Decision Analysis Chapter 15

Introduction Business analytics is about making better decisions Decision analysis can be used to develop an optimal strategy: When a decision maker is faced with several decision alternatives and an uncertain or risk-filled pattern of future events For example: The State of North Carolina used decision analysis in evaluating whether to implement a medical screening test to detect metabolic disorders in newborns A good decision analysis includes careful consideration of risk

Introduction Risk analysis helps to provide the probability information about the favorable as well as the unfavorable outcomes that may occur Decision analysis considers problems that involve reasonably few decision alternatives and reasonably few possible future events Topics to be discussed under decision analysis: Payoff tables and decision trees Sensitivity analysis Use of Bayes’ theorem

Problem Formulation Payoff Tables Decision Trees

Problem Formulation The first step in the decision analysis process is problem formulation: Create verbal statement of the problem Identify the decision alternatives: The uncertain future events, referred to as chance events The outcomes associated with each combination of decision alternative and chance event outcome

Problem Formulation Example: Construction project of Pittsburgh Development Corporation PDC commissioned preliminary architectural drawings for three different projects: One with 30 condominiums One with 60 condominiums One with 90 condominiums The financial success of the project depends on: The size of the condominium complex The chance event concerning the demand for the condominiums

Problem Formulation The statement of the PDC decision problem is to select the size of the new luxury condominium project that will lead to the largest profit given the uncertainty concerning the demand for the condominiums Given the statement of the problem, it is clear that the decision is to select the best size for the condominium complex PDC has the following three decision alternatives: d1 = a small complex with 30 condominiums d2 = a medium complex with 60 condominiums d3 = a large complex with 90 condominiums

Problem Formulation In decision analysis, the possible outcomes for a chance event are the states of nature The states of nature are mutually exclusive (no more than one can occur) and collectively exhaustive (at least one must occur) Thus one and only one of the possible states of nature will occur For PDC example: The chance event concerning the demand for the condominiums has two states of nature: s1 = strong demand for the condominiums s2 = weak demand for the condominiums

Problem Formulation Payoff Tables Table 15.1: Payoff Table for the PDC Condominium Project ($ Millions) Payoff Tables Payoff is the outcome resulting from a specific combination of a decision alternative and a state of nature Payoff table is a table showing payoffs for all combinations of decision alternatives and states of nature

Problem Formulation We will use the notation Vij to denote the payoff associated with decision alternative i and state of nature j Using Table 15.1, V31 = 20 indicates that a payoff of $20 million occurs if the decision is to build a large complex (d3) and the strong demand state of nature (s1) occurs

Problem Formulation Decision Tree A decision tree provides a graphical representation of the decision- making process Shows the natural or logical progression that will occur over time Example: The topmost payoff of 8 indicates that an $8 million profit is anticipated if PDC constructs a small condominium complex (d1) and demand turns out to be strong (s1)

Figure 15.1: Decision Tree For The PDC Condominium Project ($ Millions)

Problem Formulation The decision tree in Figure 15.1 shows: Four nodes, numbered 1–4 Nodes are used to represent decisions and chance events Squares are used to depict decision nodes, circles are used to depict chance nodes Node 1 is a decision node, and nodes 2, 3, and 4 are chance nodes

Problem Formulation The branches connect the nodes; those leaving the decision node correspond to the decision alternatives The branches leaving each chance node correspond to the states of nature The outcomes (payoffs) are shown at the end of the states-of-nature branches

Decision Analysis Without Probabilities Optimistic Approach Conservative Approach Minimax Regret Approach

Decision Analysis Without Probabilities Decision analysis without probabilities is appropriate in situations: In which a simple best-case and worst-case analysis is sufficient Where the decision maker has little confidence in his or her ability to assess the probabilities

Decision Analysis Without Probabilities Optimistic Approach The optimistic approach evaluates each decision alternative in terms of the best payoff that can occur The decision alternative that is recommended is the one that provides the best possible payoff For minimization problems, this approach leads to choosing the alternative with the smallest payoff

Decision Analysis Without Probabilities Table 15.2: Maximum Payoff For Each PDC Decision Alternative In the PDC problem, the optimistic approach would lead the decision maker to choose the alternative corresponding to the largest profit

