CHAPTER-4 DECISION THEORY mmmmmmmmmmmmmm.pptx

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1. Chapter 6 Decision Theory 7/1/2019 Decision Theory 1

2. Decision Theory Making a decision requires enumeration of feasible and viable alternatives , the consequences associated with different alternatives , and measure of effectiveness by which the most preferred alternative is identified . Decision theory provides an analytical and systematic approach to the study of decision making. It provides a method of natural decision making wherein data concerning the occurrence of different outcomes may be evaluated to enable the decision maker to identify suitable alternative. 7/1/2019 2 Decision Theory

3. cont’d Decision models useful in helping decision makers make the best possible decisions are classified according to the degree of certainty . The scale of certainty can range from complete certainty to complete uncertainty . The region which falls between this two is decision making under risk or probabilistic problems. 7/1/2019 3 Decision Theory

4. Some Important Terminologies List of alternatives: are a set of mutually exclusive and collectively exhaustive decisions that are available to the decision maker. Two or more alternatives are said to be mutually exclusive alternative if they cannot occur simultaneously. States of nature: the set of possible future conditions, or events, beyond the control of the decision maker. Payoffs: the payoffs might be profits, revenues, costs, or other measures of value. The number of payoffs depends on the number of alternative/state of nature combination. 7/1/2019 Decision Theory 4

5. Degree of certainty: the approach often used by a decision maker depends on the degree of certainty that exists. one extreme is complete certainty and the other is complete uncertainty. Between these two extreme there is a risk. Decision criteria: the decision maker’s attitude toward the decision as well as the degree of certainty that surrounds a decision. Example; maximize the expected payoffs. 7/1/2019 Decision Theory 5

6. Some decision makers are more optimistic (wants to maximize gains) and some decision makers are pessimistic (protecting against large losses). The payoff table ; a payoff table is a device that a decision maker can use to summarize information relevant to a particular decision. It includes: List of alternative Possible future state of nature The payoffs associated with each of the alternative/state of nature combination. Probabilities 7/1/2019 Decision Theory 6

7. The Payoff Table States of Nature S1 S2 S3 P1 P2 p3 Alternatives A1 V11 V12 V13 A2 V21 V22 V23 A3 V31 V32 V33 where: Ai = the ith alternative Sj = the jth states of nature Pi = probability under each state of nature Vij = the value or payoff that will be realized if alternative i is chosen and event j occurs. 7/1/2019 Decision Theory 7

8. The decision making process involves the following steps: Identification of the various possible outcomes, called state of nature or events, Ei’s for the decision problem. The events are beyond the control of the decision maker. Identification of all the course of action, Aj’s or the strategies that is available to the decision maker. The decision- maker has control over choice of these. Determination of the payoff function which describes the consequence resulting from the different combinations of the acts and events. The payoff may be designed as Vij’s . The payoff resulting from i th events and j th strategy. Choosing from among the various alternatives on the basis of some criterion, which may involve the information given in step 3 only or which may require and incorporate some additional information. 7/1/2019 Decision Theory 8

9. Example 1: suppose that a real estate developer must decide on a plan for developing a certain piece of property. After careful consideration, the developer has left with the following list of acceptable alternative. Residential proposal Commercial proposal #1 Commercial proposal #2 The main factor that will influence the profitability of the development is whether or not a shopping center built, and the size of the shopping center, if one is built. Suppose that developer views the possibilities as: No-shopping center Medium –size shopping center Large shopping center 7/1/2019 Decision Theory 9

10. States of nature No SC MS SC LS SC Alternatives R 400,000 1,600,000 1,200,000 C#1 600,000 500,000 1,400,000 C#2 -100,000 400,000 1,500,000 The following payoff table provides data about profits of the various states of nature/alternative combination in $ If the residential proposal is chosen and no shopping center is built, the developer will realize a profit of $ 400,000. If the second commercial proposal is selected and no center is built, the developer will lose $ 100,000 7/1/2019 Decision Theory 10

11. 1. Decision Making Under Certainty When decision is made under conditions of complete certainty, the attention of the decision maker is focused on the column in payoff table that corresponds to the state of nature that will occur. The decision maker then selects the alternative that yields the best payoff, given that state of nature. For example, if there is an announcement that no shopping center will be built, the developer then can focus on the first column of the payoff table. Because the commercial proposal #1 has the highest payoff in that column ($600,000), would be selected 7/1/2019 Decision Theory 11

