Framework Decisions under uncertainty

1. Framework Decisions under uncertainty Fred Wenstøp

2. Assumptions • State of nature • Environmental conditions (national or international economy, etc.) that will influence the outcome of our decisions • Nature is blind • Nature determines the future state without paying attention to our choices • if this is not the case, we face an opponent instead of nature, and we are into game theory Fred Wenstøp: Framework

3. Decision tables • Alternative actions are choices between rows • Possible states of nature are columns • Theinterior numbers are payoffs • Money or utilities • Compact form, but it can not show sequential decisions • Use decision trees... Fred Wenstøp: Framework

4. Strict uncertainty • Strict uncertainty • the decision maker has no inkling of an idea as to how likely the various states of nature are, and is therefore completely unable to asses probabilities • This is not uncommon in practise • Experiment • you are offered the choice between two envelopes • you are told that one contains twice as much money as the other • you are strictly uncertain about how much money it can be • you select one. It contains NOK 200. You may swap. Should you?? Fred Wenstøp: Framework

5. Decision rule 1:Wald’s maximin criterion • Choose the action with the highest security level • An action's security level is the worst thing that can happen under that action Fred Wenstøp: Framework

6. Decision rule 2:Hurwicz’s optimism-pessimism index • In addition to the security levels, compute the optimism levels (Maxima) • Decide on a pessimism weight a, a=0.1 • Choose the action with the highest combined index Fred Wenstøp: Framework

7. Decision rule 3:Savage’s minimax regret • Transform the outcome table into a regret table • For each column, subtract the maximum of that column from all the numbers in the column • Find the security level of each action (regretwise) • Choose the action with the lowest security level • Remember: regrets should be small Fred Wenstøp: Framework

8. Savage’s minimax regretExample Fred Wenstøp: Framework

9. Decision rule 4:Laplace's principle of insufficient reason • If you do not anything about the probabilities of the different states of nature, then all probabilities are identical • Choose the action with the highest expected outcome Fred Wenstøp: Framework

10. Reasonable properties of decision rules under strict uncertainty: I • Axiom 1: Complete ranking • A decision rule should provide a complete ranking of all the possible outcomes • This is achieved if the rule provides a performance index for each alternative • All four rules do this • Axiom 2: Independence of labelling • The decision will be the same if rows or columns in the decision table are permuted • Met by all four rules Fred Wenstøp: Framework

11. Reasonable properties of decision rules under strict uncertainty: II • Axiom 3: Independence of value scale • The decision will be the same regardless of the measure scale used to represent the outcomes (USD or NOK, etc.) • Met by all four rules • Axiom 4: Strong domination • Suppose that there are two actions A and B in the decision table and that A has higher outcomes than B for all possible states of nature in the table. • Then the decision rule should conclude that A is better than B. • Met by all four rules Fred Wenstøp: Framework

12. Reasonable properties of decision rules under strict uncertainty: III • Axiom 5: Independence of irrelevant alternatives • A decision rule has been applied to a decision table resulting in a ranking of the alternatives • Assume that a new alternative is introduced in the table and the decision rule applied • Then the original alternatives should maintain their original internal ranking • Savage's rule does not meet this requirement Fred Wenstøp: Framework

13. Savage’s violation of axiom 5 Fred Wenstøp: Framework

14. Reasonable properties of decision rules under strict uncertainty: IV • Axiom 6: Independence of an addition of a constant to a column • To add a number to all outcomes in a column affects all alternatives in the same way and should have no consequence for the ranking • Wald's and Hurwicz's rules do not satisfy this requirement Fred Wenstøp: Framework

15. Reasonable properties of decision rules under strict uncertainty: V • Axiom 7: Independence of row permutation • If two actions have exactly the same list of outcomes, but placed differently in the columns, then they should be ranked as equal • Since we have idea as to the likelihood of the columns, which outcome is in which column cannot matter • Savage's rule fails this requirement Fred Wenstøp: Framework

16. Reasonable properties of decision rules under strict uncertainty: VI • Axiom 8: Independence of column duplication • The ranking of the alternatives must be unaffected if a column is duplicated • If the likelihood of the states of nature are truly unknown, such an action cannot matter • Laplace's principle of insufficient reason obviously fails this requirement Fred Wenstøp: Framework

17. Strict uncertainty impossibility theorem • Theorem: No decision rule can satisfy axiom 1 to 8 • Axioms 7 and 8 specify what is meant by strict uncertainty • The theorem shows that strict uncertainty is void of meaning • it is seen in the envelop paradox: an assumption of strict uncertainty leads to inconsistent behaviour • We must introduce probabilities • New decision rule: • Maximise the expected outcome • This rule satisfies the first six axioms Fred Wenstøp: Framework

18. Subjective probability • Uncertainty • Can be represented as subjective probability • measured by referring to objective probabilities created by tossing of coins, dice, etc. • Example: • What is the probability that Norway will be member of EU before 2050? • Choose A or B • A: You get NOK 100 000 in 2050 if arrow stops in the yellow area • B: You get NOK 100 000 in 2050 if Norway is a member Fred Wenstøp: Framework

19. Decision trees • The branches of a tree do not grow together again • Decision points: squares • Chance nodes: circles • Outcomes: on the leaves Fred Wenstøp: Framework

20. Construction of decision trees I • What should be included? • Decisions that can be made at a later point in time, but may affect the outcome of decisions that have to made now • Future uncertain events that may influence the outcome of decisions that have to be made now • The branches from a decision node must be mutually exclusive and at the same time complete Fred Wenstøp: Framework

21. Construction of decision trees II • The time horizons • The decision tree horizon • Choose the end points so as to contain the most important decision and chance nodes for decisions that have to be made today • The cash flow horizon • This lies beyond the decision tree horizon and stretches as far as it is meaningful to predict cash flows • The residual value • This is the assessed lump value of cash flows outside the cash flow horizon • The leaf values • The discounted values of the cash flows up to the cash flow horizon plus the residual values Fred Wenstøp: Framework

22. Construction of decision trees III • Sequencing chance nodes and decisions • Subjective chronology • The decision nodes must be placed where the decisions are irrevocably made • The chance nodes must be placed where the decision maker learns about which events took place, not when they actually did take place • If there are alternative but equivalent representations of the same problem • choose the one with the highest pedagogical merit Fred Wenstøp: Framework

23. Analysis of decision trees • The roll back technique • Start at the leaves • Compute the expected values at the nearest chance nodes • at each decision point, preselect the decision leading to the chance node or leaf with the highest expected value • Write this value down at the decision point and mark the preselected decision • When you get to the root, unroll the decisions according to the preselected courses Fred Wenstøp: Framework

24. The roll back technique 400 200 400 300 180 200 0,2 0,5 0,2 0,8 250 300 250 0,5 0,5 0,3 0,5 0,5 Fred Wenstøp: Framework