value of information dr. yan liu department of biomedical, industrial & human factors...

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”.

What about Pr(A | A’) = ?

0A)|'APr(1A)|Pr(A' If the clairvoyant always correctly identifies the situation of Dow Jones index, then

1)APr(0Pr(A)1

Pr(A)1

)A)Pr(APr(A'|A)Pr(A)Pr(A'|

A)Pr(A)Pr(A'|

)A'|Pr(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 ?)'A|APr(

1)APr(1Pr(A)0

)APr(1

)A)Pr(A'|APr(A)Pr(A)'|APr(

)A)Pr(A'|APr(

)'A|APr(

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 )'A|APr( )APr(

5

Expected Value of Perfect Information (EVPI)

Stock Market ExampleAn 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

High-Risk

StockLow-Risk

StockSavings

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

Market

EMV=$580

EMV=$540

Influence Diagram

Investment

Decision

Market Activity

Payoff

Decision Tree

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 Activit

y

High-Risk StockLow-Risk StockSavings

Account

$1,500

$200

$1,000$500$100

Payoff

Up (0.5)

Flat (0.3)

High-Risk StockLow-Risk StockSavings

AccountHigh-Risk

StockLow-Risk StockSavings

Account

$500

-$100

-$1,000$500

EVPI = EMV(with perfect information) – EMV (Without information)=1000-580=$420Therefore, the investor should not pay more than $420 for the clairvoyant

EMV=$1,000

Investment

Decision

Market Activity

Payoff

8

Expected Value of Imperfect Information (EVII)

Perfect information is rarely available in real situations

0A)|'APr(1A)|Pr(A' 0)A|Pr(A'1)A|'APr(

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)

Up Flat Down

"Up" Pr(“Up”|Up)=0.80 Pr(“Up”|Flat)=0.15 Pr(“Up”|Down)=0.20

"Flat" Pr(“Flat”|Up)=0.10 Pr(“Flat”|Flat)=0.70 Pr(“Flat”|Down)=0.20

"Down" Pr(“Down”|Up)=0.10 Pr(“Down”|Flat)=0.15 Pr(“Down”|Down)=0.60

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

485.02.020.03.015.05.080.0)DownMPr()DownM|"Up"EPr(

)FlatMPr()FlatM|"Up"EPr()UpMPr()UpM|"Up"EPr()DownM"Up"EPr()FlatM"Up"EPr()UpM"Up"EPr()"Up"EPr(

If the economist says “Market Up”

Economist’s Foreca

st

High-Risk

StockLow-Risk

Stock

Savings Account

$1,500$100-

$,1000

Payoff

“Up”(?)

Up (?)Flat (?)Down (?)

$1,000$200-

$100


$500

Pr(E=“Up”) =?Pr(M=Up|E=“Up”) =?Pr(M=Flat|E=“Up”) =?Pr(M=Down|E=“Up”) =?

)"Up"E|UpMPr( 825.0485.05.080.0

")Up"EPr()UpMPr()UpM|"Up"EPr(

)"Up"E|FlatMPr( 093.0485.03.015.0

")Up"EPr()FlatMPr()FlatM|"Up"EPr(

)"Up"E|DownMPr( 082.0485.02.020.0

")"Pr()Pr()|Pr(

UpEDownMDownMUpE

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

If the economist says “Market Flat”


st

High-Risk

StockLow-Risk

Stock

Savings Account

$1,500$100-

$,1000

Payoff

“Flat”(?)


$1,000$200-

$100


$500

Pr(E=“Flat”) =?Pr(M=Up|E=“Flat”) =?Pr(M=Flat|E=“Flat”) =?Pr(M=Down|E=“Flat”) =?

3.02.020.03.070.05.010.0)DownMPr()DownM|"Flat"EPr(

)FlatMPr()FlatM|"Flat"EPr()UpMPr()UpM|"Flat"EPr()DownM"Flat"EPr()FlatM"Flat"EPr()UpM"Flat"EPr()"Flat"EPr(

133.03.02.020.0

")Flat"EPr()DownMPr()DownM|"Flat"EPr(

)"Flat"E|"Down"MPr(

)"Flat"E|FlatMPr( 7.03.03.07.0

")Flat"EPr()FlatMPr()FlatM|"Flat"EPr(

)"Flat"E|UpMPr( 167.03.05.010.0

")Flat"EPr()UpMPr()UpM|"Flat"EPr(

13


st

High-Risk

StockLow-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)$50

0

EMV= $187

EMV= $293

14

215.02.060.03.015.05.010.0)DownMPr()DownM|"Down"EPr(

)FlatMPr()FlatM|"DownPr(")UpMPr()UpM|"Down"EPr()DownM"Down"EPr(

)FlatM"Down"EPr()UpM"Down"EPr()"Down"EPr(

If the economist says “Market Down”


st

High-Risk

StockLow-Risk

Stock

Savings Account

$1,500$100-

$,1000

Payoff

Down(?)


$1,000$200-

$100


$500


st

High-Risk

StockLow-Risk

Stock

Savings Account

$1,500$100-

$,1000

Payoff

“Down”(?)


$1,000$200-

$100


$500

Pr(E=“Down”) =?Pr(M=Up|E=“Down”) =?Pr(M=Flat|E=“Down”) =?Pr(M=Down|E=“Down”) =?

)"Down"E|FlatMPr( 209.0215.03.015.0

")Down"EPr()FlatMPr()FlatM|"Down"EPr(

558.0215.02.060.0

")Down"EPr()DownMPr()DownM|"Down"EPr(

)"Down"E|DownMPr(

)"Down"E|UpMPr( 233.0215.05.010.0

")Down"EPr()UpMPr()UpM|"Down"EPr(

15


st

High-Risk

StockLow-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


st

“Up” (0.485)“Flat” (0.3)“Down” (0.215)

EMV= $1,164EMV= $500EMV= $500

EMV= $822

EVII = EMV(with imperfect information) – EMV (Without information)=822-580=$242Therefore, the investor should not pay more than $242 for the economist

value of information dr. yan liu department of biomedical, industrial & human factors...

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