Wk7 DQ - Data Analysis & Business Intelligence

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An Introduction to Decision Theory

Chapter 20

20-1

Statistical decision theory focuses on the process of making decisions, including the payoffs (or consequences) that may result from selecting a particular decision alternative. Classical statistics, like we have studied so far, does not address any payoff (or consequence) when making a decision. In this chapter, we find there are several ways to select the best decision alternative.

Learning Objectives

LO20-1 Identify and apply the three components of a decision

LO20-2 Analyze a decision using expected monetary value

LO20-3 Analyze a decision using opportunity loss

LO20-4 Apply maximin, maximax, and minimax regret strategies to make a decision

LO20-5 Compute and explain the expected value of perfect information

LO20-6 Apply sensitivity analysis to evaluate a decision subject to uncertainty

LO20-7 Use a decision tree to illustrate and analyze decision making under uncertainty

20-2

Statistical Decision Theory

Statistical decision theory is concerned with making decisions from a set of alternatives

Examples

Ford Motor Company must decide whether to purchase assembled door locks for the 2017 F-150 or to manufacture the locks themselves

If sales increase – it’s more profitable to make

If sales level off or decline – it’s more profitable to buy

GE is considering three options regarding the prices of their refrigerators next year, raise the prices 5%, 2.5%, or leave the prices as they are

The decision will be based on sales estimates and what the other refrigerator producers might do

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These examples characterize the nature of decision making. Possible decision alternatives can be listed, possible future events determined, and even probabilities established, but the decisions are made in the face of uncertainty.

Elements of a Decision

The choices available

The various courses of action are called the acts or alternatives

The states of nature

The uncontrollable future events are called the states of nature

The payoffs

The consequence of a particular decision alternative and state of nature is the payoff

We can make better decisions if we establish probabilities for the states of nature

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For instance, Ford can decide to manufacture the door locks in Louisville (one alternative), or it may decide to purchase them (another alternative). The state of nature in this example is that Ford does not know if demand will remain high for the F-150. One estimated payoff for Ford is $40,000 if they manufacture their own door locks and demand is low for the F-150; another estimated payoff is $22,000 if they purchase the locks and demand is high. Establishing probabilities can be based on historical data or made subjectively. For instance, Ford may estimate the probability of continued demand at .70.

Decision Making under Uncertainty

20-5

Here are the main elements of the decision under conditions of uncertainty.

Payoff Table

Bob Hill, a small investor, has $1,100 to invest. He has studied several common stocks and narrowed his choices to three: Kayser Chemicals, Rim Homes, and Texas Electronics. He estimates that if stock prices increase drastically (a bull market), his investment in Kayser stock would more than double. However, if

stock prices decrease (a bear market), the value of the Kayser stock could drop to $1000. His predictions for the three stocks are shown in the Payoff Table below.

20-6

The first step is to set up a payoff table. All possible combinations of decision alternatives and states of nature appear in the payoff table. The various choices are called the decision alternatives or acts and are labeled A1, A2, and A3 in this example. Whether the market turns out to be a bull market (S1) or a bear market (S2) is out of the control of Bob and as such, are the states of nature.

Expected Payoff

In the expected monetary value (EMV) criterion, the expected value for each decision alternative is computed, and the optimal one is selected

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A study of historical data shows that during the past 10 years, the stock market went up six times and went down 4 times. So Bob assigns probabilities of .60 and .40 respectively to arrive at the expected payoff of buying each of the three stocks. The top table contains the expected payoff of buying Kayser Chemicals stock. Then we use formula 20-1 to calculate the EMV, the expected monetary value for Kayser Chemicals stock. This formula can be used to calculate the EMV for the other two stocks as well; the table on the bottom of the slide has the expected payoffs for all three stocks. An analysis of the expected payoffs indicates that purchasing Kayser Chemicals would yield the greatest expected profit. Even so, the investor may still decide to buy Texas Electronics stock to minimize the risk of losing some of the $1,100 investment.

Opportunity Loss

The difference between the optimal decision and any other decision is the opportunity loss or regret due to making a decision other than the optimum

An opportunity loss table can be developed

An opportunity loss table is constructed by taking the difference between the optimal decision for each state of nature and the other decision alternatives

The expected opportunity loss (EOL) is similar to the expected monetary value

The opportunity loss is combined with the probabilities of the various states of nature for each decision alternative to determine the expected opportunity loss

20-8

See the next slide for an example with an opportunity loss table.

Opportunity Loss Table

Suppose Bob Hill had purchased Rim Homes stock and then a bull market developed. The value of the Rim Homes stock did increase from $1,100 to $2,200 as expected. But had Bob bought Kayser Chemicals stock and market values increased, the value of his stock purchase would be $2,400; so he missed making an extra $200. The $200 is the opportunity loss for not knowing the future state of the market.

