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Old Exam Decision Tree

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

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Old Exam Decision Tree

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  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(A|T)=.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(lose|testimony) = .6 • P(lose|no testimony) = .3

  7. 40 settle 30 .6 lose 60 win testimony trial 30 .1 lose .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

  18. 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”

  19. Another Old Exam Problem Two-stage decision

  20. 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

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

  23. 3 cancel .4 3+3=6 Another yr Not cancel 3+15=18 sign movie well .2 3+1=4 Not well 10 Don’t sign Comedies do well .3 2 Not well

  24. E(x) if he signs, not cancelled, and X-files movie

  25. 3 cancel .4 3+3=6 Another yr Not cancel 3+15=18 sign movie well .2 6.8 3+1=4 Not well 10 Don’t sign Comedies do well .3 2 Not well

  26. Decision Node

  27. 3 cancel .4 3+3=6 Another yr Not cancel 6.8 3+15=18 sign movie well .2 6.8 3+1=4 Not well 10 Don’t sign Comedies do well .3 2 Not well

  28. E(x) if he signs

  29. 3 cancel .4 5.28 3+3=6 Another yr Not cancel 6.8 3+15=18 sign movie well .2 6.8 3+1=4 Not well 10 Don’t sign Comedies do well .3 2 Not well

  30. E(x) if he does not sign

  31. 3 cancel .4 5.28 3+3=6 Another yr Not cancel 6.8 3+15=18 sign movie well .2 6.8 3+1=4 Not well 10 Don’t sign Comedies do well 4.4 .3 2 Not well

  32. Final Decision Node

  33. 3 cancel .4 5.28 3+3=6 Another yr Not cancel 6.8 3+15=18 sign movie well .2 6.8 5.28 3+1=4 Not well 10 Don’t 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.

  35. Post-exam update • Film “evolution” grossed $37 million

  36. 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

  39. 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

  40. 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.

  41. 10+1 = 11 positive .01 1 Run test negative 1000 die .05 .95 Do not run test positive 0 ok .01 0 negative

  42. E(x) if run test

  43. 10+1 = 11 positive .01 1.1 1 Run test negative 1000 die .05 .95 Do not run test positive 0 ok .01 0 negative

  44. E(x) if do not run test, but patient would have tested positive

  45. 10+1 = 11 positive .01 1.1 1 Run test negative 1000 die 50 .05 .95 Do not run test positive 0 ok .01 0 negative

  46. E(x) if do not run test

  47. 10+1 = 11 positive .01 1.1 1 Run test negative 1000 die 50 .05 .95 Do not run test positive 0 ok .01 .5 0 negative

  48. Decision Node

  49. 10+1 = 11 positive .01 1.1 1 Run test negative 1000 die 0.5 50 .05 .95 Do not run test positive 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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