Appendix Decision Models and Uncertainty *

This appendix explores the characteristics of uncertainty, describes an approach managers can use to make decisions in a world of uncertainty, and illustrates the insights gained when uncertainty is recognized in CVP analysis. In the face of uncertainty, managers rely on decision models to help them make the right choices.

Role of a Decision Model

Uncertainty is the possibility that an actual amount will deviate from an expected amount. In the GMAT Success example, Emma might forecast sales at 42 units, but actual sales might turn out to be 30 units or 60 units. A decision model helps managers deal with such uncertainty. It is a formal method for making a choice, commonly involving both quantitative and qualitative analyses. This appendix focuses on the quantitative analysis that usually includes the following steps:

  1. Step 1: IDENTIFY A CHOICE CRITERION. A choice criterion is an objective that can be quantified, such as maximize income or minimize costs. Managers use the choice criterion to choose the best alternative action. Emma’s choice criterion is to maximize expected operating income at the Chicago college fair.
  2. Step 2: IDENTIFY THE SET OF ALTERNATIVE ACTIONS THAT CAN BE TAKEN. We use the letter a with subscripts 1,2,and 3 to distinguish each of Emma’s three possible actions:

a1=Pay $2,000 fixed feea2=Pay $800 fixed fee plus 15% of GMAT Success revenuesa3=Pay 25% of GMAT Success revenues with no fixed fee

  1. Step 3: IDENTIFY THE SET OF EVENTS THAT CAN OCCUR. An event is a possible relevant occurrence, such as the actual number of GMAT Success packages Emma might sell at the fair. The set of events should be mutually exclusive and collectively exhaustive. Events are mutually exclusive if they cannot occur at the same time. Events are collectively exhaustive if, taken together, they make up the entire set of possible relevant occurrences (no other event can occur). Examples of mutually exclusive and collectively exhaustive events are growth, decline, or no change in industry demand and increase, decrease, or no change in interest rates. Only one event out of the entire set of mutually exclusive and collectively exhaustive events will actually occur.
  2. Suppose Emma’s only uncertainty is the number of units of GMAT Success that she can sell. For simplicity, suppose Emma estimates that sales will be either 30 or 60 units. This set of events is mutually exclusive because clearly sales of 30 units and 60 units cannot both occur at the same time. It is collectively exhaustive because under our assumptions sales cannot be anything other than 30 or 60 units. We use the letter x with subscripts 1 and 2 to distinguish the set of mutually exclusive and collectively exhaustive events:

x1=30 unitsx2=60 units  

  1. Step 4: ASSIGN A PROBABILITY TO EACH EVENT THAT CAN OCCUR. A probability is the likelihood or chance that an event will occur. The decision model approach to coping with uncertainty assigns probabilities to events. A probability distribution describes the likelihood, or the probability, that each of the mutually exclusive and collectively exhaustive set of events will occur. In some cases, there will be much evidence to guide the assignment of probabilities. For example, the probability of obtaining heads in the toss of a coin is 1/2 and that of drawing a particular playing card from a standard, well-shuffled deck is 1/52. In business, the probability of having a specified percentage of defective units may be assigned with great confidence on the basis of production experience with thousands of units. In other cases, there will be little evidence supporting estimated probabilities—for example, expected sales of a new pharmaceutical product next year. Suppose that Emma, on the basis of past experience, assesses a 60% chance, or a 6/10 probability, that she will sell 30 units and a 40% chance, or a 4/10 probability, that she will sell 60 units. Using PX as the notation for the probability of an event, the probabilities are:

P(x1)=6/10=0.60P(x2)=4/10=0.40  

  1. The sum of these probabilities must equal 1.00 because these events are mutually exclusive and collectively exhaustive.
  2. Step 5: IDENTIFY THE SET OF POSSIBLE OUTCOMES. Outcomes specify, in terms of the choice criterion, the predicted economic results of the various possible combinations of actions and events. In the GMAT Success example, the outcomes are the six possible operating incomes displayed in the decision table in Exhibit 3-6. A decision table is a summary of the alternative actions, events, outcomes, and probabilities of events.

Exhibit 3-6 

Decision Table for GMAT Success

Screenshot of a spreadsheet depicting the operating income under each possible event.

The selling price is 200 dollars, and the package cost is 120 dollars. The event x1 is; 30 units sold whose probability is 0.60 and the event x2 is; 60 units sold whose probability is 0.40.

  1. Action: Pay 2,000 dollars fixed fee

    • Fixed fee: 2,000 dollars

    • Fair revenues: 0 percent

    • Operating income under the event x1: 400 dollars

    • Operating income under the event x2: 2,800 dollars

  2. Action 2: Pay 800 dollars fixed fee plus 15 percent of revenues

    • Fixed fee: 800 dollars

    • Fair revenues: 15 percent

    • Operating income under the event x1: 700 dollars

    • Operating income under the event x2: 2,200 dollars

  3. Action 2: Pay 25 percent of revenues with no fixed fee

    • Fixed fee: 0 dollars

    • Fair revenues: 25 percent

    • Operating income under the event x1: 900 dollars

    • Operating income under the event x2: 1,800 dollars

  4. Row 10 referring to operating income, under the event x1, of 400 dollars equals the product of 200 dollars minus 120 dollars and 30, minus 2,000 dollars.

  5. Row 11 referring to operating income, under the event x2, of 2,800 dollars equals the product of 200 dollars minus 120 dollars and 60, minus 2,000 dollars.

