S-shaped Value Function Research Articles

Indistinguishability of small probabilities, subproportionality, and the common ratio effect

We show that, in the framework of Cumulative Prospect Theory, subproportionality as a property of the probability weighting function alone does not automatically imply the common ratio effect. This issue is particularly relevant for equal-mean lotteries because both risk-averse and risk-seeking behavior have to be predicted there. As a solution, we propose three simple properties of the probability weighting function which are sufficient to accommodate the empirical evidence on the common ratio effect for equal-mean lotteries for any S-shaped value function. These are (1) subproportionality, (2) indistinguishability of small probabilities, and (3) an intersection point with the diagonal lower than 0.5. While subproportionality and a fixed point lower than 0.5 are common assumptions in the literature, the property indistinguishability of small probabilities is introduced for the first time. The ratio of decision weights for infinitesimally small probabilities characterizes indistinguishability and is also an informative measure for the curvature of the probability weighting function at zero. The intuition behind indistinguishability is that, even though the ratio of probabilities stays constant, individuals tend to neglect this relative difference when probabilities get smaller.

Making regulation fit by taking irrationality into account: the case of the whistleblower

Prospect theory describes people as bounded rational decision maker. What sparked widespread discussion after its initial introduction in 1979 is today criticized for lack of applicability. I use the debate about whistleblowing laws to show that prospect theory may be applied prescriptively in economics as a tool to design effective legislation. Whistleblowing is often seen as an important way to uncover fraud, which causes billions of USD in damages annually. I first examine the fragmented legal landscape across Europe, showing that it can be framed as one favoring rewards or the prevention of losses. I conduct an experiment with 39 university students, wherein legislative incentives are evaluated under a prospect theoretical frame in a setting of ambiguity and high stakes. Results suggest that people exhibit the typical s-shaped value function and loss aversion in line with prospect theory. In addition, their intention to whistleblow is more heavily reduced by losses than increased by gains. The study adds to the scarce literature of prospect theory on decisions in ambiguous contexts—as well as to the applicability of the theory as a prescriptive instrument in designing institutional frames. For whistleblowing in particular, a protection-based approach seems most promising.

Optimal consumption sequences under habit formation and satiation

In this note, we contribute to the literature of behavioral economics by extending a result in Baucells and Sarin (2010). We find the optimal consumption sequence for many goods and many periods in an intertemporal choice framework where preferences can exhibit at the same time habit formation and satiation. For the case in which habit formation matters more than satiation, we find that if consumers must consume a good in every period, for certain types of S-shaped value function, consuming the same good in all the periods is optimal. However, if consumers can choose not to consume in some periods, then this behavior is optimal only for certain kind of piece-wise linear value functions. We also discuss briefly the case when satiation dominates habituation.

Probability weighting, stop-loss and the disposition effect

In this paper we study a continuous-time, optimal stopping model of an asset sale with prospect theory preferences under pre-commitment. We show for a wide range of value and probability weighting functions, including those of Tversky and Kahneman (1992), that the optimal prospect takes the form of a stop-loss threshold and a distribution over gains. It is skewed with a long right tail. This is consistent with both the widespread use of stop-loss strategies in financial markets, and recent experimental evidence. Moreover, our model with probability weighting in tandem with the S-shaped value function makes predictions for the disposition effect which match in magnitude that calculated by Odean (1998).

Optimal Consumption Sequence When Habituation Dominates Satiation

In this note, we answer an open problem proposed in Baucells and Sarin (2010, p. 293) to find the optimal consumption sequence when habit formation dominates satiation in an intertemporal choice framework. We find that if consumers do not abstain from consuming goods then, for certain types of S-shaped value functions, consuming the same good in all the periods is optimal. However, if consumers do abstain from consuming goods, that result holds for certain kind of piece-wise linear functions.

When payoffs look like probabilities: Separating form and content in risky choice.

