Probability Weighting Function Research Articles

Estimating Probability Weighting Functions through Option Pricing Bounds

Abstract This paper proposes a novel approach to estimating the probability weighting function (PWF) of investors in the option market. We match observed option prices to the option pricing bounds under stochastic dominance rules. Using 1-month S&P 500 index option data, we find that investors could subjectively employ an inverse S-shaped probability weighting function, which increases the weights on extreme returns and asymmetrically assigns greater weights to extremely low returns than to extremely high returns. Our findings suggest that the inverse S-shaped nature of the PWFs is robust across various estimation specifications, such as adopting an alternative methodology to construct the return distribution, and employing option data with different times to maturity. (JEL G12)

Asset pricing for the lottery-like security under probability weighting: Based on generalized Wang transform

We present a two-stage lottery model with the generalized Wang transform as a probability weighting function to formally derive investors’ demand for the lottery-like security. Probability overweight on higher expected payoffs accounts for investors’ overvaluation of the security with moderately lottery-like feature, raising the demand and driving up the price. Investors’ aggressive demand is enhanced by optimistic attitude segmented from probability weighting function and weakened by probability distortion towards extreme events if considering fundamental risks. Our model predicts overpricing in small securities with optimistic investors, and implies skewness pricing and “realized kurtosis puzzle”.

Estimation and inference for multi-kink expectile regression with nonignorable dropout

In this paper, we consider parameter estimation, kink points testing and statistical inference for a longitudinal multi-kink expectile regression model with nonignorable dropout. In order to accommodate both within-subject correlations and nonignorable dropout, the bias-corrected generalized estimating equations are constructed by combining the inverse probability weighting and quadratic inference function approaches. The estimators for the kink locations and regression coefficients are obtained by using the generalized method of moments. A selection procedure based on a modified BIC is applied to estimate the number of kink points. We theoretically demonstrate the number selection consistency of kink points and the asymptotic normality of all estimators. A weighted cumulative sum type statistic is proposed to test the existence of kink effects at a given expectile, and its limiting distributions are derived under both the null and the local alternative hypotheses. Simulation studies show that the proposed estimators and test have desirable finite sample performance in both homoscedastic and heteroscedastic errors. An application to the Nation Growth, Lung and Health Study dataset is also presented.

(No-)Betting Pareto Optima Under Rank-Dependent Utility

In a pure-exchange economy with no aggregate uncertainty, we characterize in closed form and full generality Pareto-optimal allocations between two agents who maximize (nonconcave) rank-dependent utilities (RDU). We then derive a necessary and sufficient condition for Pareto optima to be no-betting allocations (i.e., deterministic allocations or full insurance allocations). This condition depends only on the probability weighting functions of the two agents and not on their (concave) utility of wealth. Hence, with RDU preferences, it is the difference in probabilistic risk attitudes given common beliefs rather than heterogeneity or ambiguity in beliefs that is a driver of betting behavior. As by-product of our analysis, we answer the question of when sunspots matter in this economy. Funding: M. Ghossoub acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada [Grant 2018-03961].

Comparing Elicitation Methods of Risk Preferences in Personal Financial Planning: A Field Experiment among Working Adults in Malaysia

Risk preference is an important input in designing investment types or port-folios in personal finance. Although there are many reported risk elicitation methods, the risk preference measure obtained from these methods has not been associated with any behavioural or psychological reasons underlying the choices made. This association is important in providing the underlying theoretical understanding of risk preferences and establishing the robustness of a measure. The present paper attempted to test the consistency of risk preferences between two widely used elicitation methods: Grable and Lytton (1999, 2003) risk tolerance score and probability weighting function in prospect theory (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992). The risk score captures the self-reported willingness to engage in risky investment, and the probability weighting function shows overweighting or underweighting of the probability. We conducted a series of field experiments involving working adults of three main ethnic groups in Malaysia. The results showed consistency in risk preferences between the two methods: high willingness to engage in risk was complemented by optimistic probability weighting, and respondents with low willingness to engage in risky investment evaluated high probability more favourably. The study is useful to financial professionals when implementing the questionnaire to measure risk preferences.