12. 2. Decision Making Under Complete Uncertainty (Without Probabilities) The decision maker either is unable to estimate the probabilities for the occurrence of the different state of nature Approaches to decision making under complete uncertainty: Maxi-max, Maxi-min, Mini-max regret, Equal likelihood and Hurwitz, 7/1/2019 Decision Theory 12

13. Maxi mal/ maxi max (optimistic strategy) The best payoff for each alternative is defined, and the alternative with the maximum of these is the designed decision. For the above problem 7/1/2019 Decision Theory 13

14. B. Maxi-Min / Pessimistic Criteria Consists of identifying the worst (minimum) payoff for each alternative, and, then, selecting the alternative that has the best (maximum) of the worst payoffs. Many people view the maxi-min criterion as pessimistic because they believe that the decision maker must assume that the worst will occur For the previous problem: 7/1/2019 Decision Theory 14

15. C. mini-max regret For the previous problem: An approach that takes all payoffs into account. To use this approach, it is necessary to develop an opportunity loss table that reflects the difference between each payoff and the best possible payoff in a column (i.e., given a state of nature). Hence, opportunity loss amounts are found by identifying the best payoff in a column and then subtracting each of the other values in the column from that payoff. 7/1/2019 Decision Theory 15

16. For the above example: 7/1/2019 Decision Theory 16

17. C. mini-max regret For the previous problem: 7/1/2019 Decision Theory 17

18. D. Principle of Insufficient Reason/ Equal Likelihood/ Laplace States of Nature Row average S1 S2 S3 Alternatives A1 400,000 1,600,000 1,200,000 1,066,666.67 A2 600,000 500,000 1,400,000 833,333.3 A3 -100,000 400,000 1,500,000 600,000 It treats the states of nature as if each was equally likely and it focuses on the average payoff for each row, selecting the alternative that has the highest row average. 7/1/2019 Decision Theory 18

19. E. The Hurwitz Criterion The approach offers the decision maker a compromise between the maximax and the maximin criteria. Requires the decision maker to specify a degree of optimism, in the form of a coefficient of optimism α , with possible values of α ranging from 0 to 1 ( <  < 1 ). The closer the selected value of α is to 1, the more optimistic the decision maker is, and the closer the value of α is to 0, the more pessimistic the decision maker is. 7/1/2019 Decision Theory 19

20. If  = 1 , then the decision maker is said to be completely optimistic, If  = 0, then the decision maker is completely pessimistic. Given this definition, if  is coefficient of optimism, 1-  is coefficient of pessimism. The Hurwitz criterion requires that for each alternative, the maximum payoff is multiplied by  and the minimum payoff be multiplied by 1-  . 7/1/2019 Decision Theory 20

21. Example: If  = 0.4 for the above example, States of Nature S1 S2 S3 Alternatives A1 400,000 1,600,000 1,200,000 A2 600,000 500,000 1,400,000 A3 -100,000 400,000 1,500,000 A1 = (0.4x1600000) + (0.6x400000) = 880,000 A2 = (0.4x1400000) + (0.6x500000)= 860,000 A3 = (0.4x1500000) – (0.6x100000)= 540,000 7/1/2019 Decision Theory 21

22. Summary of Methods for Decision Making under Complete Uncertainty 7/1/2019 Decision Theory 22

23. 3. Decision Making Under Risk (With Probabilities) A. Expected Monetary Value (EMV)   The EMV approach provides the decision maker with a value which represents an average payoff for each alternative. The best alternative is, then, the one that has the highest EMV. The average or expected payoff of each alternative is a weighted average: 7/1/2019 Decision Theory 23

24. Cont’d Where: EMVi = The EMV for the ith alternative Pi = The probability of the ith state of nature Vij = The estimated payoff for alternative i under state of nature j. Note: the sum of the probabilities for all states of nature must be 1. Sources of probabilities: Subjective estimates, Expert opinions and Historical frequencies 7/1/2019 Decision Theory 24

25. State of Nature Expected payoff S1 S2 S3 Probability 0.20 0.50 0.30 S1 S2 S3 Alternatives A1 400000 1600000 1200000 1,240,000* max A2 600000 500000 1400000 790,000 A3 -100000 400000 1500000 630,000 For the above Example EMV (A1) = 0.20(400000) + 0.50(1600000) + 0.30(1200000) = 1,240,000,* maximum EMV (A2) = 0.20(6) + 0.50(5) + 0.30(14) = 790,000 EMV (A3) = 0.20(-1) + 0.50(4) + 0.30(15) = 630,000 7/1/2019 Decision Theory 25

