Probability Distortion Research Articles

From time-inconsistency to time-consistency for optimal stopping problems.

For optimal stopping problems with time-inconsistent preference, we measure the inherent level of time-inconsistency by taking the time required to turn the naive strategies into the sophisticated ones. In particular, when in a repeated experiment, the naive agent can observe her actual sequence of actions that is inconsistent with what she has planned at the initial time, and she would then choose her immediate action based on the observations of her later actual behavior. The procedure is repeated until her actual sequence of actions is consistent with her plan at any time. We show that for the preference value of cumulative prospect theory, in which the time-inconsistency is due to the probability distortion, the higher the degree of probability distortion, the more severe the level of time-inconsistency, and the more time required to turn the naive strategies into the sophisticated ones.

Probability distortion and non-participation

This paper explores the impact of probability distortion on investors’ participation decisions and market performance. Specifically, we construct a one-period investment model with two types of investors—probabilistic pessimistic and neutral investors. The former assign more weight to bad outcomes and tail events when making decisions. Our findings suggest that non-participation arises from these probabilistic pessimistic investors. In equilibrium, their participation decisions affect the asset premium, which can be decomposed into the risk premium and probabilistic premium.

A Budget Constraint Incentive Mechanism Based on Risk Preferences of Collaborators in Edge Computing

Mobile Edge Computing (MEC) is a new distributed computing method based on the mobile communication network. It can provide cloud services and an IT service environment for application developers and service providers at the edge of the network. Computation offloading is a crucial technology of edge computing. However, computation offloading will consume the resources of the edge devices, and therefore the edge devices will not offload computation unconditionally. In addition, the service quality of edge computing applications is related to the cooperation rate of edge devices. Therefore, it is essential to design an appropriate incentive mechanism to motivate edge devices to execute computation offloading. However, the current existing incentive mechanisms have two problems: Firstly, existing mechanisms do not account for probability distortions under uncertainty in collaborator utility valuation models. Secondly, the platform ignores the risk preferences of collaborators in multiple rounds of decision-making. To address these issues, we propose an incentive mechanism based on risk preference, IMRP. The IMRP considers the collaborator’s probability distortion, introduces an uncertain utility bonus scheme, and builds a probability distortion model to influence the collaborator’s willingness to offload tasks. The IMRP also considers the collaborator’s risk preference and builds the collaborator’s risk preference model to influence the collaborator’s bidding decision. Simulation results show that our mechanism effectively improves the cooperation rate of edge devices and the utility of the requester.

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”.

Charging/Refueling Navigation Strategies for Plug-in Hybrid Hydrogen and Electric Vehicles With Irrationalities and Energy Substitution

Electric vehicles (EVs) are believed to be effective in reducing the use of fossil fuels. However, the increasing penetration of EVs can lead to challenges in managing EV charging/refueling. Typically, when EVs simultaneously gather at the stations, long queueing lengths and waiting times might occur, which might result in EV disutility. Thus, in this paper, novel decision-making strategies considering the irrational behaviors of EVs are formulated to ensure the utility of EVs. Three types of EVs are investigated, i.e., plug-in electric vehicles (PEVs), fuel cell electric vehicles (FCEVs), and plug-in hybrid hydrogen and electric vehicles (PH2EVs). First, three typical types of irrationalities are modeled based on prospect theory and irrational herding theory, including risk attitude irrationality, probability distortion, and irrational herding behaviors. Second, decision-making strategies, including station selection, navigation, and energy purchasing, are modeled based on EV irrationalities. Additionally, when making the station selection decision, the selection ranking index is proposed to balance energy price and waiting time. Third, the energy substitution of PH2EV is investigated to determine the optimal energy-purchasing mixture of electricity and hydrogen. Simulation results show that the average waiting time and charging/refueling time of EV users of the proposed model are reduced compared with the other two cases where EV users are assumed to be rational. In addition, the overall utility of EV users is enhanced.

Cooperation with dynamic asymmetric evaluation in complex networks from a risk perspective.

