Non-probability decision rules

Dr. Yan Liu

Department of Biomedical, Industrial & Human Factors Engineering

Wright State University

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Types of Decision Making Environment

Non-Probability Decision Making Decision maker knows with certainty the consequences of every alternative or

decision choice

Decision Making under Risk Decision maker can assign the probabilities of the various outcomes

Decision Making under Uncertainty Decision maker can neither predict nor describe the probabilities of the various

outcomes

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Types of Non-Probabilistic Decision Rules

Lexicographic Ordering Satisficing Maxmax Payoff Maxmin Payoff Minmax Regret Laplace Hurwitz Principle

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Desirable Properties of Decision Rules

Transitivity If alternative A is preferred to alternative B and alternative B is preferred to

alternative C, then alternative A is preferred to alternative C

Column Linearity The preference relation between two alternatives is unchanged if a constant is

added to all entries of a column of the decision table

Addition/Deletion of Alternatives The preference relation between two alternatives is unchanged if another

alternative is added/deleted from the decision table

Addition/Deletion of Identical Columns The preference relation between two alternatives is unchanged if a column with

the same value in all alternatives is added/deleted to the decision table

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Lexicographic Ordering

V1≥V2≥ ∙∙∙≥Vn, n values are ordered in order of importance

Compare different decision alternatives on the most important value, and continue until one alternative is the best

Values

Alternatives Safety Price Reliability

A High $15k High

B Medium $11k Medium

C High $13k Medium

Non-exhaustive comparisons in values and can be efficient when there are many values

C > A > B

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Satisficing/Minimum Aspiration Level

Select any alternative which satisfies the minimum aspiration levels (the minimum acceptable criteria) of all values

Values

AlternativesSafety

≥MediumPrice≤13k

Reliability ≥Medium

A High $15k High

B Medium $11k Medium

C High $13k Medium

May not be optimal because not all alternatives will be considered as long as one satisfactory alternative is found

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Maxmax Payoff

Select the alternative which results in the maximum of maximum payoffs; an optimistic criterion

Outcomes

Alternatives O1 O2 O3

A $1,000 $1,000 $1,000

B $10,000 -$7,000 $500

C $5,000 $0 $800D $8,000 -$2,000 $700

Maximum Payoff

$1,000

$10,000

Payoff Table

$5,000

$8,000

B > D > C > A

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Outcomes

Alternatives O1 O2 O3

A $1,000 $1,000+9,000 $1,000

B $10,000 -$7,000+9,000 $500

C $5,000 $0+9,000 $800D $8,000 -$2,000+9,000 $700

Maximum Payoff

$10,000$10,000

$9,000

$8,000

A = B > C > D

Maxmax payoff violates column linearity

Payoff Table

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Outcomes

Alternatives O1 O2 O3 O4

A $1,000 $1,000 $1,000 $8,000

B $10,000 -$7,000 $500 $8,000

C $5,000 $0 $800 $8,000D $8,000 -$2,000 $700 $8,000

Payoff Table

Maximum Payoff

$8,000

$10,000

$8,000$8,000

B > A = C = D

Maxmax payoff violates addition/deletion of identical columns

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Maxmin Payoff

Select the alternative which results in the maximum of minimum payoffs; a pessimistic criterion

Outcomes

Alternatives O1 O2 O3

A $1,000 $1,000 $1,000

B $10,000 -$7,000 $500

C $5,000 $0 $800D $8,000 -$2,000 $700

Minimum Payoff

$1,000

-$7,000

Payoff Table

$0

-$2,000

A > C > D > B

Maxmin payoff violates column linearity and addition/deletion of identical columns

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Minmax Regret Select the alternative which results in the minimum of maximum regret Regret is the difference between the maximum payoff possible for a specific outcome and the payoff actually obtained when a specific alternative is chosen and that outcome is encountered