  6. Row 12 referring to operating income, under the event x1, of 700 dollars equals the product of 200 dollars minus 120 dollars minus 15 percent of 200 dollars and 30, minus 800 dollars.

  7. Row 13 referring to operating income, under the event x2, of 2,200 dollars equals the product of 200 dollars minus 120 dollars minus 15 percent of 200 dollars and 60, minus 2,000 dollars.

  8. Row 14 referring to operating income, under the event x1, of 900 dollars equals the product of 200 dollars minus 120 dollars minus 25 percent of 200 dollars and 30.

  9. Row 15 referring to operating income, under the event x2, of 2,200 dollars equals the product of 200 dollars minus 120 dollars minus 25 percent of 200 dollars and 60.

Distinguish among actions, events, and outcomes. Actions are decision choices available to managers—for example, the particular rental alternatives that Emma can choose. Events are the set of all relevant occurrences that can happen—for example, the different quantities of GMAT Success packages that may be sold at the fair. The outcome is operating income, which depends both on the action the manager selects (rental alternative chosen) and the event that occurs (the quantity of packages sold).

Exhibit 3-7 presents an overview of relationships among a decision model, the implementation of a chosen action, its outcome, and subsequent performance evaluation. Thoughtful managers step back and evaluate what happened and learn from their experiences. This learning serves as feedback for adapting the decision model for future actions.

Exhibit 3-7 A 

Decision Model and Its Link to Performance Evaluation

Cyclic flowchart shows the decision model and its link to performance evaluation.

The steps of the flowchart are as follows:

Expected Value

An expected value is the weighted average of the outcomes, with the probability of each outcome serving as the weight. When the outcomes are measured in monetary terms, expected value is often called expected monetary value. Using information in Exhibit 3-6, the expected monetary value of each booth-rental alternative denoted by E(a1),E(a2), and E(a3) is as follows:

Pay $2,000 fixed fee:E(a1)=(0.60×$400)  + (0.40×$2,800)= $1,360Pay $800 fixed fee plus15%of revenues:E(a2) = (0.60×$700) + (0.40×$2,200)= $1,300Pay 25%of revenues with no fixed fee:E(a3) =(0.60×$900) + (0.40×$1,800)= $1,260 

To maximize expected operating income, Emma should select action a1—pay the Chicago fair organizers a $2,000 fixed fee.

To interpret the expected value of selecting action a1, imagine that Emma attends many fairs, each with the probability distribution of operating incomes given in Exhibit 3-6. For a specific fair, Emma will earn operating income of either $400, if she sells 30 units, or $2,800, if she sells 60 units. But if Emma attends 100 fairs, she will expect to earn $400 operating income 60% of the time (at 60 fairs) and $2,800 operating income 40% of the time (at 40 fairs), for a total operating income of $136,000 ($400×60+$2,800×40). The expected value of $1,360 is the operating income per fair that Emma will earn when averaged across all fairs ($136,000÷100). Of course, in many real-world situations, managers must make one-time decisions under uncertainty. Even in these cases, expected value is a useful tool for choosing among alternatives.

Consider the effect of uncertainty on the preferred action choice. If Emma were certain she would sell only 30 units (that is, Px1 =1), she would prefer alternative a3—pay 25% of revenues with no fixed fee. To follow this reasoning, examine Exhibit 3-6. When 30 units are sold, alternative a3 yields the maximum operating income of $900. Because fixed costs are $0, booth-rental costs are lower, equal to $1,500 (25% of revenues=0.25×$200  per unit×30 units), when sales are low.

However, if Emma were certain she would sell 60 packages (that is, P(x2)=1), she would prefer alternative a1—pay a $2,000 fixed fee. Exhibit 3-6 indicates that when 60 units are sold, alternative a1 yields the maximum operating income of $2,800. That’s because, when 60 units are sold, rental payments under a2($800+0.15×$200  per unit×60 units=$2,600) and a3 (0.25×$200 per unit×60 units=$3,000) are more than the fixed $2,000 fee under a1. 

Despite the high probability of selling only 30 units, Emma still prefers to take action a2, which is to pay a fixed fee of $2,000. That’s because the high risk of low operating income (the 60% probability of selling only 30 units) is more than offset by the high return from selling 60 units, which has a 40% probability. If Emma were more averse to risk (measured in our example by the difference between operating incomes when 30 versus 60 units are sold), she might have preferred action a2 or a3. For example, action a2 ensures an operating income of at least $700, greater than the operating income of $400 that she would earn under action a1 if only 30 units were sold. Of course, choosing a2 limits the upside potential to $2,200 relative to $2,800 under a1, if 60 units are sold. If Emma is very concerned about downside risk, however, she may be willing to forgo some upside benefits to protect against a $400 outcome by choosing a2.*

Good Decisions and Good Outcomes

Always distinguish between a good decision and a good outcome. One can exist without the other. Suppose you are offered a one-time-only gamble tossing a coin. You will win $20 if the outcome is heads, but you will lose $1 if the outcome is tails. As a decision maker, you proceed through the logical phases: gathering information, assessing outcomes, and making a choice. You accept the bet. Why? Because the expected value is $9.50 [0.5($20)+0.5(−$1)]. The coin is tossed, and the outcome is tails. You lose. From your viewpoint, this was a good decision but a bad outcome.

A decision can be made only on the basis of information that is available at the time of evaluating and making the decision. By definition, uncertainty rules out guaranteeing that the best outcome will always be obtained. As in our example, it is possible that bad luck will produce bad outcomes even when good decisions have been made. A bad outcome does not mean a bad decision was made. The best protection against a bad outcome is a good decision.