26. B. Expected Opportunity Loss (EOL) The opportunity losses for each alternative are weighted by the probabilities of their respective state of nature to compute a long run average opportunity loss, and the alternative with the smallest expected loss is selected as the best choice. 7/1/2019 Decision Theory 26

27. Opportunity Loss Table States of Nature S1 S2 S3 0.2 0.5 0.3 Alternatives A1 600,000-400,000= 200,000 1,600,000-1,600,000 = 0 1,500,000-1,200,000 =300,000 A2 600,000-600,000=0 1,600,000-500,000 = 1100000 1,500,000-1,400,000 =100,000 A3 600,000-(-100,000) = 700,000 1,600,000 -400,000 = 1,200,000 1,500,000-1,500,000 = 0 EOL (A1) = 0.20(2) + 0.50(0) + 0.30(3) = 130,000 *minimum EOL (A2) = 0.20(0) + 0.50(11) + 0.30(1) = 580,000 EOL (A3) = 0.20(7) + 0.50(12) + 0.30(0) = 740,000 The EOL approach resulted in the same alternative as the EMV approach (Maximizing the payoffs is equivalent to minimizing the opportunity losses). 7/1/2019 Decision Theory 27

28. C. Expected Value of Perfect Information (EVPI) The EVPI is the measure of the difference between the certainty payoffs that could be realized under a condition involving risk. If the decision maker knows that S1 will occur, A2 would be chosen with a payoff of $6. Similarly for S2 $16 (for A1) and for S3, $15 (with A3) would be chosen. Hence, the expected payoff under certainty (EPC) would be: EPC= 0.20(600,000)+0.50(1,600,000) + 0.30(1,500,000) = 1,370,000 7/1/2019 Decision Theory 28

29. Cont’d 7/1/2019 Decision Theory 29 The difference between this figure and the expected payoff under risk (i.e., the EMV) is the expected value of perfect information. Thus: EVPI = EPC – EMV = 1,370,000 – 1,240,000 = 130,000 The EOL indicates the expected opportunity loss due to imperfect information, which is another way of saying the expected payoff that could be achieved by having perfect information.

30. Decision Trees Decision tree, like probability tree, is composed of squares, circles, and lines: The squares indicate decision points and Circles represent chance events (circles and squares are called nodes) The lines (branches) emanating from squares represent alternatives . The lines from circles represent states of nature The tree is read from right to left. 7/1/2019 Decision Theory 30

31. Cont’d 7/1/2019 Decision Theory 31

32. Decision Trees 7/1/2019 Decision Theory 32

33. Example: Pay off Table for Real Estate Investment Decision (Purchase) State of Nature Good Economic Conditions Poor Economic Conditions 0.60 0.40 Apartment Building 50,000 30,000 Office Building 100,000 -40,000 Warehouse 30,000 10,000 7/1/2019 Decision Theory 33

34. The decision tree for the above example will be: EV(node 2) = .60($ 50,000) + .40($ 30,000) = $42,000 EV(node 3) = .60($100,000) + .40($-40,000) = $ 44,000 EV(node 4) = .60($ 30,000) + .40($ 10,000) = $22,000 Good E conditions (0.60) Bad E conditions(0.40) Good E conditions (0.60) Bad E conditions(0.40) Apartment Warehouse Purchase Office building Bad E conditions(0.40) Good E conditions (0.60) 7/1/2019 Decision Theory 34

35. The research department of YES P.L.C. has recommended the marketing department to launch three different level waters. The marketing manager has to decide one of the three of water levels to be launched under the following estimated pay-off (in millions of birr) for various levels of sales Types of Water Estimated Levels of Sale (Unit) 15000 10000 5000 YES .5 Liter 30 10 10 YES – 1 Liter 40 15 5 YES – 1.5 Liter 55 20 3 Exercise What will be the marketing manager decision if I) maxi-min II) maxi-max III) Laplace IV) mini-max regret Suppose the research department has assigned probabilities of 0.2 for 15000 estimated sale, 0.5 for 10000 estimated sale and 0.3 for 5000 estimated sale . Which alternative will be chosen using expected monetary value decision criteria, and determine the expected value of perfect information for the problem 7/1/2019 Decision Theory 35

36. Thanks! 7/1/2019 Decision Theory 36