Cognitive Uncertainty

Abstract This article documents the economic relevance of measuring cognitive uncertainty: people’s subjective uncertainty over their ex ante utility-maximizing decision. In a series of experiments on choice under risk, the formation of beliefs, and forecasts of economic variables, we show that cognitive uncertainty predicts various systematic biases in economic decisions. When people are cognitively uncertain—either endogenously or because the problem is designed to be complex—their decisions are heavily attenuated functions of objective probabilities, which gives rise to average behavior that is regressive to an intermediate option. This insight ties together a wide range of empirical regularities in behavioral economics that are typically viewed as distinct phenomena or even as reflecting preferences, including the probability weighting function in choice under risk; base rate insensitivity, conservatism, and sample size effects in belief updating; and predictable overoptimism and -pessimism in forecasts of economic variables. Our results offer a blueprint for how a simple measurement of cognitive uncertainty generates novel insights about what people find complex and how they respond to it.

NONPARAMETRIC NUMERICAL APPROACHES TO PROBABILITY WEIGHTING FUNCTION CONSTRUCT FOR MANIFESTATION AND PREDICTION OF RISK PREFERENCES

Probability weighting function (PWF) is the psychological probability of a decision-maker for objective probability, which reflects and predicts the risk preferences of decision-maker in behavioral decisionmaking. The existing approaches to PWF estimation generally include parametric methodologies to PWF construction and nonparametric elicitation of PWF. However, few of them explores the combination of parametric and nonparametric elicitation approaches to approximate PWF. To describe quantitatively risk preferences, the Newton interpolation, as a well-established mathematical approximation approach, is introduced to task-specifically match PWF under the frameworks of prospect theory and cumulative prospect theory with descriptive psychological analyses. The Newton interpolation serves as a nonparametric numerical approach to the estimation of PWF by fitting experimental preference points without imposing any specific parametric form assumptions. The elaborated nonparametric PWF model varies in accordance with the number of the experimental preference points elicitation in terms of its functional form. The introduction of Newton interpolation to PWF estimation into decision-making under risk will benefit to reflect and predict the risk preferences of decision-makers both at the aggregate and individual levels. The Newton interpolation-based nonparametric PWF model exhibits an inverse S-shaped PWF and obeys the fourfold pattern of decision-makers’ risk preferences as suggested by previous empirical analyses.

Attribute attention and option attention in risky choice

Probability weighting is one of the most powerful theoretical constructs in descriptive models of risky choice and constitutes a central component of cumulative prospect theory (CPT). Probability weighting has been shown to be related to two facets of attention allocation: one analysis showed that differences in the shape of CPT's probability-weighting function are linked to differences in how attention is allocated across attributes (i.e., probabilities vs. outcomes); another analysis (that used a different measure of attention) showed a link between probability weighting and differences in how attention is allocated across options. However, the relationship between these two links is unclear. We investigate to what extent attribute attention and option attention independently contribute to probability weighting. Reanalyzing data from a process-tracing study, we first demonstrate links between probability weighting and both attribute attention and option attention within the same data set and the same measure of attention. We then find that attribute attention and option attention are at best weakly related and have independent and distinct effects on probability weighting. Moreover, deviations from linear weighting mainly emerged when attribute attention or option attention were imbalanced. Our analyses enrich the understanding of the cognitive underpinnings of preferences and illustrate that similar probability-weighting patterns can be associated with very different attentional policies. This complicates an unambiguous psychological interpretation of psycho-economic functions. Our findings indicate that cognitive process models of decision making should aim to concurrently account for the effects of different facets of attention allocation on preference. In addition, we argue that the origins of biases in attribute attention and option attention need to be better understood.

An EffcientNet-encoder U-Net Joint Residual Refinement Module with Tversky–Kahneman Baroni–Urbani–Buser loss for biomedical image Segmentation

