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Module C1 Decision Models Uncertainty

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What is a Decision Analysis Model? Decision Analysis Models is about making optimal decisions when the future is unknown. The model consists of: –A set of possible decisions Discrete or continuous Finite or infinte –A set of states of nature for the future –A “payoff” for each decision-state of nature combination A return when a particular state of nature occurs after making a particular decision

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Types of Decisions Discrete or continuous; Finite or infinite Examples Choose any number –continuous and infinite Choose any integer –discrete and infinite Choose any number between 1 and 5 –Continuous and finite Choose any integer between 1 and 5 –Discrete and finite

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Payoff Tables We will concentrate on models in which the set of possible decisions and the states of nature are both discrete and finite –We can list all possibilities If we can determine a payoff for each decision-state of nature combination, we can construct a payoff table

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Example -- Flipping Coins Decisions: D 1 -- Call Heads D 2 -- Call Tails States of Nature: S 1 -- Result is Heads S 2 -- Result is Tails Payoffs:$1 if right -$1 if wrong

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Simple Payoff Table $1 -$1$1 -$1 S1: Flip Is Heads S2: Flip Is Tails D1: Call Heads D2: Call Tails Decisions States of Nature Payoffs

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Example -- Tom Brown Investment Tom Brown has $1000 to invest in: –Gold -- generally moves opposite of market –Bond -- generally moves with the market –Stock -- generally moves with the market –C/D -- pays same small amount regardless of market –Option -- generally moves with the market How do we measure the market? How do we determine payoffs?

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Measuring the Market Could use DOW, NASDAQ, S&P, other, etc. –Let’s use the Dow Could list every possible outcome for the DOW between 500 and 50,000 say Here we define: –S 1 -- Large Rise (> 1500 point gain from current value) –S 2 -- Small Rise (500-1500 point gain) –S 3 -- No change (decrease or increase of < 500 points) –S 4 -- Small Fall (500 - 1200 point drop) –S 5 -- Large Fall (> 1200 point drop from current value)

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The Payoff Table Using the “rules of thumb”, the following are the estimates of the gains on a $1000 invetment -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D D5: Option$200 $60 $500 -$200 $60 -$150 $60 $150 $60 $250$100 Which decision should be made?

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Dominated Strategies Before deciding which decision should be made, we can eliminate one that will never be made. We see that sometimes Gold is better than Bond and vice versa, Stock is better than CD, etc. But let’s look at the option vs. the bond -$150-$100$150$200$250 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D2: Bond D5: Option$200-$200-$150$150 The bond is always at least as good as the option! Thus, we would never choose the option -- it is dominated by the bond decision! Eliminate the option!

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The Resulting Payoff Table -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100 No decision alternative dominates any other decision alternative!

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Decision Making Under Certainty We are never “certain” about the future, but suppose we were. Strategy: For the certain state of nature, choose the decision with the highest payoff -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100 For this certain state of nature D3-Stock $500 D3-Stock $250 D1-Gold $200 D1-Gold $300 D3-C/D $60

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Decision Making Under Uncertainty Situation: We have no idea which state of nature will occur Approach: Depends on the attitude of the decision maker –Optimistic –Pessimistic (Conservative) –Regrets Not Making Best Choice –Each state of Nature Equally Likely

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Optimistic Decision Maker MAXIMAX Criterion Optimist: No matter what decision I make, the best state of nature for that decision will occur -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100 Max Payoff List Maximum payoff for each decision $300 $250 $500 $60 Highest -- Choose D3 - Stock

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Pessimistic Decision Maker MAXIMIN Criterion Pessimist: No matter what decision I make, the worst state of nature for that decision will occur -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100 Min Payoff List Minimum payoff for each decision -$100 -$150 -$600 $60 Highest -- Choose D4 - C/D

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“I Should Have” Decision Maker MINIMAX REGRET Criterion Regret for a decision given a particular state of nature is the difference between the best payoff for that state of nature and the payoff for that decision given that state of nature

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-$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D Payoff Table Regret Table $660 250-60 $0 $600 $250 500-60 $0 $440 $100 $0$150 200-60 $50 $190$140 $500 300-60 $0 $50 $240 $210 $0 $400 60 - 0$60

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Pessimistic Regret Decision Maker MINIMAX REGRET Criterion Pessimist: No matter what decision I make, the worst regret for that decision will occur $600 $210$400$50 $250 $190 $500$660 $150$0 $60 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$440 $0 $240$0$140 $0$100 Max Regret List Maximum regret for each decision $600 $400 $660 $440 Lowest -- Choose D2 - Bond Regret Table

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Indifferent Decision Maker INSUFFICIENT REASON Criterion Indifference: Each state of nature is equally likely Average Value = Sum/5 -- best is one with highest sum -$100 -$150-$100$150$200$250 $60 -$200-$600 $100$200$300$0 S1 Lg Rise S2 Sm Rise S3 No Chg. S4 Sm Fall S5 Lg Fall D1: Gold D2: Bond D3: Stock D4: C/D$60 $500 $60 $250$100 Sum of Payoffs List Sum of payoffs for each decision $500 $350 $50 $300 Highest -- Choose D1 - Gold

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Using the Decision Template Enter Decision Names State Names Payoffs Optimal

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Summary of Decisions Optimistic ---Stock Pessimistic ---C/D Pessimistic Regret ---Bond Indifference---Gold Which one should be chosen? Depends on the decision maker!

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Module C1 Review Decision Models –Decisions, States Of Nature, Payoffs Dominated Strategies Decision Making Under Certainty Decision Making Under Uncertainty –Optimistic Maximax –PessimisticMaximin –Pessimistic RegretMinimax Regret –IndifferenceInsufficient Reason Use of Decision Template

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