Decision tree

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Bayesian Approach

DECISION TREE

D R . K A L P N A S H A R M A , D E PA RT M E N T O F M AT H E M AT I C S

M A N I PA L U N I V E R S I T Y J A I P U R

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D R . K A L P N A S H A R M A , D E P A R T M E N T O F M A T H E M A T I C S , M A N IP A L U N I V E R S I T Y J A I P U R

DECISION TREES

A decision tree is a chronological representation of the decision problem.

Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes

correspond to the decision alternatives. The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives.

At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.

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FIVE STEPS TODECISION TREE ANALYSIS

1. Define the problem.

2. Structure or draw the decision tree.

3. Assign probabilities to the states of nature.

4. Estimate payoffs for each possible combination of alternatives and states of nature.

5. Solve the problem by computing expected monetary values (EMVs) for each state of nature node.

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EXAMPLE

A developer must decide how large a luxury condominium complex to build – small, medium, or large. The profitability of this complex depends upon the future level of demand for the complex’s condominiums.

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ELEMENTS OF DECISION THEORY

States of nature: The states of nature could be defined as low demand and high demand.

Alternatives: Developer could decide to build a small, medium, or large condominium complex.

Payoffs: The profit for each alternative under each potential state of nature is going to be determined.

We develop different models for this problem on the following slides.

D R . K A L P N A S H A R M A , D E PA R T M E N T O F M A T H E M A T I CS , M A N I P A L U N I V E R S I T Y JA I P U R

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PAYOFF TABLE

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Alternatives Low HighSmall 8 8Medium 5Large -11 22

States of Nature

(payoffs in millions)

THIS IS A PROFIT PAYOFF TABLE

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DECISION TREE

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Small Complex

Medium Complex

Large Complex

Low demand

Low demand

Low demand

High demand

High demand

High demand

8

5

15

22

-11

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EXAMPLE: BURGER PRINCE

Burger Prince Restaurant is contemplating opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120. The payoff table (profits) for the three models is on the next slide.

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Payoff Table

Average Number of Customers Per Hour

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

Model C $ 6,000 $16,000 $21,000

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Expected Value Approach

Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d1, d2, d3 represent the decision alternatives of models A, B, C, and s1, s2, s3 represent the states of nature of 80, 100, and 120.

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11

.2

.4

.4

.4

.2

.4

.4

.2

.4

d1

d2

d3

s1

s1

s1

s2

s3

s2

s2

s3

s3

Payoffs

10,000

15,000

14,0008,000

18,000

12,000

6,000

16,000

21,000

22

33

44

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Expected Value For Each Decision

Choose the model with largest EV, Model C.

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d1

d2

d3

EMV = .4(10,000) + .2(15,000) + .4(14,000) = $12,600

EMV = .4(8,000) + .2(18,000) + .4(12,000) = $11,600

EMV = .4(6,000) + .2(16,000) + .4(21,000) = $14,000

Model A

Model B

Model C

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11

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EXAMPLE PROBLEM: THOMPSON LUMBER COMPANY

Thompson Lumber Company is trying to decide whether to expand its product line by manufacturing and marketing a new

product which is “backyard storage sheds.”

The courses of action that may be chosen include:

(1) large plant to manufacture storage sheds,

(2) small plant to manufacture storage sheds, or

(3) build no plant at all.

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THOMPSON LUMBER COMPANY

Probability 0.5 0.5

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Probability of favorable market is same as probability of unfavorable market. Each state of nature has a 0.50 probability.

EXPECTED MONETARY VALUEThompson Lumber Company

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CALCULATING THE EVPI

Best outcome for state of nature "favorable market" is "build a large plant" with a payoff of $200,000.

Best outcome for state of nature "unfavorable market" is "do nothing," with payoff of $0.

Therefore, Expected profit with perfect information

EPPI = ($200,000)(0.50) + ($0)(0.50) = $ 100,000

If one had perfect information, an average payoff of $100,000 could be achieved in the long run.

However, the maximum EMV (EVBEST) or expected value without perfect information, is $40,000.

Therefore, EVPI = $100,000 - $40,000 = $60,000.

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TO TEST OR NOT TO TEST

Often, companies have the option to perform market tests/surveys, usually at a price, to get additional

information prior to making decisions.However, some interesting questions need to be answered

before this decision is made:

How will the test results be combined with prior information?

How much should you be willing to pay to test?

The good news is that Bayes’ Theorem can be used to combine the information, and we can use our decision tree to find EVSI, the Expected Value of Sample Information.

In order to perform these calculations, we first need to know how reliable the potential test may be.

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MARKET SURVEY RELIABILITY IN PREDICTING ACTUAL STATES OF NATURE

Assuming that the above information is available, we can combine these conditional probabilities with our prior probabilities using Bayes’ Theorem.

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MARKET SURVEY RELIABILITY IN PREDICTING ACTUAL STATES OF NATURE

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PROBABILITY REVISIONS GIVEN POSITIVE SURVEY

Alternatively, the following table will produce the same results:

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PROBABILITY REVISIONS GIVEN NEGATIVE SURVEY

D R . K A L P N A S H A R M A , D E PA R T M E N T O F M A T H E M A T I CS , M A N I P A L U N I V E R S I T Y JA I P U R

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PLACING POSTERIOR PROBABILITIES ON THE DECISION TREE

The bottom of the tree is the “no test” part of the analysis; therefore, the prior probabilities are assigned to these events.

