1. DADSS Lecture 6: Introduction to Decision Analysis John Gasper

2. Administrative Details Homework #3 due Thursday at 5pm How’s it coming? It’s longer… Homework 4 will be posted Thursday Questions from last class? Readings on Decision Analysis are in the text, Chapter 9 (but lecture > text)

3. Decision Problem StrongModerateWeak Bonds1263 Stocks153-2 T-Bill6.5 Choices States

4. Heuristics Optimist: Maxi-Max (best of the best) StrongModerateWeakMax Bonds1263 Stocks153-2 T-Bill6.5

5. Heuristics Optimist: Maxi-Max (best of the best) StrongModerateWeakMax Bonds1263 Stocks153-2 15 T-Bill6.5

6. Heuristics Pessimist: Maxi-Min (best of the worst) StrongModerateWeakMin Bonds1263 Stocks153-2 T-Bill6.5

7. Heuristics Pessimist: Maxi-Min (best of the worst) StrongModerateWeakMin Bonds1263 3 Stocks153-2 T-Bill6.5

8. Heuristics Mixture of optimism and pessimism Hurwicz with  =percent optimist StrongModerateWeakMix Bonds1263 12  + 3(1-  ) Stocks153-2 15  - 2(1-  ) T-Bill6.5

9. Heuristics Rational (Laplace) all states equally likely: Maxi-Average (best average) StrongModerateWeakMin Bonds1263 (1/3)*12 + (1/3)*6 + (1/3)*3 Stocks153-2 (1/3)*15 + (1/3)*3 - 2(1/3) T-Bill6.5

10. Heuristics Regret: Mini-Max-Regret Transform to a regret matrix StrongModerateWeak Bonds1263 Stocks153-2 T-Bill6.5 Regret: StrongModerateWeak Bonds15-12=3.53.5 Stocks03.58.5 T-Bill8.500

11. Heuristics Regret: Mini-Max-Regret Transform to a regret matrix StrongModerateWeak Bonds1263 Stocks153-2 T-Bill6.5 Regret: StrongModerateWeakMin Bonds15-12=3.53.5 Stocks03.58.5 T-Bill8.500

12. Failures of Simple Decision Rules Laplace, Hurwicz, Regret, etc. are all useful rules in many cases, but – like all heuristics – they fail in certain cases The use of heuristic rules must include recognition of their limitations

13. Min/Max Counterexample Optimism (maxi-max) Pessimism (maxi-min) S1S2Maximax Alt 1989 Alt 210-50,00010 S1S2Maximin Alt 13110,00031 Alt 2323332

14. Regret Counterexample PayoffsRegretMinimax Alt 180  04  4 Alt 224  606 PayoffsRegretMinimax Alt 180  077 Alt 22463  6 Alt 317707 But suppose we now add another alternative…

15. Laplace Counterexample GoodBadEV Alt 1223026 Alt 24610  28 Suppose we’re uncertain about the weather: But what if we structured the problem as: Bad Weather GoodRainFogSnowEV Alt 12230  28 Alt 24610 19

16. What is Causing the Problems? These methods all give too little or too non-specific attention to probabilities Their treatment of probabilities can introduce biases and distort answers

17. Choice and Consequence A decision analysis involves four parts: Choices Have a party inside, on the porch, or outside States It will rain or not Acts I pick a party location Outcomes/Consequences The party is a success or not

18. Choices In the context of the problem, what can we do? Narrow construction is important – after all, almost anything is possible. Choice Examples: Discrete: What direction to turn? {Turn Left, Turn Right} Continuous: What temperature to set? {22, 22.1, 22.01, 22.001, …}

19. States States represent the role of chance in the environment Either it’s sunny out or it isn’t, but beforehand, all we can say is that there is a 10% chance of rain, etc. For most decisions, the uncertainty over the states causes most of the problems

20. Acts Acts are what we do when faced with choices and states If we are optimizing, our act will also happen to be the best choice The burden, then, is on determining how choices and acts can be reconciled with our limited knowledge of the states and their impact on outcomes

21. Outcomes Given an act, what happens next is the outcome Relationships between the 4 parts: If all the choices lead to the same outcome, the act is irrelevant If more than one choice produces the same outcome, those choices are equivalent (I am indifferent between them) Failures of these relationships mean the problem has not been specified correctly

22. Decisions Under Certainty If there is only 1 state possible, then decisions are very simple: If you know that the weather will be good, then you hold the party outside With certainty, all that you have to do is pick the most desirable choice and you can get it We are rarely so lucky

23. Decisions Under Uncertainty Our acts must be made even though there is an uncertain linkage between choices and outcomes This uncertain linkage is the states Suppose, for example, holding the party outside and it rains would be a disaster. Now what do I do?