Notice, that Kayser Chemicals stock would be a good investment choice in a rising (bull) market, Texas Electronics stock would be the best buy in a declining (bear) market, and Rim Homes stock is a compromise.

20-9

The opportunity losses corresponding to this example are given in the table. Each amount is the outcome of a particular combination of acts and a state of nature, that is, a stock purchase and market reaction.

Expected Opportunity Loss

The two approaches, lowest expected opportunity loss and highest expected payoff, will always lead to the same decision.

20-10

The probability of a market rise is .60 and that of a market decline is .40. These probabilities can be combined to determine the expected opportunity loss. The table at the top of the slide shows the expected opportunity loss of buying Rim Homes stock is $140. We can use formula 20-2 to calculate the expected opportunity loss for each stock, Bob’s three decision alternatives; the results are shown in the table at the bottom of the slide. The lowest expected opportunity loss is $60, meaning that he would experience the least regret on average if he purchased Kayser Chemicals. The decision to purchase Kayser stock because it offers the lowest expected opportunity loss reinforces the earlier decision to buy Kayser stock when that resulted in the highest expected payoff.

Maximin, Maximax, and Minimax Regret Strategies

Maximin

The strategy of maximizing the minimum gain

Investment advisors may urge the investor to be conservative and buy Texas Electronics stock

The investor is assured of at least $1,150

Maximax

The strategy of maximizing the maximum gain

Under this strategy, the investor would buy Kayser stock

There’s a possibility of selling the stock later for $2,400

Minimax regret

The strategy that minimizes the maximum regret

Buy Kayser, there’s a maximum opportunity loss of $150

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The investor (and others) may use maximin, maximax, and minimax regret strategies to make a decision.

Value of Perfect Information

VALUE OF PERFECT INFORMATION The difference between the maximum expected payoff under conditions of certainty and the maximum expected payoff under uncertainty.

EVPI = Expected value under certainty − Expected value under uncertainty

= $1,900 − $1,840 = $60

If Bob knew precisely what the market would do, he could maximize profit by always buying the correct stock. What if someone could supply him with information about when the market will rise or decline? What would this information be worth?

20-12

The expected value of perfect information (EVPI) is the difference between the best expected payoff under certainty and the best expected payoff under uncertainty. To explain, the maximum expected value under conditions of certainty means the investor would buy Kayser stock if a market rise was predicted and Texas Electronics stock if a market decline was imminent. Use formula 20-3 to calculate the expected value of perfect information. In this example, it would be worth up to $60.

Sensitivity Analysis

Sensitivity analysis examines the effects of various probabilities for the states of nature on the expected values

The rankings of the decision alternatives are frequently not highly sensitive to changes within a plausible range

20-13

Earlier, we used historical data to arrive at the .60 probability of a market rise and a .40 probability of a market decline. But someone could argue that future market behavior may be different from past experiences. One person, Bob’s brother, might believe instead that there’s a probability of .40 of a market rise and .60 of a market decline. Another person, Bob’s cousin, might believe that there is a 50-50 probability. Even so, the decision is the same in all three cases – buy Kayser Chemical stock.

Decision Trees

Decision trees are useful for structuring the various alternatives

They present a picture of the various courses of action and the possible states of nature

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After the decision tree has been constructed, the best decision strategy is found by backward induction. For example, suppose the investor is considering the act of purchasing Texas Electronics stock. Starting at the lower right in the chart with the anticipated payoff given a market rise ($1,900) versus a market decline ($1,150) and going backward (moving left),the probabilities are applied to give the expected payoff of $1,600. The other stocks’ probabilities would be arrived at similarly. Assuming the investor wants to maximize the expected value of the stock purchase, $1,840 is preferred over $1,760 or $1,600, so the investor draws a double bar across those two branches. The unmarked branch is the best action to follow: buy Kayser Chemicals stock.

Decision tree analysis provides an alternative method to the calculations used earlier in the chapter. Some managers find decision tree sketches helpful in following decision logic.

Chapter 20 Practice Problems

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Question 1

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The following payoff table was developed. Let P(S1) = .30, P(S2) = .50, and P(S3) = .20. Compute the expected monetary value for each of the alternatives. What decision would you recommend?

LO20-2

Question 3

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Refer to Exercise 1. Develop an opportunity loss table. Determine the opportunity loss for each decision.

LO20-3

Question 5

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Refer to Exercises 1 and 3. Compute the expected opportunity losses.

LO20-3

Question 7

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Refer to Exercises 1, 3, and 5. Compute the expected value of perfect information.

LO20-5

Question 9

20-20

Refer to Exercise 1. Revise the probabilities as follows: P(S1) = .50, P(S2) = .20, and P(S3) = .30 and use Expected Monetary Value to evaluate the decision. Does this change the decision?

LO20-6