A Stochastic Expected
Utility Theory
Pavlo R. Blavatskyy
June 2007
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
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)
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)
StEUT
• Four assumptions:
1. Stochastic expected utility of lottery
Lx1 , p1 ;... xn , pn  is
n
U L    p i u  x i    L
i 1
– Utility function u:R→R is defined over
changes in wealth (e.g. Markowitz, 1952)
– Error term ξL is independently and normally
distributed with zero mean
StEUT continued
2. 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
n
u x1    pi u  xi    L  u x n 
i 1
The normal distribution of an error term is
truncated
StEUT continued
3. The standard deviation of random errors
is higher for lotteries with a wider range
of possible outcomes (ceteris paribus)
4. The standard deviation of random errors
converges to zero for lotteries
converging to a degenerate lottery
lim  L  0, i  1,..., n
pi 1
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
The Allais paradox
• The choice pattern
  

L 0,0.9;5  10 ,0.1  L 0,0.89;10 ,0.11
L1 10 ,1  L2 0,0.01;10 ,0.89;5  10 ,0.1
6
6
6
2
6
6
1
– frequently found in experiments (e.g. Slovic
and Tversky, 1974)
– Not explainable by deterministic EUT
The Allais paradox continued
PDF of U(L2)
PDF of U(L1')
PDF of U(L2')
The fourfold pattern of risk attitudes
•
Individuals often exhibit risk aversion over:
–
–
•
The same individuals often exhibit risk seeking
over:
–
–
•
Probable gains
Improbable losses
Improbable gains
Probable losses
Simultaneous purchase of insurance and lotto
tickets (e.g. Friedman and Savage, 1948)
The fourfold pattern of risk attitudes
continued
• Calculate the certainty equivalent CE
uCE   EU L
• According to StEUT:
2
2

u

x





u

x




n
L
1
L




2
2


2 L
2 L
L
e
e
1 

CE  u  L 

2 u  x n    L   u x1    L  




n
 L   p i u  xi 
i 1
Φ(.) is c.d.f. of the normal
distribution with zero mean
and standard deviation σL
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
Fit to experimental data continued
• Utility function defined exactly as the value
function of CPT:

 x ,
x0
ux   

   x  , x  0
• Standard deviation of random errors
 L    uxn   ux1 
n
 1  p 
i
i 1
• Minimize the weighted sum of squared errors
n

WSSE   CE
i 1
StEUT
i

CEi  1
2
Fit to experimental data continued
Experimental study
WSSE, CPT
WSSE, StEUT
Tversky and Kahneman (1992)
0.5092
0.6601
0.6672
0.4889
Gonzalez and Wu (1999)
17.4612
15.4721
Wu and Gonzalez (1996)
0.2419
0.2183
Camerer and Ho (1994)
0.1895
0.1860
Bernasconi (1994)
1.3609
1.1452
Camerer (1992) large gains
Camerer (1992) small gains
Camerer (1992) small losses
0.0122
0.0122
0.0416
0.0207
0.0115
0.0262
Camerer (1989) large gains
Camerer (1989) small gains
Camerer (1989) small losses
0.1996
0.1871
0.2170
0.2359
0.1639
0.1281
Conlisk (1989)
0.0196
0.0195
Loomes and Sugden (1998)
5.6009
2.2116
The effect of monetary incentives
Best fitting parameters of StEUT
Experimental study
Type of incentives
Power of utility
function
Standard deviation of
random errors
0.7750
(0.7621)
0.7711
0.6075
hypothetical + auction
0.4416
1.4028
Wu and Gonzalez (1996)
hypothetical
0.1720
0.8185
Camerer and Ho (1994)
a randomly chosen
subject plays lottery
0.5215
0.1243
Bernasconi (1994)
random lottery incentive
scheme
0.2094
0.2766
hypothetical
0.5871
0.9123
(0.5182)
0.0868
0.0914
0.2299
hypothetical
0.3037
0.4816
0.6830
(0.6207)
0.2897
0.2252
Tversky and Kahneman (1992)
hypothetical
Gonzalez and Wu (1999)
Camerer (1992)
Camerer (1989)
random lottery incentive
scheme
Conlisk (1989)
hypothetical
0.5049
1.8580
Loomes and Sugden (1998)
random lottery incentive
scheme
0.3513
0.1382
Hey and Orme (1994)
random lottery incentive
scheme
0.7144
0.4789
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
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