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[Pages:30]19 Decision Making

USING STATISTICS @ The Reliable Fund

19.1 Payoff Tables and Decision Trees

19.2 Criteria for Decision Making Maximax Payoff Maximin Payoff

Expected Monetary Value

Expected Opportunity Loss

Return-to-Risk Ratio

19.3 Decision Making with Sample Information

19.4 Utility

THINK ABOUT THIS: RISKY BUSINESS

USING STATISTICS @ The Reliable Fund Revisited

CHAPTER 19 EXCEL GUIDE

Learning Objectives

In this chapter, you learn:

? How to use payoff tables and decision trees to evaluate alternative courses of action

? How to use several criteria to select an alternative course of action ? How to use Bayes' theorem to revise probabilities in light of sample information ? About the concept of utility

USING STATISTICS

@ The Reliable Fund

A s the manager of The Reliable Fund, you are responsible for purchasing and selling stocks for the fund. The investors in this mutual fund expect a large return on their investment, and at the same time they want to minimize their risk. At the present time, you need to decide between two stocks to purchase. An economist for your company has evaluated the potential one-year returns for both stocks, under four economic conditions: recession, stability, moderate growth, and boom. She has also estimated the probability of each economic condition occurring. How can you use the information provided by the economist to determine which stock to choose in order to maximize return and minimize risk?

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CHAPTER 19 Decision Making

In Chapter 4, you studied various rules of probability and used Bayes' theorem to revise probabilities. In Chapter 5, you learned about discrete probability distributions and how to compute the expected value. In this chapter, these probability rules and probability distributions are applied to a decision-making process for evaluating alternative courses of action. In this context, you can consider the four basic features of a decision-making situation:

? Alternative courses of action A decision maker must have two or more possible choices to evaluate prior to selecting one course of action from among the alternative courses of action. For example, as a manager of a mutual fund in the Using Statistics scenario, you must decide whether to purchase stock A or stock B.

? Events A decision maker must list the events, or states of the world that can occur and consider the probability of occurrence of each event. To aid in selecting which stock to purchase in the Using Statistics scenario, an economist for your company has listed four possible economic conditions and the probability of occurrence of each event in the next year.

? Payoffs In order to evaluate each course of action, a decision maker must associate a value or payoff with the result of each event. In business applications, this payoff is usually expressed in terms of profits or costs, although other payoffs, such as units of satisfaction or utility, are sometimes considered. In the Using Statistics scenario, the payoff is the return on investment.

? Decision criteria A decision maker must determine how to select the best course of action. Section 19.2 discusses five decision criteria: maximax payoff, maximin payoff, expected monetary value, expected opportunity loss, and return-to-risk ratio.

19.1 Payoff Tables and Decision Trees

In order to evaluate the alternative courses of action for a complete set of events, you need to develop a payoff table or construct a decision tree. A payoff table contains each possible event that can occur for each alternative course of action and a value or payoff for each combination of an event and course of action. Example 19.1 discusses a payoff table for a marketing manager trying to decide how to market organic salad dressings.

EXAMPLE 19.1

A Payoff Table for Deciding How to Market Organic Salad Dressings

You are a marketing manager for a food products company, considering the introduction of a new brand of organic salad dressings. You need to develop a marketing plan for the salad dressings in which you must decide whether you will have a gradual introduction of the salad dressings (with only a few different salad dressings introduced to the market) or a concentrated introduction of the salad dressings (in which a full line of salad dressings will be introduced to the market). You estimate that if there is a low demand for the salad dressings, your first year's profit will be $1 million for a gradual introduction and - $5 million (a loss of $5 million) for a concentrated introduction. If there is high demand, you estimate that your first year's profit will be $4 million for a gradual introduction and $10 million for a concentrated introduction. Construct a payoff table for these two alternative courses of action.

SOLUTION Table 19.1 is a payoff table for the organic salad dressings marketing example.

TABLE 19.1

Payoff Table for the Organic Salad Dressings Marketing Example (in Millions of Dollars)

EVENT, Ei

Low demand, E1 High demand, E2

ALTERNATIVE COURSE OF ACTION

Gradual, A1

1 4

Concentrated, A2

-5 10

Using a decision tree is another way of representing the events for each alternative course of action. A decision tree pictorially represents the events and courses of action through a set of branches and nodes. Example 19.2 illustrates a decision tree.

