A Stochastic Expected Utility Theory

1
A Stochastic Expected Utility Theory

• Pavlo R. Blavatskyy
• June 2007

2
Presentation overview

• Why another decision theory?
• Description of StEUT
• How StEUT explains empirical facts
• The Allais Paradox
• The fourfold pattern of risk attitudes
• Violation of betweenness
• Fit to empirical data
• Conclusions extensions

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Introduction

• Expected utility theory
• Normative theory (e.g. von Neumann Morgenstern,
  1944)
• Persistent violations (e.g. Allais, 1953)
• No clear alternative (e.g. Harless and Camerer,
  1944 Hey and Orme, 1994)
• Cumulative prospect theory as the most successful
  competitor (e.g. Tversky and Kahneman, 1992)

4
Introduction continued

• The stochastic nature of choice under risk
• Experimental evidence average consistency rate
  is 75 (e.g. Camerer, 1989 Starmer Sugden,
  1989 Wu, 1994)
• Variability of responses is higher than the
  predictive error of various theories (e.g. Hey,
  2001)
• Little emphasis on noise in the existing models
  (e.g. Harless and Camerer, 1994 Hey and Orme,
  1994)

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StEUT

• Four assumptions
• Stochastic expected utility of lottery
  is
• Utility function uR?R is defined over changes in
  wealth (e.g. Markowitz, 1952)
• Error term ?L is independently and normally
  distributed with zero mean

6
StEUT continued

• Stochastic expected utility of a lottery
• Cannot be less than the utility of the lowest
  possible outcome
• Cannot exceed the utility of the highest possible
  outcome
• The normal distribution of an error term is
  truncated

7
StEUT continued

• The standard deviation of random errors is higher
  for lotteries with a wider range of possible
  outcomes (ceteris paribus)
• The standard deviation of random errors converges
  to zero for lotteries converging to a degenerate
  lottery

8
Explanation of the stylized facts

• The Allais paradox
• The fourfold pattern of risk attitudes
• The generalized common consequence effect
• The common ratio effect
• Violations of betweenness

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The Allais paradox

• The choice pattern
• frequently found in experiments (e.g. Slovic and
  Tversky, 1974)
• Not explainable by deterministic EUT

10
The Allais paradox continued
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The fourfold pattern of risk attitudes

• Individuals often exhibit risk aversion over
• Probable gains
• Improbable losses
• The same individuals often exhibit risk seeking
  over
• Improbable gains
• Probable losses
• Simultaneous purchase of insurance and lotto
  tickets (e.g. Friedman and Savage, 1948)

12
The fourfold pattern of risk attitudes continued

• Calculate the certainty equivalent CE
• According to StEUT

F(.) is c.d.f. of the normal distribution with
zero mean and standard deviation sL
13
Fit to experimental data

• Parametric fitting of StEUT to ten datasets
• Tversky and Kahneman (1992)
• Gonzalez and Wu (1999)
• Wu and Gonzalez (1996)
• Camerer and Ho (1994)
• Bernasconi (1994)
• Camerer (1992)
• Camerer (1989)
• Conlisk (1989)
• Loomes and Sugden (1998)
• Hey and Orme (1994)

Aggregate choice pattern
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Fit to experimental data continued

• Utility function defined exactly as the value
  function of CPT
• Standard deviation of random errors
• Minimize the weighted sum of squared errors

15
Fit to experimental data continued
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The effect of monetary incentives
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StEUT in a nutshell

• An individual maximizes expected utility
  distorted by random errors
• Error term additive on utility scale
• Errors are normally distributed, internality
  axiom holds
• Variance ? for lotteries with a wider range of
  outcomes
• No error in choice between sure things
• StEUT explains all major empirical facts
• StEUT fits data at least as good as CPT
• ?Descriptive decision theory can be constructed
  by modeling the structure of an error term

18
Extensions

• StEUT (and CPT) does not explain satisfactorily
  all available experimental evidence
• Gambling on unlikely gains
  (e.g. Neilson and Stowe, 2002)
• Violation of betweenness when modal choice is
  inconsistent with betwenness axiom
• Predicts too many violations of dominance
  (e.g. Loomes and Sugden, 1998)
• There is a potential for even better descriptive
  decision theory
• Stochastic models make clear prediction about
  consistency rates