Quantitative analysis on biomedical images has been on increasing demand nowadays and for modern computer vision approaches. While recently advanced procedures have been enforced, there is still necessity in optimizing network architecture and loss functions. Inspired by the pretrained EfficientNet-B4 and the refinement module in boundary-aware problems, we propose a new two-stage network which is called EffcientNet-encoder U-Net Joint Residual Refinement Module and we create a novel loss function called the Tversky–Kahneman Baroni–Urbani–Buser loss function. The loss function is built on the basement of the Baroni–Urbani–Buser coefficient and the Jaccard–Tanimoto coefficient and reformulated in the Tversky–Kahneman probability-weighting function. We have evaluated our algorithm on the four popular datasets: the 2018 Data Science Bowl Cell Nucleus Segmentation dataset, the Brain Tumor LGG Segmentation dataset, the Skin Lesion ISIC 2018 dataset and the MRI cardiac ACDC dataset. Several comparisons have proved that our proposed approach is noticeably promising and some of the segmentation results provide new state-of-the-art results. The code is available at https://github.com/tswizzle141/An-EffcientNet-encoder-U-Net-Joint-Residual-Refinement-Module-with-TK-BUB-Loss.

Expected subjective value theory (ESVT): A representation of decision under risk and certainty

We present a descriptive model of choice derived from neuroscientific models of efficient value representation in the brain. Our basic model, a special case of Expected Utility Theory, can capture a number of behaviors predicted by Prospect Theory. It achieves this with only two parameters: a time-indexed “payoff expectation” (reference point) and a free parameter we call “predisposition”. A simple extension of the model outside the domain of Expected Utility also captures the Allais Paradox. Our models shed new light on the computational origins and evolution of risk attitudes and aversion to outcomes below reward expectation (reference point). It delivers novel explanations of the endowment effect, the observed heterogeneity in probability weighting functions, and the Allais Paradox, all with fewer parameters and higher descriptive accuracy than Prospect Theory.

The Prospect Theory and First Price Auctions: an Explanation of Overbidding

This paper attempted using the prospect theory to explain overbidding in first price auctions. The standard outlook in the literature on auctions is that bidders overbid, but the probability weighting functions are nonlinear as in the prospect theory, so they not only tend to underweight the probabilities of winning the auction but also overweight, so that there are overbidders and underbidders. This paper proves that to some extent, non-linear weighting functions do explain overbidding the risk-neutral Nash equilibrium valuation (RNNE). Furthermore, coherent risk measures, such as certainty equivalent and translation invariance, were used to show loss aversion among bidders, and in line with the prospect theory, convexity was also confirmed with sub-additivity, monotonicity and with positive homogeneity.

Loss Aversion, Risk Aversion, and the Shape of the Probability Weighting Function

Loss Aversion, Risk Aversion, and the Shape of the Probability Weighting Function

Testing Hurwicz Expected Utility

Gul and Pesendorfer (2015) propose a promising theory of decision under uncertainty, they dub Hurwicz expected utility (HEU). HEU is a special case of α‐maxmin EU that allows for preferences over sources of uncertainty. It is consistent with most of the available empirical evidence on decision under risk and uncertainty. We show that HEU is also tractable and can readily be measured and tested. We do this by deriving a new two‐parameter functional form for the probability weighting function, which fits our data well and which offers a clean separation between ambiguity perception and ambiguity aversion. In two experiments, we find support for HEU's predictions that ambiguity aversion is constant across sources of uncertainty and that ambiguity aversion and first order risk aversion are positively correlated.

Probability Weighting in Decision-Making Tasks under Risk

The analysis of alternative decisions and the choice of the optimal – in a given sense – decision is an integral part of people’s purposeful activity in all areas of their social life. Many formal approaches have been proposed to solve these problems. One such approach is expected utility theory, which correctly models individuals’ subjective preferences and attitudes to risk. For a very long time this theory was the leading approach for decision making under conditions of risk. However, numerous practical studies have shown its weakness: the theory did not explicitly use subjective perceptions of decision outcome probabilities in optimal decision-making processes. This research has led to the creation and development of approaches to explicitly consider the probabilities of outcomes in decision making. This paper provides a critical analysis of the descriptive properties of expected utility theory and presents various forms of probability weighting functions.

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. Applying the new method to the datasets of Tversky and Kahneman (1992) and Bruhin et al. (2010), we reject the convexity of the utility function in the loss domain and show 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. Furthermore, our method highlights that the overweighting of tail probabilities is more pronounced in the gain domain than in the loss domain. Overall, our results show that the utility function varies little across domains, thus suggesting that probability distortions are key to capture differences in risk attitudes in the gain and loss domains.

