1. Decision Trees Updated 2 December 2005
EMIS 7300
Decision Trees
Updated 2 December 2005

2. Example 1
There is a 0.65 probability of no growth in the investment climate and 0.35 probability of rapid growth. The payoffs are $500 for a bond investment in a no-growth state, $100 for a bond investment in a rapid-growth state, -$200 for a stock investment in a no-growth state, and a $1100 payoff for a stock investment in a rapid-growth state.
STATE OF NATURE
No Growth (0.65)
Rapid Growth (0.35)
DECISION ALTERNATIVE
Bonds Stocks
$500 -$200
$100 $1100

3. EMV(bonds) = $500(0.65) + $100(0.35) = $360
EMV Criterion
The expected monetary value for the bonds decision alternative is
EMV(bonds) = $500(0.65) + $100(0.35) = $360
The expected monetary value for the stocks decision alternative is
EMV(stocks) = -$200(0.65) + $1100(0.35) = $255
Select bonds under the EMV criterion.

4. Example 1 as a Decision Tree
Payoffs
No Growth (0.65)
Rapid Growth (0.35)
$500
Decision Node 1
Bonds
$100
No Growth (0.65)
$-200
Stocks 2
Rapid Growth (0.35)
$1100
State-of-Nature Node

5. Calculate EMV for Each Node
EMV for Node 1 = $360
Payoffs
No Growth (0.65)
Rapid Growth (0.35)
$500 1
Bonds
$100
No Growth (0.65)
$-200
Stocks 2
EMV for Node 2 = $255
Rapid Growth (0.35)
$1100

6. Completed Decision Tree
Payoffs
No Growth (0.65)
Rapid Growth (0.35)
$500
$360 1
Bonds
$100
$360
Select Bonds
No Growth (0.65)
$-200
Stocks 2
$255
Rapid Growth (0.35)
$1100

7. Expected Value with Perfect Information
Suppose that before we invest, we can consult an oracle who knows with certainty which state of nature will occur. Our investment policy will be:
If the oracle predicts no growth, then invest in bonds and receive a payoff of $500.
If the oracle predicts rapid growth, then invest in stocks and receive a payoff of $1,100.
EVwPI = (0.65)($500) + (0.35)($1,100) = $710.

8. Example 2
Banana Computer Company manufactures memory chips in batches of ten chips. From past experience, Banana knows that 80% of all batches contain 10% (1 out of 10) defective chips, and 20% of all batches contain 50% (5 out of 10) defective chips. If a good (that is, 10% defective) batch of chips is sent on to the next stage of production, processing costs of $1000 are incurred, and if a bad batch (50% defective) is sent on to the next stage of production, processing costs of $4000 are incurred. Banana also has the option of reworking a batch at a cost of $1000 before sending it to the next stage of production. A reworked batch is sure to be a good batch.

9. Example 2 Continued
Develop a decision tree for this problem and use it to determine a policy for minimizing Banana’s expected total cost per batch.

10. Decision Tree 1 2 -$2,000 Rework every batch
Good Batch (0.8)
Bad Batch (0.2)
Send Directly to next stage
-$1,000
-$4,000
EMV(2)=(0.8)(-$1,000)-(0.2)($4,000) = -$1,600
Decision Node
State of Nature Node

11. Optimal Policy Send every batch directly to the next stage
EMV = -$1,600

12. Example 3
For a cost of $100, Banana can test one chip from each batch in an attempt to determine whether the batch is defective.
Develop a decision tree for this problem and use it to determine a policy for minimizing Banana’s expected total cost per batch.
What is the expected value of the sample information (EVSI) obtained from testing chips?
What is the expected value of perfect information (EVPI) for this problem?

13. Policies to Consider
Rework every batch before sending it Stage 2. This policy will cost $2000 for every batch: $1000 to rework it and $1000 to process the reworked batch at stage 2.
Send every batch to Stage 2 without reworking it. E[cost] = (0.8)($1000) + (0.2)($4000) = $1600
Test a chip from each batch and decide what to do based on the result.

14. Expected Cost of Policy 3
Case 1: The chip tested is good (not defective).
The expected cost of sending the batch to stage 2 is ($1000)P(GB|GC) + ($4000)P(BB|GC).
Case 2: The chip is defective.
The expected cost of sending the batch to stage 2 is ($1000)P(GB|DC) + ($4000)P(BB|DC).

15. Banana Computer Solution

16. Send Directly to next stage
Rework 4
Good Batch (?)
Bad Batch (1-?)
Direct to Stage 2
-$1,100
-$4,100
-$2,100
Decision Tree
Test a Chip 3
Defective Chip (0.18)
Good Chip (0.82)
Rework 5
Good Batch (??)
Bad Batch (1-??)
-$1,100
-$4,100
-$2,100
Direct to Stage 2
Rework every batch
-$2,000 1 2
Good Batch (0.8)
Bad Batch (0.2)
Send Directly to next stage
-$1,000
-$4,000
Decision Node
State of Nature Node

17. Banana Computer Solution: Posterior Probabilities

18. Banana Computer Solution: Posterior Probabilities

19. Decision Tree 4 3 5 1 2 Rework Good Batch (0.88) Bad Batch (0.12)
Direct to Stage 2
-$1,100
-$4,100
-$2,100
Decision Tree
Test a Chip 3
Defective Chip (0.18)
Good Chip (0.82)
Rework 5
Good Batch (0.44)
Bad Batch (0.56)
-$1,100
-$4,100
-$2,100
Direct to Stage 2
EMV(4)=(0.88)(-$1,100)-(0.12)($4,100) = -$1,460
EMV(5)=(0.44)(-$1,100)-(0.56)($4,100) = -$2,780
Rework every batch
-$2,000 1
EMV(3)=(0.82)(-$1,460)-(0.18)($2,100) = -$1575.2 2
Good Batch (0.8)
Bad Batch (0.2)
Send Directly to next stage
-$1,000
-$4,000
EMV(2)=(0.8)(-$1,000)-(0.2)($4,000) = -$1,600
Decision Node
State of Nature Node

20. Optimal Policy Test one chip EMV = -$1,575.25
If the chip is good, then send the batch directly to the next stage
If the chip is defective, then rework the batch before sending it to the next stage
EMV = -$1,575.25

21. Expected Value of Sample Information (EVSI)
The optimal decision is to test a chip.
The expected payoff is -$1,
If the test were free, the expected payoff would be $1,475.25
The optimal decision without the test is to send all batches directly to the next stage
The expected payoff is -$1,600.
The expected value of the test (EVSI) is given by (-$1,475.25) – (-$1,600) = $124.75
If the test cost more than $ we wouldn’t use it.

22. EVPI
The optimal decision is to pay to test the chip.The expected payoff is -$1,575.25
The optimal decision with perfect information is
If the batch is good, then send it to the next stage.
If the batch is bad, then rework it before sending it to the next stage
EMVwPI = (-$1,000)(0.8)+(-2,000)(0.2) = -$1,200.
The expected value of perfect information is $1,200 – (- $1,575.25) = $375.25