Decision-Tree Analysis

Transcript Decision-Tree Analysis

Decision Tree Analysis



A graphical tool for describing (1) the actions available to the decision-maker, (2) the events that can occur, and (3) the relationship between the actions and events.

Constructing a Decision Tree A Company is considering marketing a new product. Once the product is introduced, there is a 70% chance of encountering a competitive product. Two options are available each situation.

Decision Points Events ( ) Probability Market

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Option 1 (with competitive product): Raise your price and see how your competitor responds. If the competitor raises price, your profit will be $60. If they lower the price, you will lose $20.

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Option 2 (without competitive product): You still two options: raise your price or lower your price. Do not market

First Decision Point

$0 Our Price

High

Competitor’s price

High (0.5)

Conditional Profit $60

(0.5) Low

-$20

Competitive Product (0.7) High (0.2)

$40

Low

No Competitive Product (0.3)

Low (0.8)

$10

High

$100

Low

$30

Second Decision Point

The conditional profits associated with each event along with the likelihood of each event is shown in the decision tree.

Rollback Procedure

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To analyze a decision tree, we begin at the end of the tree and work backward.

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For each chance node, we calculate the expected monetary value (EMV), and place it in the node to indicate that it is the expected value calculated over all branches emanating from that node.

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For each decision node, we select the one with the highest EMV (or minimum cost). Then those decision alternatives not selected are eliminated from further consideration.

Making Sequential Investment Decisions

$20

Set High Price High (0.5) (0.5) Low

$60 -$20 $44 Market $44

Competitive Product (0.7)

$20 Do not market No Competitive Product (0.3)

Low High (0.2)

$40 $16

Set High Price Low (0.8)

$10 $100 $100

Low

$0 $30

Decision Rules

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Market the new product.

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Whether or not you encounter a competitive product, raise your price.

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The expected monetary value associated with marketing the new product is $44.

Practice Problem

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A company is considering the purchase of a new labor saving machine.

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The machine’s cost will turn out to be $55 per day. Each hour of labor that is saved reduces costs by $5. However, there is some uncertainty over the number of hours that actually will be saved.

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It is judged that the hours of labor saved per day will be 10, 11, or 12, with probabilities of 0.10, 0.60, 0.30, respectively.

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Let us define “profit” as the excess of labor-cost savings over the machine cost.

Construct a Decision Tree

-$5 0.10

$1.0

10 $1 Invest 0.60

11 0 12 0.30

Do not invest $5

EMV = $1.0

Decision: Purchase the equipment

Expected Value of Perfect Information (EVPI)

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What is EVPI? This is equivalent to asking yourself how much you can improve your decision if you had perfect information.

Mathematical Relationship: EVPI = EPPI – EMV = EOL where EPPI (Expected profit with perfect information) is the expected profit you could obtain if you had perfect information, and EMV (Expected monetary value) is the expected profit you could obtain based on your own judgment. This is equivalent to expected opportunity loss ( EOL ).

Expected Value of Perfect Information (EVPI)

State of Nature 10 11 12 Best Strategy Don’t Buy Indifferent Buy Maximum Payoff 0 0 5 Probability the State of Nature Occurs 0.10

0.60

0.30

Expected Payoff or each State 0 0 1.5

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Expected Profit with Perfect Information (EPPI): (0.10)(0) + (0.60)(0) + (0.30)(5) = $1.5

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Expected Value of Perfect Information (EVPI) = EPPI – EMV $1.5 - $1 = $0.5

Bill’s Decision Problem – $50,000 to Invest

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Decision Problem:

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Buying a highly speculative stock (d three potential levels of return – High (50%), Medium (9%), and Low ( 30%). 1 ) with

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Buying a very safe U.S. Treasury bond (d 2 ) with a guaranteed 7.5% return.

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Seek advice from an expert?

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Seek professional advice before making the decision

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Do not seek professional advice – do on his own.

