Decision Making in English
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This chapter discusses decision making under uncertainty. It describes the basic steps in decision making as listing alternative actions, uncertain events, determining payoffs, and adopting decision criteria. It introduces payoff tables and decision trees as methods to display this information. Expected monetary value and expected opportunity loss are presented as decision criteria that aim to maximize expected payoff or minimize expected loss. The value of perfect information is defined as the expected gain from knowing the outcome with certainty compared to the best action under uncertainty. Finally, it discusses how to account for risk by considering the variability of payoffs through measures like variance and standard deviation.
Decision Making in English
- 1. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-1 Chapter 16 Decision Making Statistics for Managers Using Microsoft® Excel 4th Edition
- 2. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-2 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 value criterion for decision making Compute the value of perfect information Describe utility and attitudes toward risk
- 3. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-3 Steps in Decision Making List Alternative Courses of Action Choices or actions List Uncertain Events 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 Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-4 List Possible Actions or Events Payoff Table Decision Tree Two Methods of Listing
- 5. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-5 A Payoff Table A payoff table shows alternatives, states of nature, and payoffs Investment Choice (Action) Profit in $1,000’s (Events) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20
- 6. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-6 Sample Decision 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 Payoffs 200 50 -120 40 30 20 90 120 -30
- 7. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-7 Opportunity Loss Investment Choice (Action) Profit in $1,000’s (Events) Strong Economy Stable Economy Weak Economy Large factory Average factory Small factory 200 90 40 50 120 30 -120 -30 20The action “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 is the difference between an actual payoff for an action and the optimal payoff, given a particular event Payoff Table
- 8. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-8 Opportunity Loss 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 (continued) Investment Choice (Action) Opportunity Loss in $1,000’s (Events) Strong Economy Stable Economy Weak Economy Large factory Average factory 0 110 70 0 140 50 Payoff Table Opportunity Loss Table
- 9. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-9 Decision Criteria Expected Monetary Value (EMV) The expected profit for taking action Aj Expected Opportunity Loss (EOL) The expected opportunity loss for taking action Aj Expected Value of Perfect Information (EVPI) The expected opportunity loss from the best decision
- 10. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-10 Expected Monetary Value Solution The expected monetary value is the weighted average payoff, given specified probabilities for each event ∑= = N 1i iijPx)j(EMV Where EMV(j) = expected monetary value of action j xij = payoff for action j when event i occurs Pi = probability of event i Goal: Maximize expected value
- 11. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-11 The expected value is the weighted average payoff, given specified probabilities for each event 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 Suppose these probabilities have been assessed for these three events (continued) Expected Monetary Value Solution
- 12. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-12 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: 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 value Expected Monetary Value Solution
- 13. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-13 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
- 14. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-14 Add Probabilities and Payoffs Large factory Small factory Decision Average factory Uncertain Events 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)
- 15. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-15 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
- 16. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-16 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
- 17. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-17 Expected Opportunity Loss Solution The expected opportunity loss is the weighted average loss, given specified probabilities for each event ∑= = N 1i iijPL)j(EOL Where EOL(j) = expected monetary value of action j Lij = opp. loss for action j when event i occurs Pi = probability of event i Goal: Minimize expected opportunity loss
- 18. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-18 Expected Opportunity Loss Solution Investment Choice (Action) Opportunity Loss in $1,000’s (Events) Strong Economy (.3) Stable Economy (.5) Weak Economy (.2) Large factory Average factory Small factory 0 110 160 70 0 90 140 50 0 Example: EOL (Large factory) = 0(.3) + 70(.5) + (140)(.2) = 63 Expected Op. Loss (EOL) 63 43 93 Minimize expected op. loss by choosing Average factory Opportunity Loss Table Goal: Minimize expected opportunity loss
- 19. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-19 Value of Information Expected Value of Perfect Information, EVPI Expected Value of Perfect Information EVPI = Expected profit under certainty – expected monetary value of the best alternative (EVPI is equal to the expected opportunity loss from the best decision)
- 20. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-20 Expected Profit Under Certainty Expected profit 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:
- 21. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-21 Expected Profit 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 profit under certainty
- 22. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-22 Value of Information Solution 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)
- 23. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-23 Accounting for Variability 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, but what about risk?
