Decision Trees

1
Decision Trees

• A way to view complex decisions
• 1. Identify all possible actions
• 2. Identify the future events that can occur
  (i.e., states of nature)
• 3. Determine the payoffs
• 4. Construct the decision tree
• 5. Use a particular decision approach to derive
  an optimal decision

2
Decision Trees General Form
v(a1, s1)
s1
s2
v(a1, s2)
.
.
sm
.
v(a1, sm)
a1
s1
v(a2, s1)
s2
a2
v(a2, s2)
.
sm
.
.
.
.
.
v(a2, sm)
an
s1
v(an, s1)
s2
Decision node
v(an, s2)
.
sm
.
.
Event node
v(an, sm)
Terminal node
3
Solving Decision Trees

• 1. Work backwards
• 2. At each event (circular) node, calculate the
  return according to your preferred decision
  approach from the branches leaving that node
• 3. At each decision (square) node, the payoff
  corresponds to the branch with the largest return

4
Decision Trees with Multiple Decisions

5
Example 1 Houston Oil

• Houston Oil owns some land for which it must make
  a decision
• a1 Drill for oil
• Cost to drill 100,000 for an oil-producing
  well and 75,000 for a dry well
• Revenue 1.50 per barrel
• a2 Unconditional land lease
• Receive 45,000
• a3 Conditional land lease
• Receive 0.50 per barrel, provided the land
  yields at least 200,000 barrels Otherwise,
  receive nothing

6
Example 1 Houston Oil

• Geology Report
• The land can be classified in four ways with
  corresponding probabilities
• Classification Probability
• s1 500,000-barrel well 0.10
• s2 200,000-barrel well 0.15
• s3 50,000-barrel well 0.25
• s4 Dry well 0.50
• Which decision should Houston Oil make to
  maximize its profits?
• First draw a decision tree what are the possible
  payoffs?
• Its Quickie time ?

7
In-class Quickie

• a1 Drill for oil
• Cost to drill 100,000 for an oil-producing
  well and 75,000 for a dry well
• Revenue 1.50 per barrel
• a2 Unconditional land lease
• Receive 45,000
• a3 Conditional land lease
• Receive 0.50 per barrel, provided the land
  yields at least 200,000 barrels Otherwise,
  receive nothing
• State of nature Probability
• s1 500,000-barrel well 0.10
• s2 200,000-barrel well 0.15
• s3 50,000-barrel well
  0.25
• s4 Dry well
  0.50
• Draw a decision tree include the payoffs that
  occur at each terminal node

8
Houston Oil Decision Tree
s1 750K 100K 650,000
0.10
s2 300K 100K 200,000

0.15
s3 75K 100K 25,000
0.25
s4 0 75K 75,000
0.50
a1
0.10
s1 45,000
a2
0.15
s2 45,000
0.25
s3 45,000
0.50
s4 45,000
a3
0.10
s1 500,000 x 0.50 250,000
0.15
s2 200,000 x 0.50 100,000
0.25
a1 Drill a2 Unconditional land lease a3
Conditional land lease
s3 0
0.50
s4 0
9
Houston Oil Decision Tree
s1 650,000
0.10

• Maximum Likelihood

s2 200,000
0.15
75,000
0.25
s3 25,000
s4 75,000
0.50
a1
0.10
s1 45,000
45,000
a2
0.15
s2 45,000
0.25
45,000
s3 45,000
0.50
s4 45,000
a3
0.10
s1 250,000
0
0.15
s2 100,000
0.25
a1 Drill a2 Unconditional land lease a3
Conditional land lease
s3 0
0.50
Optimal Decision a2 Uncond. lease
s4 0
10
Houston Oil Decision Tree
s1 650,000
0.10

• MaxiMin

s2 200,000
0.15
75,000
0.25
s3 25,000
s4 75,000
0.50
a1
0.10
s1 45,000
45,000
a2
0.15
s2 45,000
0.25
45,000
s3 45,000
0.50
s4 45,000
a3
0.10
s1 250,000
0
0.15
s2 100,000
0.25
a1 Drill a2 Unconditional land lease a3
Conditional land lease
s3 0
0.50
Optimal Decision a2 Uncond. lease
s4 0
11
Houston Oil Decision Tree
s1 650,000
0.10

• Bayes Decision Rule

s2 200,000
0.15
51,250
0.25
s3 25,000
s4 75,000
0.50
a1
0.10
s1 45,000
45,000
a2
0.15
s2 45,000
0.25
51,250
s3 45,000
0.50
s4 45,000
a3
0.10
s1 250,000
40,000
0.15
s2 100,000
0.25
a1 Drill a2 Unconditional land lease a3
Conditional land lease
s3 0
0.50
Optimal Decision a1 Drill
s4 0
12
Example 2 Contractor Bidding

• Two competing contractors (us vs. them) bid on a
  single project. Lowest bid gets it.
• Our possible bids (in 100,000s) 10, 13, 16
• Two uncertainties Competitors bid and actual
  cost of project.
• Competitors Bid 8 11 14 17
• Probability 0.1 0.4
  0.4 0.1
• Actual Cost of Project 6 9 12 15 18
• Probability 0.2 0.2
  0.3 0.2 0.1

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Example 2 Contractor Bidding

• Competitors Bid 8 11 14 17
• Probability 0.1 0.4
  0.4 0.1
• Actual Cost of Project 6 9 12 15 18
• Probability 0.2 0.2
  0.3 0.2 0.1
• To simplify the analysis, convert competitors
  bids into probabilities of us winning or losing,
  based on our bids
• Our Bid Prob(We Win)
  Prob(We Lose)
• 10 0.9 0.1
• 13 0.5 0.5
• 16 0.1 0.9
• What should be our optimal bidding policy?

