Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial?...
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Transcript of Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial?...
![Page 1: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/1.jpg)
Old Exam Decision Tree
![Page 2: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/2.jpg)
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
![Page 3: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/3.jpg)
Probabilities
• If trial, Probability that judge allows testimony from state troopers = .1
• Conditional probability = P(A|T)=.1
![Page 4: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/4.jpg)
• If testimony, he either wins or loses• If he wins, 60 Democrats• If he loses, 30 Democrats
![Page 5: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/5.jpg)
Same outcomes if no testimony
But different probabilities
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Conditional probability that he loses
• P(lose|testimony) = .6• P(lose|no testimony) = .3
![Page 7: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/7.jpg)
30
60
30
40
60
60
settle
trialtestimony
No testimony
lose
win
lose
win
.6
.3
.1
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E(x) or EMV if testimony
Outcome x P(x) xP(x)
lose 30 .6 =P(lose|test)
18
win 60 1-.6= .4 24
E(x) 42
![Page 9: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/9.jpg)
Note we do E(x) from right to left
Draw tree from leftFind optimal decision from right
![Page 10: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/10.jpg)
E(x) if no testimony
Outcome x P(x) xP(x)
Lose 30 .3=P(lose|no testimony)
9
Win 60 1-.3= .7 42
E(x) 51
![Page 11: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/11.jpg)
30
60
30
40
60
60
settle
trial
42
51
testimony
No testimony
lose
win
lose
win
![Page 12: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/12.jpg)
E(x) if trial
Outcome x P(x) xP(x)
testimony 42 .1 4.2
No testimony
51 1-.1=.9 45.9
E(x) 50.1
![Page 13: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/13.jpg)
30
60
30
40
60
60
settle
50.1
trial
42
51
testimony
No testimony
lose
win
lose
win
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Decision Node
Action Expected Number of Democrats
Settle 40
Trial 50.1 = max
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30
60
30
50.1
40
60
60
settle
50.1
trial
42
51
testimony
No testimony
lose
win
lose
win
![Page 16: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/16.jpg)
Exam Format
• Max E(x) = 50.1• Interpretation: Bill should go to trial
![Page 17: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/17.jpg)
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
![Page 18: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/18.jpg)
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 Norton’s character describes calculation of E(x) for recall decision in film “Fight Club”
• Pop Culture: Miguel Ferrer’s character explains decision to smuggle drugs across border in film “Traffic”
![Page 19: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/19.jpg)
Another Old Exam Problem
Two-stage decision
![Page 20: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/20.jpg)
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
![Page 21: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/21.jpg)
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
![Page 23: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/23.jpg)
3
3+3=6
3+15=18
3+1=4
10
2
sign
Don’t sign
cancel
Not cancel
Another yr
movie well
Not well
Comedies do well
Not well
.4
.2
.3
![Page 24: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/24.jpg)
E(x) if he signs, not cancelled, and X-files movie
Outcome x P(x) xP(x)
X-files movie does well
18 .2 3.6
Not well 4 1-.2=.8 3.2
E(x) 6.8
![Page 25: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/25.jpg)
3
3+3=6
3+15=18
3+1=4
10
2
6.8
sign
Don’t sign
cancel
Not cancel
Another yr
movie well
Not well
Comedies do well
Not well
.4
.2
.3
![Page 26: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/26.jpg)
Decision Node
Act E(x)
Another year 6
X-files movie 6.8 = max
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3
3+3=6
3+15=18
3+1=4
10
2
6.8
6.8sign
Don’t sign
cancel
Not cancel
Another yr
movie well
Not well
Comedies do well
Not well
.4
.2
.3
![Page 28: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/28.jpg)
E(x) if he signs
Outcome x P(x) xP(x)
Cancel 3 .4 1.2
Not cancel 6.8 1-.4 = .6 4.08
E(x) 5.28
![Page 29: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/29.jpg)
3
3+3=6
3+15=18
3+1=4
10
2
6.8
6.8
5.28
sign
Don’t sign
cancel
Not cancel
Another yr
movie well
Not well
Comedies do well
Not well
.4
.2
.3
![Page 30: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/30.jpg)
E(x) if he does not sign
Outcome x P(x) xP(x)
Comedies do well
10 .3 3
Not well 2 1-.3 = .7 1.4
E(x) 4.4
![Page 31: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/31.jpg)
3
3+3=6
3+15=18
3+1=4
10
2
6.8
6.8
5.28
4.4
sign
Don’t sign
cancel
Not cancel
Another yr
movie well
Not well
Comedies do well
Not well
.4
.2
.3
![Page 32: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/32.jpg)
Final Decision Node
Act E(x)
Sign 5.28 = max
Don’t sign 4.4
![Page 33: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/33.jpg)
3
3+3=6
3+15=18
3+1=4
10
2
6.8
6.8
5.28
4.4
5.28
sign
Don’t sign
cancel
Not cancel
Another yr
movie well
Not well
Comedies do well
Not well
.4
.2
.3
![Page 34: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/34.jpg)
Exam Format
• Max E(x) = 5.28• Interpretation: He should sign the contract.
