Chapter Topics Components of Decision Making with Additional Information Chapter 12 Utility 12-1 12-2 Overview Components of Decision Making A state of nature is an actual event that may occur in the future. Previous chapters used an assumption of certainty with regards to problem parameters. This chapter relaxes the certainty assumption A payoff table is a means of organizing a decision situation, presenting the payoffs from different decisions given the various states of nature. Two categories of decision situations: Probabilities can be assigned to future occurrences Probabilities cannot be assigned to future occurrences Table 12.1 Payoff table 12-3 12-4 Decision Making Without Probabilities Table 12.2 Payoff table for the real estate investments Figure 12.1 Decision situation with real estate investment alternatives Decision-Making Criteria maximax maximin minimax minimax regret Hurwicz equal likelihood 12-5 12-6 1
Maximax Criterion In the maximax criterion the decision maker selects the decision that will result in the maximum of maximum payoffs; an optimistic criterion. Maximin Criterion In the maximin criterion the decision maker selects the decision that will reflect the maximum of the minimum payoffs; a pessimistic criterion. Table 12.3 Payoff table illustrating a maximax decision Table 12.4 Payoff table illustrating a maximin decision 12-7 12-8 Minimax Regret Criterion Regret is the difference between the payoff from the best decision and all other decision payoffs. Example: under the Good Economic Conditions state of nature, the best payoff is $100,000. The manager s regret for choosing the Warehouse alternative is $100,000-$30,000=$70,000 Minimax Regret Criterion The manager calculates regrets for all alternatives under each state of nature. Then the manager identifies the maximum regret for each alternative. Finally, the manager attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret. Table 12.5 Regret table 12-9 Table 12.6 Regret table illustrating the minimax regret decision 12-10 Hurwicz Criterion Equal Likelihood Criterion The Hurwicz criterion is a compromise between the maximax and maximin criteria. A coefficient of optimism,, is a measure of the decision maker s optimism. The Hurwicz criterion multiplies the best payoff by and the worst payoff by 1-, for each decision, and the best result is selected. Here, = 0.4. Decision Values Apartment building $50,000(.4) + 30,000(.6) = 38,000 Office building $100,000(.4) - 40,000(.6) = 16,000 The equal likelihood ( or Laplace) criterion multiplies the decision payoff for each state of nature by an equal weight, thus assuming that the states of nature are equally likely to occur. Decision Values Apartment building $50,000(.5) + 30,000(.5) = 40,000 Office building $100,000(.5) - 40,000(.5) = 30,000 Warehouse $30,000(.5) + 10,000(.5) = 20,000 Warehouse $30,000(.4) + 10,000(.6) = 18,000 12-11 12-12 2
Summary of Criteria Results Solution with QM for Windows (1 of 3) A dominant decision is one that has a better payoff than another decision under each state of nature. The appropriate criterion is dependent on the risk personality and philosophy of the decision maker. Criterion Maximax Maximin Minimax regret Hurwicz Equal likelihood Decision (Purchase) Office building Apartment building Apartment building Apartment building Apartment building Exhibit 12.1 12-13 12-14 Solution with QM for Windows (2 of 3) Solution with QM for Windows (3 of 3) Equal likelihood weight Exhibit 12.2 Exhibit 12.3 12-15 12-16 Solution with Excel Expected Value Expected value is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence. =MIN(C7,D7) =MAX(E7,E9) =MAX(C18,D18) =MAX(F7:F9) =MAX(C7:C9)-C9 =C7*C25+D7*C26 Table 12.7 Payoff table with probabilities for states of nature =C7*0.5+D7*0.5 Exhibit 12.4 12-17 EV(Apartment) = $50,000(.6) + 30,000(.4) = $42,000 EV(Office) = $100,000(.6) - 40,000(.4) = $44,000 EV(Warehouse) = $30,000(.6) + 10,000(.4) = $22,000 12-18 3
Expected Opportunity Loss The expected opportunity loss is the expected value of the regret for each decision. The expected value and expected opportunity loss criterion result in the same decision. EOL(Apartment) = $50,000(.6) + 0(.4) = 30,000 EOL(Office) = $0(.6) + 70,000(.4) = 28,000 EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000 Expected Value Problems Solution with QM for Windows Expected values Table 12.8 Regret table with probabilities for states of nature 12-19 Exhibit 12.5 12-20 Expected Value Problems Solution with Excel and Excel QM (1 of 2) Expected Value Problems Solution with Excel and Excel QM (2 of 2) Click on Add-Ins to access the Excel QM menu Expected value for apartment building Exhibit 12.6 Exhibit 12.7 12-21 12-22 Expected Value of Perfect Information EVPI Example (1 of 2) The expected value of perfect information (EVPI) is the maximum amount a decision maker would pay for additional information. EVPI equals the expected value given perfect information minus the expected value without perfect information. EVPI equals the expected opportunity loss (EOL) for the best decision. Table 12.9 Payoff table with decisions, given perfect information 12-23 12-24 4
