Statistical decision theory

Statistical decision theory and all criterion with examples.

1. Statistical Decision Theory Group 5 Akash Agarwal – 2020005 Anushree Saxena – 2020014 Shubham Bhadauria – 2020020 Girish Khatri – 2020026  Meghana Mahabal Hegde – 2020038 Rachna Gupta – 2020046 Sarthak Baluni – 2020054 Vignesh Kumar R – 2020062 

2. Introduction to Statistical Decision Theory Statistical Decision Theory – Process concerned with decision making with the presence of statistical knowledge which provides further information in the process of decision making What is a Decision? It is a conclusion of a process designed to weigh the relative utilities or merits of a set of available alternatives so that the most preferred course of action can be selected for implementation Why must decisions be made? When there is limited availability of resources and certain objectives must be met, decision theory helps in making certain decisions that can help in meeting the set objectives in the most optimal way

3. Classification of Decisions General Decisions Strategic Decisions 2. Administrative Decisions Depending on Nature of Problem Programmed Decisions 2. Non-Programmed Decisions Depending on Area of Interest Political Decisions 2. Economic Decisions 3. Scientific Decisions Can also be Classified as Static Decisions 2. Dynamic Decisions

4. Phases and Steps of Decisions Making Phases of Decision Making 1. How to formulate goals and objectives, enumerate the constraints, identify the various alternatives involved, and the relevant payoffs 2. How to choose the optimal strategy given the objectives, strategies, and payoffs Steps Involved in Decision Making Clearly define the problem List all the possible alternatives (strategies) that can be considered in the decision Identify all outcomes or the states of nature for each alternative (These are not in the control of the decision-maker) Identify the payoff and construct a payoff table for each alternative and outcome combination Use a decision modelling technique to choose an alternative

5. Types of Decision-Making Environment Decision Making Under Certainty (DMUC) Decision Making Under Risk (DMUR) Decision Making Under Uncertainty (DMUU)

6. Laplace Criterion This criterion is based on the principle of insufficient reason and was developed by Thomas Bayles and supported by Simon de Laplace As there is no objective evidence, we can assign equal probabilities to each of the state of nature This subjective assumption of equal probabilities is known as Laplace criterion, or criterion of insufficient reason in management literature If there are n states of nature, each can be assigned a probability of occurrence = 1/n. Using these probabilities, we compute the expected payoff for each course of action and the action with maximum expected value is regarded as optimal

7. Illustration – Laplace Criterion A farmer wants to decide which of the three crops he should plant on his 100 Acre farm. The profit from each is dependent on the rainfall during the growing seasons. The farmer has categorized the amount of rainfall as high, medium, low. His estimated profit for each is show in the table:  Rainfall Crop A Crop B Crop C High 8000 3500 5000 Medium 4500 4500 5000 Low 2000 5000 4000 If the farmer wishes to plant only one crop, decide which will be his choice using Laplace criterion

8. Solution As Crop A has the highest EMV (Profits), it is the optimal strategy Probabilities  State of nature ( Rainfall ) Expected Monetary Value (EMV) Profits High Medium Low 1/3 1/3 1/3 Strategies Utility or Payoffs Crop A 8000 3500 5000 8000x1/3 + 3500x1/3 + 5000x 1/3= 4833 Maximum  Crop B 3500 4500 5000 3500x1/3 + 4500x1/3 + 5000x1/3= 4333 Crop C 2000 5000 4000 2000x1/3 + 5000x1/3 + 4000x 1/3= 4666 

9. Maximin Criterion  One of the criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable In maximin approach, one looks at the worst that could happen under each action and then choose the action with the largest payoff.  It is assumed that the worst will happen, and the action will be taken by choosing the best among the worst cases, so this approach means ‘best of the worst’ It is also known as the Pessimist or Conservative approach

10. Illustration – Maximin Approach Identify where the money should be invested using the Maximin/conservative approach.  Economy Alternatives Growing  Stable  Declining Bonds 40 45 5 Stock 70 30 -13 Mutual Funds 53 45 -5

11. Solution Step 1: Identify the lowest value corresponding to each decision alternative and name the column as Worst  Economy Alternatives Growing  Stable  Declining Worst Bonds  40 45 5 5 Stock  70 30 -13 -13 Mutual Funds 53 45 -5 -5

