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Value of Information and Decision Trees in Probabilistic Decision Making, Slides of Artificial Intelligence

The concept of the value of information in decision making through examples and the use of decision trees. The calculation of the value of information in the context of buying a program, an oil company's drilling rights, and software development. Decision trees are introduced as a tool to represent all possible scenarios and outcomes in a decision-making process.

Typology: Slides

2012/2013

Uploaded on 04/24/2013

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CSCI 100
Think Like Computers
Lecture 6
Fall 2008
Is there a winning strategy?
Bet $1 in the beginning.
If lose, double the amount next time.
Does this work?
Exercise: A Decision Problem
You consider buying a program to manage
your finances that costs $100. There is a
prior probability of 0.7 that the program is
suitable in which case it will have a
positive effect on your work worth $500.
There is a probability of 0.3 that the
program is not suitable in which case it will
have no effect.
Value of information
Sometimes, an agent must actively gather
information before making a decision
Using utility theory, an agent can figure out
how much it would be willing to pay for a
piece of information
Called information value theory
Example
What is the value of knowing whether the
program is suitable before buying it?
[0.7*(500-100)+0.3(0)] –
[0.7(500-100)+0.3(0-100)] = 280 - 250
= 30
Another example
Suppose an oil company is hoping to buy one of
n blocks of ocean drilling rights.
Exactly one block contains oil worth C dollars.
The price of each block is C/n dollars.
If the company is risk-neutral, it will be indifferent
between buying a block or not.
A seismologist offers the company a survey
indicating whether block #3 contains oil.
How much should the company be willing to pay
for the information?
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CSCI 100

Think Like Computers

Lecture 6

Fall 2008

Is there a winning strategy?

• Bet $1 in the beginning.

• If lose, double the amount next time.

• Does this work?

Exercise: A Decision Problem

• You consider buying a program to manage

your finances that costs $100. There is a

prior probability of 0.7 that the program is

suitable in which case it will have a

positive effect on your work worth $500.

There is a probability of 0.3 that the

program is not suitable in which case it will

have no effect.

Value of information

• Sometimes, an agent must actively gather

information before making a decision

• Using utility theory, an agent can figure out

how much it would be willing to pay for a

piece of information

• Called information value theory

Example

• What is the value of knowing whether the

program is suitable before buying it?

• [0.7*(500-100)+0.3(0)] –

[0.7(500-100)+0.3(0-100)] = 280 - 250

Another example

• Suppose an oil company is hoping to buy one of

n blocks of ocean drilling rights.

• Exactly one block contains oil worth C dollars.

• The price of each block is C/n dollars.

• If the company is risk-neutral, it will be indifferent

between buying a block or not.

• A seismologist offers the company a survey

indicating whether block #3 contains oil.

• How much should the company be willing to pay

for the information?

The value of information cont.

• What can the company do with the information?

• Case 1: block #3 contains oil (p=1/n).

Company will buy it and make a profit of:

C - C/n = (n-1) C/n dollars.

• Case 2: block #3 contains no oil (p=(n-1)/n).

Company will buy different block and make:

C/(n-1) - C/n = C/(n (n-1)) dollars.

• Now, the overall expected profit is C/n

• What is the value of information?

Decision trees

• A decision tree is an explicit representation of all

the possible scenarios from a given state.

• Each path corresponds to decisions made by the

agent, actions taken, possible observations,

state changes, and a final outcome node.

• Similar to a game played against “nature”

Example: Software development

  • EU(make) = 0.3 ∗ $380K + 0.7 ∗ $450K = $429K
  • EU(reuse) = 0.4 ∗ $275K + 0.6 ∗ [0.2 ∗ $310K + 0.8 ∗ $490K] = $382.4K
  • EU(buy) = 0.7 ∗ $210K + 0.3 ∗ $400K = $267K

$380K

$450K

$275K

$310K

$490K

$210K

$400K

make reuse

simple P=0.

difficult P=0.

buy

minor changes P=0.

major changes P=0.

major changes P=0.

minor changes P=0.

simple P=0.

complex P=0.

Display Software

Nodes in a decision tree

• Chance nodes (ovals) represent possible

outcomes

• Decision nodes (squares) represent

options available to the decision maker

• Utility nodes (Diamonds) or value nodes

represent the overall utility

Example 2: Buying a car

  • There are two candidate cars C 1 and C 2 , each can be of good quality (+) or bad quality (−).
  • P(+C1)=0.7, P(+C2)=0.
  • There are two possible tests, T 1 (costs $50) and T (^2) (costs $20).
  • P(Pass(T1,c)|+c)=0.9, P(Pass(T1,c)|-c)=0.
  • P(Pass(T2,c)|+c)=0.8, P(Pass(T2,c)|-c)=0.
  • C 1 costs $1500 ($500 below market value) but if it is of bad quality repair cost is $700.
  • C 2 costs $1150 ($250 below market value) but if it is of bad quality repair cost is $150.
  • Buyer must buy one of the cars and can perform at most one test.

Example 2: Buying a car cont.

T 2

T 1

T 0

fail pass fail pass

C 1 C 2 C 1 C 2 C 1 C 2 C 1 C 2

C 1 C 2