P(favorable market) = P(FM) = 0.5

P(unfavorable market) = P(UM) = 0.5

The calculations here will be identical to the EMV calculations performed without a decision tree.

The top of the tree is the “test” part of the analysis; therefore, the posterior probabilities are assigned to these events.

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DECISION TREES FOR TEST/NO TEST MULTI-STAGE DECISION PROBLEMS

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DECISION TREE SOLUTION

Thompson Lumber Company

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IN-CLASS PROBLEM 3

Leo can purchase a historic home for $200,000 or land in a growing area for $50,000. There is a 60% chance the economy will grow and a 40% change it will not. If it grows, the historic home will appreciate in value by 15% yielding a $30,00 profit. If it does not grow, the profit is only $10,000. If Leo purchases the land he will hold it for 1 year to assess the economic growth. If the economy grew during the first year, there is an 80% chance it will continue to grow. If it did not grow during the first year, there is a 30% chance it will grow in the next 4 years. After a year, if the economy grew, Leo will decide either to build and sell a house or simply sell the land. It will cost Leo $75,000 to build a house that will sell for a profit of $55,000 if the economy grows, or $15,000 if it does not grow. Leo can sell the land for a profit of $15,000. If, after a year, the economy does not grow, Leo will either develop the land, which will cost $75,000, or sell the land for a profit of $5,000. If he develops the land and the economy begins to grow, he will make $45,000. If he develops the land and the economy does not grow, he will make $5,000.

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IN-CLASS PROBLEM 3: SOLUTION

1

2

3

4

5

6

7

Purchase historic home

Purchase land

Economy grows (.6)

No growth (.4)

Economy grows (.6)

No growth (.4)

Build house

Economy grows (.8)

No growth (.2)

Sell land

Develop land

Sell land

Economy grows (.3)

No growth (.7)

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IN-CLASS PROBLEM 3: SOLUTION

1

2

3

4

5

6

7

Purchase historic home

Purchase land

$35,000

$22,000 Economy grows (.6) $30,000

No growth (.4)$10,000

Economy grows (.6)

No growth (.4)

$35,000

$47,000

Build house

$47,000

Economy grows (.8) $55,000

$15,000No growth (.2)

Sell land

$15,000

$17,000

Develop land

Sell land $5,000

Economy grows (.3)

No growth (.7)

$45,000

$5,000

$17,000

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SIMPLE EXAMPLE: UTILITY THEORY

$5,000,000

$0

$2,000,000

Accept Offer

Reject Offer

Heads(0.5)

Tails(0.5)

Let’s say you were offered $2,000,000 right now on a chance to win $5,000,000. The $5,000,000 is won only if you flip a coin and get tails. If you get heads you lose and get $0. What should you do?

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Decision Trees

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Planning Tool

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DECISION TREES

Enable a business to quantify decision makingUseful when the outcomes are uncertain

Places a numerical value on likely or potential outcomes

Allows comparison of different possible decisions to be made

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DECISION TREES

Limitations:

How accurate is the data used in the construction of the tree?

How reliable are the estimates of the probabilities?

Data may be historical – does this data relate to real time?

Necessity of factoring in the qualitative factors – human resources, motivation, reaction, relations with suppliers and

other stakeholders

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Process

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THE PROCESS

Expand by opening new outlet

Maintain current status

Economic growth rises

Economic growth declines

0.7

0.3

Expected outcome£300,000

Expected outcome-£500,000

£0

A square denotes the point where a decision is made, In this example, a business is contemplating opening a new outlet. The uncertainty is the state of the economy – if the economy continues to grow healthily the option is estimated to yield profits of £300,000. However, if the economy fails to grow as expected, the potential loss is estimated at £500,000.

There is also the option to do nothing and maintain the current status quo! This would have an outcome of £0.

The circle denotes the point where different outcomes could occur. The estimates of the probability and the knowledge of the expected outcome allow the firm to make a calculation of the likely return. In this example it is:

Economic growth rises: 0.7 x £300,000 = £210,000

Economic growth declines: 0.3 x £500,000 = -£150,000

The calculation would suggest it is wise to go ahead with the decision ( a net ‘benefit’ figure of +£60,000)

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The Process

Expand by opening new outlet

Maintain current status

Economic growth rises

Economic growth declines

0.5

0.5

Expected outcome£300,000

Expected outcome-£500,000

£0

Look what happens however if the probabilities change. If the firm is unsure of the potential for growth, it might estimate it at 50:50. In this case the outcomes will be:

Economic growth rises: 0.5 x £300,000 = £150,000

Economic growth declines: 0.5 x -£500,000 = -£250,000

In this instance, the net benefit is -£100,000 – the decision looks less favourable!

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Advantages

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Disadvantages

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Thank You

Date post:	12-Nov-2014
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Decision tree

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