Making Decisions Under Uncertainty Decision Models Making Decisions Under Uncertainty

Decision Analysis Models Decision Analysis is about making optimal decisions when the future is unknown. A decision analysis model consists of: A set of possible decisions Discrete or continuous Finite or infinite 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 For discrete, finite models – these can be summarized in a payoff table.

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?

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: S1 -- Large Rise (> 1500 point gain from current value) S2 -- Small Rise (500-1500 point gain) S3 -- No change (decrease or increase of < 500 points) S4 -- Small Fall (500 - 1200 point drop) S5 -- Large Fall (> 1200 point drop from current value)

The Payoff Table Which decision should be made? Using the “rules of thumb”, the following are the estimates of the gains on a $1000 invetment -$100 -$150 $150 $200 $250 $60 -$200 -$600 $100 $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 $500 Which decision should be made?

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 and vice versa, 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 The bond is always at least as good as the option! The option is dominated by the bond decision! Eliminate the option!

The Resulting Payoff Table -$100 -$150 $150 $200 $250 $60 -$200 -$600 $100 $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 $500 No decision alternative dominates any other decision alternative!

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 $150 $200 $250 $60 -$200 -$600 $100 $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 $500 D1-Gold $200 D1-Gold $300 D3-Stock $500 D3-Stock $250 D4-C/D $60 For this certain state of nature

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

Optimistic Decision Maker MAXIMAX Criterion Optimist: No matter what decision I make, the best state of nature for that decision will occur Max Payoff List Maximum payoff for each decision -$100 -$150 $150 $200 $250 $60 -$200 -$600 $100 $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 $500 $300 $250 Highest -- Choose D3 - Stock $500 $60

Pessimistic Decision Maker MAXIMIN Criterion Pessimist: No matter what decision I make, the worst state of nature for that decision will occur Min Payoff List Minimum payoff for each decision -$100 -$150 $150 $200 $250 $60 -$200 -$600 $100 $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 $500 -$100 -$150 -$600 Highest -- Choose D4 - C/D $60

“I Should Have” Decision Maker MINIMAX REGRET Criterion The 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

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

Pessimistic Regret Decision Maker MINIMAX REGRET Criterion Pessimist: No matter what decision I make, the worst regret for that decision will occur Max Regret List Maximum regret for each decision $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 $240 $140 $100 Regret Table $600 $400 Lowest -- Choose D2 - Bond $660 $440

Indifferent Decision Maker INSUFFICIENT REASON Criterion Indifference: Each state of nature is equally likely Average Value: Sum/5 - best is one with highest sum Sum of Payoffs List Sum of payoffs for each decision -$100 -$150 $150 $200 $250 $60 -$200 -$600 $100 $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 $500 $500 Highest -- Choose D1 - Gold $350 $50 $300

Using the Decision Template Enter Decision Names, State Names, Payoffs Optimal

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

Review Decision Models Dominated Strategies Decisions, States Of Nature, Payoffs Dominated Strategies Decision Making Under Certainty Decision Making Under Uncertainty Optimistic Maximax Pessimistic Maximin Pessimistic Regret Minimax Regret Indifference Insufficient Reason Use of Decision Template