ECE 307 – Techniques for Engineering Decisions Value of Information George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved VALUE OF INFORMATION ‰ While we cannot 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 give us choices that improve chances for a good outcome ‰ We focus on the principles behind information valuation © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved SIMPLE INVESTMENT EXAMPLE market up (0.5) k c to s k ri s gh i h low-risk stock 1,700 – 200 = 1,500 flat (0.3) down (0.2) – 800 – 200 = – 1,000 up (0.5) 1,200 – 200 = 1,000 flat (0.3) 400 – 200 = down (0.2) 100 – 200 = – 100 300 – 200 = savings account 100 200 500 stock investment entails a brokerage fee of $ 200 © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved NOTION OF PERFECT INFORMATION ‰ We say that an expert’s information is perfect if it is always correct; we think of an expert as essentially a clairvoyant ‰ We can place a value on information in a decision problem by measuring the expected value of info ( EVI ) © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved NOTION OF PERFECT INFORMATION ‰ 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 have no value © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved NOTION OF PERFECT INFORMATION ‰ 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 improved results then would be the case otherwise ‰ In worst case conditions: regardless of the information, we take the same decision as © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved NOTION OF PERFECT INFORMATION without the information and consequently EVI = 0; the interpretation is that we are equally well off without an expert ‰ Cases in which we have information and in which we change the optimal decision: these lead to EVI > since we make a decision with an improved outcome using the available information © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved EVI ASSESSMENT ‰ It follows that the value of information is always nonnegative, EVI ≥ ‰ In fact, with perfect information, there is no uncertainty and the expected value of perfect information EVPI provides an upper bound for EVI EVPI ≥ EVI © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved INVESTMENT EXAMPLE: COMPUTATION OF EVPI ‰ Absent any expert information, a value – maximizing investor selects the high – risk stock investment ‰ The introduction of an expert or clairvoyant brings in perfect information since there is perfect knowledge of what the market will before the investor makes his decision and the investor’s decision is based on this information © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved COMPUTATION 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 information} © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 10 EXPECTED VALUE OF IMPERFECT INFORMATION ‰ 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; his forecasts constitute imperfect information © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 13 EXPECTED VALUE OF IMPERFECT INFORMATION conditioning event up flat down “up” 0.8 0.15 0.2 “flat” 0.1 0.7 0.2 “down” 0.1 0.15 0.6 P{ “flat”| market is flat } © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved conditional probabilities economist’s prediction true market state 14 EVII ASSESSMENT ‰ We use the decision tree approach to compute EVII ‰ For the decision tree, we evaluate probabilities using Bayes’ theorem ‰ For the imperfect information, we define with probability 0.5 ⎧ up market ⎪ M= = ⎨ flat with probability 0.3  performance ⎪ ⎩down with probability 0.2 © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 15 EVII ASSESSMENT and the forecast r.v F =  ⎧ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪⎩ "up" "flat" "down" without the knowledge of the corresponding probabilities of the two r.v.s © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 16 EVII COMPUTATION: INCOMPLETE DECISION TREE market activity up (0.5) ck to s k 80 ris = gh V hi EM low-risk stock EMV = 580 flat (0.3) down (0.2) up flat (0.5) (0.3) down (0.2) savings account market activity 1500 100 high-risk stock − 1000 1000 economist says “market up” 200 − 100 500 low-risk stock (?) up (?) flat (?) down (?) up (?) flat (?) 