Rank-dependent Utility Model Research Articles

Generalized Optimized Certainty Equivalent with Applications in the Rank-Dependent Utility Model

Generalized Optimized Certainty Equivalent with Applications in the Rank-Dependent Utility Model

Can Climate Shocks Make Vulnerable Subjects More Willing to Take Risks?

While economists in the past tended to assume that individual preferences, including risk preferences, are stable over time, a recent literature has developed and indicates that risk preferences respond to shocks, with mixed evidence on the direction of the responses. This paper utilizes a natural experiment with covariate (drought) and idiosyncratic shocks in combination with an independent field risk experiment. The risk experiment uses a Certainty Equivalent-Multiple Choice List approach and is played 1–2 years after the subjects were (to a varying degree) exposed to a covariate drought shock or idiosyncratic shocks for a sample of resource-poor young adults living in a risky semi-arid rural environment in Sub-Saharan Africa. The experimental approach facilitates a comprehensive assessment of shock effects on experimental risk premiums for risky prospects with varying probabilities of good and bad outcomes. The experiment also facilitates the estimation of the utility curvature in an Expected Utility (EU) model and, alternatively, separate estimation of probability weighting and utility curvature in three different Rank Dependent Utility models with a two-parameter Prelec probability weighting function. Our study is the first to comprehensively test the theoretical predictions of Gollier and Pratt (Econom J Econom Soc 64:1109–1123, 1996) versus Quiggin (Econ Theor 22(3):607–611, 2003). Gollier and Pratt (1996) build on EU theory and state that an increase in background risk will make subjects more risk averse while Quiggin (2003) states that an increase in background risk can enhance risk-taking in certain types of non-EU models. We find strong evidence that such non-EU preferences dominate in our sample.

On a disparity between willingness to pay and willingness to accept under the Rank-Dependent Utility model

On a disparity between willingness to pay and willingness to accept under the Rank-Dependent Utility model

Second-order stochastic dominance with respect to the rank-dependent utility model

A generalized family of partial orders is studied, which is semiparametrized by a utility function <inline-formula> <tex-math id="M1">\begin{document}$ f $\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M1.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M1.png"/> </alternatives> </inline-formula> and a distortion function <inline-formula> <tex-math id="M2">\begin{document}$ g $\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M2.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M2.png"/> </alternatives> </inline-formula>, namely, <inline-formula> <tex-math id="M3">\begin{document}$ (f,g) $\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M3.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M3.png"/> </alternatives> </inline-formula>-utility and distorted stochastic dominance ( <inline-formula> <tex-math id="M4">\begin{document}$ (f,g) $\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M4.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M4.png"/> </alternatives> </inline-formula>-UDSD). Such a family is especially suitable for representing a decision maker’s preferences in terms of risk aversion. We characterize the monotonicity of the partial order in the rank-dependent utility model, and the isotonic classes of rank-dependent utility with <inline-formula> <tex-math id="M5">\begin{document}$ (f,g) $\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M5.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M5.png"/> </alternatives> </inline-formula>-UDSD are also established. Inspired by the concept of the congruent utility class, we introduce the definition of the congruent distortion class. The characterization of the congruent utility class or distortion class of <inline-formula> <tex-math id="M6">\begin{document}$ (f,g) $\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M6.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0097_M6.png"/> </alternatives> </inline-formula>-utility and distorted stochastic dominance is investigated. Based on the main results in this paper, we unify some related results in the existing literature. As an application, we propose a general approach to develop a continuum between first-order stochastic dominance and second-order stochastic dominance based on the partial order.

Risk aversion over finite domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range (alternatively, when the payoffs are numerical and their domain is finite or convex, respectively). By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk.

Dual Moments and Risk Attitudes

The well-known Pratt–Arrow approximation, developed independently by John W. Pratt and Kenneth Arrow, provides an insightful dissection of the risk premium under the expected utility (EU) model. It is given by one-half the product of the variance of the risk and the local index of absolute risk aversion of the decision maker. Quite surprisingly, despite many important developments on “global” risk aversion in non-EU models, the “local” approach to risk aversion has received little attention outside EU. By considering the first two dual moments, mean and maxiance, on equal footing with the first two primal moments, mean and variance, the authors develop a dissection of the risk premium under the popular rank-dependent utility (RDU) model. This yields a simple approximation to the risk premium and a local index of absolute risk aversion under the RDU model.

Risk perceptions and economic incentives for mitigating windstorm damage to housing

ABSTRACT Mitigation measures such as window shutters can substantially reduce windstorm damage to housing. However, the cost-effectiveness of wind mitigation measures remains unclear when considering influencing factors such as financial protection via insurance, individual risk perceptions and planning time horizons. Economic incentives may also be provided to motivate homeowners to invest in mitigation. In this study, the installation of cyclone-rated window shutters as a mitigation measure for homeowners in Australia is used as a case study. A life-cycle cost analysis in conjunction with a rank-dependent expected utility model are employed to identify economic incentives that are financially attractive to homeowners with different perceived risks and planning time horizons. The effect of home insurance is also taken into account. The results suggest that, as expected, the economic incentives needed to promote window shutters among homeowners decrease with the perceived storm damage risk and planning time horizon. A rebate for shutter installation cost or a discounted insurance premium have been found to be feasible economic incentives considering a typical planning time horizon of ten years. Statistical surveys are needed in future work to calibrate the parameters within the rank-dependent utility model to better represent the risk perceptions of different groups of homeowners.

