Decision Analysis

■ Components of Decision Making

■ Decision Making without Probabilities

■ Decision Making with Probabilities

■ Decision Analysis with Additional Information

■ Utility

Chapter Topics

Payoff Table

■ 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.

Decision AnalysisComponents of Decision Making

Decision AnalysisDecision Making Without Probabilities

Decision-Making Criteria

maximax maximin minimax

minimax regret Hurwicz equal likelihood

Decision AnalysisDecision Making without Probabilities

In the maximax criterion the decision maker selects the decision that will result in the maximum of maximum payoffs; an optimisticcriterion.

Decision Making without ProbabilitiesMaximax Criterion

In the maximin criterion the decision maker selects the decision that will reflect the maximum of the minimum payoffs; a pessimisticcriterion.

Decision Making without ProbabilitiesMaximin 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 ProbabilitiesMinimax Regret 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 ValuesApartment building $50,000(.4) + 30,000(.6) = 38,000Office building $100,000(.4) - 40,000(.6) = 16,000Warehouse $30,000(.4) + 10,000(.6) = 18,000

Decision Making without ProbabilitiesHurwicz 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 ValuesApartment building $50,000(.5) + 30,000(.5) = 40,000Office building $100,000(.5) - 40,000(.5) = 30,000Warehouse $30,000(.5) + 10,000(.5) = 20,000

Decision Making without ProbabilitiesEqual Likelihood Criterion

■ 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 buildingMaximin Apartment buildingMinimax regret Apartment buildingHurwicz Apartment buildingEqual likelihood Apartment building

Decision Making without ProbabilitiesSummary of Criteria Results

Decision Making without ProbabilitiesSolution with QM for Windows (1 of 3)

Decision Making without ProbabilitiesSolution with QM for Windows (2 of 3)

Decision Making without ProbabilitiesSolution with QM for Windows (3 of 3)

Decision Making without ProbabilitiesSolution with Excel

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) = 42,000EV(Office) = $100,000(.6) - 40,000(.4) = 44,000EV(Warehouse) = $30,000(.6) + 10,000(.4) = 22,000

Decision Making with ProbabilitiesExpected Value

■ 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) = 30,000EOL(Office) = $0(.6) + 70,000(.4) = 28,000EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000

Decision Making with ProbabilitiesExpected Opportunity Loss

Expected Value ProblemsSolution with QM for Windows

Expected Value ProblemsSolution with Excel and Excel QM (1 of 2)

Expected Value ProblemsSolution with Excel and Excel QM (2 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.

Decision Making with ProbabilitiesExpected Value of Perfect Information

Decision Making with ProbabilitiesEVPI Example (1 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 = $28,000EOL(office) = $0(.60) + 70,000(.4) = $28,000

Decision Making with ProbabilitiesEVPI Example (2 of 2)

Decision Making with ProbabilitiesEVPI with QM for Windows

A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches).

Decision Making with ProbabilitiesDecision Trees (1 of 4)

Decision Making with ProbabilitiesDecision Trees (2 of 4)

■ The expected value is computed at each probability node:EV(node 2) = .60($50,000) + .40(30,000) = $42,000EV(node 3) = .60($100,000) + .40(-40,000) = $44,000EV(node 4) = .60($30,000) + .40(10,000) = $22,000

■ Branches with the greatest expected value are selected.

Decision Making with ProbabilitiesDecision Trees (3 of 4)

Decision Making with ProbabilitiesDecision Trees (4 of 4)

Decision Making with ProbabilitiesDecision Trees with QM for Windows

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan (1 of 4)

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan (2 of 4)

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan (3 of 4)

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan (4 of 4)

Decision Making with ProbabilitiesSequential 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 ProbabilitiesSequential Decision Trees (2 of 4)

Decision Making with ProbabilitiesSequential 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 ProbabilitiesSequential Decision Trees (4 of 4)

Sequential Decision Tree AnalysisSolution with QM for Windows

Sequential Decision Tree AnalysisSolution with Excel and TreePlan

■ 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 InformationBayesian Analysis (1 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 estateinvestment decision, forming conditional probabilities:

g = good economic conditionsp = poor economic conditionsP = positive economic reportN = negative economic reportP(P⏐g) = .80 P(N⏐G) = .20P(P⏐p) = .10 P(N⏐p) = .90

Decision Analysis with Additional InformationBayesian Analysis (2 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 realestate 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 InformationBayesian Analysis (3 of 3)

Decision Analysis with Additional InformationDecision 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 InformationDecision Trees with Posterior Probabilities (2 of 4)

Decision Analysis with Additional InformationDecision 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 InformationDecision Trees with Posterior Probabilities (4 of 4)

Decision Analysis with Additional InformationComputing Posterior Probabilities with Tables

Decision Analysis with Additional Information Computing Posterior Probabilities with Excel

■ 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 InformationExpected Value of Sample Information

Decision Analysis with Additional InformationUtility (1 of 2)

Expected Cost (insurance) = .992($500) + .008(500) = $500Expected 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 with Additional InformationUtility (2 of 2)

States of Nature

Decision Good Foreign Competitive Conditions

Poor Foreign Competitive Conditions

Expand Maintain Status Quo Sell now

$ 800,000 1,300,000 320,000

$ 500,000 -150,000 320,000

Decision Analysis Example Problem Solution (1 of 9)

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.

Step 1 (part a): Determine decisions without probabilities.Maximax Decision: Maintain status quo

Decisions Maximum PayoffsExpand $800,000Status quo 1,300,000 (maximum)Sell 320,000

Maximin Decision: ExpandDecisions Minimum PayoffsExpand $500,000 (maximum)Status quo -150,000Sell 320,000

Decision Analysis Example Problem Solution (3 of 9)

Minimax Regret Decision: ExpandDecisions Maximum RegretsExpand $500,000 (minimum)Status quo 650,000Sell 980,000

Hurwicz (α = .3) Decision: ExpandExpand $800,000(.3) + 500,000(.7) = $590,000Status quo $1,300,000(.3) - 150,000(.7) = $285,000Sell $320,000(.3) + 320,000(.7) = $320,000

Decision Analysis Example Problem Solution (4 of 9)

Equal Likelihood Decision: ExpandExpand $800,000(.5) + 500,000(.5) = $650,000Status quo $1,300,000(.5) - 150,000(.5) = $575,000Sell $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) = $710,000Status quo $1,300,000(.7) - 150,000(.3) = $865,000Sell $320,000(.7) + 320,000(.3) = $320,000

Decision Analysis Example Problem Solution (5 of 9)

Expected opportunity loss decision: Maintain status quoExpand $500,000(.7) + 0(.3) = $350,000Status quo 0(.7) + 650,000(.3) = $195,000Sell $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) = $1,060,000EV without perfect information = $1,300,000(.7) - 150,000(.3) = $865,000EVPI = $1.060,000 - 865,000 = $195,000

Decision Analysis Example Problem Solution (6 of 9)

Step 4 (part d): Develop a decision tree.

Decision Analysis Example Problem Solution (7 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 (8 of 9)

Step 6 (part f): Decision tree analysis.

Decision Analysis Example Problem Solution (9 of 9)