Statistics for Business and Economics 7th Edition Chapter 18 Statistical Decision Theory Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-1 Chapter Goals After completing this chapter, you should be able to:  Describe basic features of decision making  Construct a payoff table and an opportunity-loss table  Define and apply the expected monetary value criterion for decision making  Compute the value of sample information  Describe utility and attitudes toward risk Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-2 18.1  Steps in Decision Making List Alternative Courses of Action   List States of Nature   Possible events or outcomes Determine ‘Payoffs’   Choices or actions Associate a Payoff with Each Event/Outcome combination Adopt Decision Criteria  Evaluate Criteria for Selecting the Best Course of Action Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-3 List Possible Actions or Events Two Methods of Listing Payoff Table Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Decision Tree Ch 18-4 Payoff Table  Form of a payoff table  Mij is the payoff that corresponds to action and state of nature sj States of nature Actions s1 s2 sH a1 M11 M12 M1H a2 M21 M22 aK MK1 MK2 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall M2H MKH Ch 18-5 Payoff Table Example A payoff table shows actions (alternatives), states of nature, and payoffs Investment Choice (Action) Large factory Average factory Small factory Profit in $1,000’s (States of nature) Strong Economy Stable Economy Weak Economy 200 90 40 50 120 30 -120 -30 20 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-6 Decision Tree Example Large factory Average factory Small factory Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Strong Economy 200 Stable Economy 50 Weak Economy -120 Strong Economy 90 Stable Economy 120 Weak Economy -30 Strong Economy 40 Stable Economy 30 Weak Economy 20 Payoffs Ch 18-7 18.2 Decision Making Overview Decision Criteria No probabilities known * Probabilities are known Nonprobabilistic Decision Criteria: Decision rules that can be applied if the probabilities of uncertain events are not known  maximin criterion  minimax regret criterion Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-8 The Maximin Criterion  Consider K actions a1, a2, , aK and H possible states of nature s1, s2, , sH  Let Mij denote the payoff corresponding to the ith action and jth state of nature  For each action, find the smallest possible payoff and denote the minimum M1* where * M1 Min(M11,M12 , ,M1H )  More generally, the smallest possible payoff for action is given by Mi* (M11,M12 , ,M1H )  Maximin criterion: select the action for which the corresponding Mi* is largest (that is, the action with the greatest minimum payoff) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-9 Maximin Example The maximin criterion For each option, find the minimum payoff Investment Choice (Alternatives) Large factory Average factory Small factory Profit in $1,000’s (States of Nature) Strong Economy 200 90 40 Stable Economy 50 120 30 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Weak Economy -120 -30 20 Minimum Profit -120 -30 20 Ch 18-10 Bayes’ Theorem Example (continued) P(F1 | E1 ) .9 , P(F1 | E ) .3 P(E1 ) .7 , P(E ) .3  Revised Probabilities (Bayes’ Theorem) P(E1 )P(F1 | E1 ) (.7)(.9) P(E1 | F1 )   .875 P(F1 ) (.7)(.9)  (.3)(.3) P(E )P(F1 | E ) P(E2 | F1 )  .125 P(F1 ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-31 EMV with Revised Probabilities Pi Event Stock A xijPi Stock B xijPi 875 strong 30 26.25 14 12.25 125 weak -10 -1.25 1.00 Σ = 25.0 Revised probabilities Σ = 11.25 EMV Stock B = 11.25 EMV Stock A = 25.0 Maximum EMV Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-32 Expected Value of Sample Information, EVSI  Suppose there are K possible actions and H states of nature, s1, s2, , sH  The decision-maker may obtain sample information Let there be M possible sample results, A1, A2, , AM  The expected value of sample information is obtained as follows:   Determine which action will be chosen if only the prior probabilities were used Determine the probabilities of obtaining each sample result: P(A i ) P(A i | s1 )P(s1 )  P(A i | s2 )P(s )    P(A i | sH )P(sH ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-33 Expected Value of Sample Information, EVSI (continued)  For each possible sample result, Ai, find the difference, Vi, between the expected monetary value for the optimal action and that for the action chosen if only the prior probabilities are used  This is the value of the sample information, given that Ai was observed EVSI P(A1 )V1  P(A )V2    P(A M )VM Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-34 Expected Value of Perfect Information, EVPI Perfect information corresponds to knowledge of which state of nature will arise  To determine the expected value of perfect information:    Determine which action will be chosen if only