A neuronal prospect theory model in the brain reward circuitry

Prospect theory, arguably the most prominent theory of choice, is an obvious candidate for neural valuation models. How the activity of individual neurons, a possible computational unit, obeys prospect theory remains unknown. Here, we show, with theoretical accuracy equivalent to that of human neuroimaging studies, that single-neuron activity in four core reward-related cortical and subcortical regions represents the subjective valuation of risky gambles in monkeys. The activity of individual neurons in monkeys passively viewing a lottery reflects the desirability of probabilistic rewards parameterized as a multiplicative combination of utility and probability weighting functions, as in the prospect theory framework. The diverse patterns of valuation signals were not localized but distributed throughout most parts of the reward circuitry. A network model aggregating these signals reconstructed the risk preferences and subjective probability weighting revealed by the animals’ choices. Thus, distributed neural coding explains the computation of subjective valuations under risk.

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).

Dorsolateral prefrontal cortex plays causal role in probability weighting during risky choice

In this study, we provide causal evidence that the dorsolateral prefrontal cortex (DLPFC) supports the computation of subjective value in choices under risk via its involvement in probability weighting. Following offline continuous theta-burst transcranial magnetic stimulation (cTBS) of the DLPFC subjects (N = 30, mean age 23.6, 56% females) completed a computerized task consisting of 96 binary lottery choice questions presented in random order. Using the hierarchical Bayesian modeling approach, we then estimated the structural parameters of risk preferences (the degree of risk aversion and the curvature of the probability weighting function) and analyzed the obtained posterior distributions to determine the effect of stimulation on model parameters. On a behavioral level, temporary downregulation of the left DLPFC excitability through cTBS decreased the likelihood of choosing an option with higher expected reward while the probability of choosing a riskier lottery did not significantly change. Modeling the stimulation effects on risk preference parameters showed anecdotal evidence as assessed by Bayes factors that probability weighting parameter increased after the left DLPFC TMS compared to sham.

Risky choice: Probability weighting explains independence axiom violations in monkeys

Expected Utility Theory (EUT) provides axioms for maximizing utility in risky choice. The Independence Axiom (IA) is its most demanding axiom: preferences between two options should not change when altering both options equally by mixing them with a common gamble. We tested common consequence (CC) and common ratio (CR) violations of the IA over several months in thousands of stochastic choices using a large variety of binary option sets. Three monkeys showed consistently few outright Preference Reversals (8%) but substantial graded Preference Changes (46%) between the initial preferred gamble and the corresponding altered gamble. Linear Discriminant Analysis (LDA) indicated that gamble probabilities predicted most Preference Changes in CC (72%) and CR (88%) tests. The Akaike Information Criterion indicated that probability weighting within Cumulative Prospect Theory (CPT) explained choices better than models using Expected Value (EV) or EUT. Fitting by utility and probability weighting functions of CPT resulted in nonlinear and non-parallel indifference curves (IC) in the Marschak-Machina triangle and suggested IA non-compliance of models using EV or EUT. Indeed, CPT models predicted Preference Changes better than EV and EUT models. Indifference points in out-of-sample tests were closer to CPT-estimated ICs than EV and EUT ICs. Finally, while the few outright Preference Reversals may reflect the long experience of our monkeys, their more graded Preference Changes corresponded to those reported for humans. In benefitting from the wide testing possibilities in monkeys, our stringent axiomatic tests contribute critical information about risky decision-making and serves as basis for investigating neuronal decision mechanisms.

Moral-hazard-free insurance: mean-variance premium principle and rank-dependent utility theory

This paper investigates a Pareto-optimal insurance problem, where the insured maximizes her rank-dependent utility preference and the insurer is risk-neutral and employs the mean-variance premium principle. To eliminate potential moral hazard issues, we only consider the so-called moral-hazard-free insurance contracts that obey the incentive compatibility constraint. The insurance problem is first formulated as a non-concave maximization problem involving Choquet expectation, then turned into a concave quantile optimization problem and finally solved by the calculus of variations method. The optimal contract is expressed by a semi-linear second-order double-obstacle ordinary differential equation with nonlocal operator. An effective numerical method is proposed to compute the optimal contract assuming the probability weighting function has a density. Also, we provide an example that is analytically solved.