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Chap 21-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 21 Statistical Decision Theory Statistics for Business and Economics 6 th Edition

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-2 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-3 Steps in Decision Making List Alternative Courses of Action Choices or actions List States of Nature Possible events or outcomes Determine ‘Payoffs’ Associate a Payoff with Each Event/Outcome combination Adopt Decision Criteria Evaluate Criteria for Selecting the Best Course of Action

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-4 List Possible Actions or Events Payoff TableDecision Tree Two Methods of Listing

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-5 Payoff Table Form of a payoff table M ij is the payoff that corresponds to action a i and state of nature s j Actions States of nature s1s1 s2s2...sHsH a1a2...aKa1a2...aK M 11 M 21. M K1 M 12 M 22. M K M 1H M 2H. M KH

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-6 Payoff Table Example A payoff table shows actions (alternatives), states of nature, and payoffs Investment Choice (Action) Profit in $1,000’s (States of nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-7 Decision Tree Example Large factory Small factory Average factory Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Payoffs

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-8 Decision Making Overview No probabilities known Probabilities are known Decision Criteria Nonprobabilistic Decision Criteria: Decision rules that can be applied if the probabilities of uncertain events are not known * maximin criterion minimax regret criterion

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-9 The Maximin Criterion Consider K actions a 1, a 2,..., a K and H possible states of nature s 1, s 2,..., s H Let M ij denote the payoff corresponding to the i th action and j th state of nature For each action, find the smallest possible payoff and denote the minimum M 1 * where More generally, the smallest possible payoff for action a i is given by Maximin criterion: select the action a i for which the corresponding M i * is largest (that is, the action with the greatest minimum payoff)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Maximin Example Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory Minimum Profit The maximin criterion 1.For each option, find the minimum payoff

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Maximin Solution Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory Minimum Profit The maximin criterion 1.For each option, find the minimum payoff 2.Choose the option with the greatest minimum payoff 2. Greatest minimum is to choose Small factory (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Regret or Opportunity Loss Suppose that a payoff table is arranged as a rectangular array, with rows corresponding to actions and columns to states of nature If each payoff in the table is subtracted from the largest payoff in its column... ... the resulting array is called a regret table, or opportunity loss table

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Minimax Regret Criterion Consider the regret table For each row (action), find the maximum regret Minimax regret criterion: Choose the action corresponding to the minimum of the maximum regrets (i.e., the action that produces the smallest possible opportunity loss)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Opportunity Loss Example Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory The choice “Average factory” has payoff 90 for “Strong Economy”. Given “Strong Economy”, the choice of “Large factory” would have given a payoff of 200, or 110 higher. Opportunity loss = 110 for this cell. Opportunity loss (regret) is the difference between an actual payoff for a decision and the optimal payoff for that state of nature Payoff Table

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Opportunity Loss Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory (continued) Investment Choice (Alternatives) Opportunity Loss in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory Payoff Table Opportunity Loss Table

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Minimax Regret Example Investment Choice (Alternatives) Opportunity Loss in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory Opportunity Loss Table The minimax regret criterion: 1.For each alternative, find the maximum opportunity loss (or “regret”) 1. Maximum Op. Loss

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Minimax Regret Example Investment Choice (Alternatives) Opportunity Loss in $1,000’s (States of Nature) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory Opportunity Loss Table The minimax regret criterion: 1.For each alternative, find the maximum opportunity loss (or “regret”) 2.Choose the option with the smallest maximum loss 1. Maximum Op. Loss Smallest maximum loss is to choose Average factory (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Decision Making Overview No probabilities known Probabilities are known Decision Criteria * Probabilistic Decision Criteria: Consider the probabilities of uncertain events and select an alternative to maximize the expected payoff of minimize the expected loss maximize expected monetary value

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Payoff Table Form of a payoff table with probabilities Each state of nature s j has an associated probability P i Actions States of nature s 1 (P 1 ) s 2 (P 2 )...s H (P H ) a1a2...aKa1a2...aK M 11 M 21. M K1 M 12 M 22. M K M 1H M 2H. M KH

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Expected Monetary Value (EMV) Criterion Consider possible actions a 1, a 2,..., a K and H states of nature Let M ij denote the payoff corresponding to the i th action and j th state and P j the probability of occurrence of the j th state of nature with The expected monetary value of action a i is The Expected Monetary Value Criterion: adopt the action with the largest expected monetary value

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Expected Monetary Value Example The expected monetary value is the weighted average payoff, given specified probabilities for each state of nature Investment Choice (Alternatives) Profit in $1,000’s (States of Nature) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory Suppose these probabilities have been assessed for these states of nature

