Fourfold Pattern Of Risk Attitudes Research Articles

Nonlinear probability weighting can reflect attentional biases in sequential sampling.

Nonlinear probability weighting allows cumulative prospect theory (CPT) to account for key phenomena in decision making under risk (e.g., certainty effect, fourfold pattern of risk attitudes). It describes the impact of risky outcomes on preferences in terms of a rank-dependent nonlinear transformation of their objective probabilities. The attentional Drift Diffusion Model (aDDM) formalizes the finding that attentional biases toward an option can shape preferences within a sequential sampling process. Here we link these two influential frameworks. We used the aDDM to simulate choices between two options while systematically varying the strength of attentional biases to either option. The resulting choices were modeled with CPT. Changes in preference due to attentional biases in the aDDM were reflected in highly systematic signatures in the parameters of CPT's weighting function (curvature, elevation). In a re-analysis of a large set of previously published data, we demonstrate that attentional biases are also empirically linked to patterns in probability weighting as suggested by the simulations. Our analyses also revealed a previously overlooked link between patterns in probability weighting and response times. These findings highlight that distortions in probability weighting can arise from simple option-specific attentional biases in information search, and suggest an alternative to common interpretations of weighting-function parameters in terms of probability sensitivity and optimism. They also point to novel, attention-based explanations for empirical phenomena associated with characteristic shapes of CPT's probability-weighting function (e.g., certainty effect, description-experience gap). The results advance the integration of two prominent computational frameworks for decision making. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

A Model of Anchoring and Adjustment for Decision-Making under Risk

We introduce a general model of anchoring and adjustment for decision-making under risk. To evaluate a lottery, agents first anchor on the simple mean of the lottery's outcomes, ignoring the outcomes' probabilities. Then, they adjust the anchor (insufficiently) in the direction of the lottery's expected utility. The resulting model implies behavior which is observationally similar to established prospect theory models. However, it is not rank-dependent and naturally applies to lotteries with any given number of outcomes, including continuous lotteries. We consider a number of applications to demonstrate the model's range. For the discrete case, we show how the model can explain four commonly-observed choice anomalies - both Allais paradoxes, the St. Petersburg paradox, event-splitting effects, and the effect of probability in the fourfold pattern of risk attitudes. For the continuous case, we show how the anchoring model can be applied to the equity premium puzzle. We also use historical data on asset returns to show that equity premiums can be explained by reasonable combinations of preference parameters. The anchoring model is an attractive and flexible tool that assumes people want to simplify complex decision processes.

The Reflection Effect and Fourfold Pattern of Risk Attitudes: A Structural Econometric Analysis

The reflection effect and fourfold pattern of risk attitudes are often thought to be stylized facts about decision making under risk in lab settings. We take a closer look at this received wisdom, using 5 existing data sets from pairwise lottery choice experiments with real monetary incentives. We apply a structural econometric model that allows for within-individual correlation in risk preferences across gain and loss domains, as well as unobserved preference heterogeneity across individuals. The estimates of utility and probability weighting functions do not support the received wisdom.

A second-generation disappointment aversion theory of decision making under risk

This paper presents a new decision theory for modelling choice under risk. The new theory is a two-parameter generalization of expected utility theory. The proposed theory assumes that a decision maker: (1) behaves as if maximizing expected utility; but (2) may experience disappointment (elation) when the utility of a lottery’s outcome falls short of (exceeds) the expected utility of the lottery; and (3) may have a preference for gambling (attraction/aversion to positively/negatively skewed lotteries). The proposed theory can rationalize the fourfold pattern of risk attitudes; the common ratio effect and the reverse thereof (in certain types of decision problems); the Allais paradox in classical common consequence problems and the reverse Allais paradox—in common consequence problems with an even split of a probability mass; violations of the betweenness axiom; switching behavior in the Samuelson’s example; violations of ordinal, upper and lower cumulative independence (which falsify rank-dependent utility and cumulative prospect theory); and preference reversals between valuations and choice. In application to insurance, the theory can rationalize full insurance with an actuarially unfair premium and aversion to probabilistic insurance. In application to optimal portfolio investment, the theory can rationalize the equity premium puzzle.

