CHAPTER 14

Utility Axioms

Paradoxes & Implications

Chapter 14

• Learning objectives:

• Axioms for Expected Utility

• Paradoxes

• Implication for Utility Assessment

• Managerial and Policy Implications

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

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

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?

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

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

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 maker’s preferences

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

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

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

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.

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

Axioms for Expected Utility

• Invariance: All that is needed to determine a decision maker’s preferences among uncertain events are the payoffs (or consequences) and the associated probabilities

• Finiteness : No consequences are considered infinitely bad or infinitely good

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 individual’s risk attitude can change depending on the way the decision problem is posed

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

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.

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)

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.

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

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

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.

Implications for Utility Assessment

• McCord and De Neufville’s 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.

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