Chapter 23 Decision Analysis

23.1 Introduction We discuss the problem of selecting an alternative from a set of possible decisions. The decision may or may not depend on statistical evidence. Financial considerations are explicitly included in the decision process

23.2 Decision Problem To cover several new concepts and use terminology typical to decision analysis we need to define the following: Acts States of nature Payoff table Opportunity loss Additional definitions are mentioned later.

Acts Example 23.1 A man wants to invest $1 million for one year. Three investment alternatives considered (the acts) are: a1: guaranteed income certificate paying 10% a2: bond with a coupon value of 8% a3: well diversified portfolio of stocks

States of Nature Return on investment in the diversified portfolio depends on the behavior of interest rate next year. Interest rate during the year is expected to behave along one of the following three lines: s1: Interest rate increases s2: Interest rate stay the same s3: Interest rater decrease.

Payoff Table States of Nature Acts a1 a2 a3 s1 s2 s3 100,000 -50,000 150,000 s2 80,000 90,000 s3 180,000 40,000 The entry in each cell is the payoff for a given act when a certain state of nature takes place.

Opportunity Loss Table States of Nature Acts a1 a2 a3 s1 50,000 200,000 150K-150K 150K-100K 150K-(-50K) s2 0 20,000 10,000 States of Nature Acts a1 a2 a3 s1 100,000 -50,000 150,000 s2 80,000 90,000 s3 180,000 40,000

Opportunity Loss Table States of Nature Acts a1 a2 a3 s1 50,000 200,000 S2 20,000 10,000 S3 100,000 140,000

Decision Trees When decisions are made sequentially, a decision tree is used to identify the best sequence. A decision tree consists of Branches Nodes Decision trees are used for the analysis of multistage decision problems

Decision Trees Example 23.1 – continued Draw the decision tree that represents this decision problem. $100,000 a1 A decision node a2 A state of nature is about to occur a3

Decision Trees Example 23.1 – continued Draw the decision tree that represents this decision problem. $100,000 a1 A decision node -$50,000 s1 $150,000 a2 $80,000 s2 $90,000 a3 $180,000 s3 $40,000

Expected Monetary Value Decision In many decision problems we can assign probabilities to state of nature. In theses cases decisions can be made using the expected monetary value (EMV) approach.

Expected Monetary Value Decision The procedure: For each act ai calculate the expected payoff as follows: EMV(ai) = P(s1)Payoff(ai,s1)+P(s2)Payoff(ai,s2)+… P(ai,)Payoff(ai,sk) Choose the decision a* with the most favorable expected value.

Expected Monetary Value Decision; Example Example 23.1 – continued Use the EMV criterion to find the optimal act for the investment problem presented in this example EMV(a1) = .2(100K) + .5(100K) + .3(100K) = 100K EMV(a2) = .2(-50K) + .5(80K) + .3(180K) = 84K EMV(a3) = .2(150K) + .5(90K) + .3( 40K) = 87K The optimal act a* = a1, and EMV*= 100K

Expected Opportunity Loss (EOL) Decision; Example Example 23.1 – continued Use the EOL criterion to find the optimal act for the investment problem presented in this example EMV(a1) = .2(50K) + .5(0K) + .3(80K) = 34K EMV(a2) = .2(200K) + .5(20K) + .3(0K) = 50K EMV(a3) = .2(0K) + .5(10K) + .3(140K) = 47K The optimal act a* = a1, and EOL* = 34K.

Rollback Technique for Decision trees The probabilities are added to the decision tree on the branches associated with each state of nature $100,000 A decision node a1 -$50,000 P(s1)=.2 a2 P(s2)=.5 $80,000 P(s3)=.3 $180,000 a3 $150,000 P(s1)=.2 P(s2)=.5 $90,000 $40,000 P(s3)=.3