Decision Analysis Without Probabilities Conservative Approach The conservative approach evaluates each decision alternative in terms of the worst payoff that can occur The decision alternative recommended is the one that provides the best of the worst possible payoffs For problems involving minimization (for example, when the output measure is cost), this approach identifies the alternative that will minimize the maximum payoff

Decision Analysis Without Probabilities Table 15.3: Minimum Payoff For Each PDC Decision Alternative In the PDC problem, the conservative approach would lead the decision maker to choose the alternative that maximizes the minimum possible profit that could be obtained

Decision Analysis Without Probabilities Minimax Regret Approach Regret is the difference between the payoff associated with a particular decision alternative and the payoff associated with the decision that would yield the most desirable payoff for a given state of nature Regret is often referred to as opportunity loss Under the minimax regret approach, one would choose the decision alternative that minimizes the maximum state of regret that could occur over all possible states of nature

Decision Analysis Without Probabilities Using equation (15.1) and the payoffs in Table 15.1, the regret associated with each combination of decision alternative di and state of nature sj is computed To compute the regret, subtract each entry in a column from the largest entry in the column

Decision Analysis Without Probabilities Table 15.4: Opportunity Loss, or Regret, Table for the PDC Condominium Project ($ Millions) Table 15.5: Maximum Regret for Each PDC Decision Alternative

Decision Analysis Without Probabilities The next step in applying the minimax regret approach is to list the maximum regret for each decision alternative For the PDC problem, the alternative to construct the medium condominium complex, with a corresponding maximum regret of $6 million, is the recommended minimax regret decision

Decision Analysis with Probabilities Expected Value Approach Risk Analysis Sensitivity Analysis

Decision Analysis With Probabilities Expected Value Approach The expected value (EV) of a decision alternative is the sum of weighted payoffs for the decision alternative The weight for a payoff is the probability of the associated state of nature and therefore the probability that the payoff will occur

Figure 15.2: PDC Decision Tree with State-of-Nature Branch Probabilities

Figure 15.3: Applying the Expected Value Approach Using a Decision Tree for the PDC Condominium Project

Decision Analysis With Probabilities Select the decision branch leading to the chance node with the best expected value The decision alternative associated with this branch is the recommended decision In practice, obtaining precise estimates of the probabilities for each state of nature is often impossible, so historical data is preferred to use for estimating the probabilities for the different states of nature

Decision Analysis With Probabilities Risk Analysis Risk analysis helps the decision maker recognize the difference between the expected value of a decision alternative and the payoff that may actually occur Decision alternative and a state of nature combine to generate the payoff associated with a decision Risk profile for a decision alternative shows the possible payoffs along with their associated probabilities

Figure 15.4: Risk Profile for the Large Complex Decision Alternative for the PDC Condominium Project

Decision Analysis With Probabilities Sensitivity Analysis Sensitivity analysis determines how changes in the probabilities for the states of nature or changes in the payoffs affect the recommended decision alternative In many cases, the probabilities for the states of nature and the payoffs are based on subjective assessments Sensitivity analysis helps the decision maker understand which of these inputs are critical to the choice of the best decision alternative If a small change in the value of one of the inputs causes a change in the recommended decision alternative, the solution to the decision analysis problem is sensitive to that particular input

Decision Analysis With Probabilities Example: Suppose that, in the PDC problem, the probability for a strong demand is revised to 0.2 and the probability for a weak demand is revised to 0.8 EV(d1 ) = 0.2 (8) + 0.8 (7) = 7.2 EV(d2 ) = 0.2 (14) + 0.8 (5) = 6.8 EV(d3 ) = 0.2 (20) + 0.8 (29) = 23.2 With these probability assessments, the recommended decision alternative is to construct a small condominium complex (d1), with an expected value of $7.2 million When the probability of strong demand is large, PDC should build the large complex; when the probability of strong demand is small, PDC should build the small complex

Decision Analysis with Sample Information Expected Value of Sample Information Expected Value of Perfect Information

Decision Analysis with Sample Information Decision makers have the ability to collect additional information about the states of nature Additional information is obtained through experiments designed to provide sample information about the states of nature The preliminary or prior probability assessments for the states of nature that are the best probability values available prior to obtaining additional information Posterior probabilities are revised probabilities after obtaining additional information