19.1 Payoff Tables and Decision Trees

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EXAMPLE 19.2 Given the payoff table for the organic salad dressings example, construct a decision tree.

A Decision Tree for the Organic Salad Dressings Marketing Decision

SOLUTION Figure 19.1 is the decision tree for the payoff table shown in Table 19.1.

Low Demand

$1

Gradual

High Demand

$4

FIGURE 19.1

Decision tree for the organic salad dressings marketing example (in millions of dollars)

Concentrated

Low Demand

?$5

High Demand

$10

In Figure 19.1, the first set of branches relates to the two alternative courses of action: gradual introduction to the market and concentrated introduction to the market. The second set of branches represents the possible events of low demand and high demand. These events occur for each of the alternative courses of action on the decision tree.

The decision structure for the organic salad dressings marketing example contains only two possible alternative courses of action and two possible events. In general, there can be several alternative courses of action and events. As a manager of The Reliable Fund in the Using Statistics scenario, you need to decide between two stocks to purchase for a short-term investment of one year. An economist at the company has predicted returns for the two stocks under four economic conditions: recession, stability, moderate growth, and boom. Table 19.2 presents the predicted one-year return of a $1,000 investment in each stock under each economic condition. Figure 19.2 shows the decision tree for this payoff table. The decision (which stock to purchase) is the first branch of the tree, and the second set of branches represents the four events (the economic conditions).

TABLE 19.2

Predicted One-Year Return ($) on $1,000 Investment in Each of Two Stocks, Under Four Economic Conditions

ECONOMIC CONDITION

Recession Stable economy Moderate growth Boom

STOCK

A

B

30

- 50

70

30

100

250

150

400

FIGURE 19.2

Decision tree for the stock selection payoff table

Stock A Stock B

$30 Recession Stable Economy $70

Moderate Growth $100 Boom

$150

Recession

?$50

Stable Economy $30

Moderate Growth

Boom

$250

$400

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CHAPTER 19 Decision Making

You use payoff tables and decision trees as decision-making tools to help determine the best course of action. For example, when deciding how to market the organic salad dressings, you would use a concentrated introduction to the market if you knew that there would be high demand. You would use a gradual introduction to the market if you knew that there would be low demand. For each event, you can determine the amount of profit that will be lost if the best alternative course of action is not taken. This is called opportunity loss.

OPPORTUNITY LOSS

The opportunity loss is the difference between the highest possible profit for an event and the actual profit for an action taken.

Example 19.3 illustrates the computation of opportunity loss.

EXAMPLE 19.3

Finding Opportunity Loss in the Organic Salad Dressings Marketing Example

Using the payoff table from Example 19.1, construct an opportunity loss table.

SOLUTION For the event "low demand," the maximum profit occurs when there is a gradual introduction to the market (+ $1 million). The opportunity that is lost with a concentrated introduction to the market is the difference between $1 million and - $5 million, which is $6 million. If there is high demand, the best action is to have a concentrated introduction to the market ($10 million profit). The opportunity that is lost by making the incorrect decision of having a gradual introduction to the market is $10 million - $4 million = $6 million. The opportunity loss is always a nonnegative number because it represents the difference between the profit under the best action and any other course of action that is taken for the particular event. Table 19.3 shows the complete opportunity loss table for the organic salad dressings marketing example.

TABLE 19.3

Opportunity Loss Table for the Organic Salad Dressings Marketing Example (in Millions of Dollars)

Event

Low demand High demand

Optimum Action

Gradual Concentrated

Profit of Optimum Action

1 10

Alternative Course of Action

Gradual

Concentrated

1-1=0 10 - 4 = 6

1 - (-5) = 6 10 - 10 = 0

FIGURE 19.3

Opportunity loss analysis worksheet results for Example 19.3

Figure 19.3 displays the COMPUTE worksheet of the Opportunity Loss workbook. Create this worksheet using the instructions in Section EG19.1.

Figure 19.3 shows the opportunity loss analysis worksheet for Example 19.3.