EVPI Example (2 of 2) Decision with perfect information: $100,000(.60) + 30,000(.40) = $72,000 EVPI with QM for Windows The expected value, given perfect information, in Cell F12 Decision without perfect information: EV(office) = $100,000(.60) - 40,000(.40) = $44,000 EVPI = $72,000-44,000 = $28,000 EOL(office) = $0(.60) + 70,000(.4) = $28,000 =MAX(E7:E9) =F12-F11 Exhibit 12.8 12-25 12-26 Decision Trees (1 of 4) Decision Trees (2 of 4) A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches). Table 12.10 Payoff table for real estate investment example Figure 12.2 Decision tree for real estate investment example 12-27 12-28 Decision Trees (3 of 4) Decision Trees (4 of 4) The expected value is computed at each probability node: EV(node 2) =.60($50,000) +.40(30,000) = $42,000 EV(node 3) =.60($100,000) +.40(-40,000) = $44,000 EV(node 4) =.60($30,000) +.40(10,000) = $22,000 Branches with the greatest expected value are selected. Figure 12.3 Decision tree with expected value at probability nodes 12-29 12-30 5
Decision Trees with QM for Windows Select node to add from Number of branches from node 1 Decision Trees with Excel and TreePlan (1 of 4) Exhibit 12.10 Add branches from node 1 to 2, 3, and 4 Exhibit 12.9 12-31 12-32 Decision Trees with Excel and TreePlan (2 of 4) Decision Trees with Excel and TreePlan (3 of 4) To create another branch, click B5, then the Decision Tree menu, and select Add Branch Invoke TreePlan from the Add Ins menu Click on cell F3, then Decision Tree Select Change to Event Node and add two new branches Exhibit 12.11 Exhibit 12.12 12-33 12-34 Decision Trees with Excel and TreePlan (4 of 4) Sequential Decision Tree Analysis Solution with QM for Windows Add numerical dollar and probability values in these cells in column H Exhibit 12.13 Cell A16 contains the expected value of $44,000 These cells contain decision tree formulas; do not type in these cells in columns E and I 12-35 Exhibit 12.14 12-36 6
Sequential Decision Trees (1 of 4) Sequential Decision Trees (2 of 4) A sequential decision tree is used to illustrate a situation requiring a series of decisions. Used where a payoff table, limited to a single decision, cannot be used. The next slide shows the real estate investment example modified to encompass a ten-year period in which several decisions must be made. Figure 12.4 Sequential decision tree 12-37 12-38 Sequential Decision Trees (3 of 4) Sequential Decision Trees (4 of 4) Expected value of apartment building is: $1,290,000-800,000 = $490,000 Expected value if land is purchased is: $1,360,000-200,000 = $1,160,000 The decision is to purchase land; it has the highest net expected value of $1,160,000. 12-39 Figure 12.5 Sequential decision tree with nodal expected values 12-40 Sequential Decision Tree Analysis Solution with Excel QM Sequential Decision Tree Analysis Solution with TreePlan Exhibit 12.15 12-41 Exhibit 12.16 12-42 7
with Additional Information Bayesian Analysis (1 of 3) Bayesian analysis uses additional information to alter the marginal probability of the occurrence of an event. In the real estate investment example, using the expected value criterion, the best decision was to purchase the office building with an expected value of $444,000, and EVPI of $28,000. Table 12.11 Payoff table for real estate investment with Additional Information Bayesian Analysis (2 of 3) A conditional probability is the probability that an event will occur given that another event has already occurred. An economic analyst provides additional information for the real estate investment decision, forming conditional probabilities: g = good economic conditions p = poor economic conditions P = positive economic report N = negative economic report P(P g) =.80 P(N G) =.20 P(P p) =.10 P(N p) =.90 12-43 12-44 with Additional Information Bayesian Analysis (3 of 3) A posterior probability is the altered marginal probability of an event based on additional information. Prior probabilities for good or poor economic conditions in the real estate decision: P(g) =.60; P(p) =.40 Posterior probabilities by Bayes rule: (g P) = P(P G)P(g)/[P(P g)p(g) + P(P p)p(p)] = (.80)(.60)/[(.80)(.60) + (.10)(.40)] =.923 Posterior (revised) probabilities for decision: P(g N) =.250 P(p P) =.077 P(p N) =.750 with Additional Information Decision Trees with Posterior Probabilities (1 of 4) Decision trees with posterior probabilities differ from earlier versions in that: Two new branches at the beginning of the tree represent report outcomes. Probabilities of each state of nature are posterior probabilities from Bayes rule. 12-45 12-46 with Additional Information Decision Trees with Posterior Probabilities (2 of 4) with Additional Information Decision Trees with Posterior Probabilities (3 of 4) EV (apartment building) = $50,000(.923) + 30,000(.077) = $48,460 EV (strategy) = $89,220(.52) + 35,000(.48) = $63,194 Figure 12.6 Decision tree with posterior probabilities 12-47 12-48 8
