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Probabilities If trial, Probability that judge allows testimony from state troopers =.1 Conditional probability = P(A|T)=.1
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Old Exam Decision Tree
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
Probabilities
• If trial, Probability that judge allows testimony from state troopers = .1
• Conditional probability = P(A|T)=.1
• If testimony, he either wins or loses• If he wins, 60 Democrats• If he loses, 30 Democrats
Same outcomes if no testimony
But different probabilities
Conditional probability that he loses
• P(lose|testimony) = .6• P(lose|no testimony) = .3
30
60
30
40
60
60
settle
trialtestimony
No testimony
lose
win
lose
win
.6
.3
.1
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
Note we do E(x) from right to left
Draw tree from leftFind optimal decision from right
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
30
60
30
40
60
60
settle
trial
42
51
testimony
No testimony
lose
win
lose
win
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
30
60
30
40
60
60
settle
50.1
trial
42
51
testimony
No testimony
lose
win
lose
win
Decision Node
Action Expected Number of Democrats
Settle 40
Trial 50.1 = max
30
60
30
50.1
40
60
60
settle
50.1
trial
42
51
testimony
No testimony
lose
win
lose
win
Exam Format
• Max E(x) = 50.1• Interpretation: Bill should go to trial
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
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”
Another Old Exam Problem
Two-stage decision
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
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
Probabilities
• P(X-Files cancelled) = .4• P(X-Files movie does well) = .2• P(Comedy movies do well) = .3
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
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
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
Decision Node
Act E(x)
Another year 6
X-files movie 6.8 = max
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
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
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
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
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
Final Decision Node
Act E(x)
Sign 5.28 = max
Don’t sign 4.4
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
Exam Format
• Max E(x) = 5.28• Interpretation: He should sign the contract.
If not cancelled, he should do the X-files movie.
Post-exam update
• Film “evolution” grossed $37 million
Decision Tree: MINIMIZE Cost
Managed Health Care Example
Decision Maker: HMO physician
• MD must decide whether or not to run test to determine if patient has disease
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
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
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.
10+1 = 11
1
1000
0
0
positive
negative
die
ok
negative
positive
Run test
Do not run test
.01
.01
.05.95
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
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
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
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
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
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
Decision Node
Act E(x)
Run test 1.1
Do not run test 0.5=min
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
Exam Format
• Min E(x) = 0.5 from tree• Interpretation: MD should not run test, for
expected cost of $ 500
EVPI if minimizing cost
Simplified version of previous problem
Payoff Table
Act Test positive Test negative
Run test 11 1
Do not run test 50 0
OL = |Best – Actual|
• Here: Best = MIN in col• OL = |MIN – Actual|
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
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
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
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
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
MIN EOL
Act EOL
Run test .99
Don’t run test .39 = min EOL
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
Decision Making Without Probability
MINIMAX
• Return to OL Table
Opportunity Loss Table
Act Test positive Test negative
Run test 0 1 = MAX IN ROW
Do not run test 39 = MAX IN ROW
0
MINIMAX
MINImum of MAXimum OL
Minimax
Act MAX OL
Run Test 1 = MIN
Do not run test 39
Minimax Decision: Run Test
