Decision ModelsDecision Models

Making Decisions

Under Uncertainty

Decision Analysis ModelsDecision 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 tablepayoff table.

Example -- Tom Brown Example -- Tom Brown InvestmentInvestment

• 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 MarketMeasuring 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 sayHere 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 TableThe 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

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D

D5: Option $200

$60

$500

-$200

$60

-$150

$60

$150 $150

$60

$250 $100

Which decision should be made?

Dominated StrategiesDominated 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

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D2: BondD5: Option $200 -$200 -$150$150 $150

The bond is always at least as good as the option! The option is dominated by the bond decision! Eliminate the option!Eliminate the option!

The Resulting Payoff TableThe Resulting Payoff Table

-$100

-$150-$100$150$200$250

$60

-$200 -$600

$100 $200 $300 $0

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D $60

$500

$60 $60$60

$250 $100

No decision alternative dominates any other decision alternative!

Decision Making Under CertaintyDecision 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

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D $60

$500

$60 $60$60

$250 $100

For this certainstate of nature

D3-Stock$500

D3-Stock$250

D1-Gold$200

D1-Gold$300

D4-C/D$60

Decision Making Under Uncertainty Decision Making Under Uncertainty • Situation:Situation: We have no idea which state of

nature will occur

• Approach: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 MakerOptimistic Decision MakerMAXIMAXMAXIMAX Criterion Criterion

Optimist: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

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D $60

$500

$60 $60$60

$250 $100

MaxPayoff

List Maximum payoff for each decision

$300

$250

$500

$60

Highest -- Choose D3 - Stock

Pessimistic Decision MakerPessimistic Decision MakerMAXIMINMAXIMIN Criterion Criterion

Pessimist: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

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D $60

$500

$60 $60$60

$250 $100

MinPayoff

List Minimum payoff for each decision

-$100

-$150

-$600

$60

Highest -- Choose D4 - C/D

““I Should Have” Decision I Should Have” Decision MakerMaker

MINIMAX REGRETMINIMAX REGRET Criterion Criterion

The regret 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

-$100

-$150-$100$150$200$250

$60

-$200 -$600

$100 $200 $300 $0

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D $60

$500

$60 $60$60

$250 $100

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D

Payoff TablePayoff Table

Regret TableRegret 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

Pessimistic Regret Decision MakerPessimistic Regret Decision MakerMINIMAX REGRETMINIMAX REGRET Criterion Criterion

Pessimist:Pessimist: No matter what decision I make, the worst regret for that decision will occur

$600

$210$400$50$50$250

$190

$500 $660

$150 $0 $0 $60

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D $440

$0

$240 $0$140

$0 $100

MaxRegret

List Maximum regret for each decision

$600

$400

$660

$440

Lowest -- Choose D2 - Bond

Regret TableRegret Table

Indifferent Decision MakerIndifferent Decision MakerINSUFFICIENT REASONINSUFFICIENT REASON Criterion CriterionIndifference:Indifference: Each state of nature is equally likely

Average Value:Average Value: Sum/5 - best is one with highest sum

-$100

-$150-$100$150$200$250

$60

-$200 -$600

$100 $200 $300 $0

S1Lg Rise

S2Sm Rise

S3No Chg.

S4Sm Fall

S5Lg Fall

D1: Gold

D2: Bond

D3: Stock

D4: C/D $60

$500

$60 $60$60

$250 $100

Sum ofPayoffs

List Sum of payoffs for each decision

$500

$350

$50

$300

Highest -- Choose D1 - Gold

Using the Decision TemplateUsing the Decision Template

Enter Decision Names, State Names, Payoffs

Optimal

Summary of DecisionsSummary of Decisions

• Optimistic --- Stock

• Pessimistic --- C/D

• Pessimistic Regret --- Bond

• Indifference --- Gold

Which one should be chosen?

Depends on the approachDepends on the approachof the decision maker!of the decision maker!

ReviewReview• Decision Models– Decisions, States Of Nature, Payoffs

• Dominated Strategies• Decision Making Under Certainty• Decision Making Under Uncertainty– Optimistic MaximaxMaximax– Pessimistic MaximinMaximin– Pessimistic Regret Minimax RegretMinimax Regret– Indifference Insufficient ReasonInsufficient Reason

• Use of Decision Template