the prior probabilities P(s1), P(s2), , P(sH) are used For each possible state of nature, si, find the difference, Wi, between the payoff for the best choice of action, if it were known that state would arise, and the payoff for the action chosen if only prior probabilities are used This is the value of perfect information, when it is known that si will occur Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-35 Expected Value of Perfect Information, EVPI (continued)  The expected value of perfect information (EVPI) is EVPI P(s1 )W1  P(s2 )W2    P(sH )WH  Another way to view the expected value of perfect information Expected Value of Perfect Information EVPI = Expected monetary value under certainty – expected monetary value of the best alternative Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-36 Expected Value Under Certainty  Expected value under certainty = expected value of the best decision, given perfect information Profit in $1,000’s (Events) Investment Choice (Action) Strong Economy (.3) Large factory Average factory Small Value offactory best decision for each event: 200 90 40 200 Stable Economy (.5) Weak Economy (.2) 50 120 30 -120 -30 20 120 20 Example: Best decision given “Strong Economy” is “Large factory” Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-37 Expected Value Under Certainty (continued) Profit in $1,000’s (Events) Investment Choice (Action)  Now weight these outcomes with their probabilities to find the expected value: Large factory Average factory Small factory Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) 200 90 40 50 120 30 -120 -30 20 200 200(.3)+120(.5)+20(.2) = 124 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 120 20 Expected value under certainty Ch 18-38 Expected Value of Perfect Information Expected Value of Perfect Information (EVPI) EVPI = Expected profit under certainty – Expected monetary value of the best decision Recall: Expected profit under certainty = 124 EMV is maximized by choosing “Average factory”, where EMV = 81 so: EVPI = 124 – 81 = 43 (EVPI is the maximum you would be willing to spend to obtain perfect information) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-39 18.5  Utility Analysis Utility is the pleasure or satisfaction obtained from an action  The utility of an outcome may not be the same for each individual  Utility units are arbitrary Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-40 Utility Analysis (continued)  Example: each incremental $1 of profit does not have the same value to every individual:  A risk averse person, once reaching a goal, assigns less utility to each incremental $1  A risk seeker assigns more utility to each incremental $1  A risk neutral person assigns the same utility to each extra $1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-41 Utility Utility Utility Three Types of Utility Curves $ Risk Aversion $ Risk Seeker Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall $ Risk-Neutral Ch 18-42 Maximizing Expected Utility  Making decisions in terms of utility, not $    Translate $ outcomes into utility outcomes Calculate expected utilities for each action Choose the action to maximize expected utility Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-43 The Expected Utility Criterion    Consider K possible actions, a1, a2, , aK and H states of nature Let Uij denote the utility corresponding to the i th action and jth state and Pj the probability of occurrence of the jth state of nature Then the expected utility, EU(ai), of the action is H EU(ai ) P1Ui1  P2Ui2    PHUiH  PjUij j 1  The expected utility criterion: choose the action to maximize expected utility  If the decision-maker is indifferent to risk, the expected utility criterion and expected monetary value criterion are equivalent Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-44 Chapter Summary    Described the payoff table and decision trees Defined opportunity loss (regret) Provided criteria for decision making      If no probabilities are known: maximin, minimax regret When probabilities are known: expected monetary value Introduced expected profit under certainty and the value of perfect information Discussed decision making with sample information and Bayes’ theorem Addressed the concept of utility Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18-45 ... criterion for decision making  Compute the value of sample information  Describe utility and attitudes toward risk Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 18- 2 18. 1... Expected Return: 18. 0 Stock A has a higher EMV 12.2 Ch 18- 28 Bayes’ Theorem Example (continued) Prior Probability  Permits revising old probabilities based on new information New Information Revised... Publishing as Prentice Hall Ch 18- 29 Bayes’ Theorem Example (continued) Additional Information: Economic forecast is strong economy  When the economy was strong, the forecaster was correct 90% of