ECE 307 Techniques for Engineering Decisions

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1 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 1 ECE 307 Techniques for Engineering Decisions 15. Value of Information George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign

2 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 2 VLUE OF INFORMTION While we cannot do away with uncertainty, there is always a desire to attempt to reduce the uncertainty about future outcomes The reduction in uncertainty about future outcomes may provide us with choices that strongly improve the chances for a good outcome We focus on the principles behind information valuation

3 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 3 SIMPLE INVESTMENT EXMPLE low-risk stock market up (0.5) flat (0.3) down (0.2) up (0.5) flat (0.3) 1, = 1, = = 1,000 1, = 1, = 200 invest $ 500 down (0.2) = 100 savings account 500 stock investment entails a brokerage fee of $200

4 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 4 NOTION OF PERFECT INFORMTION We say that an expert s information is perfect if it is always correct; we may view an expert as a clairvoyant We can quantify the value of information in a decision problem by measuring the expected value of information ( EVI )

5 NOTION OF PERFECT INFORMTION We consider the role of perfect information in the simple investment example In this decision problem, the optimal policy is to invest in high risk stock since it has the highest returns Suppose an expert predicts that the market goes up: this implies the investor still chooses the high risk stock investment and consequently the perfect information of the expert appears to be of no value George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 5

6 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 6 NOTION OF PERFECT INFORMTION On the other hand, suppose the expert predicts a market decrease or a flat market: under this information, the investor s choice is the savings account and the perfect information has value because it leads to a changed outcome with better performance than would be the case otherwise In worst case conditions: regardless of the information, we take the same decision as

7 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 7 NOTION OF PERFECT INFORMTION without the information and consequently EVI = 0; the interpretation is that we are equally well off without the expert Cases in which we have information and we change to a different optimal decision lead to EVI > 0 since we make a decision that improves the outcome using the available information

8 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 8 EVI SSESSMENT It follows that the value of information is always nonnegative, EVI 0 Indeed, perfect information removes all uncertainty, and the expected value of perfect information EVPI provides an upper bound for EVI EVI < EVPI

9 INVESTMENT EXMPLE: COMPUTTION OF EVPI bsent any expert information, a value maximi zing investor selects the high risk stock option The introduction of an expert or clairvoyant brings in perfect information since there is perfect a priori knowledge of what the market will do before the investor makes his decision and the investor s decision is based on this information George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 9

10 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 10 COMPUTTION OF EVPI We use a decision tree approach to compute EVPI by reversing the decision and uncertainty order: we view the value of information in an a priori sense and define EVPI = E {decision with perfect information} E {decision without additional information}

11 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 11 COMPUTTION OF EVPI low-risk stock EMV = 540 up (0.5) flat (0.3) down (0.2) up (0.5) flat (0.3) down (0.2) savings account 500 high-risk stock 1500 low-risk stock 1000 consult clairvoyant EMV = 1000 market flat (0.3) savings account high-risk stock low-risk stock savings account high-risk stock 1000 low-risk stock 100 savings account 500

12 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 12 COMPUTTION OF EVPI For the investment problem, EVPI = 1, = 420 We may view EVPI to represent the maximum amount that the investor is willing to pay the expert for the perfect information resulting in the improved outcome

13 EXPECTED VLUE OF IMPERFECT INFORMTION In practice, we cannot obtain perfect information; rather, the information is imperfect since there are no clairvoyants We evaluate the expected value of imperfect information, EVII For example, we engage an economist to fore cast the future stock market trends; the economist s forecasts constitute imperfect information: the track record based on past performance is George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 13

14 EXPECTED VLUE OF IMPERFECT INFORMTION conditioning event economist s prediction actual market state up flat down up flat down P{ flat market is flat } conditional probabilities George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 14

15 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 15 EVII SSESSMENT We use the decision tree approach to compute EVII For the decision tree, we evaluate probabilities with ayes theorem For the imperfect information, we define the r.v.

