential payoff; the actual payoff cannot be less than this amount. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 11–5 Maximax Solution for Real Estate Problem
The maximax approach is the opposite of the previous one: The best payoff for each alternative is identified, and the alternative with the maximum of these is the designated decision. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 11–6 Payoff Table with Similar Maximum Payoffs
Minimax Regret An approach that takes all payoffs into account. To use this approach, it is necessary to develop an opportunity loss table that reflects the difference between each payoff and the best possible payoff in a column (i.e., given a state of nature). Hence, opportunity loss amounts are found by identifying the best payoff in a column and then subtracting each of the other values in the column from that payoff. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 11–7 Opportunity Loss Table for Real Estate Problem

Table 11–8 Identifying the Minimax Regret Alternative

Table 11–9 Minimax Regret Can Lead in a Poor Decision

The Hurwicz (Realism) Criterion (Weighted Average or Realism Criterion)
The approach offers the decision maker a compromise between the maximax and the maximin criteria. Requires the decision maker to specify a degree of optimism, in the form of a coefficient of optimism α, with possible values of α ranging from 0 to 1.00. The closer the selected value of α is to 1.00, the more optimistic the decision maker is, and the closer the value of α is to 0, the more pessimistic the decision maker is. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 11–10 Equal Likelihood Criterion

Table 11–11 Summary of Methods for Decision Making under Complete Uncertainty

Exhibit 11-1 Using Excel to Make Decisions under Complete Uncertainty

Decision Making under Risk
Decision making under partial uncertainty Distinguished by the present of probabilities for the occurrence of the various states of nature under partial uncertainty. The term risk is often used in conjunction with partial uncertainty. Sources of probabilities Subjective estimates Expert opinions Historical frequencies Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 11–12 Real Estate Payoff Table with Probabilities
Expected Monetary Value (EMV) approach Provides the decision maker with a value that represents an average payoff for each alternative. The best alternative is, then, the one that has the highest expected monetary value. The average or expected payoff of each alternative is a weighted average: the state of nature probabilities are used to weight the respective payoffs. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Approaches to Incorporating Probabilities in the Decision Making Process
Expected Monetary Value (EMV) approach Provides the decision maker with a value that represents an average payoff for each alternative. Expected Opportunity Loss (EOL) The opportunity losses for each alternative are weighted by the probabilities of their respective states of nature to compute a long-run average opportunity loss, and the alternative with the smallest expected loss is selected as the best choice. Expected Value of Perfect Information (EVPI) A measure of the difference between the certain payoff that could be realized under a condition of certainty and the expected payoff under a condition involving risk. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 11-2 Using Excel to Make Decisions under Risk

Figure 11–1 Decision Tree Format
Decision trees are used by decision makers to obtain a visual portrayal of decision alternatives and their possible consequences. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 11–2 Decision Tree for Real Estate Developer Problem

Figure 11–3 Real Estate Problem with a Second Possible Decision

Exhibit 11–3 Initial TreePlan Dialog Box

Exhibit 11–4 Decision Tree Initially Developed by TreePlan

Exhibit 11–5 TreePlan Dialog Box to Add Branches, Decision Nodes, or Events

Exhibit 11–6 Modified Decision Tree with Three Branches

Exhibit 11–7 TreePlan Dialog Box to Add or Change Decision Nodes or Events

Exhibit 11–8 Modified Decision Tree with Three Branches and the Added Event Node with Three Nodes

Exhibit 11–9 Excel Solution to the Real Estate Developer Decision Tree Problem

Figure 11–4. Sequential Decision Tree for Unicom Inc
Figure 11–4 Sequential Decision Tree for Unicom Inc. (Example 11-3, part a) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 11–10. Excel Solution to the Unicom Inc
Exhibit 11–10 Excel Solution to the Unicom Inc. Sequential Decision Tree Problem Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 11–5 Conceptual Portrayal of Market Test Example

Figure 11–6 Summary of Analysis of Market Test Example

Table 11–13 Reliability of Market Test

Table 11–14 Probability Calculations Given the Market Test Indicates a Strong Market

Table 11–15 Probability Calculations Given the Market Test Indicates a Weak Market
Conditional probabilities express the reliability of the sampling device (e.g., market test) given the condition of actual market type. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 11–11 Calculation of the Revised Probabilities for the Market Test Example

Figure 11–7 Format of Graph for Sensitivity Analysis
Sensitivity Analysis enables decision makers to identify a range of probabilities over which a particular alternative would be optimal. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 11–8 The Expected Value Line for Alternative a.

Figure 11–9 Example of Finding the Expected Value for Alternative a when P(#2) Is .50

Figure 11–10 All Three Alternatives Are Plotted on a Single Graph

Figure 11–11 The Line with the Highest Expected Profit Is Optimal for a Given Value of P(#2)

Utility Utility (of a payoff) Risk Risk Averters Risk Takers
A measure of the personal satisfaction associated with a payoff. Risk A decision problem in which the states of nature have probabilities associated with their occurrence. Risk Averters Individuals that avoid taking risks. The decision maker has less utility for greater risk. Risk Takers Individuals that like taking risks and that have a greater utility for the potential winnings even though their chances of winning are very low. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 11–12 Converting P(#2) Ranges into P(#1) Ranges

Exhibit 11–12 Solved Problem 1: Decision Making under Complete Uncertainty—A Profit Maximization Problem (Part f) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 11–13 Solved Problem 2: Decision Making under Complete Uncertainty—A Cost Minimization Problem (part f) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 11–14 Calculation of the Revised Probabilities and Expected Value of Perfect Information for Solved Problem 3 (part c) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 11–15 Calculation of the Revised Probabilities for Solved Problem 5 (part c)

Exhibit 11–16 TreePlan Dialog Box to Add Branches, Decision Nodes, or Events
Exhibit 11–17 TreePlan Dialog Box to Add or Change Decision Nodes or Events Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 11–18 Decision Tree for Solved Problem 5 (part c)