Rollback Technique for Decision trees Calculate the expected monetary value for all the round nodes, beginning at the end of the tree $100,000 A decision node a1 (.5)($80,000) (.3)($180,000) (.2)(-$50,000) (+) (.5)($80,000) (.3)($180,000) (.2)(-$50,000) (+) (.5)($80,000) (.3)($180,000) (.2)(-$50,000) (+) (.5)($80,000) (.3)($180,000) (.2)(-$50,000) (+) a2 (.5)($80,000) (.3)($180,000) (.2)(-$50,000) (+) $84,000 a3 (.2)($150,000) (.5)($90,000) $87,000 (+) (.3)($40,000)

Rollback Technique for Decision trees Select the branch with the most favorable EMV at each square node $100,000 A decision node a1 a2 $ 84,000 a3 $ 87,000

23.3 Acquiring, Using, and Evaluating Additional Information Additional information can be acquired and incorporated in the decision process. Additional information has attendant costs. The usefulness of the additional information is weighed against its costs.

Expected payoff with Perfect Information (EPPI) EPPI is the maximum price a decision maker should be willing to pay for any information. States of Nature Acts a1 a2 a3 s1 100,000 -50,000 150,000 s2 80,000 90,000 s3 180,000 40,000 150,000 If we knew that state of nature s1 were certain to occur, (perfect information) we would decide to act according to a3.

Expected payoff with Perfect Information (EPPI) EPPI is the maximum price a decision maker should be willing to pay for any information. States of Nature Acts a1 a2 a3 s1 100,000 -50,000 150,000 s2 80,000 90,000 s3 180,000 40,000 100,000 If we knew that state of nature s2 were certain to occur, (perfect information) we would decide to act according to a1.

Expected payoff with Perfect Information (EPPI) EPPI is the maximum price a decision maker should be willing to pay for any information. If we knew that state of nature s3 were certain to occur, (perfect information) we would decide to act according to a2. States of Nature Acts a1 a2 a3 s1 100,000 -50,000 150,000 s2 80,000 90,000 s3 180,000 40,000 180,000

Expected payoff with Perfect Information (EPPI) Since the future state of nature is uncertain, we can only calculate the expected payoff with perfect information: EPPI = .2(150,000)+.5(100,000)+ .3(180,000)= = $134,000 Since the investor could expect $100,000 (EMV*) without perfect information…

Expected Value of Perfect Information (EVPI) The EVPI is calculated by EVPI = EPPI – EMV* In our example EVPI = 134,000 – 100,000 = $34,000

Decision Making with Additional Information Purchasing additional information may reduce the amount of uncertainty existing in a given decision problem. Question: What is the maximum a decision maker should be willing to pay for a given source of additional information?

Decision Making with Additional Information Example 23.1 – continued The investor can hire a consulting firm (IMC) for a fee of $5,000. To evaluate IMC’s potential contribution to his decision making capability, the investor receives “likelihood probabilities” from IMC. The form of the information is P(IMC had predicted a state of nature, given that the state of nature has occurred)

Decision Making with Additional Information Example 23.1 – continued Likelihood probabilities are conditional probabilities of the form P(Ii | sj), where Ii = an indicator variable that represents the prediction about the occurrence of si provided by the source of new information. Ii in our problem: I1= IMC predicts an interest rate increase (s1) I2= IMC predicts interest rate stay the same (s2) I3= IMC predicts an interest rate decrease (s3)

Decision Making with Additional Information Example 23.1 – continued The likelihood probability table: I1 (predict s1) I2 (predict s2) I3 (predict s3) s1 P(I1|s1)=.60 P(I2|s1)=.30 P(I3|s1)=.10 s2 P(I1|s2)=.10 P(I2|s2)=.80 P(I3|s2)=.10 s3 P(I1|s3)=.10 P(I2|s3)=.20 P(I3|s3)=.70 Interpretation: When s1 actually did occur in the past IMC correctly predicted s1 60% of the time.

How to use the additional Information It is preferable to incorporate the additional information with the decision maker’s own evaluation. In our problem: It is preferable to combine the likelihood probabilities with the decision maker’s subjective probabilities. The mechanism used is Bayes’ law.

