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Decision Making as Constrained Optimization

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Decision Making as Constrained Optimization Specification of Objective Function Decision Rule Identification of Constraints Where do decision rules come from? They ... – PowerPoint PPT presentation

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Title: Decision Making as Constrained Optimization


1
(No Transcript)
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 xgty or ygtx
  • Transitivity
  • If xgty and ygtz, then xgtz
  • Substitution Axiom or Sure-thing principle If
    xgty, then (x,p,z) gt (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) gt 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
  • Sophies 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 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

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
  • Whats 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 n4 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
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