Value of Information

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

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. 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”. If the clairvoyant always correctly identifies the situation of Dow Jones index, then What about Pr(A | A’) = ? In other words, Pr (A | A’) is equal to 1 regardless of the priori probability Pr(A)

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

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

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

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.

9. Economist’s Forecast Market Activity Investment Decision Payoff 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

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

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

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

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

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

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

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