1. Value of Information Dr. Yan Liu Department of Biomedical, Industrial & Human Factors Engineering Wright State University

2. 2 Introduction Before Acquiring New Information, We Need to Know  How reliable the information is perfect information, imperfect information  How much we should be willing to pay for the information monetary cost, additional time

3. 3 Probability and Perfect Information A piece of information is said to be perfect if it is always correct You are considering investing in a company. Before the investment, however, you want to know whether the Down Jones index will go up, which will affect the payoff of your investment, so you decide to consult a clairvoyant on this problem. Let A=“Dow Jones index goes up”, and A’=“clairvoyant says Dow Jones index goes up”. What about Pr(A | A’) = ? If the clairvoyant always correctly identifies the situation of Dow Jones index, then In other words, Pr (A | A’) is equal to 1 regardless of the priori probability Pr(A)

4. 4 Probability and Perfect Information What about The above conclusions indicate that after the clairvoyant with perfect information is consulted, no uncertainty remains about the event In other words, is equal to 1 regardless of the priori probability

5. 5 Expected Value of Perfect Information (EVPI) Stock Market Example An investor has some funds available to invest in one of three choices: a high-risk stock, a low-risk stock, or a savings account that pays a sure $500. If he invests in the stock, he must pay a brokerage fee of $200. If the market goes up, he will earn $1,700, $1,200 from the high-risk and low-risk stocks, respectively. If the market stays at the same level, his payoffs for the high-risk and low-risk stocks will be $300 and $400, respectively. Finally, if the market goes down, he will lose $800 with the high-risk stock but still gain $100 with the low-risk stock. The probabilities that the market goes up, stays at the same level, and goes down are 0.5, 0.3, and 0.2, respectively.

6. 6 High-Risk Stock Low-Risk Stock Savings Account Up (0.5) Flat (0.3) Down (0.2) Up (0.5) Flat (0.3) Down (0.2) $1,500 $100 -$1,000 $1,000 $200 -$100 $500 Payoff Market EMV=$580 EMV=$540 Influence Diagram Investment Decision Market Activity Payoff Decision Tree

7. 7 Now, suppose the investor can consult a clairvoyant who can reveal exactly what the market will do before making the investment decision The arrow from the Market Activity node to the decision node indicates the outcome of the chance node is known before the decision is made Down (0.2) Market Activity High-Risk Stock Low-Risk Stock Savings Account $1,500 $200 $1,000 $500 $100 Payoff Up (0.5) Flat (0.3) High-Risk Stock Low-Risk Stock Savings Account High-Risk Stock Low-Risk Stock Savings Account $500 -$100 -$1,000 $500 EVPI = EMV(with perfect information) – EMV (Without information)=1000-580=$420 Therefore, the investor should not pay more than $420 for the clairvoyant EMV=$1,000 Investment Decision Market Activity Payoff

8. 8 Expected Value of Imperfect Information (EVII) Perfect information is rarely available in real situations Stock Market Example (Cont.) Suppose the investor hires an economist who specializes in forecasting stock market trends. His economist, however, can make mistakes, and his performance given the market state is as follows. Economist's Prediction (E) True Market State (M) UpFlatDown "Up"Pr( “ Up ” |Up)=0.80Pr( “ Up ” |Flat)=0.15Pr( “ Up ” |Down)=0.20 "Flat"Pr( “ Flat ” |Up)=0.10Pr( “ Flat ” |Flat)=0.70Pr( “ Flat ” |Down)=0.20 "Down"Pr( “ Down ” |Up)=0.10Pr( “ Down ” |Flat)=0.15Pr( “ Down ” |Down)=0.60

9. 9 The arrow from the Market Activity node to the Economist’s Forecast node indicates the outcome of market activity affects the outcome of economist’s forecast Investment Decision Market Activity Economist’s Forecast Payoff

10. 10 If the economist says “Market Up” Economist’s Forecast High-Risk Stock Low-Risk Stock Savings Account $1,500 $100 -$,1000 Payoff “Up”(?) Up (?) Flat (?) Down (?) $1,000 $200 -$100 Up (?) Flat (?) Down (?) $500 Pr(E=“Up”) =? Pr(M=Up|E=“Up”) =? Pr(M=Flat|E=“Up”) =? Pr(M=Down|E=“Up”) =?

11. 11 Economist’s Forecast High-Risk Stock Low-Risk Stock Savings Account $1,500 $100 -$,1000 Payoff “Up”(0.485) Up (0.825) Flat (0.093) Down (0.082) $1,000 $200 -$100 Up (0.825) Flat (0.093) Down (0.082) $500 EMV= $1,164 EMV= $835

12. 12 If the economist says “Market Flat” Economist’s Forecast High-Risk Stock Low-Risk Stock Savings Account $1,500 $100 -$,1000 Payoff “Flat”(?) Up (?) Flat (?) Down (?) $1,000 $200 -$100 Up (?) Flat (?) Down (?) $500 Pr(E=“Flat”) =? Pr(M=Up|E=“Flat”) =? Pr(M=Flat|E=“Flat”) =? Pr(M=Down|E=“Flat”) =?

13. 13 Economist’s Forecast High-Risk Stock Low-Risk Stock Savings Account $1,500 $100 -$,1000 Payoff “Flat”(0.3) Up (0.167) Flat (0.7) Down (0.133) $1,000 $200 -$100 Up (0.167) Flat (0.7) Down (0.133) $500 EMV= $187 EMV= $293

14. 14 If the economist says “Market Down” Economist’s Forecast High-Risk Stock Low-Risk Stock Savings Account $1,500 $100 -$,1000 Payoff Down(?) Up (?) Flat (?) Down (?) $1,000 $200 -$100 Up (?) Flat (?) Down (?) $500 Economist’s Forecast High-Risk Stock Low-Risk Stock Savings Account $1,500 $100 -$,1000 Payoff “Down”(?) Up (?) Flat (?) Down (?) $1,000 $200 -$100 Up (?) Flat (?) Down (?) $500 Pr(E=“Down”) =? Pr(M=Up|E=“Down”) =? Pr(M=Flat|E=“Down”) =? Pr(M=Down|E=“Down”) =?

15. 15 Economist’s Forecast High-Risk Stock Low-Risk Stock Savings Account $1,500 $100 -$,1000 Payoff “Down”(0.215) Up (0.233) Flat (0.209) Down (0.558) $1,000 $200 -$100 $500 Up (0.233) Flat (0.209) Down (0.558) EMV= -$188 EMV= $219

16. 16 Economist’s Forecast “Up” (0.485) “Flat” (0.3) “Down” (0.215) EMV= $1,164 EMV= $500 EMV= $822 EVII = EMV(with imperfect information) – EMV (Without information)=822-580=$242 Therefore, the investor should not pay more than $242 for the economist