200 down (?) − 100 savings account consult the economist high-risk stock economist says “market flat” low-risk stock (?) flat (?) up (?) flat (?) down (?) (?) savings account high-risk stock up (?) flat (?) down (?) economist says “market down” low-risk stock up (?) flat (?) down (?) (?) savings account 100 − 1000 1000 500 up down (?) economist’s forecast 1500 1500 100 − 1000 1000 200 − 100 500 1500 100 − 1000 1000 200 − 100 500 © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 17 COMPUTATION OF REVERSE CONDITIONAL PROBABILITIES P { M = down F = "up"} =   P {F = "up" M = down} P { M = down}    [ P {F = "up" M = down} P { M = down} + P {F = "up"  P {F = "up"  M = down} P { M = up}   M = flat} P { M = flat}   + ] 0.2 ( 0.2 ) P {F = "up"} =  0.2 ( 0.2 ) + 0.5 ( 0.3 ) + 0.8 ( 0.5 ) we flip the probabilities in this way © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 18 EVII COMPUTATION: FLIPPING THE “market down” (0.10) “market up” (0.15) market flat “market flat” (0.70) (0.3) “market down” (0.15) m n (0 2) “market flat” (0.20) what we have (?) market flat (?) market down (?) market up (?) market flat (?) market up (?) market flat (?) ?) “market down” (0.60) ( n” ow td w (0.20) market up market down (?) ke ar “m t ke ar “market up” “market flat” (?) actual market performance market down (?) what we need © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved conditional probabilities with the conditioning on the economists’ forecast (0.10) (? ) “market flat” p” (0.80) ar ke tu “market up” economist’s forecast “m economist’s forecast m ar ke t up (0 5) actual market performance conditional probabilities with the conditioning on the actual market performance PROBABILITY TREE 19 POSTERIOR PROBABILITIES economist’s prediction market up market flat market down “up” 0.8247 0.0928 0.0825 “flat” 0.1667 0.7000 0.1333 “down” 0.2325 0.2093 0.5581 © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved conditional probabilities on economists forecast posterior probability for: 20 EVII COMPUTATION ‰ We use conditional probabilities in the table to build the posterior probabilities ‰ For example P {market up economist predicts "up"} = 0.8247 ‰ We then compute P {F = "up"} = 0.485  P {F = "flat"} = 0.300  P {F = "down"} = 0.215  University of Illinois at Urbana-Champaign, All Rights Reserved © 2006 - 2009 George Gross, 21 EXPECTED VALUE OF IMPERFECT INFORMATION market activity up (0.5) 1500 flat (0.3) 100 down (0.2) −1000 up (0.5) 1000 economist says flat (0.3) “market up” (0.485) 200 down (0.2) −100 ck to s k ris = 58 gh V h i EM low-risk stock EMV = 540 savings account 500 consult economist EMV=822 economist says “market flat” (0.300) market activity high-risk stock up (0.8247) flat (0.0928) 1500 low-risk stock 100 down (0.0825) − 1000 up (0.8247) 1000 flat (0.0928) 200 EMV=835 down (0.0825) EMV=1164 − 100 500 savings account up (0.1667) high-risk stock flat (0.7000) EMV=187 down (0.1333) up (0.1667) low-risk stock flat (0.7000) EMV=293 down (0.1333) 1500 100 − 1000 1000 200 − 100 savings account economist says “market down”(0.215) up (0.2325) high-risk stock flat (0.2093) EMV=188 down (0.5581) up (0.2325) low-risk stock flat (0.2093) EMV=219 down (0.5581) 500 1500 100 − 1000 savings account © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 1000 200 − 100 500 22 EVII COMPUTATION ‰ 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 = 822 – 580 = 242 ‰ This value represents the upper limit on the worth of the economist’s forecast © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 23 EXAMPLE OF VALUE OF INFORMATION ‰ We consider the following decision tree 0.1 0.2 E A 0.6 0.1 0.7 B F 0.3 20 10 – 10 –1 with the events at E and F as independent ‰ We perform a number of valuations of EVPI for this simple decision problem © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 24 EVPI FOR F ONLY EMV (A) = 3.0 A E EMV (B) = 3.2 B F 0.1 20 0.2 10 0.6 0.1 −10 0.7 0.3 perfect information EMV (info about F) = about F 4.4 EVPI (info about F) = EMV (info about F) – EMV (B) = 4.4 – 3.2 = 1.2 0.1 0.2 E −1 B=5 0.7 0.6 0.1 B 0.2 E B = −1 0.3 10 − 10 0.1 F 20 20 10 0.6 0.1 −10 B © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved −1 25 EVPI FOR E ONLY 0.1 0.2 EMV (A) = 3.0 A E 0.6 0.1 EMV (B) = 3.2 B F 0.7 0.3 20 10 A A = 20 0.1 E EVPI (info about E) = EMV (info) – EMV (B) = 6.24 – 3.20 = 3.04 0.7 F 0.3 −10 −1 A A = 10 0.2 perfect information about E EMV (info about E) = 6.24 B 20 F 0.3 A A=0 0.6 F 0.3 A A = −10 0.1 F −1 −1 – 10 0.7 B 0.7 B −1 10 0.7 B 0.3 © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved −1 26 EVPI FOR E AND F ONLY EMV (A) = 3.0 A E 0.1 20 0.2 10 0.6 0.1 EMV (B) = 3.2 B F 0.7 0.3 A = 20 F 0.1 B=5 0.7 B = −1 0.3 −10 −1 A = 10 F 0.2 perfect information about E and F EMV (info about E and F) = 6.42 E EVPI (info about E and F) = EMV (info) – EMV (B) = 42 – 3.20 = 3.22 A=0 F 0.6 A = −10 F 0.1 B=5 0.7 B = −1 0.3 B=5 0.7 B = −1 0.3 B=5 0.7 B = −1 0.3 © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved A 20 B A 20 B −1 A 10 B A 10 B −1 A B A B −1 A −10 B A −10 B −127 ... 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved INVESTMENT EXAMPLE: COMPUTATION OF EVPI ‰ Absent any expert information, a value – maximizing investor... amount that the investor should be willing to pay the expert for the perfect information resulting in the improved outcome © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All... savings account 500 high-risk stock − 1000 low-risk stock − 100 savings account 500 © 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 12 EXPECTED VALUE