16 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 16 and the forecast r.v. EVII SSESSMENT We have no knowledge of the probabilities of the forecast r.v.; all we know is the prior probabilities of given

17 EVII COMPUTTION: INCOMPLETE DECISION TREE low-risk stock EMV = 580 consult the economist market activity up (0.5) 1500 flat (0.3) 100 down (0.2) 1000 up (0.5) 1000 flat (0.3) 200 down (0.2) 100 savings account 500 economist says market up (?) high-risk stock low-risk stock savings account market activity high-risk stock flat (?) down (?) 100 economist s 1000 economist says forecast up (?) market flat 1000 low-risk stock flat (?) 200 (?) down (?) 100 savings account 500 up (?) 1500 high-risk stock flat (?) 100 economist says down (?) 1000 market down up (?) 1000 low-risk stock flat (?) 200 (?) down (?) 100 savings account George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 17 up flat down up flat down up (?) (?) (?) (?) (?) (?) (?)

18 COMPUTTION OF REVERSE CONDITIONL PROILITIES We flip the probabilities in this way George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 18

19 EVII COMPUTTION: FLIPPING THE PROILITY TREE actual market performance market flat (0.3) economist s forecast market up (0.80) market flat (0) market down (0) market up (5) market flat (0.70) market down (5) market up (0.20) market flat (0.20) market down (0.60) what we have conditional probabilities with the conditioning on the actual market performance economist s forecast market flat (?) actual market performance market up (?) market flat (?) market down (?) market up (?) market flat (?) market down (?) market up (?) market flat (?) market down (?) what we need conditional probabilities with the conditioning on the economists forecast George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 19

20 POSTERIOR PROILITIES posterior probability for: economist s prediction market up market flat market down up conditional probabilities on economists forecast flat down George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 20

21 George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 21 EVII COMPUTTION We use conditional probabilities in the table to build the posterior probabilities For example We then compute

22 EXPECTED VLUE OF IMPERFECT INFORMTION market activity up (0.5) low-risk flat (0.3) stock EMV = 540 down (0.2) savings account consult economist EMV=822 flat (0.3) down (0.2) up (0.5) economist says market up (0.485) economist says market flat (0.300) economist says market down (0.215) high-risk stock EMV=1164 low-risk stock EMV=835 savings account high-risk stock EMV=187 low-risk stock EMV=293 savings account high-risk stock EMV=188 low-risk stock EMV=219 market activity up (0.8247) flat (0.0928) down (0.0825) up (0.8247) flat (0.0928) Down (0.0825) up (667) flat (0.7000) down (333) up (667) flat (0.7000) down (333) up (0.2325) flat (0.2093) down (0.5581) up (0.2325) flat (0.2093) down (0.5581) savings account George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved

23 EVII COMPUTTION The expected mean value for the decision made with the economist information is EMV economist = 1,164(0.485) + 500(0.515) = 822 The expected mean value without information is 580 Consequently, EVII = = 242 This value represents the bound on the worth of the economist s forecast George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 23

24 EXMPLE OF VLUE OF INFORMTION We consider the following decision tree with the events at E and F as independent We perform a number of valuations of EVPI for this simple decision problem E F George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 24

25 EVPI FOR F ONLY E EMV () = EMV () = F E perfect = 5 information EMV (info. about F) = 0.7 about F 4.4 EVPI (info. about F) = EMV (info. about F) EMV () = = F = George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 25 E

26 EVPI FOR E ONLY EMV () = 3.0 E EMV () = 3.2 F perfect information about E EMV (info. about E) = 6.24 EVPI (info. about E) = EMV (info about E) EMV () = = 3.04 E = 20 = = = George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 26 F F F F

27 EMV () = 3.0 E EMV () = 3.2 F EVPI FOR OTH E ND F perfect information about E and F EMV (info. about E and F) = 6.42 EVPI (info. about E and F) = EMV (info) EMV () = = 3.22 E = 20 = = = George Gross, University of Illinois at Urbana-Champaign, ll Rights Reserved. 27 F F F F = = = = = = = =