12. Solution Step 2: From the column worst, choose the maximum possible value  So, the money must be invested in Bonds   Economy Alternatives Growing  Stable  Declining Worst Bonds  40 45 5 5 Stock  70 30 -13 -13 Mutual Funds 53 45 -5 -5

13. Hurwicz Criterion The Hurwicz criterion attempts to find a middle ground between the extremes posed by the optimist and pessimist criteria. It incorporates a measure of pessimism and optimism by assigning a certain percentage weight to optimism and the balance to pessimism A weighted average can be computed for every action alternative with an alpha‐weight α , called the coefficient of realism The term Realism here means that the unbridled optimism of maximax is replaced by an attenuated optimism as denoted by the α. Note that 0 ≤ α ≤ 1 Thus, a better name for the coefficient of realism is coefficient of optimism. An α=1 implies absolute optimism ( maximax ), while an α = 0 implies absolute pessimism (maximin) Selecting a value for α simultaneously produces a coefficient of pessimism 1 ‐ α, which reflects the decision maker's aversion to risk

14. Hurwicz Criterion A Hurwicz weighted average a can now be computed for every action alternative a i in a as below:       If payoff v(a i , θ j ) represent profits or income (positive flow payoffs) then a max alternative is selected Where a max = αmax{v(a i , θ j )} + (1‐α)min{ v(a i , θ j )}       If payoff v(a i , θ j ) represent loss or cost (negative flow payoffs), then a min alternative is selected Where a min = αmin{v(a i , θ j )} + (1‐α)max{ v(a i , θ j )}    

15. Illustration 1 – Hurwicz Criterion A company should decide on the number of supplies to be purchased to meet customer needs during the holidays. The exact number of customers is unknown but is expected to belong to one of the following categories: 200, 250, 300 or 400 customers. Four supply levels are proposed, with level i   being ideal if the number of customers meets category i . Deviations from the ideal category, lead to additional costs, either because they keep extra supplies that are not needed, or because the demand is not met. The following Table shows the payoff values (costs) for the supplies, where states θ 1 , θ 2 , θ 3 , θ 4 corresponds to 200, 250, 300 and 400 costumers while a 1 , a 2 , a 3 , a 4 corresponds to supply levels of 200, 250, 300 and 400 pieces.  Assume that the company’s managers have agreed to assess their level of optimism at 60% and select the best alternative.  States Alternatives θ 1 θ 2 θ 3 θ 4 a 1 5 10 18 25 a 2 8 7 8 23 a 3 21 18 12 21 a 4 30 22 19 15

16. Solution 1 Procedure:   From the question it is inferred that the coefficient of optimism α is 0.6 For every action alternative compute, the minimum and maximum payoff value For every action alternative compute, the Hurwicz weighted average using the weights α and (1-α) The payoff represents cost, which is a negative flow pay off Thus, the formula used is, a min = αmin{v(a i , θ j )} + (1‐α)max{ v(a i , θ j )} Choose the alternative with the Minimum Hurwicz value Solution:   The Hurwicz Value is calculated as follows:  For a 1 a min = 0.6 x min(a i , θ j ) + (1-0.6) x max(a i , θ j )         = 0.6 x 5 + 0.4 x 25 = 13  Similarly, for a 2 , a min = 13.4 Similarly, for a 3 , a min = 15.6 Similarly, for a 4 , a min = 21 Therefore, the best alternative to follow is a 1 .  

17. Illustration 2 – Hurwicz Criterion The following matrix gives the payoff in rupees of different strategies (alternatives) A, B , and C against conditions (events) W, X, Y and Z . Identify the decision taken under the Hurwicz criterion. The decision maker’s degree of optimism (α) being 0.7 Events Alternatives W ( θ 1 ) X ( θ 2 ) Y ( θ 3 ) Z ( θ 4 ) A (a 1 ) 4000 -100 6000 18000 B (a 2 ) 20000 5000 400 C (a 3 ) 20000 15000 -2000 1000

