Consistent choice

1. Consistent choice Fred Wenstøp Fred Wenstøp: Consistent choice

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: Consistent choice

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: Consistent choice

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: Consistent choice

5. Strict uncertainty: decision rule 1Wald’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: Consistent choice

6. Strict uncertainty: decision rule 2Hurwicz’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: Consistent choice

7. Strict uncertainty: decision rule 3Savage’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: Consistent choice

8. Savage’s minimax regretExample Fred Wenstøp: Consistent choice

9. Savage’s inconsistency Fred Wenstøp: Consistent choice

10. Strict uncertainty: decision rule 4Laplace'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: Consistent choice

11. Strict uncertainty impossibility theorem • Theorem: No decision rule can satisfy a reasonable set of consistency axioms • 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 Fred Wenstøp: Consistent choice

12. 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 green area • B: You get NOK 100 000 in 2050 if Norway is a member Fred Wenstøp: Consistent choice

13. Probabilistic uncertainty: decision rule 1 • Maximize expected value Fred Wenstøp: Consistent choice

14. The St. Petersburg Paradox • Daniel Bernoulli 1738 • Suppose you are offered the following lottery • A coin is tossed until tails turn up the first time • If it happens at toss #1, you get kr 2 • If it happens at toss #2, you get kr 4 • If it happens at toss #3, you get kr 8 • etc..... • How much are you willing to pay to participate? Fred Wenstøp: Consistent choice

15. von Neumann-Morgenstern's axioms for preferential consistency I • Axiom 1. Complete ordering • All prices and lotteries can be ordered by the decision maker according to his preferences • No prices or lotteries can be incomparable • This is an uncontroversial axiom • To be able to speak about decision making, one must be able to decide • However, in practise, many decision maker's try to chicken out Fred Wenstøp: Consistent choice

16. von Neumann-Morgenstern's axioms for preferential consistency II • Axiom 2: Transitivity • If the decision maker prefers • x to y • and • y to z • He must prefer x to z • Uncontroversial • Does not hold for football teams etc. Fred Wenstøp: Consistent choice

17. von Neumann-Morgenstern's axioms for preferential consistency III • Axiom 3: Continuity • Suppose that • x is preferred to y • y is preferred to z • You get a choice between • y for certain • or • x with probability p • z with probability 1-p • Then there must exist a value of p which makes you indifferent between the choices • Controversial in many situations where z is very bad • for instance irreversible environmental damages Fred Wenstøp: Consistent choice

18. von Neumann-Morgenstern's axioms for preferential consistency IV • Axiom 4: Reduction of compound lotteries • The only things that matter for the decision maker are the final prices and their probabilities • Therefore compound lotteries are identical to reduced lotteries • This means that fun of gambling is ruled out • The process whereby we arrive at the final prices is irrelevant Fred Wenstøp: Consistent choice

19. von Neumann-Morgenstern's axioms for preferential consistency V • Axiom 5: Substitutability • Suppose that you are indifferent between a price b and a lottery c • Then c and b can substitute each other in any compound lottery without affecting its attractiveness • Often violated in practise, because the decision maker tries to avoid regret Fred Wenstøp: Consistent choice

20. Measuring a utility function • Select the working domain • As narrow as possible, still wide enough to contain all values you might want to analyse • Let the utility of the worst point be 0, and of the best point 1.0 • Offer a choice: • Either a 50/50 lottery between the end points • Or a certain outcome xc • Change xc until it is equivalent to the lottery • Then U(xc)=0.5 • Repeat the process as many times as needed • Use the new xc'sas new end points Fred Wenstøp: Consistent choice

21. Terminology • Certainty equivalent • Certain price which is equally attractive as a lottery • Risk premium • The difference between the certainty equivalent and the expected price • Risk averse utility functions are concave • Risk prone utility functions are convex • Risk neutral utility functions are linear Fred Wenstøp: Consistent choice

22. Insurance vs. betting • Insurance companies make their living from people who pay a risk premium to avoid uncertainty • Gambling organisations make their living from people who pay to achieve uncertainty • These are the same people! • How can it be explained? Fred Wenstøp: Consistent choice

23. Multicriteria decision analysis • Paradigme • To help the decision maker formulate goals, weight them and making them operational • To structure complex decision problems • Paying attention to several objectives at the same time • Main reference • Keeney and Raiffa's "Multi Criteria Decision Making" (MCDM) from 1976 Fred Wenstøp: Consistent choice

24. The MCDM Method: Problem structuring Fred Wenstøp: Consistent choice

25. Terminology • Criteria • operationalize the goals (objectives, ends) • Scores • the resulting values of the criteria when a decision is implemented • Weights • express the importance of the criteria and reflect the decision maker's subjective values (preferences) • Goal hierarchy (value tree) • structure of the decision maker's objectives • Option (decision alternative) • an action which influences the scores of the criteria Fred Wenstøp: Consistent choice

26. Common preference models • w:weights, x: scores, u: utility function • Additive utility function • Multiplicative utility function • Multilinear utility function Fred Wenstøp: Consistent choice