DECISION MAKING WITHOUT STATE PROBABILITIESThis is called DMUU—Decision Making Under Uncertainty

Decision Criteria• no state probabilities??--Your model is one of

UNCERTAINTY• use UNCERTAINTY criteria

• got state probabilities??--Your model is one of RISK• use RISK criteria

• (probabilities—probability information)

UNCERTAINTY Criteria—used when we don’t know the state probabilities

• Based on DM’s attitude toward the risk• Pessimist, also called maximin• Optimist, also called maximax• in-betweenist• Insufficient reason• Regrettist, also called minimax regret

RISK Criteria• Based on expected or probabilistic considerations• Expected payoff or value• Expected Regret

Other RISK-related measures• Expected payoff of perfect information• Expected value of perfect information

• Expected payoff of sample (imperfect) information• Expected value of sample [imperfect) information

Scenario• Consider the needs of a program manager who must

decide which among several projects to bid on.• Due to resource constraints only one of the projects can

be bid on• There are two future states—WIN the bid or LOSE the bid

The Payoff Table

WIN LOSE

Bid Project 1 $40,000 -$2,000

Bid Project 2 $100,000 -$10,000

Bid Project 3 $50,000 -$8,000

Bid Project 4 $60,000 -$15,000

Don’t bid on anything 0 0

Table 6.3. Payoff Table Involving Choice of one of five decision alternatives, each withuncertain future states.

Assume we don’t know the win/lose probabilities• So we have to use the UNCERTAINTY criteria

• Pessimist criterion• Optimist criterion• Regrettist criterion• In-betweenist criterion• Insufficient reason criterion

PESSIMIST CRITERION1) For each row, find the smallest payoff in the row and record that in a column to the right, labeled ROW MINIMUM

2) Examine the column to the right labeled ROW MINIMUM and pick the alternative with the largest payoff in that column.

Pessimist Criterion--best of all of the worst-case scenarios

• Winner is…

PAYOFF TABLE WIN LOSE ROW MINIMUM

Bid Project 1 $40,000 -$2,000 -$2,000

Bid Project 2 $100,000 -$10,000 -$10,000

Bid Project 3 $50,000 -$8,000 -$8,000

Bid Project 4 $60,000 -$15,000 -$15,000

Do Nothing 0 0 0

Table 6.4. Payoff Table Involving Choice of one of five decision alternatives, using thePESSIMIST criterion

OPTIMIST CRITERION1) For each row, find the largest payoff in the row and record that in a column to the right, labeled ROW MAXIMUM

2) Examine the column to the right labeled ROW MAXIMUM and pick the alternative with the largest payoff in that column.

Optimist Criterion--best of all of the best-case scenarios

• Winner is…

PAYOFF TABLE WIN LOSE ROW MAXIMUM

Bid Project 1 $40,000 -$2,000 $40,000

Bid Project 2 $100,000 -$10,000 $100,000

Bid Project 3 $50,000 -$8,000 $50,000

Bid Project 4 $60,000 -$15,000 $60,000

Do Nothing 0 0 0

IN-BETWEENIST CRITERION1) For each row, combine the smallest payoff in the row with the largest payoff in the row using the formula: *ROW MIN + (1 - )*ROW MAXrecord that in a column to the right, labeled COMBINED;

2) Examine the column labeled COMBINED to the right and pick the alternative with the largest payoff in that column.

In-Betweenist Criterion—alpha= .5

• Winner is…

PAYOFF TABLE

WIN LOSE ROW MIN ROW MAX COMBINED

Bid Project 1 $40,000 -$2,000 -$2,000 $40,000 $19,000 Bid Project 2 $100,000 -$10,000 -$10,000 $100,000 $45,000 Bid Project 3 $50,000 -$8,000 -$8,000 $50,000 $21,000 Bid Project 4 $60,000 -$15,000 -$15,000 $60,000 $22,000 Do Nothing 0 0 0 0 0

REGRETTIST CRITERION1) Form the regret table.

1) For each row in the regret table, find the largest regret number in the row and record that in a column to the right labeled ROW MAXIMUM.

2) Examine the column labeled ROW MAXIMUM to the right and pick the alternative with the smallest regret in that column.