Decision Tree for Bill’s Investment Problem

Evaluating Options in Bill’s Investment Problem

• • Option 1: 1) Period 0: (-$50,000 - $100) = -$50,100 Period 1: (+$75,000 - $100) - 0.20($24,800) =$69,940 2) 3) PW(5%)=-$50,100 + $69,940 (

P/F

, 5%, 1) = $16,510 Period 0: (-$50,000 - $100)= -$50,100 Period 1: (+$54,500 - $100)- (0.20)($4,300) = $53,540 PW(5%) = -$50,100 + $53,540 (

P/F

, 5%, 1) = $890 Period 0: (-$50,000 - $100) = -$50,100 Period 1: (+$35,000 - $100) – (0.20)(-$14,800) = $37,940 PW(5%)= - $50,100 + $37,940 (

P/F

, 5%, 1) = -$13,967 Option 2: Period 0: (- $50,000 - $150) = -$50,150 Period 1: (+$53,750 - $150) = $53,600 PW (5%)= -$50,150 + $53,600 (

P/F

Expected Value of Perfect Information

Decision Option Potential Return Level High (A) Medium (B) Probability 0.25

0.40

Option1: Invest in Stock $16,510 890 (Prior Optimal) Option 2: Invest in Bonds $898 898 Low(C) 0.35

-13,967 898 EMV -$405 $898 EPPI = (0.25)($16,510) + (0.40)($898) + (0.35)($898) = $4,801 EVPI = EPPI – EV = $4,801 - $898 = $ 3,903 Optimal Choice with Perfect Information Stock Bond Bond Opportunity Loss Associated with Investing in Bonds $15,612 0 $3,903 EOL = (0.25)($15,612) + (0.40)(0) + (0.35)(0) = $ 3,903 0

Bill’s Investment Problem with an Option of Getting Professional Advice Updating Conditional Profit (or Loss) after Paying a Fee to the Expert (Fee = $200) Revised Decision Tree

Conditional Probabilities of the Expert’s Prediction, Given a Potential Return on the Stock What the Report Will Say Favorable (F) Unfavorable (UF) Given Level of Stock Performance High (A) 0.80

0.20

Medium (B) 0.65

0.35

Low (C) 0.20

0.80

Nature’s Tree: Conditional Probabilities and Joint Probabilities

Nature’s Tree Joint & Marginal Probabilities

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P(A,F) = P(F|A)P(A) = (0.80)(0.25) = 0.20

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P(A,UF|A)P(A) = (0.20)(0.25) = 0.05

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P(B,F) = P(F|B)P(B) = (0.65)(0.40) = 0.26

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P(B,UF) = P(UF|B)P(B) = (0.35)(0.40) = 0.14

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P(F) = 0.20 + 0.26 + 0.07 = 0.53

P(UF) = 1 – P(F) = 1 – 0.53 = 0.47

Joint and Marginal Probabilities

What the Report Will Say Joint Probabilities When Potential Level of Return is Given High (A) Favorable (F) 0.20

Unfavorable (UF) 0.05

Medium (B) Low (C) Marginal Probabilities 0.26

0.07

0.53

0.14

0.28

0.47

Marginal Probabilities of Return Level 0.25

0.40

0.35

1.00

Determining Revised Probabilities

P(A|F) = P(A,F)/P(F) = 0.20/0.53 = 0.38

P(B|F) = P(B,F)/P(F) = 0.26/0.53 = 0.49

P(C|F) = P(C,F)/P(F) = 0.07/0.53 = 0.13

P(A|UF) = P(A,UF)/P(UF) = 0.05/0.47 = 0.11

P(B|UF) + P(B,UF)/P(UF) = 0.14/0.47 = 0.30

P(C|UF) = P(C,UF)/P(UF) = 0.28/0.47 = 0.59

Decision Making after Having Imperfect Information - $6,319

Decision-Tree Analysis

Transcript Decision-Tree Analysis