with Additional Information Decision Trees with Posterior Probabilities (4 of 4) with Additional Information Computing Posterior Probabilities with Tables Table 12.12 Computation of posterior probabilities Figure 12.7 Decision tree analysis for real estate investment 12-49 12-50 with Additional Information Computing Posterior Probabilities with Excel with Additional Information Expected Value of Sample Information The expected value of sample information (EVSI) is the difference between the expected value with and without information: For example problem, EVSI = $63,194-44,000 = $19,194 The efficiency of sample information is the ratio of the expected value of sample information to the expected value of perfect information: efficiency = EVSI /EVPI = $19,194/ 28,000 =.68 Exhibit 12.17 12-51 12-52 with Additional Information Utility (1 of 2) Table 12.13 Payoff table for auto insurance example with Additional Information Utility (2 of 2) Expected Cost (insurance) =.992($500) +.008(500) = $500 Expected Cost (no insurance) =.992($0) +.008(10,000) = $80 The decision should be do not purchase insurance, but people almost always do purchase insurance. Utility is a measure of personal satisfaction derived from money. Utiles are units of subjective measures of utility. Risk averters forgo a high expected value to avoid a lowprobability disaster. Risk takers take a chance for a bonanza on a very lowprobability event in lieu of a sure thing. 12-53 12-54 9
Example Problem Solution (1 of 9) A corporate raider contemplates the future of a recent acquisition. Three alternatives are being considered in two states of nature. The payoff table is below. Decision Expand Maintain Status Quo Sell now Good Foreign Competitive Conditions $ 800,000 1,300,000 320,000 States of Nature Poor Foreign Competitive Conditions $ 500,000-150,000 320,000 Example Problem Solution (2 of 9) a. Determine the best decision without probabilities using the 5 criteria of the chapter. b. Determine best decision with probabilities assuming.70 probability of good conditions,.30 of poor conditions. Use expected value and expected opportunity loss criteria. c. Compute expected value of perfect information. d. Develop a decision tree with expected value at the nodes. e. Given the following, P(P g) =.70, P(N g) =.30, P(P p) = 20, P(N p) =.80, determine posterior probabilities using Bayes rule. f. Perform a decision tree analysis using the posterior probability obtained in part e. 12-55 12-56 Example Problem Solution (3 of 9) Example Problem Solution (4 of 9) Step 1 (part a): Determine decisions without probabilities. Maximax Decision: Maintain status quo Decisions Maximum Payoffs Expand $800,000 Status quo 1,300,000 (maximum) Sell 320,000 Maximin Decision: Expand Decisions Minimum Payoffs Expand $500,000 (maximum) Status quo -150,000 Sell 320,000 Minimax Regret Decision: Expand Decisions Expand Status quo 650,000 Sell 980,000 Maximum Regrets $500,000 (minimum) Hurwicz ( =.3) Decision: Expand Expand $800,000(.3) + 500,000(.7) = $590,000 Status quo $1,300,000(.3) - 150,000(.7) = $285,000 Sell $320,000(.3) + 320,000(.7) = $320,000 12-57 12-58 Example Problem Solution (5 of 9) Example Problem Solution (6 of 9) Equal Likelihood Decision: Expand Expand $800,000(.5) + 500,000(.5) = $650,000 Status quo $1,300,000(.5) - 150,000(.5) = $575,000 Sell $320,000(.5) + 320,000(.5) = $320,000 Step 2 (part b): Determine Decisions with EV and EOL. Expected value decision: Maintain status quo Expand $800,000(.7) + 500,000(.3) = $710,000 Status quo $1,300,000(.7) - 150,000(.3) = $865,000 Sell $320,000(.7) + 320,000(.3) = $320,000 Expected opportunity loss decision: Maintain status quo Expand $500,000(.7) + 0(.3) = $350,000 Status quo 0(.7) + 650,000(.3) = $195,000 Sell $980,000(.7) + 180,000(.3) = $740,000 Step 3 (part c): Compute EVPI. EV given perfect information = EV without perfect information = 1,300,000(.7) + 500,000(.3) = $1,060,000 $1,300,000(.7) - 150,000(.3) = $865,000 EVPI = $1,060,000-865,000 = $195,000 12-59 12-60 10
Example Problem Solution (7 of 9) Step 4 (part d): Develop a decision tree. Example Problem Solution (8 of 9) Step 5 (part e): Determine posterior probabilities. P(g P) = P(P g)p(g)/[p(p g)p(g) + P(P p)p(p)] = (.70)(.70)/[(.70)(.70) + (.20)(.30)] =.891 P(p P) =.109 P(g N) = P(N g)p(g)/[p(n g)p(g) + P(N p)p(p)] = (.30)(.70)/[(.30)(.70) + (.80)(.30)] =.467 P(p N) =.533 12-61 12-62 Example Problem Solution (9 of 9) Step 6 (part f): Decision tree analysis. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. 12-63 12-64 11