Non-probability decision rules Dr. Yan Liu Department of Biomedical, Industrial & Human Factors Engineering Wright State University
Dec 14, 2015
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