The choice of strategy exposes individuals to the risk of betrayal. This induces individuals' irrational tendencies in strategy selection, which further influences the emergence of cooperative behavior. However, the underlying mechanisms connecting risk perception and the emergence of cooperation are still not fully understood. To address this, the classic evolutionary game model on complex networks is extended. We depict the interaction between strategy imitation and payoff evaluation from two perspectives: dynamic adjustment and irrational assessment. Specifically, the probability distortion involved in the dynamic selection of imitative reference points, as well as the asymmetric psychological utility associated with reference point dependence, is emphasized. Monte Carlo simulations demonstrate that individual irrational cognition induced by the risk of strategy selection can promote the emergence of cooperative behavior. Among them, the risk sensitivity within psychological utility has the most significant moderating effect. Moreover, the promoting effect of strong heterogeneity and high clustering in the network topology on cooperation under risk scenarios has been clarified. Additionally, the influence of initial states on the emergence of cooperation follows a step-like pattern. This research offers valuable insights for further exploring the cooperation mechanisms among irrational agents, even in scenarios involving the regulation of group cooperation behavior in risky situations.

Optimal investment problem under behavioral setting: A Lagrange duality perspective

In this paper, we consider the optimal investment problem with both probability distortion/weighting and general non-concave utility functions with possibly finite number of inflection points, and apply a Lagrange duality based relaxation approach for solving this problem. Existing literature has shown the equivalent relationships (strong duality) between the relaxed problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, combining both factors, we prove that the absence of concavity in the utilities may result in strictly positive gaps, thus the strong duality may not hold unconditionally. The necessary and sufficient conditions on eliminating such gaps have been provided under this circumstance. We have applied the solution method to obtain closed-form solutions for the optimal terminal wealth and corresponding investment strategies in a special cumulative prospect theory (CPT) example with an appropriately selected inverse S-shaped probability distortion function. Based on the solutions, the joint effects of non-concave utilities combined with the probability weighting on the trading behaviors can be explicitly characterized. We show that, under this particular example, the co-existence of both factors may water down the loss aversion effect induced by only S-shaped utility or probability distortion when the agents are more cash-strapped in the initial budgets. In addition, we find that the optimal strategies derived from distorted beliefs shall converge to a constant that can be expressed by the standard Merton ratio multiplied by an inflation factor which we name as “distorted” Merton ratio. More importantly, the inflation factor is solely dependent on the probability distortion rather than the features of the utility function.

Boundedly rational bidding decision for land auctions during the transformation of real estate markets

Real estate developers in China exhibit different boundedly rational behaviors when bidding in different land auction formats, resulting in deviation from fully rational bidding, and the transformation of China’s real estate industry has intensified this deviation. To address this problem, this study establishes a reference utility model and a subjective decision probability model under the framework of incomplete information game to characterize developers’ utility distortion and winning probability distortion in open-bid auctions and sealed-bid auctions for land, respectively. The results show that the deviation between bounded rational bidding and fully rational bidding increases as developer competition intensifies. The equilibrium bidding of cost-disadvantaged developers in the boundedly rational model deviates less from the standard model. Moreover, in sealed-bid auctions, the "hiding degree" of cost-advantaged developers is greater, showing size effects, while the bids in open-bid auctions are more complex and affected by market conditions and developers’ risk preferences. Thus, this study characterizes boundedly rational bidding in land auctions and interprets the deviation from fully rational bidding, which can provide a more real basis for land auction mechanism design.

Probability distortions, collectivism, and international stock prices

There are substantial differences in the return premia due to probability distortions in individualist and collectivist cultures. Consistent with the substantially lesser degree of probabilistic thinking in collectivist cultures documented by the psychology literature, probability-distortion-related return premia are substantially higher there than in individualist cultures. Our methodology applies a novel composite probability distortion (CPD) score based on cumulative prospect theory and salience theory. This measure is priced among all size groups in the cross-section of international stock returns: low-CPD-score stocks are underpriced while those with high scores appear overpriced. Collectivism is the main driver of differences in the CPD premium across countries and U.S. states.

