Non-probability decision rules

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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

• Decision maker knows with certainty the
  consequences of every alternative or decision
  choice (Non-probability decision making)
• Decision maker can assign the probabilities of
  the various outcomes (decision making under risk)
• Decision maker can neither predict nor describe
  the probabilities of the various outcomes
  (decision making under uncertainty)

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Types of Non-Probabilistic Decision Rules

• Lexicographic Ordering
• Satisficing
• Maxmax Payoff
• Maximin Payoff
• Minimax 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 (value) 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

• V1V2 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 Values Values
Alternatives Safety Price Reliability
A High 15k High
B Medium 11k Medium
C High 13k Medium
C gt A gt B
Non-exhaustive comparisons and can be efficient
when there are many values
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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 Values Values
Alternatives Safety Medium Cost 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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Maximax Payoff

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

Payoff Table
Outcomes Outcomes 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 Payoff
1,000
10,000
5,000
8,000
B gt D gt C gt A
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Payoff Table
Outcomes Outcomes Outcomes
Alternatives O1 O2 O3
A 1,000 1,0009,000 1,000
B 10,000 -7,0009,000 500
C 5,000 09,000 800
D 8,000 -2,0009,000 700
Maximum Payoff
10,000
10,000
9,000
8,000
A B gt C gt D
Maximax payoff violates column linearity
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Payoff Table
Outcomes Outcomes Outcomes 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 8,000
D 8,000 -2,000 700 8,000
Maximum Payoff
8,000
10,000
8,000
8,000
B gt A C D
Maximax payoff violates addition/deletion of
identical columns
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Maximin Payoff

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

Payoff Table
Outcomes Outcomes 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
Minimum Payoff
1,000
-7,000
0
-2,000
A gt C gt D gt B
Maximin 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

Regret Table
Payoff Table
Outcomes Outcomes Outcomes
O1 O2 O3
9,000 0 0
0 8,000 500
5,000 1,000 200
2,000 3,000 300
Outcomes Outcomes 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
9,000
8,000
5,000
3,000
D gt C gt B gt A
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Regret Table
Payoff Table
Outcomes Outcomes 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
Outcomes Outcomes 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
Maximum Regret
9,000
11,000
5,000
6,000
11,000
C gt D gt A gt B
Minmax regret violates addition/deletion of
alternatives
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Laplace

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

Payoff Table
Outcomes Outcomes 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
Average
1,000
1,166.7
1,933.3
2,233.3
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Hurwicz Principle

• Select 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- ?

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

Payoff Table
? 0.4
Outcomes Outcomes 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
Hurwicz Score
1,000
10,0000.4(-7,000)0.6 - 200
5,0000.400.6 2,000
8,0000.4(-2,000)0.6 2,000
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Hurwicz Scores of Alternatives with Respect to a

a Alternative Alternative Alternative Alternative
a 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
A Hurwicz score 1000
B Hurwicz score 10000a (-7000)(1-a)
17000a-7000
C Hurwicz score 5000a 0(1-a) 5000a
D Hurwicz score 8000a (-2000) (1-a)
10000a-2000
Hurwicz score Max. payoff a Min. payoff
(1-a)
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a5/70.71
a0.2
a0.4
When 0alt0.2, A is the best alternative When
0.2a0.4, C is the best alternative When
0.4a5/7, D is the best alternative When agt5/7,
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 Simple overly optimistic ignore intermediate outcomes (IIO) violates column linearity, addition/deletion of identical columns
Maxmin Payoff Simple overly pessimistic IIO violates column linearity, addition/deletion of identical columns
Minmax Regret Column linearity violates addition/deletion of alternatives
Laplace Column linearity considers all outcomes Equal weight assumption may be inappropriate
Hurwicz Models risk attitude IIO violates column linearity, addition/deletion of identical columns
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