Rank-dependent Utility Research Articles

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.

A rank-dependent utility approach to model intra- and inter-individual heterogeneity in risky choice behaviours

ABSTRACT Using a stated preferences survey, the objective of this paper is to investigate the intra- and inter-individual heterogeneity of mode choice, when travel time is subject to variability. By‘inter-individual heterogeneity’ is meant that people are different in terms of attitude to risk and have different utility functions. By ‘intra-individual heterogeneity’ is meant that the behaviour may be different even when performed by the same individual when faced with a different mode of transport. Based on Rank-Dependent Utility Theory, the paper shows that the occurrence of delays associated with train trips is overestimated whereas they are underestimated for car trips. A latent-class logit model offers a somewhat different perspective: if, overall, car users are more likely to perceive possible delays for train trips than for car trips, train users tend to consider the objective occurrence of delays as they are presented in the survey and adopt a risk neutral choice behaviour.

Bilateral Risk Sharing with Heterogeneous Beliefs and Exposure Constraints

This paper studies bilateral risk sharing under no aggregate uncertainty, where one agent has Expected-Utility preferences and the other agent has Rank-Dependent Utility preferences with a general probability distortion function. We impose exogenous constraints on the risk exposure for both agents, and we allow for any type or level of belief heterogeneity. We show that Pareto-optimal risk-sharing contracts can be obtained via a constrained utility maximization under a participation constraint of the other agent. This allows us to give an explicit characterization of optimal risk-sharing contracts. In particular, we show that an optimal risk-sharing contract contains allocations that are monotone functions of the likelihood ratio, where the latter is obtained from Lebesgue's Decomposition Theorem.

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.

Consistent Investment of Sophisticated Rank-Dependent Utility Agents in Continuous Time

We study portfolio selection in a complete continuous-time market where the preference is dictated by the rank-dependent utility. As such a model is inherently time inconsistent due to the underlying probability weighting, we study the investment behavior of sophisticated consistent planners who seek (subgame perfect) intra-personal equilibrium strategies. We provide sufficient conditions under which an equilibrium strategy is a replicating portfolio of a final wealth. We derive this final wealth profile explicitly, which turns out to be in the same form as in the classical Merton model with the market price of risk process properly scaled by a deterministic function in time. We present this scaling function explicitly through the solution to a highly nonlinear and singular ordinary differential equation, whose existence of solutions is established. Finally, we give a necessary and sufficient condition for the scaling function to be smaller than 1 corresponding to an effective reduction in risk premium due to probability weighting.

All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

Eliciting all the components of prospect theory –curvature of the utility function, weighting function and loss aversion– remains an open empirical challenge. We develop a semi-parametric method that keeps the tractability of parametric methods while providing more precise estimates. Using the data of Tversky and Kahneman (1992), we revisit their main parametric results. We reject the convexity of the utility function in the loss domain, find lower probability weighting, and confirm loss aversion. We also report that the probability weighting function does not exhibit duality and equality across domains, in line with cumulative prospect theory and in contrast with original prospect and rank dependent utility theories.

BILATERAL RISK SHARING WITH HETEROGENEOUS BELIEFS AND EXPOSURE CONSTRAINTS

AbstractThis paper studies bilateral risk sharing under no aggregate uncertainty, where one agent has Expected-Utility preferences and the other agent has Rank-dependent utility preferences with a general probability distortion function. We impose exogenous constraints on the risk exposure for both agents, and we allow for any type or level of belief heterogeneity. We show that Pareto-optimal risk-sharing contracts can be obtained via a constrained utility maximization under a participation constraint of the other agent. This allows us to give an explicit characterization of optimal risk-sharing contracts. In particular, we show that an optimal risk-sharing contract contains allocations that are monotone functions of the likelihood ratio, where the latter is obtained from Lebesgue’s Decomposition Theorem.

On comparison of the principles of equivalent utility and its applications

An insurance premium principle is a way of assigning to every risk, represented by a non-negative bounded random variable on a given probability space, a non-negative real number. Such a number is interpreted as a premium for the insuring risk. In this paper the implicitly defined principle of equivalent utility is investigated. Using the properties of the quasideviation means, we characterize a comparison in the class of principles of equivalent utility under Rank-Dependent Utility, one of the important behavioral models of decision making under risk. Then we apply this result to establish characterizations of equality and positive homogeneity of the principle. Some further applications are discussed as well.

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.

Estimating uncertainty aversion using the source method in stylized tasks with varying degrees of uncertainty

The source method provides a tractable way to analyze uncertainty aversion using source functions. In this analytical framework, uncertainty aversion is defined using the index of pessimism and the index of likelihood insensitivity. The focus of this paper is to use the source method to examine how uncertainty aversion differs across events that have the same underlying objective probabilities but are presented under varying degrees of uncertainty. In a controlled laboratory setting, subjects are presented with three lottery tasks that are ranked in order of increasing uncertainty. Given the choices observed in each task, a “source function” is estimated jointly with risk attitudes assuming Rank-Dependent Utility Theory under different probability weighting specifications for the source function. Results from the Prelec (Prelec (1998)) specification suggest that, as the degree of uncertainty increases, subjects display increasing pessimism; in contrast, the Tversky-Kahneman (Tversky-Kahneman (1992)) and the Power specifications detect no such difference. Thus, conclusions regarding uncertainty aversion are contingent upon which probability weighting specification is assumed for the source function, and caution should be taken when selecting specifications for the source function.

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.