How Do Risk Preferences Affect Golf Course Superintendents’ Adoption of Precision Irrigation Technologies? Implications from Prospect Theory

AbstractPrecision irrigation is a potential viable strategy for water use reductions on golf courses by making variable or site-specific irrigation applications. A group of US golf course superintendents were surveyed to examine whether and how superintendents’ risk preferences (attitudes) affect the adoption decisions of precision irrigation technologies on their golf courses. Under the prospect theory (PT) framework, a lottery experiment was used to elicit the measures of three risk attitudes, that is, risk curvature, probability distortion, and loss aversion. Using these three measures and other questions in the survey, we found that risk curvature has a significant positive effect on the precision irrigation technologies adoption on golf courses, while probability distortion affects the adoption negatively. Compared to the golf course in low precipitation areas, superintendents’ risk attitudes are more likely to affect the precision irrigation technologies adoption in the golf course in high precipitation areas. Additionally, risk curvature dominates the adoption decisions for newer technologies, while probability distortion dominates the older technologies adoption decisions. Our research enriches the literature on the decision-making behaviors of managers by considering how probability distortion, a factor typically ignored by other studies, affects technology adoption decisions and adds to the literature on examining the technology adoption behaviors under PT by focusing on golf course superintendents, a group that has not been studied.

An Improved TOPSIS Model Based on Cumulative Prospect Theory: Application to ESG Performance Evaluation of State-Owned Mining Enterprises

The aim of this study is to provide a comprehensive decision-making method that can be applied to investment decisions based on the Environmental, Social and Governance (ESG) performance. The study contributes to the existing literature by introducing the CPT-TOPSIS model, a pioneering approach that incorporates the effect of non-rational factors on decision-making results in uncertain conditions by combining cumulative prospect theory (CPT) with the classic TOPSIS model. Moreover, by conducting an application to ESG evaluation on five state-owned mining enterprises in China, the study provides evidence of the effectiveness and improvement offered by the new model in comparison with the classic TOPSISI model and prospect theory TOPSIS (PT-TOPSIS) model. The results suggest that the CPT-TOPSIS model considers risk preferences and probability distortion in the decision-making process, narrows the gap between ESG scores, and makes ESG performance evaluation more realistic.

Risky business: Effects of stress on risky decision making

Decision-making under risk differs for potential gains compared to potential losses. The discrepancy in how gains and losses are assessed is best described by cumulative prospect theory, which postulates that individual risk attitude is more risk seeking in the face of potential losses (i.e., gambling to avoid a loss), more risk averse in the face of potential gains (i.e., opting for a smaller, more secure payout), and more likely to overweight low-probability events (e.g., buying a lottery ticket). Stress is known to modulate decision-making; however, how stress affects individual parameters of risk attitude remains unclear. Participants randomly assigned to the stress condition completed a Zoom-adapted Trier Social Stress Test, and those assigned to the control condition completed a corresponding control task. Participants then completed a task designed to estimate cumulative prospect theory parameters of risky decision-making through a series of gambles. Bayesian hierarchical modeling was used to estimate individual preferences for risk aversion, loss aversion, and probability distortion. Stressed participants exhibited less risk averse behaviors than control participants, with their behavior approaching risk-neutral, p=0.019. Additionally, stressed participants had increased probability distortion than control participants, indicating more overweighting of extreme events, p=.005 Stressed participants were also less loss averse than control participants, p<.001. We found that stress altered risk attitudes by altering risk aversion, probability distortion, and loss aversion. By understanding which domains of risk attitude are altered by stress, better therapeutic practices can be developed to aid individuals make better decisions while under stress.

Classification From Positive and Biased Negative Data With Skewed Labeled Posterior Probability.

The binary classification problem has a situation where only biased data are observed in one of the classes. In this letter, we propose a new method to approach the positive and biased negative (PbN) classification problem, which is a weakly supervised learning method to learn a binary classifier from positive data and negative data with biased observations. We incorporate a method to correct the negative influence due to a skewed confidence, which is represented by the posterior probability that the observed data are positive. This reduces the distortion of the posterior probability that the data are labeled, which is necessary for the empirical risk minimization of the PbN classification problem. We verified the effectiveness of the proposed method by synthetic and benchmark data experiments.