- 24. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-24 Stock Choice (Action) Percent Return (Events) Strong Economy (.7) Weak Economy (.3) Stock A 30 -10 Stock B 14 8 Variance: 336.0 7.56 Calculate the variance and standard deviation for Stock A and Stock B: Expected Return: 18.0 12.2 0.336)3(.)1810()7(.)1830()X(Pμ)X(σ 22 N 1i i 2 i 2 A =−−+−=−= ∑= Example: Standard Deviation: 18.33 2.75 Accounting for Variability (continued)
- 25. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-25 Calculate the coefficient of variation for each stock: %83.101100% 0.18 33.18 100% EMV σ CV A A A =×=×= (continued) %54.22100% 2.12 75.2 100% EMV σ CV B B B =×=×= Stock A has much more relative variability Accounting for Variability
- 26. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-26 Return-to-Risk Ratio Return-to-Risk Ratio (RTRR): jσ EMV(j) RTRR(j) = Expresses the relationship between the return (expected payoff) and the risk (standard deviation)
- 27. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-27 Return-to-Risk Ratio jσ EMV(j) RTRR(j) = You might want to consider Stock B if you don’t like risk. Although Stock A has a higher Expected Return, Stock B has a much larger return to risk ratio and a much smaller CV 982.0 33.18 0.18 σ EMV(A) RTRR(A) A === 436.4 75.2 2.12 σ EMV(B) RTRR(B) B ===
- 28. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-28 Decision Making in PHStat PHStat | decision-making | expected monetary value Check the “expected opportunity loss” and “measures of valuation” boxes
- 29. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-29 Decision Making with Sample Information Permits revising old probabilities based on new information New Information Revised Probability Prior Probability
- 30. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-30 Revised Probabilities Example 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
- 31. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-31 Revised Probabilities Example Revised Probabilities (Bayes’ Theorem) 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)
- 32. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-32 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
- 33. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-33 EOL Table with Revised Probabilities EOL Stock A = 2.25 EOL Stock B = 14.00 Revised probabilities Pi Event Stock A xijPi Stock B xijPi .875 strong 0 0 16 14.00 .125 weak 18 2.25 0 0 Σ = 2.25 Σ = 14.00 Minimum EOL
- 34. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-34 Stock Choice (Action) Percent Return (Events) Strong Economy (.875) Weak Economy (.125) Stock A 30 -10 Stock B 14 8 Variance: 175.0 3.94 Calculate the variance and standard deviation for Stock A and Stock B: Expected Return: 25.0 13.25 0.175)125(.)2510()875(.)2530()X(Pμ)Xi(σ 22 N 1i i 22 A =−−+−=−= ∑= Example: Standard Deviation: 13.229 1.984 Accounting for Variability with Revised Probabilities
- 35. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-35 The coefficient of variation for each stock using the results from the revised probabilities: %92.52100% 0.25 229.13 100% EMV σ CV A A A =×=×= (continued) %97.14100% 25.13 984.1 100% EMV σ CV B B B =×=×= Accounting for Variability with Revised Probabilities
- 36. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-36 Return-to-Risk Ratio with Revised Probabilities With the revised probabilities, both stocks have higher expected returns, lower CV’s, and larger return to risk ratios 890.1 229.13 0.25 σ EMV(A) RTRR(A) A === 011.7 984.1 25.13 σ EMV(B) RTRR(B) B ===
- 37. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-37 Utility Utility is the pleasure or satisfaction obtained from an action. The utility of an outcome may not be the same for each individual.
- 38. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-38 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.
- 39. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-39 Three Types of Utility Curves Utility $ $ $ Utility Utility Risk Averter Risk Seeker Risk-Neutral
- 40. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-40 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
- 41. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 16-41 Chapter Summary Described the payoff table and decision trees Opportunity loss Provided criteria for decision making Expected monetary value Expected opportunity loss Return to risk ratio Introduced expected profit under certainty and the value of perfect information Discussed decision making with sample information Addressed the concept of utility