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Chap18 statistical decision theory

This chapter discusses statistical decision theory and making decisions under uncertainty. It covers constructing payoff tables and decision trees to list possible actions and outcomes. It introduces several decision criteria for selecting the best action, including maximin, minimax regret, and expected monetary value. Expected monetary value calculates the weighted average payoff using the probabilities of different outcomes. Bayes' theorem allows revising probabilities based on new sample information. The key goals are describing decision making processes and criteria for evaluating options under uncertainty.

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Chapter 18 Statistical Decision Theory Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Statistics for Business and Economics 7th Edition 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
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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-3 18.1
List Possible Actions or Events Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Payoff Table Decision Tree Two Methods of Listing Ch. 18-4
Payoff Table  Form of a payoff table  Mij is the payoff that corresponds to action ai and state of nature sj Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Actions States of nature s1 s2 . . . sH a1 a2 . . . aK M11 M21 . . . MK1 M12 M22 . . . MK2 . . . . . . . . . . . . M1H M2H . . . MKH Ch. 18-5
Payoff Table Example A payoff table shows actions (alternatives), states of nature, and payoffs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-6
Decision Tree Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-7
Decision Making Overview Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-8 18.2
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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall )M,,M,Min(MM 1H1211 * 1  )M,,M,(MM 1H1211 * i  Ch. 18-9
Maximin Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-10
Maximin Solution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 18-11
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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-12
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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-13
Opportunity Loss Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-14
Opportunity Loss Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-15
Minimax Regret Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-16
Minimax Regret Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 18-17
Decision Making Overview Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-18 18.3
Payoff Table  Form of a payoff table with probabilities  Each state of nature sj has an associated probability Pi Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Actions States of nature s1 (P1) s2 (P2) . . . sH (PH) a1 a2 . . . aK M11 M21 . . . MK1 M12 M22 . . . MK2 . . . . . . . . . . . . M1H M2H . . . MKH Ch. 18-19
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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall   H 1j ijjiHHi22i11i MPMPMPMP)EMV(a  1P H 1j j  Ch. 18-20
Expected Monetary Value Example  The expected monetary value is the weighted average payoff, given specified probabilities for each state of nature Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-21
Expected Monetary Value Solution Example: EMV (Average factory) = 90(.3) + 120(.5) + (-30)(.2) = 81 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Expected Values (EMV) 61 81 31 Maximize expected value by choosing Average factory (continued) Payoff Table: Goal: Maximize expected monetary value Ch. 18-22
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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-23
Add Probabilities and Payoffs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 200 50 -120 40 30 20 90 120 -30 (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2) Ch. 18-24
Fold Back the Tree Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-25
Make the Decision Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-26
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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall ))P(ss|P(A))P(ss|P(A))P(ss|P(A ))P(ss|P(A P(A) ))P(ss|P(A A)|P(s HH2211 iiii i    Ch. 18-27 18.4
Bayes’ Theorem Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-28
Bayes’ Theorem Example  Permits revising old probabilities based on new information Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall New Information Revised Probability Prior Probability (continued) Ch. 18-29
Bayes’ Theorem Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 18-30
Bayes’ Theorem Example  Revised Probabilities (Bayes’ Theorem) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3.)E|F(P,9.)E|F(P 2111  3.)E(P,7.)E(P 21  875. )3)(.3(.)9)(.7(. )9)(.7(. )F(P )E|F(P)E(P )F|E(P 1 111 11    125. )F(P )E|F(P)E(P )F|E(P 1 212 12  (continued) Ch. 18-31
EMV with Revised Probabilities Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 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(ss|P(A))P(ss|P(A))P(ss|P(A)P(A HHi22i11ii   Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-33
Expected Value of Sample Information, EVSI  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 MM2211 )VP(A)VP(A)VP(AEVSI   Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall (continued) 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
 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 Expected Value of Perfect Information, EVPI HH2211 )WP(s)WP(s)WP(sEVPI   Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall  The expected value of perfect information (EVPI) is (continued) Ch. 18-36
Expected Value Under Certainty  Expected value under certainty = expected value of the best decision, given perfect information Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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: Ch. 18-37
Expected Value Under Certainty  Now weight these outcomes with their probabilities to find the expected value: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) 200(.3)+120(.5)+20(.2) = 124 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 18-39
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 18.5
Utility Analysis  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 (continued) Ch. 18-41
Three Types of Utility Curves Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall $ $ $ Risk Aversion Risk Seeker 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 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 1j ijjiHHi22i11i UPUPUPUP)EU(a  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

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Overview of Chapter 18 on Statistical Decision Theory from 'Statistics for Business and Economics'.

