Axiomatic Theory of Probabilistic Decision Making under Risk

1
Axiomatic Theory of Probabilistic Decision Making
under Risk

• Pavlo R. Blavatskyy
• University of Zurich
• April 21st, 2007

2
Outline

• Introduction
• Framework
• Axioms
• Representation Theorem
• Implications
• Conclusions

3
Introduction

• Experimental studies of repeated decision making
  under risk gt individual choices are often
  contradictory
• Camerer (1989) reports that 31.6 of subjects
  reversed their choices
• Starmer and Sugden (1989) find that 26.5 of all
  choices are reversed
• Hey and Orme (1994) report an inconsistency rate
  of 25
• Wu (1994) finds that 5 to 45 of choice
  decisions are reversed
• Ballinger and Wilcox (1997) report a median
  switching rate of 20.8

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Introduction continued

• Majority of decision theories are deterministic
• Exception Machina (1985) and Chew et al. (1991)
• They predict that repeated choice is always
  consistent (except for decision problems where an
  individual is exactly indifferent)
• Common approach is to embed a deterministic
  decision theory into a model of stochastic choice
• tremble model of Harless and Camerer (1994)
• Fechner model of random errors (e.g. Hey and
  Orme, 1994)
• Random utility model (e.g. Loomes and Sugden,
  1995)

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This Paper

• Individuals do not have a unique preference
  relation on the set of risky lotteries
• Individuals possess a probability measure that
  captures the likelihood of one lottery being
  chosen over another lottery
• A related axiomatization of choice probabilities
• Debreu (1958)
• Fishburn (1978)

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Framework

• A finite set of all possible outcomes
  (consequences)
• Outcomes are not necessarily monetary payoffs
• A risky lottery is a probability distribution
  on X
• A compound lottery
• The set of all risky lotteries is denoted by ?

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Framework, continued

• An individual possesses a probability measure on
  
• Choice probability
  denotes a likelihood that an individual
  chooses L1 over L2 in a repeated binary choice
• A deterministic preference relation can be easily
  converted into a choice probability
• If an individual strictly prefers L1 over L2,
  then
• If an individual strictly prefers L2 over L1,
  then
• If an individual is exactly indifferent, then

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Axioms

• Axiom 1 (Completeness) For any two lotteries
  there exist a choice probability
  and a choice probability
• for any
• Only two events are possible either choose L1
  or choose L2

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Axioms, continued

• Axiom 2 (Strong Stochastic Transitivity) For any
  three lotteries if and then
• Axiom 3 (Continuity) For any three lotteries
  the sets
• and
• are closed

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Axioms, continued

• Axiom 4 (Common Consequence Independence) For any
  four lotteries and any probability
  
• If two risky lotteries yield identical chances of
  the same outcome (or, more generally, if two
  compound lotteries yield identical chances of the
  same risky lottery) this common consequence does
  not affect the choice probability

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Axioms, continued

• Axiom 5 (Interchangeability) For any three
  lotteries if then
• If an individual chooses between two lotteries at
  random then he or she does not mind which of the
  two lotteries is involved in another decision
  problem

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Representation Theorem

• Theorem 1 (Stochastic Utility Theorem)
  Probability measure on satisfies Axioms
  1-5 if and only if there exist an assignment of
  real numbers to every outcome ,
  , and there exist a non-decreasing
  function such that for any two
  risky lotteries

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Implications

• Function has to satisfy a restriction
  for every , which immediately implies
  that
• If a vector and function
  represent a probability measure on then a
  vector and a function
  represent the same probability measure for any
  two real numbers a and b,

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Special cases

• Fechner model of random errors
• function is a cumulative distribution
  function of the normal distribution with mean
  zero and constant standard deviation
• Luce choice model
• function is a cumulative distribution
  function of the logistic distribution
• where is constant
• Tremble model of Harless and Camerer (1994)
• function is the step function

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Empirical paradoxes

• Unlike expected utility theory, stochastic
  utility theory is consistent with systematic
  violations of betweenness and a common ratio
  effect
• but cannot explain a common consequence effect

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Conclusions

• Individuals often make contradictory choices
• Either individuals have multiple preference
  relations on ? (random utility model)
• or individuals have a probability measure on
• Choice probabilities admit a stochastic utility
  representation if and only if they are complete,
  strongly transitive, continuous, independent of
  common consequences and interchangeable
• Special cases Fechner model of random errors,
  Luce choice model and a tremble model of Harless
  and Camerer (1994)