Giao trinh bai tap ve hinh hoc lt

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.

1


VALUE OF INFORMATION
‰ 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 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.

2


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.

3

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.

4


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.

5


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.

6


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 > 0 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.

7


EVI ASSESSMENT
‰ It follows that the value of information is always
nonnegative, EVI ≥ 0
‰ 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.

8


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 do 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.

9


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


COMPUTATION OF EVPI
‰ For the investment problem,
EVPI = 1,000 – 580 = 420
‰ We may view EVPI to represent the maximum
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 Rights Reserved.

11


COMPUTATION OF EVPI
ck
to
s
0
k
ris = 58
gh V
hi EM

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

1500
100
−1000
1000
200
−100

500

up
t
ke
ar .5)
m
(0

consult clairvoyant
EMV = 1000

market flat
(0.3)
m

ar
ke
(0 t d
.2 ) o w
n

high-risk stock

1500

low-risk stock

1000

savings account
high-risk stock

500
100

low-risk stock

200

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 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 = 5
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
do

(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
0

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
0
– 10

5
–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

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

5

0.2
E

−1
B=5
0.7

0.6
0.1

B

0.2
E

B = −1
0.3

10
0
− 10

5
0.1

F

20

20
10

0.6

0

0.1

−10

B

© 2006 - 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

−1

25