18. Solution 2 Procedure:   From the question it is inferred that the coefficient of optimism α is 0.7 For every action alternative compute, the minimum and maximum payoff value For every action alternative compute, the Hurwicz weighted average using the weights α and ( 1-α) The payoff data is not given, so assume it is a positive flow pay off Thus, the formula used is, a max = αmax{v(a i , θ j )} + (1‐α)min{ v(a i , θ j )} Choose the alternative with the Maximum Hurwicz value Solution:   The Hurwicz Value is calculated as follows:  For a 1 : a max = 0.7 x max(a i , θ j ) + (1-0.7) x min(a i , θ j )         = 0.7 x 18000 + 0.3 x -100 = 12570  Similarly, for a 2 , a max = 14000 Similarly, for a 3 , a max = 13400 Under Hurwicz rule, alternative B is the optimal strategy as it gives highest payoff

19. Savage Criterion The decision-maker might experience regret after the decision has been made and the states of nature i.e., events occurred. Thus, the decision-maker should attempt to minimize regret before selecting a particular alternative (strategy). The Opportunity Loss = Regret Under Savage Criterion, decision making is based on opportunistic loss. Regret/Opportunity Loss 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 Thus, regret represents how much potential payoff one would forgo by selecting a particular decision alternative given that a specific state of nature will occur. Therefore, regret is often referred to as opportunity loss Under the regret approach to decision making, one would choose the decision alternative that minimizes the maximum state of regret that could occur over all possible states of nature  The basic steps involved in this criterion are:  Determine the amount of regret corresponding to each event for every alternative. The regret for the event corresponding to i th alternative is given by:  i th regret = (maximum payoff - i th payoff) for the j th event Determine the maximum regret amount for each alternative Choose the alternative which corresponds to the minimum regrets

20. Illustration -  Savage Criterion A steel manufacturing company is concerned with the possibility of a strike. It will cost an extra 20,000 to acquire an adequate stockpile. If there is a strike and the company has not stockpiled, management estimates an additional expense of 60,000 on account of lost sales. Should the company stockpile or not if it is to use Savage Criterion? Conditional Cost Table Alternatives States of Nature Strike No-Strike Stockpile -20000 -20000 Do Not Stockpile -60000

21. Solution First, we will construct the conditional regret table  Find the Maximum payoff for Strike (S 1 ) and subtract it from each payoff in S 1 Similarly, for No-Strike (S 2 ) column we find the maximum payoff and subtract it from each payoff in S 2 We get the conditional regret table Maximum regret for alternative Stockpile , is 20,000 and for Do not Stockpile it is 40,000 Therefore, company should choose alternative Stockpile, with minimax regret of 20,000 Conditional Regret Table Alternatives States of Nature Maximum of Row Strike (S 1 ) No-Strike (S 2 ) Stockpile (A 1 ) 20,000 20,000 Do Not Stockpile (A 2 ) 40,000 40,000

22. Expected Monetary Value Expected monetary value  is a statistical technique used to quantify risks and calculate the contingency reserve This technique is used in medium to high-cost projects, where the stakes are too high to risk the project failing EMV = ∑(Probability of outcome) x (Payoff of outcome) Benefit :  It helps you with a make or buy decision during the plan procurement process. Drawback :  This technique involves expert opinions to finalize the probability and impact of the risk. Therefore, personal bias may affect the result.

23. Illustration -  Expected Monetary Value  A marketing manager of an insurance company has kept complete records of the sales effort of the sales personnel. These records contain data regarding the number of insurance policies sold and net revenues received by the company as a function of four different sales strategies. The manager has constructed the conditional payoff matrix given below, based on his records. (The state of nature refers to the number of policies sold). The number within the table represents sales. Suppose you are a new salesperson and that you have access to the original records as well as the payoff matrix. Which strategy would you follow?   State of Nature N1 N2 N3 Probability 0.2 0.5 0.3 Strategies Sales S1 40 60 100 S2 60 50 90 S3 20 100 80 S4 90 30 70

24. Solution State of Nature N1 N2 N3 Probability 0.2 0.5 0.3 Strategies Sales Expected Monetary Value S1 40 60 100 40*0.2+ 60*0.5 + 100*0.3 = 68 S2 60 50 90 60*0.2 + 50*0.5 + 90*0.3 = 64 S3 20 100 80 20*0.2 + 100*0.5 + 80*0.3 = 78 S4 90 30 70 90*0.2 + 30*0.5 + 70*0.3 = 54 As the decision is to be made under risk, multiplying the probability and utility and summing them up give the expected utility for the strategy As the third strategy gives highest expected monetary value than other three, strategy three is optimal