On the indeterminacy of the representation of beliefs by probabilities

This paper examines the indeterminacy of the representations of beliefs by subjective probabilities in non-expected utility theories of decision making under uncertainty. These theories include the rank-dependent utility models, probabilistically sophisticated choice and probabilistic sophistication based on event exchangeability.

A Critical Comparison of Three Notions of Fractional Stochastic Dominance

Two notions of fractional stochastic dominance are recently proposed by Muller et al. (2017) and Huang et al. (2020), respectively. Our main objective is to understand the comparative advantages of the two notions, as well as their suitability in different contexts, by establishing several new technical results. For a more comprehensive comparison, we further include a third natural notion of fractional stochastic dominance based on the coefficient of relative risk aversion. Among the three notions, it turns out that one can be seen as a logarithmic version of second-order stochastic dominance (SSD), another one can be seen as a power version of SSD, whereas there does not exist a transformation to associate the last one with SSD. We find that these notions of fractional stochastic dominance are naturally connected to five classes of risk measures, including Value-at-Risk, Expected Shortfall, expectiles, entropic risk measures, and loss certainty equivalents. The three notions are further characterized in the contexts of the rank-dependent utility model and the cumulative prospect theory. We make some recommendations on which notion to use in specific situations, as they all have their own merits.

Smallholder Farmer Risk Preferences in Northern Ghana:Evidence from a Controlled Field Experiment

We conduct a controlled field experiment to elicit risk preferences among maize farmers in Northern Ghana. Farmers participating in the experiment were asked to choose from a menu of lotteries representing different hypothetical probability distributions over yields produced by ‘traditional’ and ‘high yield’ maize varieties. We estimate a Rank-Dependent Utility Model (RDU) with an Expo-Power utility function, allowing for systematic subjective underweighting or overweighting of outcome probabilities and non-constant relative risk aversion. Based on our estimates, we cannot reject the hypotheses that decisions made by farmers in our study can be uniformly characterised by conventional Von Neumann–Morgenstern expected utility theory (EUT), but reject the hypothesis that farmers exhibit constant relative risk aversion.

Generalized Optimized Certainty Equivalent with Applications in the Rank-dependent Utility Model

In this paper, we introduce a class of optimized certainty equivalent based on the variational preference, give its dual representation based on ϕ-divergence, and study its equivalent characterization of positive homogeneity and coherence. As applications, we investigate the properties of optimized certainty equivalent based on the rank-dependent utility (RDU) model. The dual representation of RDU-based shortfall risk measure proposed by Mao and Cai (2018) is also presented.

On Extendability of the Principle of Equivalent Utility

An insurance premium principle is a way of assigning to every risk a real number, interpreted as a premium for insuring risk. There are several methods of defining the principle. In this paper, we deal with the principle of equivalent utility under the rank-dependent utility model. The principle, generated by utility function and probability distortion function, is based on the assumption of the symmetry between the decisions of accepting and rejecting risk. It is known that the principle of equivalent utility can be uniquely extended from the family of ternary risks. However, the extension from the family of binary risks need not be unique. Therefore, the following problem arises: characterizing those principles that coincide on the family of all binary risks. We reduce the problem thus to the multiplicative Pexider functional equation on a region. Applying the form of continuous solutions of the equation, we solve the problem completely.

Decisions under uncertainty in social contexts

This paper theoretically and experimentally studies decision-making in risky and social environments. We explore the interdependence of two decisive behavioral determinants in such environments: risk attitudes and social preferences in the form of inequality aversion. Our model and the data demonstrate that individual risk aversion is attenuated when lagging behind peers, whereas it is amplified under favorable income inequality. Moreover, people's choices are not only context-dependent, but they are sensitive to their degree of inequality aversion. The majority of our experimental findings cannot be rationalized by rank-dependent utility models or cumulative prospect theory. Our results contribute to the basic understanding of the underlying motives of social incentives in firms, private households' saving decisions, employees' career-track choices or charitable giving under uncertainty.

