LECTURE TWELVE Decision-Making UNDER UNCERTAINITY

Introduction Decision analysis provides a framework and methodology for rational decision making when the outcomes are uncertain. Example: A company plans to determine the best location to startup a new plant from a choice of several locations. Each location offers a different cost scenario. Decision Analysis techniques can be used to determine the best decision.

Payoff Table States of Nature refer to future events which may occur and the values in a Payoff Table refer to either Profit or Cost. Example: s1, s2 & s3 could refer to possible scenarios (i.e. Moderate, Strong & Weak market demand) States of Nature s1 s2 s3 d1 4 4 -2 Decisions d2 0 3 -1 d3 1 5 -3

Simple Decision Making without Probabilities Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: the Optimistic approach (Maximax or Minimin) the Conservative approach (Maximin or Minimax) the Minimax Regret approach.

Optimistic Approach The optimistic approach would be used by an optimistic decision maker. The decision with the largest possible payoff is chosen. If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.

Example: Optimistic Approach Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits: States of Nature s1 s2 s3 d1 4 4 -2 Decisions d2 0 3 -1 d3 1 5 -3

Example: Optimistic Approach The Optimistic Approach is to make decision based on the maximum of the largest profits. For each decision, d, the largest profits are identified (i.e. 4, -1 & 5) Formula Spreadsheet

Example: Optimistic Approach Solution Spreadsheet

Conservative Approach The conservative approach would be used by a conservative decision maker. For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected.

Example: Conservative Approach Based on the same table with three decision alternatives and three states of nature with the following payoff table representing profits: States of Nature s1 s2 s3 d1 4 4 -2 Decisions d2 0 3 -1 d3 1 5 -3

Example: Conservative Approach Formula Spreadsheet

Example: Conservative Approach Solution Spreadsheet

Minimax Regret Approach The minimax regret approach requires the construction of a regret table or an opportunity loss table. This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature. Then, using this regret table, the maximum regret for each possible decision is listed. The decision chosen is the one corresponding to the minimum of the maximum regrets.

Example: Minimax Regret Approach Compute a regret table by subtracting each payoff in a column from the largest payoff in that column. Add a Max Regret Col and make decision based on the Minimum of Maximum Regret. Example: 4, 0, 1 is subtracted by 4 giving OL 0, 4, 3, etc. s1 OL s2 OL s3 OL Max Regret d1 4 0 4 1 -2 1 1 d2 0 4 3 2 -1 0 4 d3 1 3 5 0 -3 2 3

Example: Minimax Regret Approach Formula Spreadsheet

Example: Minimax Regret Approach Solution Spreadsheet

Decision Making with Probabilities Expected Value Approach If probabilities regarding the states of nature is available, we may use the expected value (EV) approach. Decision is based on maximising EV. The expected value (EV) of decision alternative di is defined as: where: N = the number of states of nature P(sj ) = the probability of state of nature sj Vij = the payoff corresponding to decision alternative di and state of nature sj

Example: Expected Value Approach ABC Restaurant Average Number of Customers Per Hour s1 = 80 s2 = 100 s3 = 120 Model A $10,000 $15,000 $14,000 Model B $ 8,000 $18,000 $12,000 Model C $ 6,000 $16,000 $21,000 Probability 0.4 0.2 0.4 Given s1, s2 & s3 have the probabilities: 0.4, 0.2 & 0.4

Example: Expected Value Approach Formula Spreadsheet

Example: Expected Value Approach Solution Spreadsheet

Expected Value of Perfect Information The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur. EVPI = EVwPI – Max EV where EVwPI = ∑ Pi * Max Vi where Vi is the payoffs and EVwPI = Expected Value with Perfect Information

Expected Value of Perfect Information Spreadsheet

Decision Tree Payoffs 10,000 2 15,000 d1 14,000 8,000 d2 1 3 18,000 d3 Average Number of Customers Per Hour s1 = 80 s2 = 100 s3 = 120 Model A $10,000 $15,000 $14,000 Model B $ 8,000 $18,000 $12,000 Model C $ 6,000 $16,000 $21,000 Probabilities 0.4 0.2 0.4 Payoffs s1 .4 10,000 s2 .2 2 15,000 s3 .4 d1 14,000 .4 s1 8,000 d2 1 3 s2 .2 18,000 d3 s3 .4 12,000 s1 .4 6,000 4 s2 .2 16,000 s3 .4 21,000

Decision Tree d1 2 Model A Model B d2 Choose the model with largest EV, Model C. EMV = .4(10,000) + .2(15,000) + .4(14,000) = $12,600 d1 2 Model A EMV = .4(8,000) + .2(18,000) + .4(12,000) = $11,600 Model B d2 1 3 d3 EMV = .4(6,000) + .2(16,000) + .4(21,000) = $14,000 Model C 4

Decision Tree with Tree Plan Open ‘tree164e.xla’ and enable Macro first.

Decision Tree with Tree Plan

Decision Tree with Tree Plan

Decision Tree with Tree Plan

TOM BROWN INVESTMENT DECISION Tom Brown has inherited $1000. He has to decide how to invest the money for one year. A broker has suggested five potential investments. Gold Junk Bond Growth Stock Certificate of Deposit Stock Option Hedge

TOM BROWN The return on each investment depends on the (uncertain) market behavior during the year. Tom would build a payoff table to help make the investment decision

The Payoff Table DJA is down more than 800 points DJA moves within [-300,+300] DJA is up [+300,+1000] DJA is up more than1000 points

Task Evaluate each investment alternative using: Maximax approach Maximin approach Minimax Regret approach EV approach And construct Decision Tree for EV approach Calculate EVPI

QUESTIONS

Review Questions: 1. What is meant by ‘decision-making under uncertainty’? 2. Why should sequential decisions be considered differently from a series of separate decisions? 3. How can you identify the best decisions in a decision tree?