24. Decisions Under Uncertainty I need to know: Something about the likelihood of the various states Something about my preferences for the various outcomes: I prefer a outdoor sunny party to an indoor rainy party I prefer an indoor rainy party to a porch rainy party

25. Decisions Under Uncertainty Problem Representation Here is the party problem in a table, which is how it might appear in Excel Sun 40% Rain 60% Outdoors 1000 Porch 9020 Indoors 4050 Choices States

26. Representing Uncertain Choice Decision Trees  will represent choice nodes  will represent chance nodes (possible states) Branches of choice nodes must be exhaustive (contain all possible choices) Branches of chance nodes must add to 1 (they will be probabilities)

27. Decision Analysis Outside Porch Inside.4.6.4.6.4.6 Sun Rain S R S R 100 0 90 20 40 50 The Party Problem Sun 40% Rain 60% Out 1000 Porc h 9020 In 4050

28. Working with Decision Trees Folding back the trees look forward and reason backwards Start at the outcomes and incorporate any uncertainty Rational decision makers maximize expected outcomes

29. O P I.4.6.4.6.4.6 S R S R S R 100 0 90 20 40 50 The Party Problem 40=.4(100)+.6(0) 48=.4(90)+.6(20) 46=.4(40)+.6(50) 48 Decision Analysis

30. The Party Problem Tree Is the tree structure valid? Choices: Are Outdoors, Porch and Indoors the only choices available for the problem? States: Is P(Sun) + P(Rain) = 1? What do we really know about the numbers in the problem? Does P(Sun) really equal 0.4?

31. O P I p 1-p p p S R S R S R 100 0 90 20 40 50 100p 20+70p 50-10p Problem Given Various Probabilities of Sun

32. Strategy Regions Indoors 50-10p Porch 20+70p Prior 1.00.20.40.60.80.30.50.70.90.1 20 50 100 Outdoors 100 90 40 p(Sun) Expected Value of Alternative 100p 0 0.667 100p=20+70p 30p=20 0.375 20+70p=50-10p 60p=30 IndoorsPorch Outdoors

33. Indoors Sun Cloudy Rain Sun Cloudy Rain 100 70 0 40 50 60.4.5.1.4.5.1 75=.4(100)+.5(70)+.1(0) 47=.4(40)+.5(50)+.1(60) 75 Multiple Outcome States

34. Outdoors Indoors Sun Cloudy Rain Sun Cloudy Rain 100 70 0 40 50 60 p q 1-p-q p q 40p + 50q + 60(1-p-q) 100p + 70q Multiple Unknowns

35. p= p(Sunny day) q = p(Cloudy day) 0.501.0.50 0.75.25.75 Strategy Regions 100p + 70q = 40p + 50q + 60(1-p-q) 120p + 80q = 60 If p=0 then q=0.75 If q=0 then p=0.50

36. p= p(Sunny day) q = p(Cloudy day) 0.501.0.50 0.75.25.75 Strategy Regions 100p + 70q = 40p + 50q + 60(1-p-q) 120p + 80q = 60 If p=0 then q=0.75 If q=0 then p=0.50

37. p= p(Sunny day) q = p(Cloudy day) 0.501.0.50 0.75.25.75 Indoors Outdoors Strategy Regions 100p + 70q = 40p + 50q + 60(1-p-q) 120p + 80q = 60 If p=0 then q=0.75 If q=0 then p=0.50

38. Outdoors Indoors 100 30 0 40 50 60 Sun Rain.4.6 Good Mood Bad Mood Good Mood Bad Mood Sun Rain.4.6 Good Mood Bad Mood Good Mood Bad Mood.2.8.2.8.2.8.2.8 60 10 68 =.2(100)+.8(60) 6 =.2(30)+.8(0) 30.8 =.4(68)+.6(6) 12 =.2(40)+.8(10) 52 =.2(60)+.8(50) 36 =.4(12)+.6(52) 36 Two-Stage

39. Outdoors Indoors 100 30 0 40 50 60 Sun Rain p 1-p Good Mood Bad Mood Good Mood Bad Mood Sun Rain p 1-p Good Mood Bad Mood Good Mood Bad Mood 1-q60 10 q 1-q q q q 10pq+60p+30q 20pq-40p+10q+50 Parametric Analysis

40. Strategy Regions 10pq+60p+30q = 20pq-40p+10q+50 100p+20q-10pq = 50 If q=0 then p=0.50 If q=1 then p=0.33

41. Decision Analysis O P I.4.6.4.6.4.6 S R S R S R 100 0 90 20 40 50 The Party Problem O P I p 1-p p p S R S R S R 100 0 90 20 40 50 100p 20+70p 50-10p Problem Given Various Probabilities of Sun 1.00.20.40.60.80.30.50.70.90.1 20 50 100 Outdoors Porch Indoors 100 90 40 0.375 0.667 p(Sun) Expected Value of Alternative 50-10p 20+70p 100p Prior

42. Multiple Outcome States Outdoors Indoors Sun Cloudy Rain Sun Cloudy Rain 100 70 0 40 50 60.4.5.1.4.5.1 Outdoors Indoors Sun Cloudy Rain Sun Cloudy Rain 100 70 0 40 50 60 p q 1-p-q p q 40p + 50q + 60(1-p-q) 100p + 70q p= p(Sunny day) q = p(Cloudy day) 0.501.0.50 0.75.25.75 Indoors Outdoors

43. Two Layers of Uncertainty Outdoors Indoors 100 30 0 40 50 60 Sun Rain.4.6 Fun Date Bad Date Fun Date Bad Date Sun Rain.4.6 Fun Date Bad Date Fun Date Bad Date.2.8.2.8.2.8.2.8 60 10 Outdoors Indoors 100 30 0 40 50 60 Sun Rain p 1-p Fun Date Bad Date Fun Date Bad Date Sun Rain p 1-p Fun Date Bad Date Fun Date Bad Date 1.q60 10 q 1.q q q q 10pq+60p+30q 20pq-40p+10q+50