Decision Making as Constrained Optimization

2. Decision Making as Constrained Optimization • Specification of Objective Function  Decision Rule • Identification of Constraints

3. Where do decision rules come from? • They are learned • by experience • “learning by getting hurt” • by instruction • “learning by being told” • They are induced • using logic, mathematics

4. Historical Example:The St. Petersburg Paradox • Game: • You get to toss a fair coin for as many times as you need to score a “head” (H) (on toss n, n from 1 to infinity) • Payoff: • You get $2 n • If you score H on toss 1, you get $2 • If you score H on toss 2, you get $4 • If you score H on toss 3, you get $8 • If you score H on toss 4, you get $16, etc. • Question: • How much are you willing to pay me in order to play this game for one round? • How do you decide???

5. Expected Value of Game • How much do you think you can expect to win in this game? • EV(X) = Sum over all i {xi p(x i)} • Expected Utility of Game • Daniel Bernoulli (1739) • Utility of wealth is not linear, but logarithmic • EU(X) = Sum over all i {u(xi) p(x i)} • Other decision rules??? • Minimum return (pessimist) rule: • pay no more than you can expect to get back in the worst case • Expectation heuristic (Treisman, 1986): • Figure on what trial you can expect to get the first H and pay no more than you will get on that trial • Single vs. multiple games • Does it make a difference?

6. Expected Utility Theory • Generally considered best “objective function” since axiomatization by von Neumann & Morgenstern (1947)

7. Expected-Utility Axioms(Von Neumann & Morgenstern,1947) • Connectedness x>=y or y>=x • Transitivity If x>=y and y>=z, then x>=z • Substitution Axiom or Sure-thing principle If x>=y, then (x,p,z) >= (y,p,z) for all p and z • If you “buy into” all axioms, then you will choose X over Y • if and only if EU(X) > EU(Y), where EU(X) = Sum over all i {u(xi) p(x i)} and EU(Y) = Sum over all i {u(yi) p(y i)}

8. Violation of Connectedness • Sophie’s Choice • Trading money for human life/human organs • In general • there are some dimensions between which some people are uncomfortable making tradeoffs or for which they find tradeoffs unethical

9. Violations of Transitivity • Example A • Choice 1: rose soap ($2) vs. jasmine soap ($2.30) • Choice 2: jasmine soap ($2.30) vs. honeysuckle soap ($2.60) • Choice 3: rose soap ($2) vs. honeysuckle soap ($2.60) • Example B • Choice 1: large apple vs. orange • Choice 2: orange vs. small apple • Choice 3: large apple vs. small apple

10. Violation of Substitution • Allais’ paradoxDecision I: A: Sure gain of $3,000 B: .80 chance of $4,000Decision II: C: .25 chance of $3,000 D: .20 chance of $4,000

11. Examples of EV as a good decision rule • Pricing insurance premiums • Actuaries are experts at getting the relevant information that goes into calculating the expected value of a particular policy • Testing whether slot machines follow state laws about required payout • Bloodtesting • Test each sample individually or in batches of, say, 50? • Incidence of disease is 1/100 • If group test comes back negative, all 50 samples are negative • If group test comes back positive, all samples are tested individually • What is expected number of tests you will have to conduct if you test in groups of 50?

12. Another normative model • Choosing a spouse • What’s your decision rule for saying yes/no to a marriage proposal? • Tradeoff • Say yes too early, and you may miss the best person • Say no to a “good” one, you may be sorry later • “Optimal” algorithm • Estimate the number of offers you will get over your lifetime • Say “no” to the first 37 % • Then say “yes” to the first one who is better than all previous ones

13. Objective function • Maximize the probability of getting No.1 as a function of the cutoff percentage (i.e., % after which you start saying “yes”) • Example • Say n=4 suitors • Reject first 37% • Pass up first (25%) and pick the one after that who is better than all previous ones • Gets the “best” in 11 out of 24 cases: 47% • Suitors may come in all 24 rank orders: • 1234 1243 1342 1423 • 1432 2134(*) 2143(*) 2314(*) • 2341(*) 2413(*) 2431(*) 3124(*) • 3142 (*) 3214 3241 3412(*) • 3421 4123(*) 4132(*) 4213 • 4231 4312 4321 1324 • * means that she got the “best” one, with a rank of 1

14. Assumptions underlying normative model for spousal selection • You can estimate n • You have to “sample” sequentially • You have no second chances

15. A final normative model: Multi-Attribute Utility Theory (MAUT) • Model of riskless choice • Choice of consumer products, restaurants, etc. • Need to specify • Dimensions of choice alternatives that enter into decision • Value of each alternative on those dimensions • Importance weights of dimensions given ranges (acceptable tradeoff) • Tradeoffs • Willingness to interchange x units of dim1 for y units of dim2 • Computer programs can help you with utility assessment and tradeoff assessment

16. What do normative/prescriptive models provide? • Consistency in choices • Structure for decision making process • Transparency of reasons for choice • Justifiability • “Education” of other choice processes