25. Expected opportunity loss (EOL)  A statistical calculation used primarily in the business field to help determine optimal courses of action. Doing business is full of decision making. Any decision consists of a choice between two or more events. For each event, there are two or more possible courses of action that you might take. Calculating the EOL is an organized way of using a mathematical model to compare these choices and outcomes, to make the most profitable decision EOL (Expected Opportunity Loss) can be computed by multiplying the probability of each of state of nature with the appropriate loss value and adding the resulting products

26. Illustration - Expected Opportunity Loss State of nature  Probability  A1 A2 A3 A4 S1=1 0.4 1 -1 -2 S2=2 0.3 1 2 1 S3=3 0.2 1 2 3 2 S4=4 0.1 1 2 3 4 Step 1 - Find the highest value in each row (s1, s2 etc.)  Step 2 - Subtract each column in that row with the identified number Regret table State of nature  Probability  A1 A2 A3 A4 S1=1 0.4 1 2 3 S2=2 0.3 1 1 2 S3=3 0.2 2 1 1 S4=4 0.1 3 2 1 Step 3 - Multiply the probability with the respective outcomes row wise Step 4 - Add the all the figures obtained for each act, the act with the lowest E.O.L is the act with least expected loss State of nature  Probability  A1 A2 A3 A4 S1=1 0.4 0.4 0.8 1.2 S2=2 0.3 0.3 0.3 0.6 S3=3 0.2 0.4 0.2 0.2 S4=4 0.1 0.3 0.2 0.1 Expected Opportunity Loss 1 0.8 1.2 2

27. Expected Value of Perfect Information (EVPI)  The amount by which perfect information would increase our expected payoff. In decision theory, the expected value of perfect information (EVPI) is the price that one would be willing to pay in order to gain access to perfect information. It provides an upper bound on what to pay for additional information In general, the Expected Value of Perfect Information (EVPI) is computed as follows:  EVPI = | EVwPI – EMV|   Where,  EVPI = Expected value of perfect information  EVwPI = Expected value with perfect information about the states of nature  EMV = Expected Monetary Value

28. Expected Value of Perfect Information (EVPI)  Expected Value With Perfect Information ( EVwPI ) : The expected payoff of having perfect information before making a decision.       EVwPI = ∑ (Best payoff of outcome) x (Probability of outcome)   Expected Monetary Value (EMV) uses the probabilities to calculate the average payoff for each alternative.      EMV (For alternative i ) = ∑ (Probability of outcome) x (Payoff of outcome)  

29. Illustration – Expected Value of Perfect Information  Suppose we have an option of 3 plants (Large, Small and No plant) and their respective Outcomes (Demand) is divided into 3 categories High, Moderate and Low with their probability of outcomes known, we can calculate EMV,  EVwPI  and then arrive at EVPI ​ Alternatives Outcomes (Demand) High Moderate Low EMV Large plant 2,00,000 1,00,000 -1,20,000 86,000 Small plant 90,000 50,000 -20,000 48,000 No plant Probability of outcome 0.3 0.5 0.2  

30. Solution Solution: Step 1 :  Calculate EMV   EMV (For alternative  i ) = ∑ (Probability of outcome) x (Payoff of outcome)   EMV for Large plant= (2,00,000 * 0.3) + (1,00,000 * 0.5) + (-1,20,000 * 0.2) = 86,000  EMV for Small plant= (90,000 * 0.3) + (50,000 * 0.5) + (-20,000 * 0.2) = 48,000  EMV for No plant= 0  Choose the large plant, as it has highest EMV

31. Solution Step 2: Calculate  EVwPI   Payoffs in green would be chosen based on perfect information (knowing demand level)   EVwPI  = ∑ (Best payoff of outcome) x (Probability of outcome) ​ EVwPI  for large plant= (2,00,000 * 0.3) + (1,00,000 * 0.5) + (0*0.2) = 1,10,000​ Step 3: Calculate EVPI ​ EVPI = | EVwPI  – EMV| = 1,10,000 – 86,000 =24,000​ In other words, the “perfect information” increases the expected value by 24,000​ Alternatives Outcomes (Demand) High Moderate  Low Large plant 2,00,000 1,00,000 -1,20,000 Small plant 90,000 50,000 -20,000 No plant Probability of outcome 0.3 0.5 0.2

32. Thank You