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

3. 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)

5. 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 ξ L is independently and normally distributed with zero mean

6. 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

7. 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

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

9. 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

11. 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: Φ(.) is c.d.f. of the normal distribution with zero mean and standard deviation σ L

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

14. 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 Experimental studyWSSE, CPTWSSE, StEUT Tversky and Kahneman (1992)0.5092 0.6601 0.6672 0.4889 Gonzalez and Wu (1999)17.461215.4721 Wu and Gonzalez (1996)0.24190.2183 Camerer and Ho (1994)0.18950.1860 Bernasconi (1994)1.36091.1452 Camerer (1992) large gains Camerer (1992) small gains Camerer (1992) small losses 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)0.01960.0195 Loomes and Sugden (1998)5.60092.2116

16. The effect of monetary incentives Experimental studyType of incentives Best fitting parameters of StEUT Power of utility function Standard deviation of random errors Tversky and Kahneman (1992)hypothetical 0.7750 (0.7621) 0.7711 0.6075 Gonzalez and Wu (1999)hypothetical + auction0.44161.4028 Wu and Gonzalez (1996)hypothetical0.17200.8185 Camerer and Ho (1994) a randomly chosen subject plays lottery 0.52150.1243 Bernasconi (1994) random lottery incentive scheme 0.20940.2766 Camerer (1992)hypothetical 0.5871 0.9123 (0.5182) 0.0868 0.0914 0.2299 Camerer (1989) hypothetical0.30370.4816 random lottery incentive scheme 0.6830 (0.6207) 0.2897 0.2252 Conlisk (1989)hypothetical0.50491.8580 Loomes and Sugden (1998) random lottery incentive scheme 0.35130.1382 Hey and Orme (1994) random lottery incentive scheme 0.71440.4789

17. 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