Utility Axioms

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CHAPTER 14

• Utility Axioms
• Paradoxes Implications

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Chapter 14

• Learning objectives
• Axioms for Expected Utility
• Paradoxes
• Implication for Utility Assessment
• Managerial and Policy Implications

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Chapter 14

• People expected to make choices consistent with
  maximizing of expected utility
• Cognitive psychologist have noted that people do
  not always behave according to expected utility
  theory
• Decision Analysis depends on foundational axioms

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Chapter 14

• The following example previews some of the issues
  we will consider
• The United States is preparing for an outbreak of
  an 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

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Utility Axioms Paradoxes

• Program A (Tried and True) 400 people will be
  saved
• Program B (Experimental) There is an 80 chance
  that 600 people will be saved and a 20 chance
  that no one will be saved
• Now consider the following two programs
• Program C 200 people will die
• Program D There is a 20 chance that 600 people
  will die and an 80 chance that no one will die.
• Would you prefer C or D?

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Axioms for Expected Utility

• Axioms or assumption relate to the consistency
  with which an individual expresses preferences
  from among a series of risky prospects
• We can call these axioms or rules for clear
  thinking
• In the following examples axioms are presented to
  clarify their meaning

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Axioms for Expected Utility

• Ordering and Transitivity A decision maker can
  order (establish preference or indifferences) any
  two alternatives and the ordering is transitive.
• Example Given any alternatives A1, A2 and A3
  either A1 is preferred to A2, A2 is preferred to
  A1 or the decision maker is indifferent between
  A1 and A2.
• Transitivity means that if A1 is preferred to A2
  and A2 is preferred to A3, then A1 is preferred
  to A3
• If a person preferred Amsterdam to London and
  London to Paris, then he or she would prefer
  Amsterdam to Paris

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Axioms for Expected Utility

• Reduction of compound uncertain events
• A decision maker is indifferent between a
  compound uncertain events ( a complicated mixture
  of gambles or lotteries) and a simple uncertain
  events as determined by reduction using standard
  probability manipulations.
• The assumption says that we can perform the
  reduction without affecting the decision makers
  preferences

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Axioms for Expected Utility

• Continuity A decision maker is indifferent
  between a consequence A (for example, win 100)
  and some uncertain event involving only two basic
  consequences A1 and A2
• This simply says that we can construct a
  reference gamble with some probability p, for
  which the decision maker will be indifferent
  between the reference gamble and A

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Axioms for Expected Utility

• Example for continuity
• Suppose you are in court as a plaintiff
• Court will award you either 5000 or nothing
• Defendant offers to pay you 1500 to drop the
  charges
• According to continuity axiom, there must be some
  probability P of winning 5000 for which you
  would be indifferent between taking or rejecting
  the settlement offer

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Axioms for Expected Utility

• Substitutability A decision maker is indifferent
  between any original uncertain event that
  includes outcome A and one formed by substituting
  for A an uncertain event that is judged to be its
  equivalent.
• This axioms allows the substitution of uncertain
  reference gambles into a decision for their
  certainty equivalents and is just reverse of the
  reduction axioms

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Axioms for Expected Utility

• Example for Substitutability
• Suppose you are interested in playing the
  lottery, and you are just barely willing to pay
  50 cents for a ticket
• If I owe you 50 cents, then you should be just
  as willing to accept a lottery ticket as the 50
  cents in cash.

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Axioms for Expected Utility

• 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
• Example Two different car dealerships each can
  order the new car that you want. Both dealers
  offer the same price, delivery, warranty, and
  financing, but one is more likely to provide good
  service than the other

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Axioms for Expected Utility

• Invariance All that is needed to determine a
  decision makers preferences among uncertain
  events are the payoffs (or consequences) and the
  associated probabilities
• Finiteness No consequences are considered
  infinitely bad or infinitely good

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PARADOXES

• People do not make necessarily make choices in
  accordance with utility theory
• Von Winterfeldt and Edwards (1986) and Hogarth
  (1987)
• Framing effects are among the most pervasive
  paradoxes in choice behavior
• How an individuals risk attitude can change
  depending on the way the decision problem is posed

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PARADOXES

• A good example is the influenza-outbreak problem
  at the beginning of the chapter
• The program A is the same as C and that B is the
  same as D.
• It all depends on whether you think in terms of
  deaths or lives saved
• The reason for inconsistent choices appears to be
  that different points of reference are used to
  frame the problem in two different ways

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PARADOXES

• In program A and B the reference point is that
  600 people are expected to die, but some may be
  saved. We think about gains in terms of numbers
  of lives saved.
• On the other hand, in program C and D, the
  reference point is that no people would be
  expected to die without the disease, in this case
  we tend to think about lives lost.

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PARADOXES

• Psychologists Kahneman and Tversky have
  discovered that people tend to be risk-averse in
  dealing with gains, but risk-seeking in deciding
  about losses
• Income-tax refund check
• Gamblers continue to throw more money
• Allais paradox (Allais 1953)

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PARADOXES

• Certainty Effect Whereby individuals tend to
  place too much weight on a certain outcome
  relative to uncertain outcomes
• In the Allais Paradox, the certainty effect would
  tend to make individuals overvalue A in decision
  1, possibly leading to an inconsistent choice.

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IMPLICATIONS

• People do not behave according to the behavioral
  axioms
• This fact has distinct implications for the
  practice of decision analysis
• Implication
• -Implications for Utility Assessment
• Managerial and Policy Implications

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Implications for Utility Assessment

• There are several approaches to the assessment of
  utilities
• In chapter 13 we introduced the
  certainty-equivalent approach and
  probability-equivalent approach
• the CE approach , expert argue, contain a
  built-in bias

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Implications for Utility Assessment

• Expert suggest that it would be more appropriate
  to assess utilities by comparing lotteries
• Example suppose that A is the best outcome and C
  is the worst, and we would like to assess U(B),
  which is somewhere between A and C.

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Implications for Utility Assessment

• McCord and De Neufvilles technique would have
  the decision maker assess the probability P that
  produces indifference between the two lotteries
  in decision tree (fig. 14.11)
• The McCord-De Neufville approach does indeed
  lead to utility assessments that are less
  risk-averse than those make with certainty
  equivalents.

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Managerial and Policy Implications

• The most fundamental issues has to do with
  reference point or status quo
• example of few cases about status quo
• Sunk cost
• Seat belt case
• Environmental protection