Decision Analysis with Sample Information Example: PDC management is considering a 6-month market research study designed to learn more about potential market acceptance of the PDC condominium project, anticipating two results: Favorable report: A substantial number of the individuals contacted express interest in purchasing a PDC condominium Unfavorable report: Very few of the individuals contacted express interest in purchasing a PDC condominium

Figure 15.5: The PDC Decision Tree Including the Market Research Study

Figure 15.6: The PDC Decision Tree With Branch Probabilities

Figure 15.7: PDC Decision Tree after Computing Expected Values at Chance Nodes 6 to 14

Figure 15.8: PDC Decision Tree after Choosing Best Decisions at Nodes 3, 4, And 5

Figure 15.9: PDC Decision Tree Reduced to Two Decision Branches

Decision Analysis with Sample Information If the market research is favorable, construct the large condominium complex If the market research is unfavorable, construct the medium condominium complex 

Decision Analysis with Sample Information Expected Value of Sample Information From Figure 15.9 we can conclude that the difference, 15.93 – 14.20 = 1.73, is the expected value of sample information (EVSI)

Decision Analysis with Sample Information Expected Value of Perfect Information A special case of gaining additional information related to a decision problem is when the sample information provides perfect information on the states of nature

Decision Analysis with Sample Information Table 15.6: Payoff Table for the PDC Condominium Project ($ Millions) We can state PDC’s optimal decision strategy when the perfect information becomes available as follows: If s1, select d3 and receive a payoff of $20 million If s2, select d1 and receive a payoff of $7 million

Decision Analysis with Sample Information The original probabilities for the states of nature: P(s1) = 0.8 and P(s2) = 0.2 From equation (12.2) the expected value of the decision strategy that uses perfect information is 0.8(20) + 0.2(7) = 17.4 (i.e., expected value with perfect information (EVwPI)) Earlier, we found the expected value approach is decision alternative d3 $14.2 million; this is referred to as the expected value without perfect information (EVwoPI)

Decision Analysis with Sample Information Example for PDF: Expected value of the perfect information (EVPI) is $17.4 – $14.2 = $3.2 million In general, expected value for perfect information (EVPI) is computed as:

Computing Branch Probabilities with Bayes’ Theorem

Computing Branch Probabilities with Bayes’ Theorem Bayes’ theorem can be used to compute branch probabilities for decision trees The notation | in P(s1|F) and P(s2|F) is read as “given” and indicates a conditional probability because we are interested in the probability of a particular state of nature “conditioned” on the fact that we receive a favorable market report P(s1|F) and P(s2|F) are referred to as posterior probabilities because they are conditional probabilities based on the outcome of the sample information

Figure 15.10: The PDC Decision Tree F = favorable market research report U = unfavorable market research report S1 = strong demand (state of nature 1) S2 = weak demand (state of nature 2)

Computing Branch Probabilities with Bayes’ Theorem In performing the probability computations, we need to know PDC’s assessment of the probabilities of the two states of nature, P(s1) and P(s2) We must know the conditional probability of the market research outcomes given each state of nature To carry out the probability calculations, we need conditional probabilities for all sample outcomes given all states of nature

Computing Branch Probabilities with Bayes’ Theorem In the PDC problem we assume that the following assessments are available for these conditional probabilities: Bayes’ Theorem restated is:

Utility Theory Utility and Decision Analysis Utility Functions Exponential Utility Function

Utility Theory When monetary value does not necessarily lead to the most preferred decision, expressing the value (or worth) of a consequence in terms of its utility will permit the use of expected utility to identify the most desirable decision alternative Utility is a measure of the total worth or relative desirability of a particular outcome Reflects the decision maker’s attitude toward a collection of factors such as profit, loss, and risk

Utility Theory Example of a situation in which utility can help in selecting the best decision alternative: Swofford Inc. currently has two investment opportunities that require approximately the same cash outlay The cash requirements necessary prohibit Swofford from making more than one investment at this time Consequently, three possible decision alternatives may be considered

Utility Theory The three decision alternatives are: d1 = make investment A d2 = make investment B d3 = do not invest The states of nature are: s1 = real estate prices go up s2 = real estate prices remain stable Table 15.7: Payoff Table for Swofford, Inc.

