An Excel TUTORIAL for Introductory Statistics Applications

Decision analytics is 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.
• Always includes a careful consideration of risk

Risk analysis helps provide the profitability information about the favorable as well as the unfavorable outcomes that may occur. Topics include payoff tables, decision trees, sensitivity analysis, and the use of Bayes’ theorem.

Problem formulation is the first step in the decision analysis process

• Create a verbal statement of the problem.
• Identify the decision alternatives such as uncertain future events, and the outcomes associated with each alternative and the probability of that event outcome

The possible outcomes for each chance event are known as states of nature. States of nature are mutually exclusive (no more than one can occur) and collectively exhaustive (at least one must occur)

Subsection: 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.

Subsection: Decision Trees

Decision Tree: provides a graphical representation of the decision making process

Decision Trees consist of nodes which are used to represent decisions and chance events. Consist of branches that 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. Example to be added

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

Subsection: 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

Subsection: 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

Subsection: 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
• 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 \[ R_{ij} \left\vert V^{\ast}_j - V_{ij} \right\vert , \] where
  • R_ij = the regret associated with decision alternative d_i and state of nature s_j.
  • V^*_ij == the payoff value corresponding to the best decision for the state pf nature s_j.
  • V_ij = the payoff corresponding to decision alternative d_i and state of nature s_j.

Subsection: Expected Value Approach

Expected Value Approach

• Expected Value of a decision alternative is the sum of weighted payoffs for the decision alternative
• The weight for a payoff is the probability of the state of nature and the probability the outcome will occur \[ EV(d) = \sum_{j=1}^N \Pr [s_j ]\, V_{ij} \]

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

Subsection: Risk Analysis

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

Subsection: 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 analytics

Subsection: Expected Value of Sample Information

• Decision makers have 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

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

Decision Analysis with Sample Information:

• If the market research is favorable, construct the large condominium comple
• If the market research is unfavorable, construct the medium condominium complex

Example: 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) \[ \mbox{EVSI} = \left\vert \mbox{EVwSI} - \mbox{EVwoSI} \right\vert , \] where
EVSI = expected value of sample information
EVwSI = expected value with sample information about of states of nature
EVwoSI = expected value without sample information about of states of nature

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 d₃ and receive a payoff of $7 million
The original probabilities for the states of nature: \[ \Pr [s_1 ] = 0.8 \qquad \mbox{and} \qquad \Pr [s_2 ] = 0.2 . \]
• 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)

Subsection: Expected Value of Perfect Information

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

Example: The PDC decision tree
figure ?
F = Favorable market research report U = Unfavorable market research report S₁ = Strong demand (state of nature 1) S₂ = Weak demand (state of nature 2)

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

In the PDC problem we assume that the following assessments are available for these conditional probabilities:

State of Nature	Favorable, F	Unfavorable, U
Strong demandm s1	Pr[ F | s1 ] = 0.90	Pr[ U | s1 ] = 0.10
Weak demand, s2	Pr[ F | s2 ] = 0.25	Pr[ U | s2 ] = 0.75

Bayes’ Theorem

\[ \Pr \left[ A)i \,|\, B \right] = \frac{\Pr [A_i ]\.\Pr [B\, |\, A_i ]}{\Pr [A_1 ]\.\Pr [B\, |\, A_1 ] + \Pr [A_2 ]\.\Pr [B\, |\, A_2 ] + \cdots } . \]

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. It reflects the decision maker’s attitude toward a collection of factors such as profit, loss, and risk

Example of a situation where 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 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)
• Step 3: For every other monetary value min the original payoff table, do the following to determine its utility:
  1. Define the lottery such that there is a probability pof the best payoff and a probability (1 - p) of the worst payoff
  2. Determine the value of p such that the decision maker is indifferent between a guaranteed payoff of mand the lottery defined in step 3(a)
  3. Calculate the utility of mas follows: \[ U(M) = p\,U(\mbox{best payoff}) + \left( 1-p \right) U(\mbox{Worst payoff}) . \]
• Steo 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

Subsection: 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

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

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

• 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)

Subsection: 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

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)