Utility Axioms, paradoxes, and implications

1. Utility Axioms, paradoxes, and implications Dr. Yan Liu Department of Biomedical, Industrial & Human Factors Engineering Wright State University

2. Axioms for Expected Utility • Assumptions related to the consistency with which an individual expresses preferences from among a series of risky prospects • Axiom 1: Ordering and Transitivity • Decision maker can order any two alternatives, the ordering is transitive • Axiom 2: Reduction of Compound Uncertain Events • Decision maker is indifferent between a compound uncertain event (a complicated mixture of gambles or lotteries) and a single uncertain event as determined by reduction using probability manipulations

3. p C1 1-p A C2 B C Axioms for Expected Utility (Cont.) • Axiom 3: Continuity • If , then a decision maker can “interpret” C1 and C2 with some probability p to get an outcome that is equally desirable to C

4. p C1 C 1-p C2 A D A D B E B E Decision Tree 1 Axioms for Expected Utility (Cont.) • Axiom 4: Substitutability • A decision maker is indifferent between any original uncertain event that includes C and one formed by substituting for C an uncertain event that is judged to be its equivalent  Decision Tree 2

5. (0.8) $50 A (0.2) $1,000 (0.7) $50 B (0.3) $1,000 Axioms for Expected Utility (Cont.) • Axiom 5: Monotonicity • Given two reference gambles with the same possible outcomes, a decision maker prefers the one with the higher probability of winning the preferred outcome

6. Axioms for Expected Utility (Cont.) • Axiom 6: Invariance • A decision maker’s preferences among gambles are determined only by their payoffs and associated probabilities • Axiom 7: Finiteness • No consequences are considered infinitely bad or infinitely good

7. Axioms for Expected Utility (Cont.) • These axioms do not always hold Each outcome occurs with probability 1/4 Which do you prefer? Each outcome occurs with probability 1/4

8. Axioms for Expected Utility (Cont.) • Proposition • For decision problems involving uncertain events, if all the previous seven axioms hold, then 1) it is possible to find a utility function for you to evaluate your preferences for the uncertain consequences, and 2) you should be making decisions in a way that is consistent with maximizing expected utility

9. (0.6) $40 (0.5) In A’: Pr($40) = 0.5*0.6 + 0.5*0.36 =0.48 Pr($10) =1- Pr($40) = 0.52 (0.4) $10 (0.36) $40 (0.5) A’ (0.64) $10 (1.00) $40 (0.3) (0) B’ $10 (0) $40 (0.7) (1.00) $10 Decision Tree 2 1. you are indifferent between $15 and a reference gamble: Win $40 with probability 0.36 Win $10 with probability 0.64 2. You are indifferent between $20 and another reference gamble: Win $40 with probability 0.6 Win $10 with probability 0.4 (0.5) $20 (0.5) A $15 (0.3) $40 B (0.7) $10 Decision Tree 1  Substituability Axiom

10. (0.5) (0.48) $40 $20 A’’ A (0.52) (0.5) $15 $10 (0.3) (0.3) $40 $40 B B’’ (0.7) (0.7) $10 $10 According to the reduction of compound uncertain events axiom, decision tree 2  decision tree 3 (monotonicity axiom) Decision Tree 3 Show a higher expected utility implies a preferred alternative? Decision Tree 1

11. (0.6) $40 (0.5) (0.4) $10 (0.36) $40 A’ (0.5) (0.64) $10 (1.00) $40 (0.3) (0) $10 B’ (0) $40 $10 (0.7) (1.00) MA’ ~ $20 NA’ ~$15 MB’ NB’ Decision Tree 2 Set U($40) =1 and U($10) =0 EU(MA’) =0.6(1)+0.4(0)=0.6, EU(MB’) =1(1) + 0(0) = 1 EU(NA’) =0.36(1)+0.64(0)=0.36, EU(NB’) =0(1) + 1(0) = 0

12. (0.5) (0.5) 0.60 $20 A (0.5) (0.5) A 0.36 $15 (0.3) (0.3) $40 1.00 B B (0.7) (0.7) $10 0  Decision Tree 1 EU(A) =0.5(0.6)+0.5(0.36)=0.48 EU(B) =0.3(1)+0.7(0)=0.3 12

