Decision Analysis Chapter Topics Components of Decision Making Decision Making without Probabilities Decision Making with Probabilities Decision Analysis with Additional Information Utility
Decision Analysis Components of Decision Making A state of nature is an actual event that may occur in the future. A payoff table is a means of organizing a decision situation, presenting the payoffs from different decisions given the various states of nature. Payoff Table Decision Analysis Decision Making Without Probabilities
Decision Analysis Decision Making without Probabilities Decision-Making Criteria maximax maximin minimax minimax regret Hurwicz equal likelihood Decision Making without Probabilities Maximax Criterion In the maximax criterion the decision maker selects the decision that will result in the maximum of maximum payoffs; an optimistic criterion.
Decision Making without Probabilities Maximin Criterion In the maximin criterion the decision maker selects the decision that will reflect the maximum of the minimum payoffs; a pessimistic criterion. Decision Making without Probabilities Minimax Regret Criterion Regret is the difference between the payoff from the best decision and all other decision payoffs. The decision maker attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret.
Decision Making without Probabilities Hurwicz Criterion The Hurwicz criterion is a compromise between the maximax and maximin criterion. A coefficient of optimism, α, is a measure of the decision maker s optimism. The Hurwicz criterion multiplies the best payoff by α and the worst payoff by 1- α., for each decision, and the best result is selected. Decision Values Apartment building $50,000(.4) + 30,000(.6) = 38,000 Office building $100,000(.4) - 40,000(.6) = 16,000 Warehouse $30,000(.4) + 10,000(.6) = 18,000 Decision Making without Probabilities Equal Likelihood Criterion 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
Decision Making without Probabilities Summary of Criteria Results 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 Decision (Purchase) Maximax Office building Maximin Apartment building Minimax regret Apartment building Hurwicz Apartment building Equal likelihood Apartment building Decision Making without Probabilities Solution with QM for Windows (1 of 3)
Decision Making without Probabilities Solution with QM for Windows (2 of 3) Decision Making without Probabilities Solution with QM for Windows (3 of 3)
Decision Making without Probabilities Solution with Excel Decision Making with Probabilities Expected Value Expected value is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence. 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
Decision Making with Probabilities 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 Value Problems Solution with Excel and Excel QM (1 of 2) Expected Value Problems Solution with Excel and Excel QM (2 of 2)
Decision Making with Probabilities Expected Value of Perfect Information 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. Decision Making with Probabilities EVPI Example (1 of 2)
Decision Making with Probabilities EVPI Example (2 of 2) Decision with perfect information: $100,000(.60) + 30,000(.40) = $72,000 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 Decision Making with Probabilities EVPI with QM for Windows
Decision Making with Probabilities Decision Trees (1 of 4) A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches). Decision Making with Probabilities Decision Trees (2 of 4)
Decision Making with Probabilities Decision Trees (3 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. Decision Making with Probabilities Decision Trees (4 of 4)
Decision Making with Probabilities Decision Trees with QM for Windows Decision Making with Probabilities Decision Trees with Excel and TreePlan (1 of 4)
Decision Making with Probabilities Decision Trees with Excel and TreePlan (2 of 4) Decision Making with Probabilities Decision Trees with Excel and TreePlan (3 of 4)
Decision Making with Probabilities Decision Trees with Excel and TreePlan (4 of 4) Decision Making with Probabilities Sequential Decision Trees (1 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. Real estate investment example modified to encompass a ten-year period in which several decisions must be made:
Decision Making with Probabilities Sequential Decision Trees (2 of 4) Decision Making with Probabilities Sequential Decision Trees (3 of 4) Decision is to purchase land; highest net expected value ($1,160,000). Payoff of the decision is $1,160,000.
Decision Making with Probabilities Sequential Decision Trees (4 of 4) Sequential Decision Tree Analysis Solution with QM for Windows
Sequential Decision Tree Analysis Solution with Excel and TreePlan Decision Analysis 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 real estate investment example, using expected value criterion, best decision was to purchase office building with expected value of $444,000, and EVPI of $28,000.
Decision Analysis 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. Economic analyst provides additional information for 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 Decision Analysis 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 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
Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (1 of 4) Decision tree with posterior probabilities differ from earlier versions in that: Two new branches at beginning of tree represent report outcomes. Probabilities of each state of nature are posterior probabilities from Bayes rule. Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (2 of 4)
Decision Analysis 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 Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (4 of 4)
Decision Analysis with Additional Information Computing Posterior Probabilities with Tables Decision Analysis with Additional Information Computing Posterior Probabilities with Excel
Decision Analysis 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 Decision Analysis with Additional Information Utility (1 of 2)
Decision Analysis 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 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 low-probability disaster. Risk takers take a chance for a bonanza on a very low-probability event in lieu of a sure thing. Decision Analysis Example Problem Solution (1 of 9) 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
Decision Analysis 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 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. Decision Analysis Example Problem Solution (3 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
Decision Analysis Example Problem Solution (4 of 9) Minimax Regret Decision: Expand Decisions Maximum Regrets Expand $500,000 (minimum) Status quo 650,000 Sell 980,000 Hurwicz (α =.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 Decision Analysis Example Problem Solution (5 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
Decision Analysis Example Problem Solution (6 of 9) 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 = 1,300,000(.7) + 500,000(.3) = $1,060,000 EV without perfect information = $1,300,000(.7) - 150,000(.3) = $865,000 EVPI = $1.060,000-865,000 = $195,000 Decision Analysis Example Problem Solution (7 of 9) Step 4 (part d): Develop a decision tree.
Decision Analysis 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 Decision Analysis Example Problem Solution (9 of 9) Step 6 (part f): Decision tree analysis.