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Newbold_chap21.ppt
1.
Chap 21-1 Statistics for
Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 21 Statistical Decision Theory Statistics for Business and Economics 6th Edition
2.
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
3.
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
4.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-4 List Possible Actions or Events Payoff Table Decision Tree Two Methods of Listing
5.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-5 Payoff Table Form of a payoff table Mij is the payoff that corresponds to action ai and state of nature sj Actions States of nature s1 s2 . . . sH a1 a2 . . . aK M11 M21 . . . MK1 M12 M22 . . . MK2 . . . . . . . . . . . . M1H M2H . . . MKH
6.
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 200 90 40 50 120 30 -120 -30 20
7.
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 200 50 -120 40 30 20 90 120 -30
8.
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
9.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-9 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 More generally, the smallest possible payoff for action ai is given by Maximin criterion: select the action ai for which the corresponding Mi * is largest (that is, the action with the greatest minimum payoff) ) M , , M , Min(M M 1H 12 11 * 1 ) M , , M , (M M 1H 12 11 * i
10.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-10 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 200 90 40 50 120 30 -120 -30 20 1. Minimum Profit -120 -30 20 The maximin criterion 1. For each option, find the minimum payoff
11.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-11 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 200 90 40 50 120 30 -120 -30 20 1. Minimum Profit -120 -30 20 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)
12.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-12 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
13.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-13 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)
14.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-14 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 200 90 40 50 120 30 -120 -30 20 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
15.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-15 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 200 90 40 50 120 30 -120 -30 20 (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 0 110 160 70 0 90 140 50 0 Payoff Table Opportunity Loss Table
16.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-16 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 0 110 160 70 0 90 140 50 0 Opportunity Loss Table The minimax regret criterion: 1. For each alternative, find the maximum opportunity loss (or “regret”) 1. Maximum Op. Loss 140 110 160
17.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-17 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 0 110 160 70 0 90 140 50 0 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 140 110 160 2. Smallest maximum loss is to choose Average factory (continued)
18.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-18 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
19.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-19 Payoff Table Form of a payoff table with probabilities Each state of nature sj has an associated probability Pi Actions States of nature s1 (P1) s2 (P2) . . . sH (PH) a1 a2 . . . aK M11 M21 . . . MK1 M12 M22 . . . MK2 . . . . . . . . . . . . M1H M2H . . . MKH
20.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-20 Expected Monetary Value (EMV) Criterion Consider possible actions a1, a2, . . ., aK and H states of nature Let Mij denote the payoff corresponding to the ith action and jth state and Pj the probability of occurrence of the jth state of nature with The expected monetary value of action ai is The Expected Monetary Value Criterion: adopt the action with the largest expected monetary value H 1 j ij j iH H i2 2 i1 1 i M P M P M P M P ) EMV(a 1 P H 1 j j
21.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-21 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 200 90 40 50 120 30 -120 -30 20 Suppose these probabilities have been assessed for these states of nature
22.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-22 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 200 90 40 50 120 30 -120 -30 20 Example: EMV (Average factory) = 90(.3) + 120(.5) + (-30)(.2) = 81 Expected Values (EMV) 61 81 31 Maximize expected value by choosing Average factory (continued) Payoff Table: Goal: Maximize expected monetary value Expected Monetary Value Solution
23.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-23 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
24.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-24 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) Payoffs Probabilities 200 50 -120 40 30 20 90 120 -30 (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2)
25.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-25 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 200 50 -120 40 30 20 90 120 -30 (.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
26.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-26 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 200 50 -120 40 30 20 90 120 -30 (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2) EV=61 EV=81 EV=31 Maximum EMV=81
27.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-27 Bayes’ Theorem Let s1, s2, . . ., sH 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 si will occur, given that A occurs, by P(si|A) , and the probability of A , given si , by P(A|si) Bayes’ Theorem states that the conditional probability of si, given A, can be expressed as In the terminology of this section, P(si) is the prior probability of si and is modified to the posterior probability, P(si|A), given the sample information that event A has occurred ) )P(s s | P(A ) )P(s s | P(A ) )P(s s | P(A ) )P(s s | P(A P(A) ) )P(s s | P(A A) | P(s H H 2 2 1 1 i i i i i
28.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-28 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
29.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-29 Expected Value of Perfect Information, EVPI The expected value of perfect information (EVPI) is H H 2 2 1 1 )W P(s )W P(s )W P(s EVPI (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
30.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-30 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 200 90 40 50 120 30 -120 -30 20 Example: Best decision given “Strong Economy” is “Large factory” 200 120 20 Value of best decision for each event:
31.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-31 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 200 90 40 50 120 30 -120 -30 20 200 120 20 (continued) Now weight these outcomes with their probabilities to find the expected value: 200(.3)+120(.5)+20(.2) = 124 Expected value under certainty
32.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-32 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)
33.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-33 Bayes’ Theorem Example Stock Choice (Action) Percent Return (Events) Strong Economy (.7) Weak Economy (.3) Stock A 30 -10 Stock B 14 8 Consider the choice of Stock A vs. Stock B Expected Return: 18.0 12.2 Stock A has a higher EMV
34.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-34 Permits revising old probabilities based on new information New Information Revised Probability Prior Probability Bayes’ Theorem Example (continued)
35.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-35 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 F1 = strong forecast F2 = weak forecast E1 = strong economy = 0.70 E2 = weak economy = 0.30 P(F1 | E1) = 0.90 P(F1 | E2) = 0.30 (continued) Bayes’ Theorem Example
36.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-36 Revised Probabilities (Bayes’ Theorem) 3 . ) E | F ( P , 9 . ) E | F ( P 2 1 1 1 3 . ) E ( P , 7 . ) E ( P 2 1 875 . ) 3 )(. 3 (. ) 9 )(. 7 (. ) 9 )(. 7 (. ) F ( P ) E | F ( P ) E ( P ) F | E ( P 1 1 1 1 1 1 125 . ) F ( P ) E | F ( P ) E ( P ) F | E ( P 1 2 1 2 1 2 (continued) Bayes’ Theorem Example
37.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-37 EMV with Revised Probabilities EMV Stock A = 25.0 EMV Stock B = 11.25 Revised probabilities Pi Event Stock A xijPi Stock B xijPi .875 strong 30 26.25 14 12.25 .125 weak -10 -1.25 8 1.00 Σ = 25.0 Σ = 11.25 Maximum EMV
38.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-38 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: ) ( ) | ( ) ( ) | ( ) ( ) | ( ) ( 2 2 1 1 H H i i i i s P s A P s P s A P s P s A P A P
39.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-39 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 M M 2 2 1 1 )V P(A )V P(A )V P(A EVSI Expected Value of Sample Information, EVSI (continued)
40.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-40 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
41.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-41 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)
42.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-42 Three Types of Utility Curves $ $ $ Risk Aversion Risk Seeker Risk-Neutral
43.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-43 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
44.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-44 The Expected Utility Criterion Consider K possible actions, a1, a2, . . ., aK and H states of nature. Let Uij denote the utility corresponding to the ith 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 ai 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 H 1 j ij j iH H i2 2 i1 1 i U P U P U P U P ) EU(a
45.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 21-45 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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