13. 400 people will be saved 200 people will die A C (0.2) (0.8) 600 people will die 600 people will be saved D B (0.8) (0.2) No one will be saved No one will die Paradoxes • Framing Effects • An individual’s risk attitude can change depending on the way the decision problem is posed The United States is preparing for an outbreak of unusual Asian strain of influenza. Experts expect 600 people to die from the disease. Two programs are available that could be used to combat the disease, but only one can be implemented due to limited resources. Which program will you choose, A or B? Which program will you choose, C or D?

14. Utility Gains Losses Status quo In the influenza-outbreak example, program A is the same as C, and program B is the same as D. However, research has shown that many people prefer A in the first decision and D in the second decision. So what causes such an inconsistency? In programs A and B, we think in terms of lives saved, and the reference point is that 600 people are expected to die. In programs C and D, on the other hand, we think in terms of lives lost, and the reference point is no one would die without the disease. An important general principle: people tend to be risk-averse in dealing with gains but risk-seeking in deciding about loss.

15. Paradoxes (Cont.) • Framing Effects (Cont.) • The reference point or status quo can be quite flexible in some situations • e.g. Many people change their financial status quo as soon as they file income-tax return in anticipation of a refund; they spend their refund, usually in the form of credit, long before the check arrives in mail. • People may maintain a particular reference point far longer than they should • e.g. Some managers maintain a commitment to a project that has obviously gone sour; they typically argue that backing out of a failed project amounts to a waste of the resources already spent

16. (0.11) $1M (0.89) C 0 (0.10) $5M D (0.90) 0 Paradoxes (Cont.) • Allais Paradox $1M A (0.1) $5M (0.89) B $1M (0.01) 0 Experiments show that as many as 82% of subjects prefer A over B and 83% prefer D over C. EU(A) = U($1M) EU(B) = 0.1*U($5M)+0.89*U($1M)+0.01*U($0) = 0.1+0.89U($1M) EU(A) > EU(B) U($1M)>0.1+0.89U($1M) U($1M) >0.91 EU(C) = 0.11*U($1M)+0.89*U($0)=0.11U($1M) EU(D) = 0.1*U($5M)+0.90*U(0)=0.1 EU(C) < EU(D) 0.11U($1M)<0.1 U($1M) <0.91 Choosing A and D is not consistent with expected utility This paradox may be attributed to the certainty effect

17. (p) G A (1-p) L B CE Implications • Utility Assessment In utility assessment using CE, G,L and p are fixed, and CE is to be adjusted In probability-equivalent (PE) assessment, G,L and CE are fixed, and p is to be adjusted The use of CE approach tends to result in more risk-averse responses than the use of PE approach when consequences are gains. When the consequences are losses, on the other hand, CE approach results in more risk-seeking behavior. When using PE approach, many people appear to exhibit certain forms of probability distortion. People tend to deal best with 50-50 chances The nature of the decision maker’s responses and hence the deduced risk attitude can depend on the way that questions have been posed in the assessment procedure.

18. (0.5) B (0.5) A1 C EU(A1) = 0.5*U(B)+0.5*U(C) = 0.5U(B) (p) A EU(A2) = p*U(A)+(1-p)*U(C) = p A2 (1-p) EU(A1) = EU(A2) U(B) = 2p C Implications (Cont.) • Utility Assessment (Cont.) A is the best outcome; C is the worst outcome; B is between A and C, so U(B)=? Assess probability p that makes the decision maker indifferent between A1 and A2 McCord-De Neufville Utility-assessment method

19. Implications • Managerial and Policy • Reference point or status quo plays an important role • “Sunk Cost” • Managers frequently remain committed to a project that obviously has gone bad • The real status quo is that the project is unlikely to yield the anticipated benefits instead of the amount that has been invested • “Seat belts are viewed as inconvenient and uncomfortable” • Their status quo is the level of comfort when unbuckled • An alternative status quo is the safety of people