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Contractor Bidding Decision Tree
AC 6, Prob 0.2, Payoff 4
AC 9, Prob 0.2, Payoff 1

0.9(W)
AC 12, Prob 0.3, Payoff 2
AC 15, Prob 0.2, Payoff 5
0.1(L)
0
AC 18, Prob 0.1, Payoff 8
10
AC 6, Prob 0.2, Payoff 7
0.5(W)
AC 9, Prob 0.2, Payoff 4
13
AC 12, Prob 0.3, Payoff 1
0.5(L)
AC 15, Prob 0.2, Payoff 2
16
0
AC 18, Prob 0.1, Payoff 5
0.1(W)
AC 6, Prob 0.2, Payoff 10
AC 9, Prob 0.2, Payoff 7
AC 12, Prob 0.3, Payoff 4
0.9(L)
AC 15, Prob 0.2, Payoff 1
0
AC 18, Prob 0.1, Payoff 2
15
Contractor Bidding Decision Tree
AC 6, Prob 0.2, Payoff 4
Bayes Decision Rule
1.4
AC 9, Prob 0.2, Payoff 1

0.9(W)
AC 12, Prob 0.3, Payoff 2
1.26
AC 15, Prob 0.2, Payoff 5
0.1(L)
0
AC 18, Prob 0.1, Payoff 8
10
AC 6, Prob 0.2, Payoff 7
1.6
0.5(W)
0.8
AC 9, Prob 0.2, Payoff 4
0.8
13
AC 12, Prob 0.3, Payoff 1
0.5(L)
AC 15, Prob 0.2, Payoff 2
16
0
AC 18, Prob 0.1, Payoff 5
0.46
0.1(W)
AC 6, Prob 0.2, Payoff 10
4.6
AC 9, Prob 0.2, Payoff 7
AC 12, Prob 0.3, Payoff 4
0.9(L)
AC 15, Prob 0.2, Payoff 1
Expected payoff 80,000
0
AC 18, Prob 0.1, Payoff 2
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Contractor Bidding A Complication

• Oh no! A 2nd project is coming up!
• Our possible bids (in 100,000s) 13, 16
• Two uncertainties Competitors bid and actual
  cost of project.
• Competitors Bid 14 18
• Probability 0.5 0.5
• Actual Cost of Project 8 11 15
• Probability 0.4 0.5
  0.1
• Bidding for the 2nd project begins after the 1st
  project is awarded.
• Our firm is too small to handle both projects.
• What should be our optimal bidding policy?

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Contractor Bidding A Complication

• Competitors Bid 14 18
• Probability 0.5 0.5
• Actual Cost of Project 8 11 15
• Probability 0.4 0.5
  0.1
• To simplify the analysis, convert competitors
  bids into probabilities of us winning or losing,
  based on our bids
• Our Bid Prob(We Win) Prob(We Lose)
• 13 1.0 0.0
• 16 0.5 0.5

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Contractor Bidding Decision Tree 2

• 2nd Project
• Bayes Decision Rule

AC 8, Prob 0.4, Payoff 5
2.8
AC 11, Prob 0.5, Payoff 2
1.0(W)
2.8
AC 15, Prob 0.1, Payoff 2
0.0(L)
13
0
AC 8, Prob 0.4, Payoff 8
5.8
2.9
0.5(W)
2.9
16
AC 11, Prob 0.5, Payoff 5
0.5(L)
AC 15, Prob 0.1, Payoff 1
0
Expected payoff 290,000
19
Contractor Bidding Decision Tree 3

• Must now include possibilities that we will NOT
  bid on 1st or 2nd project
• Bayes Decision Rule

1.4
0.9(W)
0.97
0.1(L)
10
2.9 (Tree 2)
3.07
0.5(W)
1.6
2.25
13
Bid on 1st proj.
0.5(L)
16
3.07
2.9 (Tree 2)
3.07
0.1(W)
2.9
No bid on 1st proj.
No bid on 2nd proj.
4.6
0
0.9(L)
Bid on 2nd proj.
2.9 (Tree 2)
2.9 (Tree 2)
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Contractor Bidding Solution

• Optimal Bidding Policy
• We will bid 16 (i.e. 1,600,000) on the 1st
  project.
• If we lose, we will bid 16 (i.e. 1,600,000) on
  the 2nd project.
• Expected Payoff 307,000
• (divided among 26 of us 11,807.69 per person
  ?)