An Introduction to
When you have completed this chapter, you will be able to:
1 Define the terms state of nature, event, decision alternative, and payoff.
2 Organize information in a payoff table or a decision tree.
3 Find the expected payoff of a decision alternative.
4 Compute opportunity loss and expected opportunity loss.
5 Assess the expected value of information.
Blackbeard’s Phantom Fireworks is considering introducing two new bottle rockets. The company can add both to the current line, neither, or just one of the two. The success of these products depends on consumers’ reactions. These reactions can be summarized as good, fair, or poor. The company’s revenues are estimated in the payoff table in Exercise 11. Compute the expected monetary value for each decision. (Exercise 11a, Goal 3)
A branch of statistics called statistical decision theory that uses probability has been developed since the early 1950s. As the name implies, the focus is on the process of making decisions and explicitly includes the payoffs that may result. In contrast, classical statistics focuses on estimating a parameter, such as the population mean, constructing a confidence interval, or conducting a hypothesis test.
Classical statistics does not address the financial consequences.
Statistical decision theory is concerned with determining which decision, from a set of possible alternatives, is optimal for a particular set of conditions. Consider the following examples of decision-theory problems.
• Ford Motor Company must decide whether to purchase assembled door locks for the 2008 Ford F-150 Harley-Davidson truck or to manufacture and assemble the door locks at its Sandusky, Ohio, plant. If sales of the F-150 truck continue to increase, it will be more profitable to manufacture and assemble the parts. If sales level off or decline, it will be more profitable to purchase the door locks assembled. Should it make or buy the door locks?
• Banana Republic developed a new line of summer rain jackets that are very popular in the cold-weather regions of the country. It would like to purchase commercial television time during the upcoming NCAA basketball final. If both teams that play in the game are from warm parts of the country, it estimates that only a small proportion of the viewers will be interested in the jackets. However, a matchup between two teams who come from cold climates would reach a large proportion of viewers who wear jackets. Should it purchase commercial television time?
• General Electric is considering three options regarding the prices of refrigerators for next year. GE could (1) raise the prices 5 percent, (2) raise the prices
2.5 percent, or (3) leave the prices as they are. The final decision will be based on sales estimates and on GE’s knowledge of what other refrigerator manufacturers might do.
In each of these cases the decision is characterized by several alternative courses of action and several factors not under the control of the decision maker.
For example, Banana Republic has no control over which teams reach the NCAA basketball final. These cases 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
There are three components to any decision: (1) the choices available, or alternatives; (2) the states of nature, which are not under the control of the decision maker; and (3) the payoffs. These concepts will be explained in the following paragraphs.
The alternatives, or acts, are the choices available to the decision maker. Ford can decide to manufacture and assemble the door locks in Sandusky, or it can decide to purchase them. To simplify our presentation, we assume the