Outcomes

Alternatives O1 O2 O3

A $1,000 $1,000 $1,000

B $10,000 -$7,000 $500

C $5,000 $0 $800

D $8,000 -$2,000 $700

Maximum Regret

Payoff Table

Outcomes

O1 O2 O3

$9,000 $0 $0

$0 $8,000 $500

$5,000 $1,000 $200

$2,000 $3,000 $300

Regret Table

$9,000

$8,000

$5,000

$3,000

D > C > B > A

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Outcomes

Alternatives O1 O2 O3

A $1,000 $1,000 $1,000

B $10,000 -$7,000 $500

C $5,000 $0 $800

D $8,000 -$2,000 $700

E -$1,000 $4,000 $0

Payoff Table

Outcomes

O1 O2 O3

$9,000 $3,000 $0

$0 $11,000 $500

$5,000 $4,000 $200

$2,000 $6,000 $300

$11,000 $0 $1,000

Regret Table

Maximum Regret$9,000

$11,000

$5,000

$6,000$11,000

C > D > A > B

Minmax regret violates addition/deletion of alternatives

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LaplaceCalculate the average of each alternative by assuming that the outcomes are equally likely to occur, and select the alternative with the largest average

Average

$1,000

$1,166.7

Outcomes

Alternatives O1 O2 O3

A $1,000 $1,000 $1,000

B $10,000 -$7,000 $500

C $5,000 $0 $800D $8,000 -$2,000 $700

Payoff Table

$1,933.3$2,233.3

D > C > B > A

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Hurwicz PrincipleSelect the alternative that has the largest weighted average of its maximum and minimum payoffs; the weight of the maximum payoff is , referred to as the coefficient of optimism, and the weight of the minimum payoff is 1-

=0.4

Hurwicz Score

$1,000

10,000*0.4+(-7,000)*0.6 = - $200

if =1, then Hurwicz criterion is the same as Maxmax payoff if =0, then Hurwicz criterion is the same as Maxmin payoff

Outcomes

Alternatives O1 O2 O3

A $1,000 $1,000 $1,000

B $10,000 -$7,000 $500

C $5,000 $0 $800

D $8,000 -$2,000 $700

Payoff Table

5,000*0.4+0*0.6 = $2,000

8,000*0.4+(-2,000)*0.6 = $2,000

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Hurwicz score = Max. payoff ∙α + Min. payoff ∙(1-α)

αAlternative

A B C D

0 1000 -7000 0 -2000

0.1 1000 -5300 500 -1000

0.2 1000 -3600 1000 0

0.3 1000 -1900 1500 1000

0.4 1000 -200 2000 2000

0.5 1000 1500 2500 3000

0.6 1000 3200 3000 4000

0.7 1000 4900 3500 5000

0.8 1000 6600 4000 6000

0.9 1000 8300 4500 7000

1 1000 10000 5000 8000

Hurwicz Scores of Alternatives with Respect to α

A: Hurwicz score = 1000

B: Hurwicz score = 10000∙α + (-7000)∙(1-α) = 17000α-7000

C: Hurwicz score = 5000∙α + 0∙(1-α) = 5000α

D: Hurwicz score = 8000∙α + (-2000) ∙(1-α) = 10000α-2000

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α=0.2 α=0.4 α=5/7≈0.71

When 0≤α<0.2, A is the best alternativeWhen 0.2≤α≤0.4, C is the best alternativeWhen 0.4≤α≤5/7, D is the best alternativeWhen α>5/7, B is the best alternative

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Summary of Non-Probabilistic Decision Rules

Each has advantages and disadvantages

Decision Rules Advantages Disadvantages

Maxmax Payoff Simpleoverly optimistic; ignore intermediate

outcomes (IIO); violates column linearity, addition/deletion of identical columns

Maxmin Payoff Simpleoverly pessimistic; IIO; violates column

linearity, addition/deletion of identical columns

Minmax Regret Column linearity violates addition/deletion of alternatives

LaplaceColumn linearity;

considers all outcomesEqual weight assumption may be inappropriate

Hurwicz Models risk attitudeIIO; violates column linearity, addition/deletion

of identical columns