Target-Adjusted Utility Functions and Expected-Utility Paradoxes

Experimental tests of expected-utility theory (EU) have accumulated empirical observations in which the predictions of EU are systematically violated. The cumulative prospect theory (CPT) explains violations such as the Allais paradoxes and fourfold pattern of risk attitudes as resulting from nonlinear probability transformations. Here we show that the classical paradoxes for decisions under risk can be explained with preferences that are linear in probabilities for any choice set and that maximize an expected-utility function with respect to an endogenous target return. We introduce the maximin payoff as a plausible and even natural target return from a choice set and show that the resulting target-adjusted utility (TAU) model explains additional empirical observations such as the scale dependence of the Allais paradox that cannot be explained by standard specifications of CPT. Further, using data from three prominent laboratory experiments, we find that TAU is effective in explaining observed behaviors. This paper was accepted by James Smith, decision analysis.

Expected utility theory and inner and outer measures of loss aversion

We introduce a weak rank dependent utility (RDU) model, with one extra parameter compared to the canonical expected utility (EUT) model, which makes many of the same predictions as cumulative prospect theory (CPT). The model extends a set of nonconvex preferences to its maximal inner convex subset, satisfies stochastic dominance principles, resolves the Allais paradox, predicts CPT fourfold pattern of risk attitudes, and characterizes reference dependent preferences. Unlike extant RDU models that transform probability weighting functions (pwfs), our model transforms ranked choice sets while leaving objective probabilities intact. So pwfs are hidden in our model. CPT’s loss aversion index is a special case of the interior solution for the extra parameter for unconstrained utility maximization, and it is driven by tail probabilities in our model. We provide several examples to show how popular formulae for the loss aversion index can be classified into inner and outer measures of loss aversion via an approximate Radon–Nikodym formulation of the model. This resolves sources of disparity in estimating the loss aversion index with experimental data. We show that under extant approaches, the loss aversion index is best estimated by a mixture of inner and outer measures of itself. Furthermore, we identify a CPT paradox: The utility loss aversion index is unmeasurable under CPT nonexpected utility framework for mixed lotteries; but measurable for same under the expected utility paradigm adapted to our model.

Responsibility effects in decision making under risk

We explore situations in which a decision maker bears responsibility for somebody else’s outcomes as well as for her own. We study such choices for gains and losses, and for different gain probabilities. For 50–50 lotteries over gains we find that being responsible for somebody else’s payoffs increases risk aversion. In the loss domain, on the other hand, we find significantly different behavior relative to gains, with slightly more risk seeking under responsibility. In a second experiment, we replicate the finding of increased risk aversion for large probabilities of a gain, while for small probability gains we find increased risk seeking under conditions of responsibility relative to large probabilities. This discredits hypotheses of a ‘cautious shift’ under responsibility, and may indicate an accentuation of the fourfold pattern of risk attitudes usually found for individual choices.

Risk attitudes in a changing environment: An evolutionary model of the fourfold pattern of risk preferences.

A striking feature of human decision making is the fourfold pattern of risk attitudes, involving risk-averse behavior in situations of unlikely losses and likely gains, but risk-seeking behavior in response to likely losses and unlikely gains. Current theories to explain this pattern assume particular psychological processes to reproduce empirical observations, but do not address whether it is adaptive for the decision maker to respond to risk in this way. Here, drawing on insights from behavioral ecology, we build an evolutionary model of risk-sensitive behavior, to investigate whether particular types of environmental conditions could favor a fourfold pattern of risk attitudes. We consider an individual foraging in a changing environment, where energy is needed to prevent starvation and build up reserves for reproduction. The outcome, in terms of reproductive value (a rigorous measure of evolutionary success), of a one-off choice between a risky and a safe gain, or between a risky and a safe loss, determines the risk-sensitive behavior we should expect to see in this environment. Our results show that the fourfold pattern of risk attitudes may be adaptive in an environment in which conditions vary stochastically but are autocorrelated in time. In such an environment the current options provide information about the likely environmental conditions in the future, which affect the optimal pattern of risk sensitivity. Our model predicts that risk preferences should be both path dependent and affected by the decision maker's current state.