Inconsistent choices over prospect theory lottery games: Evidence from field experiments

Expected utility theory and classroom experiments have been commonly used to test the consistency and/or stability of elicited risk preferences among individuals. Usually conducted in developed countries, these studies have shown that risk preferences are inconsistent or unstable. However, the literature falls short when assessing the consistency of risk preferences under the Prospect Theory (PT) paradigm. This study reports the results of lottery games, played during two consecutive experimental sessions, to test the consistency of PT risk preferences among smallholder farmers in Ecuador, a developing country. I find evidence of consistency in the farmers’ risk preferences at the sample level. Nonetheless, I find inconsistency in risk preferences (risk aversion, probability distortion, and loss-aversion) at the individual level. I assess whether there is heterogeneity in the results across farmers’ demographic and socioeconomic attributes. Evidence suggests that relative consistency in the degree of loss aversion is higher among farmers who are female, own larger farms, and are more educated.

The composition of the choice set modulates probability weighting in risky decisions.

Probability distortion-the tendency to underweight larger probabilities and overweight smaller ones-is a robust empirical phenomenon and an important driver of suboptimal choices. We reveal a novel contextual effect on probability distortion that depends on the composition of the choice set. Probability distortion was larger in a magnitude-diverse choice set (in which participants encountered more unique magnitudes than probabilities) but declined, resulting in more veridical weighting, in a probability-diverse choice set(more unique probabilities than magnitudes). This effect was consistent in two, large, independent datasets (N = 481, N = 100) and held for a subset of lotteries that were identical in the two contexts. It also developed gradually as a function of exposure to the choice set, was independent of attentional biases to probability versus magnitude information, and was specific to probability weighting, leaving risk attitudes unaffected. The results highlight the importance of context when processing probabilistic information.

Modelling refugee migration under cognitive biases: Experimental evidence and policy

In this paper, we develop a model for refugee migration. As refugees’ migration choices are made in a risk-laden environment, we compare two different theoretical frameworks of decision making under risk, namely Expected Utility Theory (EUT) and Cumulative Prospect Theory (CPT). This last framework accounts for a reference point, loss aversion, and probability distortion. We estimate refugees’ risk and time preference parameters using field experimental data and show CPT better explains refugees’ risk behaviour on average. We also investigate policy implications based on simulations. We show that, under CPT, compared to standard EUT, the value of migrating is consistently lower and the migration decision is more sensitive to policy changes. Our results suggest refugees may self-select based on their risk preferences, those exhibiting more loss aversion or less probability sensitivity being more likely to renounce migration as a reaction to migration policies.

A Duality Between Utility Transforms and Probability Distortions

A Duality Between Utility Transforms and Probability Distortions

Second-best probability weighting

Non-linear probability weighting is an integral part of descriptive theories of choice under risk such as prospect theory. But why do these objective errors in information processing exist? Should we try to help individuals overcome their mistake of overweighting small and underweighting large probabilities? In this paper, we argue that probability weighting can be seen as a compensation for preexisting biases in evaluating payoffs. In particular, inverse S-shaped probability weighting is a flipside of S-shaped payoff valuation. Probability distortions may thus have survived as a second-best solution to a fitness maximization problem, and it can be counter-productive to correct them while keeping the value function unchanged.

Distorted probability operator for dynamic portfolio optimization in times of socio-economic crisis

A robust optimal control of discrete time Markov chains with finite terminal T and bounded costs or wealth using probability distortion is studied. The time inconsistency of these distortion operators and hence its lack of dynamic programming are discussed. Due to that, dynamic versions of these operators are introduced, and its availability for dynamic programming is demonstrated. Based on dynamic programming algorithm, existence of the optimal policy is justified and an application of the theory to portfolio optimization along with a numerical study is also presented.

Correlation between Violations of Environmental Legislation and Probable Distortion of Financial Reporting

ABSTRACT This work seeks to statistically verify the hypothesis that there is a link between violations of environmental legislation and distortions in the financial reporting of economic entities. For this purpose, the authors studied the reporting documents of companies that are operating in Russia and have violated environmental legislation. The financial reporting of the latter was analyzed based on the authors’ methodology. In order to determine the probability of distortion, companies were typologized and a probit model was constructed. The results confirmed the working hypothesis that economic entities’ opportunistic behavior in the area of environmental protection is, as a rule, accompanied by distortions of financial information about their activities.