Decision Making as Constrained Optimization

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Decision Making as Constrained Optimization

• Specification of Objective Function ? Decision
  Rule
• Identification of Constraints

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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

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Historical ExampleThe St. Petersburg Paradox

• Game
• You get to toss a fair coin for as many times as
  you need to score a head (H)
• n is the toss on which the first H appears 1 lt
  n lt 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???

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• 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?

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Expected Utility Theory

• Generally considered best objective function
  since axiomatization by von Neumann Morgenstern
  (1947)

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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)

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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

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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

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Violation of Substitution

• Allais paradoxDecision I A Sure gain of
  3,000 B .80 chance of 5,000Decision II
  C .25 chance of 3,000 D .20 chance of
  5,000

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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?

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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

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• 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

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Assumptions underlying normative model for
spousal selection

• You can estimate n
• You have to sample sequentially
• You have no second chances

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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

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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