You can develop an opportunity loss table for the stock selection problem in the Using Statistics scenario. Here, there are four possible events or economic conditions that will affect the one-year return for each of the two stocks. In a recession, stock A is best, providing a return

19.1 Payoff Tables and Decision Trees

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of $30 as compared to a loss of $50 from stock B. In a stable economy, stock A again is better than stock B because it provides a return of $70 compared to $30 for stock B. However, under conditions of moderate growth or boom, stock B is superior to stock A. In a moderate growth period, stock B provides a return of $250 as compared to $100 from stock A, while in boom conditions, the difference between stocks is even greater, with stock B providing a return of $400 as compared to $150 for stock A. Table 19.4 summarizes the complete set of opportunity losses.

TABLE 19.4

Opportunity Loss Table ($) for Two Stocks Under Four Economic Conditions

Event

Recession Stable economy Moderate growth Boom

Optimum Action

A A B B

Profit of Optimum

Action

30 70 250 400

Alternative Course of Action

A

B

30 - 30 = 0 70 - 70 = 0 250 - 100 = 150 400 - 150 = 250

30 - (-50) = 80 70 - 30 = 40 250 - 250 = 0 400 - 400 = 0

Problems for Section 19.1

LEARNING THE BASICS 19.1 For this problem, use the following payoff table:

EVENT

1 2

A ($)

50 200

ACTION

B ($)

100 125

a. Construct an opportunity loss table. b. Construct a decision tree.

19.2 For this problem, use the following payoff table:

EVENT

1 2 3

ACTION

A ($)

50 300 500

B ($)

10 100 200

a. Construct an opportunity loss table. b. Construct a decision tree.

APPLYING THE CONCEPTS

19.3 A manufacturer of designer jeans must decide whether to build a large factory or a small factory in a particular location. The profit per pair of jeans manufactured is estimated as $10. A small factory will incur an annual cost of $200,000, with a production capacity of 50,000 pairs of jeans per year. A large factory will incur an annual cost of $400,000, with a production capacity of 100,000 pairs of jeans per year. Four levels of manufacturing demand are considered likely: 10,000, 20,000, 50,000, and 100,000 pairs of jeans per year.

a. Determine the payoffs for the possible levels of production for a small factory.

b. Determine the payoffs for the possible levels of production for a large factory.

c. Based on the results of (a) and (b), construct a payoff table, indicating the events and alternative courses of action.

d. Construct a decision tree. e. Construct an opportunity loss table.

19.4 An author is trying to choose between two publishing companies that are competing for the marketing rights to her new novel. Company A has offered the author $10,000 plus $2 per book sold. Company B has offered the author $2,000 plus $4 per book sold. The author believes that five levels of demand for the book are possible: 1,000, 2,000, 5,000, 10,000, and 50,000 books sold. a. Compute the payoffs for each level of demand for com-

pany A and company B. b. Construct a payoff table, indicating the events and alter-

native courses of action. c. Construct a decision tree. d. Construct an opportunity loss table.

19.5 The DellaVecchia Garden Center purchases and sells Christmas trees during the holiday season. It purchases the trees for $10 each and sells them for $20 each. Any trees not sold by Christmas day are sold for $2 each to a company that makes wood chips. The garden center estimates that four levels of demand are possible: 100, 200, 500, and 1,000 trees. a. Compute the payoffs for purchasing 100, 200, 500, or

1,000 trees for each of the four levels of demand. b. Construct a payoff table, indicating the events and alter-

native courses of action. c. Construct a decision tree. d. Construct an opportunity loss table.

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CHAPTER 19 Decision Making

19.2 Criteria for Decision Making

After you compute the profit and opportunity loss for each event under each alternative course of action, you need to determine the criteria for selecting the most desirable course of action. Some criteria involve the assignment of probabilities to each event, but others do not. This section introduces two criteria that do not use probabilities: the maximax payoff and the maximin payoff.

This section presents three decision criteria involving probabilities: expected monetary value, expected opportunity loss, and the return-to-risk ratio. For criteria in which a probability is assigned to each event, the probability is based on information available from past data, from the opinions of the decision maker, or from knowledge about the probability distribution that the event may follow. Using these probabilities, along with the payoffs or opportunity losses of each event?action combination, you select the best course of action according to a particular criterion.