Utility Theory Utility and Decision Analysis A decision maker who would choose a guaranteed payoff over a lottery with a superior expected payoff is a risk avoider The following steps state in general terms the procedure used to solve the Swofford investment problem: Step 1. Develop a payoff table using monetary values Step 2. Identify the best and worst payoff values in the table and assign each a utility, with u(best payoff)> u(worst payoff)

U(M) = pU(best payoff) + (1 - p)U(worst payoff) Utility Theory Step 3. For every other monetary value m in the original payoff table, do the following to determine its utility: Define the lottery such that there is a probability p of the best payoff and a probability (1 - p) of the worst payoff Determine the value of p such that the decision maker is indifferent between a guaranteed payoff of m and the lottery defined in step 3(a) Calculate the utility of m as follows: U(M) = pU(best payoff) + (1 - p)U(worst payoff)

Utility Theory Step 4. Convert each monetary value in the payoff table to a utility Step 5. Apply the expected utility approach to the utility table developed in Step 4 and select the decision alternative with the highest expected utility We can compute the expected utility (EU) of the utilities in a similar fashion as we computed expected value

Utility Theory Table 15.8: Utility of Monetary Payoffs for Swofford, Inc. Table 15.9: Utility Table for Swofford, Inc.

Utility Theory Utility Functions Different decision makers may approach risk in terms of their assessment of utility A risk taker is a decision maker who would choose a lottery over a guaranteed payoff when the expected value of the lottery is inferior to the guaranteed payoff

Utility Theory Analyze the decision problem faced by Swofford from the point of view of a decision maker who would be classified as a risk taker Compare the conservative point of view of Swofford’s president (a risk avoider) with the behavior of a decision maker who is a risk taker

Table 15.11: Payoff Table for Swofford, Inc. Utility Theory Table 15.10: Revised Utilities for Swofford, Inc., Assuming a Risk Taker Table 15.11: Payoff Table for Swofford, Inc.

Utility Theory Using the state-of-nature probabilities P(s1) = 0.3, P(s2) = 0.5, and P(s3) = 0.2, the expected utility for decision alternative is: EU(d2 ) = 0.3 (10) + 0.5 (1.5 ) + 0.2 (1.0 ) = 3.95 EU(d1 ) = 3.50 EU(d3 ) = 2.50 The analysis recommends investment B, with the highest expected utility of 3.95 Table 15.12: Utility Table of a Risk Taker for Swofford, Inc.

Utility Theory Utility function for a risk avoider shows a diminishing marginal return for money Utility function for a risk taker shows an increasing marginal return These values can be plotted on a graph (Figure 15.11) as the utility function for money Top curve is utility function for risk avoider Bottom curve is utility function for risk taker Utility function for a decision maker neutral to risk shows a constant return (middle line)

Figure 15.11: Utility Function for Money for Risk-Avoider, Risk-Taker, and Risk-Neutral Decision Makers

Utility Theory The following characteristics are associated with a risk-neutral decision maker: The utility function can be drawn as a straight line connecting the “best” and the “worst” points The expected utility approach and the expected value approach applied to monetary payoffs result in the same action

Utility Theory Exponential Utility Function Used as an alternative to assume that the decision maker’s utility is defined when decision maker provides enough indifference values to create a utility function All the exponential utility functions indicate that the decision maker is risk averse

Figure 15.12: Exponential Utility Functions with Different Risk Tolerance (R) Values

Utility Theory The R parameter in equation (15.7) represents the decision maker’s risk tolerance; it controls the shape of the exponential utility function Larger R values create flatter exponential functions, indicating that the decision maker is less risk averse (closer to risk neutral) Smaller R values indicate that the decision maker has less risk tolerance (is more risk averse) Example: If the decision maker is comfortable accepting a gamble with a 50 percent chance of winning $2,000 and a 50 percent chance of losing $1,000, but not with a gamble with a 50 percent chance of winning $3,000 and a 50 percent chance of losing $1,500, then we would use R = $2,000 in equation (12.7)