Regret CriterionBid Project 1 $40,000 -$2,000

Bid Project 2 $100,000 -$10,000

Bid Project 3 $50,000 -$8,000

Bid Project 4 $60,000 -$15,000

Don’t bid on anything 0 0

PAYOFF TABLE WIN LOSECOLUMN MAX $100,000 0

REGRET TABLE WIN LOSE ROW MAXIMUM

Bid Project 1 $60,000 $2,000 $60,000

Bid Project 2 $0 $10,000 $10,000

Bid Project 3 $50,000 $8,000 $50,000

Bid Project 4 $40,000 $15,000 $40,000

Do Nothing $100,000 0 $100,000

Regret TablePAYOFF TABLE WIN LOSE COLUMN MAX $100,000 0

REGRET TABLE WIN LOSE ROW MAXIMUM (Maximum regret)

Bid Project 1 $60,000 $2,000 $60,000

Bid Project 2 $0 $10,000 $10,000

Bid Project 3 $50,000 $8,000 $50,000

Bid Project 4 $40,000 $15,000 $40,000

Do Nothing $100,000 0 $100,000

The Winner is…

Two Environments

• Decision-making under uncertainty• When we don’t know the state probabilities

• Optimist, pessimist, in-betweenist, insufficient reason, regrettist

• Decision-making under risk• When we do know the state probabilities

• Expected value or payoff• Expected regret

Decision Making under Risk• Criteria

• Expected value or payoff• Expected regret

• Measures• Expected payoff of perfect information• Expected value of perfect information• Expected payoff of sample information• Expected value of sample information

DMUR Criterion

DMUR (Decision-Making Under risk) Expected Value Criterion1) For each row,calculate the product of the column probability with the payoff in that column and add up all of the products, recording the result in the column labeled EXPECTED VALUE to the right of the payoff table

2) Examine the column labeled EXPECTED VALUE to the right and pick the alternative with the largest payoff in that column.

DMUR--Expected ValuePAYOFF TABLE WIN LOSE Expected Value

Probability DECISION ALTERNATIVES

.7 .3

Bid Project 1 $40,000 -$2,000 $27,400

Bid Project 2 $100,000 -$10,000 $67,000

Bid Project 3 $50,000 -$8,000 $32,600

Bid Project 4 $60,000 -$15,000 $37,500

Do Nothing 0 0 0

EXPECTED REGRET Criterion1) Form the regret table

2) For each row calculated its expected regret by taking the product of each state probability with the regret number in that column, summing all such products

3) Pick the alternative (i.e., row) whose expected regret is smallest

This will always be the same alternative that gets chosen by the expected value or payoff criterion

Expected RegretREGRET TABLE WIN LOSE Expected Regret

Probabilities DECISION ALTERNATIVES

.7 .3

Bid Project 1 $60,000 -$2,000 $42,600

Bid Project 2 $0 -$10,000 $3,000

Bid Project 3 $50,000 -$8,000 $37,400

Bid Project 4 $40,000 -$15,000 $32,500

Do Nothing $100,000 0 $70,000

The Winner is…

Expected Value & Regret

REGRET TABLE WIN LOSE Expected Regret

Probabilities DECISION ALTERNATIVES

.7 .3

Bid Project 1 $60,000 -$2,000 $42,600

Bid Project 2 $0 -$10,000 $3,000

Bid Project 3 $50,000 -$8,000 $37,400

Bid Project 4 $40,000 -$15,000 $32,500

Do Nothing $100,000 0 $70,000

PAYOFF TABLE WIN LOSE Expected Value

ProbabilityDECISIONALTERNATIVES

.7 .3

Bid Project 1 $40,000 -$2,000 $27,400

Bid Project 2 $100,000 -$10,000 $67,000

Bid Project 3 $50,000 -$8,000 $32,600

Bid Project 4 $60,000 -$15,000 $37,500

Do Nothing 0 0 0

Notes• For any alternative, the expected value and expected

regret numbers sum to the expected payoff of perfect information

• The expected value and expected regret criteria always select the same alternative, because when the former is maximized, the latter is minimized

Expected Payoff of Perfect Information, EPPI• Calculated by finding the largest payoff in each column

and then taking the products with the column probabilities and summing these products

• The EPPI is the best we could do if we had perfect information

• $70,000 for this problem

Expected Value of Perfect Information, EVPI• by definition, EVPI = EPPI - EV*• EVPI = $70,000 - $67,000 = $3,000 • The EVPI is the value to us of the additional information• The value is the “best we could do with the additional

information” minus the “best we could do without the additional information”

More Notes• The minimum expected regret is also the EXPECTED

VALUE OF PERFECT INFORMATION—THE ABSOLUTE MAXIMUM WE WOULD BE WILLING TO PAY FOR ADDITIONAL INFORMATION

• PROOF??

RULE OF INSUFFICIENT REASON1) For each row, add-up all of the payoffs in that row and record the result in a column to the right labeled ROW SUM.

2) Examine the column labeled ROW SUM to the right and pick the alternative with the largest payoff in that column.

Insufficient Reason

• Winner is…

PAYOFF TABLE WIN LOSE ROW SUM

Bid Project 1 $40,000 -$2,000 $38,000

Bid Project 2 $100,000 -$10,000 $90,000

Bid Project 3 $50,000 -$8,000 $42,000

Bid Project 4 $60,000 -$15,000 $45,000

Do Nothing 0 0 0