Maximax Payoff

The maximax payoff criterion is an optimistic payoff criterion. Using this criterion, you do the following:

1. Find the maximum payoff for each action. 2. Choose the action that has the highest of these maximum payoffs.

Example 19.4 illustrates the application of the maximax criterion to the organic salad dressings marketing example.

EXAMPLE 19.4

Finding the Best Course of Action According to the Maximax Criterion for the Organic Salad Dressings Marketing Example

TABLE 19.5

Using the Maximax Criterion for the Organic Salad Dressings Marketing Example (in Millions of Dollars)

Return to Table 19.1, the payoff table for deciding how to market organic salad dressings. Determine the best course of action according to the maximax criterion.

SOLUTION First you find the maximum profit for each action. For a gradual introduction to the market, the maximum profit is $4 million. For a concentrated introduction to the market, the maximum profit is $10 million. Because the maximum of the maximum profits is $10 million, you choose the action that involves a concentrated introduction to the market. Table 19.5 summarizes the use of this criterion.

EVENT, Ei

High demand, E1 High demand, E2 Maximum profit for each action

ALTERNATIVE COURSE OF ACTION

Gradual, A1

1 4 4

Concentrated, A2

-5 10 10*

TABLE 19.6

Using the Maximax Criterion for the Predicted One-Year Return ($) on $1,000 Investment in Each of Two Stocks, Under Four Economic Conditions

As a second application of the maximax payoff criterion, return to the Using Statistics scenario and the payoff table presented in Table 19.2. Table 19.6 summarizes the maximax payoff criterion for that example.

ECONOMIC CONDITION

Recession Stable economy Moderate growth Boom Maximum profit for each action

STOCK

A

B

30

- 50

70

30

100

250

150

400

150

400

19.2 Criteria for Decision Making

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Because the maximum of the maximum profits is $400, you choose stock B.

Maximin Payoff

The maximin payoff criterion is a pessimistic payoff criterion. Using this criterion, you do the following:

1. Find the minimum payoff for each action. 2. Choose the action that has the highest of these minimum payoffs.

Example 19.5 illustrates the application of the maximin criterion to the organic salad dressing marketing example.

EXAMPLE 19.5

Finding the Best Course of Action According to the Maximin Criterion for the Organic Salad Dressings Marketing Example

Return to Table 19.1, the payoff table for deciding how to market organic salad dressings. Determine the best course of action according to the maximin criterion.

SOLUTION First, you find the minimum profit for each action. For a gradual introduction to the market, the minimum profit is $1 million. For a concentrated introduction to the market, the minimum profit is - $5 million. Because the maximum of the minimum profits is $1 million, you choose the action that involves a gradual introduction to the market. Table 19.7 summarizes the use of this criterion.

TABLE 19.7

Using the Maximin Criterion for the Organic Salad Dressings Marketing Example (in Millions of Dollars)

EVENT, Ei

Low demand, E1 High demand, E2 Minimum profit for each action

ALTERNATIVE COURSE OF ACTION

Gradual, A1

1 4 1

Concentrated, A2

-5 10 -5

As a second application of the maximin payoff criterion, return to the Using Statistics scenario and the payoff table presented in Table 19.2. Table 19.8 summarizes the maximin payoff criterion for that example.

TABLE 19.8

Using the Maximin Criterion for the Predicted One-Year Return ($) on $1,000 Investment in Each of Two Stocks, Under Four Economic Conditions

ECONOMIC CONDITION

Recession Stable economy Moderate growth Boom Minimum profit for each action

STOCK

A

B

30

- 50

70

30

100

250

150

400

30

- 50

Because the maximum of the minimum profits is $30, you choose stock A.

Expected Monetary Value

The expected value of a probability distribution was computed in Equation (5.1) on page 163. Now you use Equation (5.1) to compute the expected monetary value for each alternative course of action. The expected monetary value (EMV ) for a course of action, j, is the payoff (Xij) for each combination of event i and action j multiplied by Pi, the probability of occurrence of event i, summed over all events [see Equation (19.1)].

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