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

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

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???

Expected Value of Game How much do you think you can expect to win in this game? EV(X) = Sum over all i {x i 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(x i ) 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?

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

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(x i ) p(x i )} and EU(Y) = Sum over all i {u(y i ) p(y i )}

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

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

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

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?

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

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: (*)2143(*)2314(*) 2341(*)2413(*)2431(*)3124(*) 3142 (*) (*) (*)4132(*) * means that she got the “best” one, with a rank of 1

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

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

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