Prior probabilities and Posterior probabilities Before IMC is contracted to provide consulting services, the estimated probabilities (prior probabilities) are: P(s1) = .2; P(s2) = .5; P(s3) = .3 To re-evaluate these probabilities when including the new information we need to find the posterior probabilities: P(s1|Ii); P(s2|Ii); P(s3|Ii)

Finding Posterior Probabilities Using a Probability Tree Let us find P(sj|I1). · P(s1 and I1)=.2(.6) =.12 P(s2 and I1=.5(.05) =.05 P(s3 and I1=.3(.1) = .03 P(I1)= .20 P(I1|s1)=.60 P(I1|s2)=. 1 P(I1|S3)=.1 · P(s1)=.2 P(s2)=.5 P(s3)=.3 ·

Finding Posterior Probabilities - a table form Calculating P(sj|I1) sj Probabilities P(sj) P(I1|sj) P(sj and I1) P(sj|I1) s1 .2 .60 .2(.60)=.12 .12/.20=.60 s2 .5 .10 .5(.10)=.05 .05/.20=.25 s3 .3 .3(.10)=.03 .03/.20=.15 P(I1)=.20

Finding Posterior Probabilities - a table form Calculating P(sj|I2) sj Probabilities P(sj) P(I2|sj) P(sj and I2) P(sj| I2) s1 .2 .30 .2(.30)=.06 .06/.52=.115 s2 .5 .80 .5(.80)=.40 .40/.52=.770 s3 .3 .20 .3(.20)=.06 P(I1)=.52

Finding Posterior Probabilities - a table form Calculating P(sj|I3) sj Probabilities P(sj) P(I3|sj) P(sj and I3) P(sj| I3) s1 .2 .10 .2(.10)=.02 .02/.28=.071 s2 .5 .5(.10)=.05 .05/.28=.179 s3 .3 .70 .3(.70)=.21 .21/.28=.750 P(I1)=.28

Using Posterior Probabilities to calculate EMV The posterior probabilities are revised prior probabilities. They replace the prior probabilities in the calculation of EMV. For each possible prediction made by the information source (IMC) we calculate EMV

Using Posterior Probabilities to calculate EMV Finding EMV(ai| I1) from the payoff table and the posterior probabilities P(sj| I1) I1 I1 I1 I1 States of Nature Acts a1 a2 a3 s1 100,000 -50,000 150,000 s2 80,000 90,000 s3 180,000 40,000 P(sj|I1) .6 .25 .15 EMV(ai|I1) $100,000 $17,000 $118,500

Using Posterior Probabilities to calculate EMV Finding EMV(ai| I2) from the payoff table and the posterior probabilities P(sj| I2) States of Nature Acts a1 a2 a3 s1 100,000 -50,000 150,000 s2 80,000 90,000 s3 180,000 40,000 P(sj|I2) .115 .770 EMV(ai|I2) $100,000 $76,550 $91,150

Using Posterior Probabilities to calculate EMV Finding EMV(ai| I3) from the payoff table and the posterior probabilities P(sj| I3) States of Nature Acts a1 a2 a3 s1 100,000 -50,000 150,000 s2 80,000 90,000 s3 180,000 40,000 P(sj|I3) .071 .179 .750 EMV(ai|I3) $100,000 $145,770 $56,760

Making a Decision Summary of the EMV calculations Actions a1 a2 a3 I1 ICM predicts Actions a1 a2 a3 I1 $100,000 $17,000 $118,500 I2 $76,550 $91,150 I3 $145,770 $56,760 Optimal Act a3 a1 a2

Preposterior Analysis Should the decision maker pay the price quoted for the additional information? We evaluate the improvement of the EMV when purchasing the new information. The added value of the EMV is then compared to the price quoted for the additional information.

Expected Value of Sampling Information (EVSI) 116,516 EMV’ = .20(118,500) + .52(100,000) + .28(145,770) = a2 a3 118,500 P(I1)=.20 a1 100,000 P(I2)=.52 Hire IMC a2 a3 Expected Value of Sample Information = EVSI = EMV’ – EMV* = 116.516 – 100,000 = $16,516 P(I3)=.28 a1 a2 145,770 Do not hire IMC a3 a1 EMV* = 100,000 a2 a3