Abstract
This paper proposes a new decision theory of how individuals make random errors when they compute the expected utility of risky lotteries. When distorted by errors, the expected utility of a lottery never exceeds (falls below) the utility of the highest (lowest) outcome. This assumption implies that errors are likely to overvalue (undervalue) lotteries with expected utility close to the utility of the lowest (highest) outcome. Proposed theory explains many stylized empirical facts such as the fourfold pattern of risk attitudes, common consequence effect (Allais paradox), common ratio effect and violations of betweenness. Theory fits the data from ten well-known experimental studies at least as well as cumulative prospect theory.
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Notes
Computational errors occur for a variety of reasons (Hey and Orme 1994). An individual may not be sufficiently motivated to make a balanced decision. A subject can get tired during a long experiment and pay less attention (especially if lotteries do not involve losses). A subject can simply press a wrong key by accident or inertia. Wu (1994, p.50) suggests that subjects can suffer from fatigue and hurry up with their responses at the end of the experiment.
For example, in the Allais paradox, this condition is satisfied when the gain of one million starting from zero wealth position brings a higher increase in utility than the gain of an additional four million.
Bernasconi (1994) finds the common ratio effect when θ = 0.8 and θ = 0.75. Loomes and Sugden (1998) find evidence of the common ratio effect when \( \theta \in {\left\{ {0.6,2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3,0.8} \right\}} \), and no such evidence when θ = 0.4 and θ = 0.5.
Tversky and Kahneman (1992) also used eight decision problems involving mixed lotteries with positive and negative outcomes. Unfortunately, Richard Gonzalez, who conducted the experiment for Tversky and Kahneman (1992) could not find the raw data on these mixed lotteries and no reexamination was possible.
Non-linear unconstrained optimization was implemented in the Matlab 6.5 package (based on the Nelder–Mead simplex algorithm).
Specifically, the utility of lottery L(x 1, p 1;...x n , p n ) with outcomes \( x_{1} < \ldots < x_{m} < 0 \leqslant x_{{m + 1}} < \ldots < x_{n} \) is \( \widetilde{u}{\left( L \right)} = {\sum\nolimits_{i = 1}^m {u^{ - } {\left( {x_{i} } \right)}{\left( {w^{ - } {\left( {{\sum\nolimits_{j = 1}^i {p_{j} } }} \right)} - w^{ - } {\left( {{\sum\nolimits_{j = 1}^{i - 1} {p_{j} } }} \right)}} \right)}} } + {\sum\nolimits_{i = m + 1}^n {u^{ + } {\left( {x_{i} } \right)}{\left( {w^{ + } {\left( {{\sum\nolimits_{j = i}^n {p_{j} } }} \right)} - w^{ + } {\left( {{\sum\nolimits_{j = i + 1}^n {p_{j} } }} \right)}} \right)}} } \), where \( u^{ - } {\left( x \right)} = - \lambda {\left( { - x} \right)}^{\beta } \), \( u^{ + } {\left( x \right)} = x^{\alpha } \), \( w^{ + } {\left( p \right)} = {p^{\gamma } } \mathord{\left/ {\vphantom {{p^{\gamma } } {{\left( {p^{\gamma } + {\left( {1 - p} \right)}^{\gamma } } \right)}^{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} }}} \right. \kern-\nulldelimiterspace} {{\left( {p^{\gamma } + {\left( {1 - p} \right)}^{\gamma } } \right)}^{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} } \) and \( w^{ - } {\left( p \right)} = {p^{\delta } } \mathord{\left/ {\vphantom {{p^{\delta } } {{\left( {p^{\delta } + {\left( {1 - p} \right)}^{\delta } } \right)}^{{1 \mathord{\left/ {\vphantom {1 \delta }} \right. \kern-\nulldelimiterspace} \delta }} }}} \right. \kern-\nulldelimiterspace} {{\left( {p^{\delta } + {\left( {1 - p} \right)}^{\delta } } \right)}^{{1 \mathord{\left/ {\vphantom {1 \delta }} \right. \kern-\nulldelimiterspace} \delta }} } \).
I also estimated CPT with a stochastic choice model \( {\text{prob}}{\left( {S \succ R} \right)} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp {\left\{ {\tau \cdot {\left( {\widetilde{u}{\left( R \right)} - \widetilde{u}{\left( S \right)}} \right)}} \right\}}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left\{ {\tau \cdot {\left( {\widetilde{u}{\left( R \right)} - \widetilde{u}{\left( S \right)}} \right)}} \right\}}} \right)}} \), τ = const, proposed by Luce and Suppes (1965, p.335) and used by Camerer and Ho (1994) and Wu and Gonzalez (1996). The result of this estimation was nearly identical to the estimation of CPT with the Fechner model.
Kagel et al. (1990) allowed the subjects to express indifference but do not report how many subjects actually used this possibility. Camerer (1989) allowed indifference in one experimental session. Camerer (1989) reports that three subjects revealed indifference in almost every decision problem, and the rest never expressed indifference.
There is also a practical constraint why the reexamination of individual choice patterns is not feasible. Many of the experimental studies reexamined in this section were conducted over a decade ago and several authors, whom I contacted, could not find raw experimental data.
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Acknowledgments
I am grateful to Colin Camerer, Christian Ewerhart, John Hey, Wolfgang Köhler and Andreas Ortmann as well as the participants of the research seminars in IEW (Zurich, March 31, 2005) and CERGE-EI (Prague, April 14, 2005), the 20th Biennial Conference on Subjective Probability, Utility and Decision Making (Stockholm, August 22, 2005) and the 20th Annual Congress of the European Economic Association (Amsterdam, August 25, 2005) for their extensive comments. I also would like to thank the editor W. Kip Viscusi and one anonymous referee for their helpful suggestions. Richard Gonzalez, George Wu, John Hey and Chris Orme generously provided their experimental data.
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Blavatskyy, P.R. Stochastic expected utility theory. J Risk Uncertainty 34, 259–286 (2007). https://doi.org/10.1007/s11166-007-9009-6
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DOI: https://doi.org/10.1007/s11166-007-9009-6