Key objectives include decision making features, payoff tables, opportunity-loss tables, EMV application, and risk attitudes.

Outline of decision making steps: list alternatives, states of nature, determine payoffs, and evaluation criteria.

Methods to list possible actions or events through payoff tables and decision trees.

Definition and structure of a payoff table showing payoffs for different actions and states of nature.

Example of a payoff table demonstrating profits in various economic states related to investment choices.

Illustration of a decision tree corresponding to different factory sizes and economic states with payoffs.

Overview of decision criteria when probabilities are unknown including maximin and minimax regret.

Explanation of the maximin criterion for selecting actions based on the smallest possible payoffs.

Example of application of the maximin criterion to determine the best alternative investment.

Steps to find the maximin solution focusing on minimum profits for each investment choice.

Definition of the regret table created by subtracting payoffs from the best possible outcomes.

Criteria to minimize maximum regrets associated with various decision actions.

Example demonstrating opportunity loss calculations based on different investment choices and outcomes.

Continued details on calculating opportunity losses associated with various economic states.

Example showcasing the minimax regret criterion through assessment of opportunity losses.

Further application of minimax regret criterion to select the investment option with minimal loss.

Introduction to decision criteria when probabilities are known, focusing on maximizing expected monetary values.

Visualization of a payoff table that incorporates probabilities associated with different states of nature.

Definition and formula of the expected monetary value to guide decision making based on probabilities.

Example illustrating EMV calculations based on economic states and associated probabilities.

Illustration of the process to calculate expected monetary values for investment choices.

Introduction to using decision trees to visualize decision processes and outcomes.

Process of incorporating probabilities and associated payoffs into a decision tree.

Explanation of folding back the decision tree to compute expected values for alternative actions.

Selecting the optimal investment based on folded back decision tree calculations.

Definition and application of Bayes’ Theorem for updating probabilities based on new information.

Example showcasing the calculation of expected return for two stocks using Bayes' Theorem.

Discussion on adjusting prior probabilities with new information through Bayes’ Theorem.

Utilization of Bayes’ Theorem with forecasts to update probabilities for economic states.

Results from applying Bayes’ Theorem to find revised probabilities based on observed data.

Calculation of expected monetary values using updated probabilities for stock choices.

Definition of EVSI in the context of sample results affecting decision-making actions.

Process to compute the expected value of sample information including possible results.

Concept of EVPI, offering insight into benefits of perfect information on decision-making outcomes.

EVPI as the essential value of certainty versus expected monetary value in decision-making.

Calculation of expected outcomes under certainty based on perfect information scenarios.

Steps to find the expected value based on various economic outcomes and their probabilities.

Calculation showing the expected value of perfect information derived from previous analyses.

Introduction to utility in decision-making as the satisfaction derived from actions.

Explanation of how utility can vary among individuals based on risk attitudes.

Overview of different utility curves representing risk-averse, risk-seeking, and neutral attitudes.

Overview of decision-making strategies based on maximizing expected utility over monetary outcomes.

Defined expected utility with a formula for maximizing expected utility in decision making.

Recap of key topics: payoff tables, decision criteria, and utility considerations in decision making.

