Old Exam Decision Tree

1
Old Exam Decision Tree
2
Decision Should Bill settle lawsuit with Paula?

• Actions settle or trial?
• Objective Maximize number of Democrats in Senate
  in 1999
• If he settles, 40 Dems

3
Probabilities

• If trial, Probability that judge allows testimony
  from state troopers .1
• Conditional probability P(AT).1

4

• If testimony, he either wins or loses
• If he wins, 60 Democrats
• If he loses, 30 Democrats

5
Same outcomes if no testimony

• But different probabilities

6
Conditional probability that he loses

• P(losetestimony) .6
• P(loseno testimony) .3

7
40
settle
30
.6
lose
60
win
testimony
trial
30
lose
.1
.3
60
No testimony
win
60
8
E(x) or EMV if testimony
9
Note we do E(x) from right to left

• Draw tree from left
• Find optimal decision from right

10
E(x) if no testimony
11
40
settle
30
lose
42
60
win
testimony
trial
30
lose
51
60
No testimony
win
60
12
E(x) if trial
13
40
settle
30
lose
42
60
win
testimony
trial
30
lose
51
50.1
60
No testimony
win
60
14
Decision Node
15
40
settle
30
lose
42
60
50.1
win
testimony
trial
30
lose
51
50.1
60
No testimony
win
60
16
Exam Format

• Max E(x) 50.1
• Interpretation Bill should go to trial

17
Post-exam Update

• New Objective Maximize number of electoral votes
  for Al Gore in 2000
• If Bill had settled case, scandal would have been
  forgotten by Nov 2000
• Gore might have won his home state of Tenn (and
  Arkansas?) if no impeachment trial

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Unethical Decision Trees

• Ford used decision tree to decide NOT to recall
  Pinto after gas tanks exploded
• Firestone used decision tree to decide NOT to
  recall tires after SUV rollovers
• Pop Culture Ed Nortons character describes
  calculation of E(x) for recall decision in film
  Fight Club
• Pop Culture Miguel Ferrers character explains
  decision to smuggle drugs across border in film
  Traffic

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Another Old Exam Problem

• Two-stage decision

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Should David sign contract to do X-Files 2001-02?

• Objective maximize expected monetary value (all
  numbers in millions of dollars)
• If he signs, he earns 3
• If cancelled after 2002, no further income
• If not cancelled, a second decision in 2002
  decide between another year on TV for another 3,
  or an X-Files movie
• If movie does well, an additional 15, otherwise
  an additional 1

21
If he does NOT sign contract,

• He does comedy movies
• If they do well, he earns 10
• If they do not do well, he earns 2

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Probabilities

• P(X-Files cancelled) .4
• P(X-Files movie does well) .2
• P(Comedy movies do well) .3

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3
cancel
.4
336
Another yr
Not cancel
31518
sign
movie
well
.2
314
Not well
10
Dont sign
Comedies do well
.3
2
Not well
24
E(x) if he signs, not cancelled, and X-files movie
25
3
cancel
.4
336
Another yr
Not cancel
31518
sign
movie
well
.2
6.8
314
Not well
10
Dont sign
Comedies do well
.3
2
Not well
26
Decision Node
27
3
cancel
.4
336
Another yr
Not cancel
6.8
31518
sign
movie
well
.2
6.8
314
Not well
10
Dont sign
Comedies do well
.3
2
Not well
28
E(x) if he signs
29
3
cancel
.4
336
5.28
Another yr
Not cancel
6.8
31518
sign
movie
well
.2
6.8
314
Not well
10
Dont sign
Comedies do well
.3
2
Not well
30
E(x) if he does not sign
31
3
cancel
.4
336
5.28
Another yr
Not cancel
6.8
31518
sign
movie
well
.2
6.8
314
Not well
10
Dont sign
Comedies do well
4.4
.3
2
Not well
32
Final Decision Node
33
3
cancel
.4
336
5.28
Another yr
Not cancel
6.8
31518
sign
movie
well
.2
6.8
5.28
314
Not well
10
Dont sign
Comedies do well
4.4
.3
2
Not well
34
Exam Format

• Max E(x) 5.28
• Interpretation He should sign the contract. If
  not cancelled, he should do the X-files movie.

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Post-exam update

• Film evolution grossed 37 million

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Decision Tree MINIMIZE Cost

• Managed Health Care Example

37
Decision Maker HMO physician

• MD must decide whether or not to run test to
  determine if patient has disease

38
If MD runs test

• Cost of test 1000
• If test is positive, assume patient wants
  treatment, which costs 10,000
• On tree, write in thousands of dollars
• Test 1
• Treatment 10

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If MD does not run test

• If patient had disease, was diagnosed too late,
  and died, survivors win lawsuit, and HMO pays out
  1,000,000
• Tree 1000

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Probabilities

• P(test positive) .01
• P(patient diestest positive but no treatment)
  .05
• P(patient oktest positive but no treatment)
  .95
• This problem assumes only 2 outcomes dead or ok.
  In real life, several branches.

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101 11
positive
.01
1
negative
Run test
1000
die
.05
.95
positive
Do not run test
0
ok
.01
0
negative
42
E(x) if run test
43
101 11
positive
.01
1.1
1
negative
Run test
1000
die
.05
.95
positive
Do not run test
0
ok
.01
0
negative
44
E(x) if do not run test, but patient would have
tested positive
45
101 11
positive
.01
1.1
1
negative
Run test
1000
die
50
.05
.95
positive
Do not run test
0
ok
.01
0
negative
46
E(x) if do not run test
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101 11
positive
.01
1.1
1
negative
Run test
1000
die
50
.05
.95
positive
Do not run test
0
ok
.01
.5
0
negative
48
Decision Node
49
101 11
positive
.01
1.1
1
negative
Run test
1000
die
0.5
50
.05
.95
positive
Do not run test
0
ok
.01
.5
0
negative
50
Exam Format

• Min E(x) 0.5 from tree
• Interpretation MD should not run test, for
  expected cost of 500

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EVPI if minimizing cost

• Simplified version of previous problem

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Payoff Table
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OL Best Actual

• Here Best MIN in col
• OL MIN Actual

54
Payoff Table
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Interpretation

• If MD knew test would come out positive, best
  decision is to run test
• If MD knew test would come out negative, best
  decision is to NOT run test

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Opportunity Loss Table
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Expected EOL if MD runs test
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Expected OL if MD does not run test
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MIN EOL
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Interpretation

• Do NOT run test
• EVPI Expected Value of Perfect Information
  MIN EOL .39
• MD would pay up to 390 for perfect information
  about test result before running test

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Decision Making Without Probability
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MINIMAX

• Return to OL Table

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Opportunity Loss Table
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MINIMAX

• MINImum of MAXimum OL

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Minimax