DADSS Lecture 6: Introduction to Decision Analysis John Gasper

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)

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

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

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

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

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

Heuristics Mixture of optimism and pessimism Hurwicz with  =percent optimist StrongModerateWeakMix Bonds  + 3(1-  ) Stocks  - 2(1-  ) T-Bill6.5

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

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

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

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

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

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

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

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

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

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, , …}

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

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

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

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

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?

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

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

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)

Decision Analysis Outside Porch Inside Sun Rain S R S R The Party Problem Sun 40% Rain 60% Out 1000 Porc h 9020 In 4050

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

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

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?

O P I p 1-p p p S R S R S R p 20+70p 50-10p Problem Given Various Probabilities of Sun

Strategy Regions Indoors 50-10p Porch 20+70p Prior Outdoors p(Sun) Expected Value of Alternative 100p p=20+70p 30p= p=50-10p 60p=30 IndoorsPorch Outdoors

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

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

p= p(Sunny day) q = p(Cloudy day) 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

p= p(Sunny day) q = p(Cloudy day) 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

p= p(Sunny day) q = p(Cloudy day) 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

Outdoors Indoors 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(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

Outdoors Indoors 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

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

Decision Analysis O P I S R S R S R The Party Problem O P I p 1-p p p S R S R S R p 20+70p 50-10p Problem Given Various Probabilities of Sun Outdoors Porch Indoors p(Sun) Expected Value of Alternative 50-10p 20+70p 100p Prior

Multiple Outcome States Outdoors Indoors Sun Cloudy Rain Sun Cloudy Rain Outdoors Indoors Sun Cloudy Rain Sun Cloudy Rain p q 1-p-q p q 40p + 50q + 60(1-p-q) 100p + 70q p= p(Sunny day) q = p(Cloudy day) Indoors Outdoors

Two Layers of Uncertainty Outdoors Indoors Sun Rain.4.6 Fun Date Bad Date Fun Date Bad Date Sun Rain.4.6 Fun Date Bad Date Fun Date Bad Date Outdoors Indoors 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