A Theory of Decision-Making Under Risk as a Tradeoff between Expected Utility, Expected Utility Deviation and Expected Utility Skewness

This paper presents a new decision theory for modelling choice under risk. The new theory is a two-parameter generalization of expected utility theory. The proposed theory assumes that a decision maker: 1) behaves as if maximizing expected utility; but 2) may experience disappointment (elation) when the utility of a lottery’s outcome falls short of (exceeds) the expected utility of the lottery; and 3) may have a preference for gambling (attraction/aversion to positively/negatively skewed lotteries). The proposed theory can rationalize the fourfold pattern of risk attitudes; the common ratio effect and the reverse thereof (in certain types of decision problems); the Allais paradox in classical common consequence problems and the reverse Allais paradox — in common consequence problems with an even split of a probability mass; violations of the betweenness axiom; switching behavior in the Samuelson’s example; violations of ordinal, upper and lower cumulative independence (which falsify rank-dependent utility and cumulative prospect theory); and preference reversals between valuations and choice. Behavioral characterization (axiomatization) of the theory is provided. In application to insurance, the theory can rationalize full insurance with an actuarially unfair premium and aversion to probabilistic insurance. In application to optimal portfolio investment, the theory can rationalize the equity premium puzzle.

On mental transformations.

The paper presents an alternative interpretation of the experimental data published by Kahneman and Tversky in their 1992 study Advances in Prospect Theory”, which describes the Cumulative version of their Prospect Theory from 1979. It was assumed that, apart from the operations made during the initial stage of problem resolution, which Prospect Theory defines as Editing (here generalized as Mental Adaptation), other mental transformations such as Prospect Scaling (resulting from Focused Attention) and Logarithmic Perception of Financial Stimuli should be considered when analyzing the experimental data. This led to the design of an explicit, simple and symmetric solution without the use of the probability weighting function. The double S-type function obtained (the aspiration function) resembles the utility curve specified by the Markowitz hypothesis (1952) and substitutes the fourfold pattern of risk attitudes introduced by Cumulative Prospect Theory. The results presented constitute a basis for negating Prospect Theory as a theory which correctly describes how decisions are made under conditions of risk and may signal a return to a description of people’s behavior that only relies on the utility-like function.

Measuring the time stability of Prospect Theory preferences

Prospect Theory (PT) is widely regarded as the most promising descriptive model for decision making under uncertainty. Various tests have corroborated the validity of the characteristic fourfold pattern of risk attitudes implied by the combination of probability weighting and value transformation. But is it also safe to assume stable PT preferences at the individual level? This is not only an empirical but also a conceptual question. Measuring the stability of preferences in a multi-parameter decision model such as PT is far more complex than evaluating single-parameter models such as Expected Utility Theory under the assumption of constant relative risk aversion. There exist considerable interdependencies among parameters such that allegedly diverging parameter combinations could in fact produce very similar preference structures. In this paper, we provide a theoretic framework for measuring the (temporal) stability of PT parameters. To illustrate our methodology, we further apply our approach to 86 subjects for whom we elicit PT parameters twice, with a time lag of 1 month. While documenting remarkable stability of parameter estimates at the aggregate level, we find that a third of the subjects show significant instability across sessions.

Solving the St. Petersburg Paradox in cumulative prospect theory: the right amount of probability weighting

Cumulative Prospect Theory (CPT) does not explain the St. Petersburg Paradox. We show that the solutions related to probability weighting proposed to solve this paradox, (Blavatskyy, Management Science 51:677–678, 2005; Rieger and Wang, Economic Theory 28:665–679, 2006) have to cope with limitations. In that framework, CPT fails to accommodate both gambling and insurance behavior. We suggest replacing the weighting functions generally proposed in the literature by another specification which respects the following properties: (1) to solve the paradox, the slope at zero has to be finite. (2) to account for the fourfold pattern of risk attitudes, the probability weighting has to be strong enough.

A dual system model of preferences under risk.

This article presents a dual system model (DSM) of decision making under risk and uncertainty according to which the value of a gamble is a combination of the values assigned to it independently by the affective and deliberative systems. On the basis of research on dual process theories and empirical research in Hsee and Rottenstreich (2004) and Rottenstreich and Hsee (2001) among others, the DSM incorporates (a) individual differences in disposition to rational versus emotional decision making, (b) the affective nature of outcomes, and (c) different task construals within its framework. The model has good descriptive validity and accounts for (a) violation of nontransparent stochastic dominance, (b) fourfold pattern of risk attitudes, (c) ambiguity aversion, (d) common consequence effect, (e) common ratio effect, (f) isolation effect, and (g) coalescing and event-splitting effects. The DSM is also used to make several novel predictions of conditions under which specific behavior patterns may or may not occur.

