Value of Information - Human Decision Making - Lecture Slides, Slides of Human-Computer Interaction Design

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Value of Information

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

4

Probability and Perfect Information

What about

Pr(A |A')?

1

0 Pr(A) 1 Pr(A)

1 Pr(A)

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

Pr( A'|A)Pr(A)

 

Pr(A|A')

The above conclusions indicate that after the clairvoyant with perfect information

is consulted, no uncertainty remains about the event

In other words, Pr(A|A')is equal to 1 regardless of the priori probability Pr(A)

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.

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

Activity

High-Risk Stock

Low-Risk Stock

Savings Account

Payoff

Up (0.5)

Flat (0.3)

High-Risk Stock

Low-Risk Stock

Savings Account

High-Risk Stock

Low-Risk Stock

Savings Account

EVPI = EMV(with perfect information) – EMV (Without information)=1000-580=$

Therefore, the investor should not pay more than $420 for the clairvoyant

EMV=$1,

Investment

Decision

Market

Activity

Payoff

8

Expected Value of Imperfect Information (EVII)

 Perfect information is rarely available in real situations

Pr(A'| A) 1 Pr(A'|A) 0

Pr(A '|A) 1 Pr(A'|A) 0

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.

"Flat" Pr(“Flat”|Up)=0.10 Pr(“Flat”|Flat)=0.70 Pr(“Flat”|Down)=0.

"Down" Pr(

Down

|Up)=0.10 Pr(

Down

|Flat)=0.15 Pr(

Down

|Down)=0.

10

Pr(E "Up"|M Down)Pr(M Down)

Pr(E "Up"|M Up)Pr(M Up) Pr(E "Up"|M Flat)Pr(M Flat)

Pr(E "Up") Pr(E "Up" M Up) Pr(E "Up" M Flat) Pr(E "Up" M Down)

If the economist says “Market Up”

Economist’s

Forecast

High-Risk Stock

Low-Risk Stock

Savings Account

Payoff

“Up”(?)

Up (?)

Flat (?)

Down (?)

Up (?)

Flat (?)

Down (?)

Pr(E=“Up”) =?

Pr(M=Up|E=“Up”) =?

Pr(M=Flat|E=“Up”) =?

Pr(M=Down|E=“Up”) =?

Pr(M Up|E"Up")

Pr(E "Up")

Pr( E "Up"|M Up)Pr(M Up)

Pr(M Flat|E"Up")

1. 093

2. 485

3. 150. 3

Pr(E "Up")

Pr( E "Up"|M Flat)Pr(M Flat)

 





  

Pr(M Down|E"Up")

1. 082

2. 485

3. 200. 2

Pr( " ")

Pr( | )Pr( )

 





  

E Up

E UpM Down M Down

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Economist’s

Forecast

High-Risk Stock

Low-Risk Stock

Savings Account

Payoff

“Up”(0.485)

Up (0.825)

Flat (0.093)

Down (0.082)

Up (0.825)

Flat (0.093)

Down (0.082)

EMV= $1,

EMV= $

13

Economist’s

Forecast

High-Risk Stock

Low-Risk Stock

Savings Account

Payoff

“Flat”(0.3)

Up (0.167)

Flat (0.7)

Down (0.133)

Up (0.167)

Flat (0.7)

Down (0.133)

EMV= $

EMV= $

14

1. 10 0. 5 0. 15 0. 3 0. 60 0. 2 0. 215

Pr(E "Down"|M Down)Pr(M Down)

Pr(E "Down"|M Up)Pr(M Up) Pr("Down"|M Flat)Pr(M Flat)

Pr(E "Down" M Down)

Pr(E "Down") Pr(E "Down" M Up) Pr(E "Down" M Flat)

      

  

       

  

         

If the economist says “Market Down”

Economist’s

Forecast

High-Risk Stock

Low-Risk Stock

Savings Account

Payoff

Down(?)

Up (?)

Flat (?)

Down (?)

Up (?)

Flat (?)

Down (?)

Economist’s

Forecast

High-Risk Stock

Low-Risk Stock

Savings Account

Payoff

“Down”(?)

Up (?)

Flat (?)

Down (?)

Up (?)

Flat (?)

Down (?)

Pr(E=“Down”) =?

Pr(M=Up|E=“Down”) =?

Pr(M=Flat|E=“Down”) =?

Pr(M=Down|E=“Down”) =?

Pr(M Flat|E"Down") 0. 209

1. 215

2. 150. 3

Pr(E "Down")

Pr( E "Down"|M Flat)Pr(M Flat)

 





  

1. 558

2. 215

3. 600. 2

Pr(E "Down")

Pr( E "Down"|M Down)Pr(M Down)

 





  

Pr(M Down|E"Down")

Pr(M Up|E"Down")

1. 233

2. 215

3. 100. 5

Pr(E "Down")

Pr( E "Down"|M Up)Pr(M Up)

 





  

16

Economist’s

Forecast

“Up” (0.485)

“Flat” (0.3)

“Down” (0.215)

EMV= $1,

EMV= $

EMV= $

EMV= $

EVII = EMV(with imperfect information) – EMV (Without information)=822-580=$

Therefore, the investor should not pay more than $242 for the economist