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

22 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

33 Types of Non-Probabilistic Decision Rules Lexicographic Ordering Satisficing Maxmax Payoff Maxmin Payoff Minmax Regret Laplace Hurwitz Principle

44 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

55 Lexicographic Ordering V 1 ≥V 2 ≥ ∙∙∙≥V n, 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 AlternativesSafetyPriceReliability AHigh$15kHigh BMedium$11kMedium CHigh$13kMedium Non-exhaustive comparisons in values and can be efficient when there are many values C > A > B

66 Satisficing/Minimum Aspiration Level Select any alternative which satisfies the minimum aspiration levels (the minimum acceptable criteria) of all values Values Alternatives Safety ≥Medium Price ≤13k Reliability ≥Medium AHigh$15kHigh BMedium$11kMedium CHigh$13kMedium May not be optimal because not all alternatives will be considered as long as one satisfactory alternative is found

77 Maxmax Payoff Select the alternative which results in the maximum of maximum payoffs; an optimistic criterion Outcomes Alternatives O1O2O3 A$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 Payoff Table $5,000 $8,000 B > D > C > A

88 Outcomes Alternatives O1O2O3 A$1,000$1,000+9,000$1,000 B$10,000-$7,000+9,000$500 C$5,000$0+9,000$800 D$8,000-$2,000+9,000$700 Maximum Payoff $10,000 $9,000 $8,000 A = B > C > D Maxmax payoff violates column linearity Payoff Table

99 Outcomes Alternatives O1O2O3O4 A$1,000 $8,000 B$10,000-$7,000$500$8,000 C$5,000$0$800$8,000 D -$2,000$700$8,000 Payoff Table Maximum Payoff $8,000 $10,000 $8,000 B > A = C = D Maxmax payoff violates addition/deletion of identical columns

10 Maxmin Payoff Select the alternative which results in the maximum of minimum payoffs; a pessimistic criterion Outcomes Alternatives O1O2O3 A$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 Payoff Table $0 -$2,000 A > C > D > B Maxmin payoff violates column linearity and addition/deletion of identical columns

11 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 O1O2O3 A$1,000 B$10,000-$7,000$500 C$5,000$0$800 D$8,000-$2,000$700 Maximum Regret Payoff Table Outcomes O1O2O3 $9,000$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

12 Outcomes Alternatives O1O2O3 A$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 O1O2O3 $9,000$3,000$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

13 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 Average $1,000 $1,166.7 Outcomes Alternatives O1O2O3 A$1,000 B$10,000-$7,000$500 C$5,000$0$800 D$8,000-$2,000$700 Payoff Table $1,933.3 $2,233.3 D > C > B > A

14 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-   =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 O1O2O3 A$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

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

16 α=0.2α=0.4 α=5/7 ≈0.71 When 0≤α<0.2, A is the best alternative When 0.2≤α≤0.4, C is the best alternative When 0.4≤α≤5/7, D is the best alternative When α>5/7, B is the best alternative

17 Summary of Non-Probabilistic Decision Rules Each has advantages and disadvantages Decision RulesAdvantagesDisadvantages 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 linearityviolates 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