If not cancelled, he should do the X-files movie.
![Page 35: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/35.jpg)
Post-exam update
• Film “evolution” grossed $37 million
![Page 36: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/36.jpg)
Decision Tree: MINIMIZE Cost
Managed Health Care Example
![Page 37: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/37.jpg)
Decision Maker: HMO physician
• MD must decide whether or not to run test to determine if patient has disease
![Page 38: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/38.jpg)
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
![Page 39: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/39.jpg)
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
![Page 40: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/40.jpg)
Probabilities
• P(test positive) = .01• P(patient dies|test positive but no treatment)
= .05 • P(patient ok|test positive but no treatment)
= .95• This problem assumes only 2 outcomes:
dead or ok. In real life, several branches.
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10+1 = 11
1
1000
0
0
positive
negative
die
ok
negative
positive
Run test
Do not run test
.01
.01
.05.95
![Page 42: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/42.jpg)
E(x) if run test
Outcome x P(x) xP(x)
Test positive
11 .01 .11
Test negative
1 1-.01 = .99 .99
1.1
![Page 43: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/43.jpg)
10+1 = 11
1
1000
0
0
1.1
positive
negative
die
ok
negative
positive
Run test
Do not run test
.01
.01
.05.95
![Page 44: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/44.jpg)
E(x) if do not run test, but patient would have tested positive
Outcome x P(x) xP(x)
Die 1000 .05 50
Ok 0 .95 0
E(x) 50
![Page 45: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/45.jpg)
10+1 = 11
1
1000
0
0
50
1.1
positive
negative
die
ok
negative
positive
Run test
Do not run test
.01
.01
.05.95
![Page 46: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/46.jpg)
E(x) if do not run test
Outcome x P(x) xP(x)
Test positive
50 .01 0.5
Test negative
0 1-.01=.99 0
0.5
![Page 47: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/47.jpg)
10+1 = 11
1
1000
0
0
50
.5
1.1
positive
negative
die
ok
negative
positive
Run test
Do not run test
.01
.01
.05.95
![Page 48: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/48.jpg)
Decision Node
Act E(x)
Run test 1.1
Do not run test 0.5=min
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10+1 = 11
1
1000
0
0
50
.5
1.1
0.5
positive
negative
die
ok
negative
positive
Run test
Do not run test
.01
.01
.05.95
![Page 50: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/50.jpg)
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
Act Test positive Test negative
Run test 11 1
Do not run test 50 0
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OL = |Best – Actual|
• Here: Best = MIN in col• OL = |MIN – Actual|
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Payoff Table
Act Test positive Test negative
Run test 11=MIN in col = Best in col
1
Do not run test 50 0 = min in col = Best in col
![Page 55: Old Exam Decision Tree. Decision: Should Bill settle lawsuit with Paula? Actions: settle or trial? Objective: Maximize number of Democrats in Senate in.](https://reader035.fdocuments.us/reader035/viewer/2022062311/5a4d1b0a7f8b9ab05998a87a/html5/thumbnails/55.jpg)
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
Act Test positive Test negative
Run test 11-11 = 0 |0-1| = 1
Do not run test |11-50| = 39 0-0 = 0
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Expected EOL if MD runs test
Outcome OL P(OL) OL*P(OL)
Test positive
0 .01 0
Test negative
1 .99 .99
EOL .99
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Expected OL if MD does not run test
Outcome OL P(OL) OL*P(OL)
Test positive
39 .01 0.39
Test negative
0 .99 0
EOL 0.39
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MIN EOL
Act EOL
Run test .99
Don’t run test .39 = 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
Act Test positive Test negative
Run test 0 1 = MAX IN ROW
Do not run test 39 = MAX IN ROW
0
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MINIMAX
MINImum of MAXimum OL
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Minimax
Act MAX OL
Run Test 1 = MIN
Do not run test 39
Minimax Decision: Run Test