Characterizing Households' Large (and Small) Stakes Decisions: Evidence from Flood Insurance

We examine the flood insurance decisions of over 100,000 households, using standard expected utility models and rank dependent utility models that incorporate probability distortions. Consumers’ insurance choices provide important insights into their risk attitudes. Previous research has typically examined modest stakes choices, such as appliance warranties and deductibles, leaving important questions about larger stakes decisions. Features of U.S. flood insurance allow us to model risk attitudes over large stakes from consumers’ coverage limits. We find that consumers are typically willing to pay premiums well above the expected value of their contracts, though consumers’ decisions vary substantially over large stakes. Explaining these choices with standard expected utility models requires massive variation in risk aversion. In contrast, models incorporating probability distortions greatly improve the ability to predict households’ decisions. These models explain consumers’ choices through their overweighting of small probabilities and can more easily accommodate the observed variation in consumers’ decisions. Our large-stakes choice analyses reveal new insights on models of consumers’ risk preferences.

Risk Aversion Over Finite Domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range. By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk. This general insight, which we explain to raise several interesting questions for further research, is our main contribution.

Asset Integration and Attitudes toward Risk: Theory and Evidence

Measures of risk attitudes derived from experiments are often questioned because they are based on small stakes bets and do not account for the extent to which the decision-maker integrates the prizes of the experimental tasks with personal wealth. We exploit the existence of detailed information on individual wealth of experimental subjects in Denmark, and directly estimate risk attitudes and the degree of asset integration consistent with observed behavior. The behavior of the adult Danes in our experiments is consistent with partial asset integration: they behave as if some small fraction of personal wealth is combined with experimental prizes in a utility function, and that this combination entails less than perfect substitution. Our subjects do not perfectly asset integrate. The implied risk attitudes from estimating these specifications imply risk premia and certainty equivalents that are a priori plausible under expected utility theory or rank dependent utility models. These are reassuring and constructive solutions to payoff calibration paradoxes. In addition, the rigorous, structural modeling of partial asset integration points to a rich array of neglected questions in risk management and policy evaluation in important field settings.

Inverse S-shaped probability weighting and its impact on investment

In this paper we analyze how changes in inverse S-shaped probability weighting influence optimal portfolio choice in a rank-dependent utility model. We derive sufficient conditions for the existence of an optimal solution of the investment problem, and then define the notion of a more inverse S-shaped probability weighting function. We show that an increase in inverse S-shaped weighting typically leads to a lower allocation to the risky asset, regardless of whether the return distribution is skewed left or right, as long as it offers a non-negligible risk premium. Only for lottery stocks with poor expected returns and extremely positive skewness does an increase in inverse S-shaped probability weighting lead to larger portfolio allocations.

Investigating Gender Differences under Time Pressure in Financial Risk Taking.

There is a significant gender imbalance on financial trading floors. This motivated us to investigate gender differences in financial risk taking under pressure. We used a well-established approach from behavior economics to analyze a series of risky monetary choices by male and female participants with and without time pressure. We also used second to fourth digit ratio (2D:4D) and face width-to-height ratio (fWHR) as correlates of pre-natal exposure to testosterone. We constructed a structural model and estimated the participants' risk attitudes and probability perceptions via maximum likelihood estimation under both expected utility (EU) and rank-dependent utility (RDU) models. In line with existing research, we found that male participants are less risk averse and that the gender gap in risk attitudes increases under moderate time pressure. We found that female participants with lower 2D:4D ratios and higher fWHR are less risk averse in RDU estimates. Males with lower 2D:4D ratios were less risk averse in EU estimations, but more risk averse using RDU estimates. We also observe that men whose ratios indicate a greater prenatal exposure to testosterone exhibit a greater optimism and overestimation of small probabilities of success.

Inverse S-Shaped Probability Weighting and Its Impact on Investment

In this paper we analyze how changes in inverse S-shaped probability weighting influence optimal portfolio choice in a rank-dependent utility model. We derive sufficient conditions for the existence of an optimal solution of the investment problem, and then define the notion of a more inverse S-shaped probability weighting function. We show that an increase in inverse S-shaped weighting typically leads to a lower allocation to the risky asset, regardless of whether the return distribution is skewed left or right, as long as it offers a non-negligible risk premium. Only for lottery stocks with poor expected returns and extremely positive skewness does an increase in inverse S-shaped probability weighting lead to larger portfolio allocations.

The empirical adequacy of cumulative prospect theory and its implications for normative assessment

Much behavioral welfare economics assumes that expected utility theory (EUT) does not accurately describe most human choice under risk. A substantial literature instead evaluates welfare consequences by taking cumulative prospect theory (CPT) as the natural default alternative, at least where description is concerned. We present evidence, based on a review of previous literature and new experimental data, that the most empirically adequate hypothesis about human choice under risk is that it is heterogeneous, and that where EUT does not apply, more choice is characterized by rank-dependent utility models than by CPT. Most of the apparently loss-averse choice behavior results from probability weighting rather than from direct disutility experienced when an outcome is framed as a loss against an idiosyncratic reference point. We then consider implications of this finding for methodological debates about how to model welfare effects of policies, and argue that abandonment of a dogmatic belief in CPT as the correct theory of risk human choice exposes a conceptual error that is widespread in behavioral welfare economics. We provide concluding reflections on second-order, philosophical issues around the grounding of normative commitments in policy-focused economics.