Forward Rank-Dependent Performance Criteria: Time-Consistent Investment Under Probability Distortion

We introduce the concept of forward rank-dependent performance criteria, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time-consistent nature of forward performance criteria with the time-inconsistency stemming from probability distortions. For this, we first propose two distinct definitions, one based on the preservation of performance value and the other on the time-consistency of policies and, in turn, establish their equivalence. We then fully characterize the viable class of probability distortion processes and provide the following dichotomy: it is either the case that the probability distortions are degenerate in the sense that the investor would never invest in the risky assets, or the marginal probability distortion equals to a normalized power of the quantile function of the pricing kernel. We also characterize the optimal wealth process, whose structure motivates the introduction of a new, “distorted” measure and a related market. We then build a striking correspondence between the forward rank-dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection also provides a direct construction method for forward rank-dependent criteria. Finally, our results on forward rank-dependent performance criteria motivate us to revisit the classical (backward) setting. We follow the so-called dynamic utility approach and derive conditions for existence and a construction of dynamic rank-dependent utility processes.

Adam Smith on lotteries: an interpretation and formal restatement

The paper concerns a neglected aspect of the Wealth of Nations (with the notable exception of D. Levy 1999), dealing directly with decision under risk. In a few pages from book I, chapter 10, Adam Smith explicitly named “lotteries” various objects of choice (possible occupations, or investment opportunities, for instance) and provided an analysis which standard expected utility glasses would hardly fit. Taking this into account allows a better understanding of the part played by typical characters like the “projector” or the “sober man”, in such matters as Smith’s conception of entrepreneurship or of the credit market. The use of some modern concepts in decision analysis (inverse stochastic dominance, rank dependent utility, prudence toward risk), is a means to show the existence, in Smith’s work, of an original theory from decision under risk, where his analysis of lotteries in the Wealth of Nations is consistent with statements from his moral philosophy on asymmetric sensitivity to gains and losses and to the regulating part played by the impartial spectator.

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.

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.

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.

Beliefs and preferences for food-safety policies: a discrete choice model under uncertainty

AbstractOutcomes of food policies are highly uncertain. Therefore, the public’s support for these policies depends on individuals’ beliefs and the provision of scientific information. Using data collected from a discrete choice experiment survey, we explore whether new information regarding a food-safety policy influences respondents’ support, while controlling for risk and time preferences. Additionally, we examine if support depends on whether information is perceived as either good or bad news. Results from the estimation of parametric error component logit models, based on expected utility theory and rank dependent utility theory, suggest that good and bad news affects preferences and welfare measures.

Aversion to risk of regret and preference for positively skewed risks

We assume that the ex-post utility of an agent facing a menu of lotteries depends upon the actual payoff together with its forgone best alternative, thereby allowing for the expost emotion of regret. An increase in the risk of regret is obtained when the actual payoff and its forgone best alternative are statistically less concordant in the sense of Tchen (1980). The aversion to any such risk of regret is thus equivalent to the supermodularity of the bivariate utility function. We show that more regret-risk-averse agents are more willing to choose the risky act in a one-risky-one-safe menu, in particular when the payoff of the risky choice is highly skewed. This is compatible with the possibility effect that is well documented in prospect theory. Symmetrically, we define the aversion to elationrisk that can prevail when the ex-post utility is alternatively sensitive to the forgone worst payoff. We show that elation-risk-seeking is compatible with the certainty effect. We finally show that regret-risk-averse and elation-risk-seeking people behave as if they had rank-dependent utility preferences with an inverse-S shaped probability weighting function that reproduces estimates existing in the literature.

Optimal insurance under rank‐dependent utility and incentive compatibility

AbstractBernard, He, Yan, and Zhou (Mathematical Finance, 25(1), 154–186) studied an optimal insurance design problem where an individual's preference is of the rank‐dependent utility (RDU) type, and show that in general an optimal contract covers both large and small losses. However, their results suffer from the unrealistic assumption that the random loss has no atom, as well as a problem of moral hazard that provides incentives for the insured to falsely report the actual loss. This paper addresses these setbacks by removing the nonatomic assumption, and by exogenously imposing the “incentive compatibility” constraint that both indemnity function and insured's retention function are increasing with respect to the loss. We characterize the optimal solutions via calculus of variations, and then apply the result to obtain explicitly expressed contracts for problems with Yaari's dual criterion and general RDU. Finally, we use numerical examples to compare the results between ours and Bernard et al.

Accounting for ambiguity and trust in partial outsourcing: A behavioral real options perspective

Concerned with the hidden costs of outsourcing, this paper examines the role of ambiguity and trust in partial outsourcing decisions from the perspective of real options theory. We study pricing and quantity dynamics between an ambiguity averse vendor and a less (more) trusting client in a leader-follower framework with fixed timing. We find that a client's partial outsourcing quantity increases with the vendor's ambiguity if outsourcing is meant for cost-saving purposes. Meanwhile, the effect of trust on outsourcing quantity is jointly moderated by the vendor's ambiguity and quality of shared information forecasts when cost advantages are exaggerated. In terms of pricing effects, the vendor increases (decreases) their threshold with increasing ambiguity for long-term (short-term) contracts. These insights hold under the multiple-priors and worst-case ambiguity specification. When Choquet ambiguity and rank-dependent utility are considered, more complex and subtle dynamics are obtained. Ambiguity has additional non-linear effects on outsourcing quantity due to heterogeneity in ambiguity preferences (seeking versus aversion) and probability weighting. The vendor's price not only increases (decreases) with increasing ambiguity-seeking for long-term (short-term) contracts, but also with ambiguity aversion when specific risk-return conditions are met. Trust effects are qualitatively similar under both ambiguity specifications.