The Fourfold Pattern of Risk Attitudes in Choice and Pricing Tasks

We examine the robustness of the fourfold pattern of risk attitudes under two elicitation procedures. We find that individuals are, on average, risk-seeking over low-probability gains and high-probability losses and risk-averse over high-probability gains and low-probability losses when we elicit prices for the gambles. However, a choice-based elicitation procedure, where participants choose between a gamble and its expected value, yields individual decisions that are indistinguishable from random choice. Sensitivity to elicitation procedure holds between and within participants, and remains when participants are allowed to review and change decisions. The price elicitation procedure is more complex; this finding may be further evidence that an increase in cognitive load exacerbates behavioural anomalies.

On Mental Transformations

This paper presents an alternative interpretation of the experimental data published by Kahneman and Tversky in their 1992 paper describing Cumulative Prospect Theory. It was assumed that mental transformations such as mental adaptation, prospect scaling, and logarithmic perception should be considered when analyzing the experimental data. This led to the design of a solution that did not require the probability weighting function. The double S-type function obtained (the decision utility) resembles the utility curve specified by the Markowitz hypothesis (1952) and substitutes the fourfold pattern of risk attitudes introduced by Cumulative Prospect Theory.

How to Solve the St Petersburg Paradox in Rank-Dependent Models?

The Cumulative Prospect Theory, as it was specified by Tversky and Kahneman (1992) does not explain the St Petersburg Paradox. This study shows that the solutions proposed in the literature (Blavatskky, 2005; Rieger and Wang, 2006) to guarantee, under rank dependant models, finite subjective utilities for any prospects with finite expected values have to cope with many limitations. In that framework, CPT fails to accommodate both gambling and insurance behavior. We suggested to replace the weighting function generally proposed in the literature with another specification which respects the following properties. 1) In order to guarantee finite subjective values for all prospects with finite expected values, the slope at zero should be finite. 2)To account for the fourfold pattern of risk attitudes, the probability weighting should be strong enough to overcome the concavity of the value function.

Risk attributes theory: Decision making under risk

In this paper, some limitations in the conventional theories and models of the decision making under risk are presented from a new angle. To eliminate those limitations, a theoretical framework is proposed, in which the risk preference relation is constructed on a set of risk attributes instead of risky prospects. The axioms are presented to ensure the existence of the representation function for this risk preference relation through employing the Multi-attribute Utility Theory (MAUT). In our framework, we construct a risk decision model with four risk attributes. One of these attributes measured by a newly defined attribute measurement, the Relative Maximum Absolute Value (RMAV), can be employed in our model to explain the experimental result more intuitively compared with the overweighting of small probability proposed by Kahneman and Tversky in their Prospect Theory. In addition, the validity of the model is testified by applying it to explain the Allais Paradox and the Fourfold Pattern of Risk Attitudes. The advantages of our theory are summarized to complete this paper. Since our model is based on an axiomatized preference relation and has a good descriptive ability, it can either serve as a normative or a descriptive decision model under risk.

Stochastic expected utility theory

This paper proposes a new decision theory of how individuals make random errors when they compute the expected utility of risky lotteries. When dis- torted 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.

Stochastic Choice Under Risk

An individual makes random errors when evaluating the expected utility of a risky lottery. Errors are symmetrically distributed around zero as long as an individual does not make transparent mistakes such as choosing a risky lottery over its highest possible outcome for certain. This stochastic decision theory explains many well-known violations of expected utility theory such as the fourfold pattern of risk attitudes, the discrepancy between certainty equivalent and probability equivalent elicitation methods, the preference reversal phenomenon, the generalized common consequence effect (the Allais paradox), the common ratio effect and the violations of the betweenness.

A Stochastic Expected Utility Theory

This paper proposes a new model that explains the violations of expected utility theory through the role of random errors. The paper analyzes decision making under risk when individuals make random errors when they compute expected utilities. Errors are drawn from the normal distribution, which is truncated so that the stochastic utility of a lottery cannot be greater (lower) than the utility of the highest (lowest) possible outcome. The standard deviation of random errors is higher for lotteries with a wider range of possible outcomes. It converges to zero for lotteries converging to a degenerate lottery. The model explains all major stylized empirical facts such as the Allais paradox and the fourfold pattern of risk attitudes. The model fits the data from ten well-known experimental studies at least as good as cumulative prospect theory.