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

is

– 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

nn pxpxL ,;..., 11

L

n

iii xupLU

1

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

The normal distribution of an error term is truncated

nL

n

iii xuxupxu

1

1

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

niLpi

,...,1,0lim1

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

– frequently found in experiments (e.g. Slovic and Tversky, 1974)

– Not explainable by deterministic EUT

1.0,105;89.0,10;01.0,01,10 662

61 LL

11.0,10;89.0,01.0,105;9.0,0 61

62 LL

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:– 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)

The fourfold pattern of risk attitudes continued

• Calculate the certainty equivalent CE

• According to StEUT: LUECEu

LLn

xuxu

LL xuxu

eeuCE

L

Ln

L

L

1

221

2

2

2

21

2

n

iiiL xup

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:

• Standard deviation of random errors

• Minimize the weighted sum of squared errors

0,

0,

xx

xxxu

n

iinL pxuxu

11 1

n

ii

StEUTi CECEWSSE

1

21

Fit to experimental data continuedExperimental study WSSE, CPT WSSE, StEUTTversky 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

Experimental study Type of incentives

Best fitting parameters of StEUT

Power of utility function

Standard deviation of random errors

Tversky and Kahneman (1992) hypothetical0.7750

(0.7621)0.77110.6075

Gonzalez and Wu (1999) 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

Camerer (1992) hypothetical0.58710.9123

(0.5182)

0.08680.09140.2299

Camerer (1989)

hypothetical 0.3037 0.4816

random lottery incentive scheme

0.6830(0.6207)

0.28970.2252

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 CPTDescriptive 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