Non-probability decision rules

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

2. 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) 2

3. Types of Non-Probabilistic Decision Rules Lexicographic Ordering Satisficing Maxmax Payoff Maximin Payoff Minimax Regret Laplace Hurwitz Principle 3

4. 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 4

5. 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 C > A > B Non-exhaustive comparisons and can be efficient when there are many values 5

6. Satisficing/Minimum Aspiration Level Select any alternative which satisfies the minimum aspiration levels (the minimum acceptable criteria) of all values May not be optimal because not all alternatives will be considered as long as one satisfactory alternative is found 6

7. Maximax Payoff Select the alternative which results in the maximum of maximum payoffs; an optimistic criterion Payoff Table Maximum Payoff $1,000 $10,000 $5,000 $8,000 B > D > C > A 7

8. Payoff Table Maximum Payoff $10,000 $10,000 $9,000 $8,000 A =B > C > D Maximax payoff violatescolumn linearity 8

9. Payoff Table Maximum Payoff $8,000 $10,000 $8,000 $8,000 B > A = C = D Maximax payoff violatesaddition/deletion of identical columns 9

10. Maximin Payoff Select the alternative which results in the maximum of minimum payoffs; a pessimistic criterion Payoff Table Minimum Payoff $1,000 -$7,000 $0 -$2,000 A > C > D > B Maximin payoff violatescolumn linearity and addition/deletion of identical columns 10

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 Regret Table Payoff Table Maximum Regret $9,000 $8,000 $5,000 $3,000 D > C > B > A 11

12. Regret Table Payoff Table Maximum Regret $9,000 $11,000 $5,000 $6,000 $11,000 C > D > A > B Minmax regret violates addition/deletion of alternatives 12

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 Payoff Table Average $1,000 $1,166.7 $1,933.3 $2,233.3 13

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-  • 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 Hurwicz Score $1,000 10,000*0.4+(-7,000)*0.6 = - $200 5,000*0.4+0*0.6 = $2,000 8,000*0.4+(-2,000)*0.6 = $2,000 14

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

16. α=5/7≈0.71 α=0.2 α=0.4 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 17