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Investment Choice (Action) Profit in $1,000’s (States of nature) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory Example: EMV (Average factory) = 90(.3) + 120(.5) + (-30)(.2) = 81 Expected Values (EMV) Maximize expected value by choosing Average factory (continued) Payoff Table: Goal: Maximize expected monetary value Expected Monetary Value Solution

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Decision Tree Analysis A Decision tree shows a decision problem, beginning with the initial decision and ending will all possible outcomes and payoffs Use a square to denote decision nodes Use a circle to denote uncertain events

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Add Probabilities and Payoffs Large factory Small factory Decision Average factory States of nature Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy (continued) PayoffsProbabilities (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Fold Back the Tree Large factory Small factory Average factory Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2) EMV=200(.3)+50(.5)+(-120)(.2)= 61 EMV=90(.3)+120(.5)+(-30)(.2)= 81 EMV=40(.3)+30(.5)+20(.2)= 31

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Make the Decision Large factory Small factory Average factory Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy Strong Economy Stable Economy Weak Economy (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2) EV= 61 EV= 81 EV= 31 Maximum EMV= 81

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Bayes’ Theorem Let s 1, s 2,..., s H be H mutually exclusive and collectively exhaustive events, corresponding to the H states of nature of a decision problem Let A be some other event. Denote the conditional probability that s i will occur, given that A occurs, by P(s i |A), and the probability of A, given s i, by P(A|s i ) Bayes’ Theorem states that the conditional probability of s i, given A, can be expressed as In the terminology of this section, P(s i ) is the prior probability of s i and is modified to the posterior probability, P(s i |A), given the sample information that event A has occurred

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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(s 1 ), P(s 2 ),..., P(s H ) are used For each possible state of nature, s i, find the difference, W i, 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 s i will occur

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Expected Value of Perfect Information, EVPI The expected value of perfect information (EVPI) is (continued) 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Expected Value Under Certainty Expected value under certainty = expected value of the best decision, given perfect information Investment Choice (Action) Profit in $1,000’s (Events) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory Example: Best decision given “Strong Economy” is “Large factory” Value of best decision for each event:

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Expected Value Under Certainty Investment Choice (Action) Profit in $1,000’s (Events) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory (continued) Now weight these outcomes with their probabilities to find the expected value: 200(.3)+120(.5)+20(.2) = 124 Expected value under certainty

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Expected Value of Perfect Information Expected Value of Perfect Information (EVPI) EVPI = Expected profit under certainty – Expected monetary value of the best decision so: EVPI = 124 – 81 = 43 Recall: Expected profit under certainty = 124 EMV is maximized by choosing “Average factory”, where EMV = 81 (EVPI is the maximum you would be willing to spend to obtain perfect information)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Bayes’ Theorem Example Stock Choice (Action) Percent Return (Events) Strong Economy (.7) Weak Economy (.3) Stock A Stock B 148 Consider the choice of Stock A vs. Stock B Expected Return: Stock A has a higher EMV

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Permits revising old probabilities based on new information New Information Revised Probability Prior Probability Bayes’ Theorem Example (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Additional Information: Economic forecast is strong economy When the economy was strong, the forecaster was correct 90% of the time. When the economy was weak, the forecaster was correct 70% of the time. Prior probabilities from stock choice example F 1 = strong forecast F 2 = weak forecast E 1 = strong economy = 0.70 E 2 = weak economy = 0.30 P(F 1 | E 1 ) = 0.90 P(F 1 | E 2 ) = 0.30 (continued) Bayes’ Theorem Example

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Revised Probabilities (Bayes’ Theorem) (continued) Bayes’ Theorem Example

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap EMV with Revised Probabilities EMV Stock A = 25.0 EMV Stock B = Revised probabilities PiPi EventStock Ax ij P i Stock Bx ij P i.875strong weak Σ = 25.0 Σ = Maximum EMV

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Expected Value of Sample Information, EVSI Suppose there are K possible actions and H states of nature, s 1, s 2,..., s H The decision-maker may obtain sample information. Let there be M possible sample results, A 1, A 2,..., A M 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:

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap For each possible sample result, A i, find the difference, V i, 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 A i was observed Expected Value of Sample Information, EVSI (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Utility 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Utility 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 (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Three Types of Utility Curves Utility $$$ Risk Aversion Risk SeekerRisk-Neutral

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The Expected Utility Criterion Consider K possible actions, a 1, a 2,..., a K and H states of nature. Let U ij denote the utility corresponding to the i th action and j th state and P j the probability of occurrence of the j th state of nature Then the expected utility, EU(a i ), of the action a i is 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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

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