Chap18 statistical decision theory

  • 1.
    Chapter 18 Statistical DecisionTheory Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Statistics for Business and Economics 7th Edition Ch. 18-1
  • 2.
    Chapter Goals After completingthis 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
  • 3.
    Steps in DecisionMaking  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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-3 18.1
  • 4.
    List Possible Actionsor Events Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Payoff Table Decision Tree Two Methods of Listing Ch. 18-4
  • 5.
    Payoff Table  Formof a payoff table  Mij is the payoff that corresponds to action ai and state of nature sj Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Actions States of nature s1 s2 . . . sH a1 a2 . . . aK M11 M21 . . . MK1 M12 M22 . . . MK2 . . . . . . . . . . . . M1H M2H . . . MKH Ch. 18-5
  • 6.
    Payoff Table Example Apayoff table shows actions (alternatives), states of nature, and payoffs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-6
  • 7.
    Decision Tree Example Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-7
  • 8.
    Decision Making Overview Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-8 18.2
  • 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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall )M,,M,Min(MM 1H1211 * 1  )M,,M,(MM 1H1211 * i  Ch. 18-9
  • 10.
    Maximin Example Copyright ©2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-10
  • 11.
    Maximin Solution Copyright ©2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 18-11
  • 12.
    Regret or OpportunityLoss  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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-12
  • 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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-13
  • 14.
    Opportunity Loss Example Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-14
  • 15.
    Opportunity Loss Copyright ©2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-15
  • 16.
    Minimax Regret Example Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-16
  • 17.
    Minimax Regret Example Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 18-17
  • 18.
    Decision Making Overview Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-18 18.3
  • 19.
    Payoff Table  Formof a payoff table with probabilities  Each state of nature sj has an associated probability Pi Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Actions States of nature s1 (P1) s2 (P2) . . . sH (PH) a1 a2 . . . aK M11 M21 . . . MK1 M12 M22 . . . MK2 . . . . . . . . . . . . M1H M2H . . . MKH Ch. 18-19
  • 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall   H 1j ijjiHHi22i11i MPMPMPMP)EMV(a  1P H 1j j  Ch. 18-20
  • 21.
    Expected Monetary Value Example The expected monetary value is the weighted average payoff, given specified probabilities for each state of nature Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-21
  • 22.
    Expected Monetary Value Solution Example:EMV (Average factory) = 90(.3) + 120(.5) + (-30)(.2) = 81 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Expected Values (EMV) 61 81 31 Maximize expected value by choosing Average factory (continued) Payoff Table: Goal: Maximize expected monetary value Ch. 18-22
  • 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-23
  • 24.
    Add Probabilities andPayoffs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 200 50 -120 40 30 20 90 120 -30 (.3) (.5) (.2) (.3) (.5) (.2) (.3) (.5) (.2) Ch. 18-24
  • 25.
    Fold Back theTree Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-25
  • 26.
    Make the Decision Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-26
  • 27.
    Bayes’ Theorem  Lets1, 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall ))P(ss|P(A))P(ss|P(A))P(ss|P(A ))P(ss|P(A P(A) ))P(ss|P(A A)|P(s HH2211 iiii i    Ch. 18-27 18.4
  • 28.
    Bayes’ Theorem Example Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-28
  • 29.
    Bayes’ Theorem Example Permits revising old probabilities based on new information Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall New Information Revised Probability Prior Probability (continued) Ch. 18-29
  • 30.
    Bayes’ Theorem Example Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 18-30
  • 31.
    Bayes’ Theorem Example Revised Probabilities (Bayes’ Theorem) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3.)E|F(P,9.)E|F(P 2111  3.)E(P,7.)E(P 21  875. )3)(.3(.)9)(.7(. )9)(.7(. )F(P )E|F(P)E(P )F|E(P 1 111 11    125. )F(P )E|F(P)E(P )F|E(P 1 212 12  (continued) Ch. 18-31
  • 32.
    EMV with Revised Probabilities Copyright© 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 18-32
  • 33.
    Expected Value of SampleInformation, 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(ss|P(A))P(ss|P(A))P(ss|P(A)P(A HHi22i11ii   Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-33
  • 34.
    Expected Value of SampleInformation, EVSI  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 MM2211 )VP(A)VP(A)VP(AEVSI   Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall (continued) Ch. 18-34
  • 35.
    Expected Value of PerfectInformation, 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
  • 36.
     Another wayto 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 Expected Value of Perfect Information, EVPI HH2211 )WP(s)WP(s)WP(sEVPI   Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall  The expected value of perfect information (EVPI) is (continued) Ch. 18-36
  • 37.
    Expected Value UnderCertainty  Expected value under certainty = expected value of the best decision, given perfect information Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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: Ch. 18-37
  • 38.
    Expected Value UnderCertainty  Now weight these outcomes with their probabilities to find the expected value: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) 200(.3)+120(.5)+20(.2) = 124 Expected value under certainty Ch. 18-38
  • 39.
    Expected Value of PerfectInformation Expected Value of Perfect Information (EVPI) EVPI = Expected profit under certainty – Expected monetary value of the best decision Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 18-39
  • 40.
    Utility Analysis  Utilityis 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 18.5
  • 41.
    Utility Analysis  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 (continued) Ch. 18-41
  • 42.
    Three Types ofUtility Curves Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall $ $ $ Risk Aversion Risk Seeker Risk-Neutral Ch. 18-42
  • 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-43
  • 44.
    The Expected UtilityCriterion  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 1j ijjiHHi22i11i UPUPUPUP)EU(a  Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-44
  • 45.
    Chapter Summary  Describedthe 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