Paralleling research in perception, behavioral models of risky choice posit "psychophysical" transformations of material outcomes and probabilities. Prospect theory assumes a value function that is concave for gains and convex for losses, and an inverse S-shaped probability weighting function. But in typical experiments, form and content are confounded: Probabilities are represented on a bounded numerical scale, whereas representations of monetary gains (losses) are unbounded above (below). To unconfound form and content, we conducted experiments using a probability-like representation of outcomes and an outcome-like representation of probability. We show that interchanging numerical representations can interchange the resulting psychophysical functions: A proportional (rather than absolute) representation of outcomes leads to an inverse S-shaped value function for gains. This alternative value function generates novel framing effects, a common ratio effect for bounded gains, and a "framing interaction," where gain-loss framing matters less for proportional outcomes. In addition, we show that an absolute (rather than proportional) representation of probability reduces sensitivity to large probabilities. These findings highlight the deeply constructive nature of the psychophysics of risky choice, and suggest that traditional models may reflect subjective reactions to numerical form rather than substantive content. (PsycINFO Database Record

Efficient optimisation for portfolio selections under prospect theory

The prospect theory (PT) is one of the most useful tools for portfolio optimisation. The main concept of PT is to use a S-shaped value function to depict how human beings' mental behaviour affect their investment decisions under different risk levels. However, the complexity in the theory results in the difficulty in the proposal of efficient algorithms for global optimisation. In order to make improvements, we first approximate the S-shaped value function in PT with a piecewise linear (PWL) surrogate model, and equivalently transform the resulted problem into a continuous concave piecewise linear maximisation problem. Despite of the non-smoothness and non-convexity of the problem, we propose two local search algorithms based on the interior point method, and present the theoretical analysis on the convergence. Moreover, we propose a global search algorithm based on the proposed local search algorithms and the γ valid cut method in concave optimisation. The numerical experiments on the historical data of different assets obtained from Yahoo concern the comparisons of the proposed algorithms and the existing methods in the literature. The results confirm the performances of the proposed algorithm on efficiency and accuracy.

Efficient optimisation for portfolio selections under prospect theory

The prospect theory (PT) is one of the most useful tools for portfolio optimisation. The main concept of PT is to use a S-shaped value function to depict how human beings' mental behaviour affect their investment decisions under different risk levels. However, the complexity in the theory results in the difficulty in the proposal of efficient algorithms for global optimisation. In order to make improvements, we first approximate the S-shaped value function in PT with a piecewise linear (PWL) surrogate model, and equivalently transform the resulted problem into a continuous concave piecewise linear maximisation problem. Despite of the non-smoothness and non-convexity of the problem, we propose two local search algorithms based on the interior point method, and present the theoretical analysis on the convergence. Moreover, we propose a global search algorithm based on the proposed local search algorithms and the γ valid cut method in concave optimisation. The numerical experiments on the historical data of different assets obtained from Yahoo concern the comparisons of the proposed algorithms and the existing methods in the literature. The results confirm the performances of the proposed algorithm on efficiency and accuracy.

Evaluating the Role of Three Basic Factors of Prospect Theory in Decision Making

The main aim of this paper is to empirically evaluate the role of three significant factors of the Prospect Theory: the S-shaped value function, the loss aversion, and the distortion of probability, in decision making. In order to do this, a general behavioral reward-risk model is firstly setup and an empirical evaluation about the role of each of these factor, separately and in interaction, on the optimal solutions of the problem follows. For the analysis, well known US equity portfolios consisting by stocks listed in NYSE, AMEX, and NASDAQ formed on investment style are employed. The findings indicate that agents differentiate their behavior according to their type of preferences and their loss aversion level but they seem to always prefer high positively skewed assets such as small and value stocks. The attractiveness of positively skewed assets is re-enforced when probability distortion is introduced in the model. The introduction of probability distortion also affects the optimal perspective values of the problem increasing significantly their magnitude. After that, results show that as loss aversion increases agents tend to follow more conservative strategies, with and without probability distortion, while the value functional form has also its role in the model; bounded value functions as the negative exponential function drives agents to more conservative behaviors while unbounded value functions as the piecewise power function give the incentive to agents to undertake great risks and follow more aggressive strategies. The examination of the interaction of these factors indicate that the combination of an unbounded value functional form with a large loss aversion index may reduce agents' aggressiveness and limit (but not alter) the value functional form effect on optimal solutions.

A Generalized PROMETHEE III with Risk Preferences on Losses and Gains

This study aims to generalize the Preference Ranking Organization METHod for Enrichment Evaluations (PROMETHEE) III model by introducing risk preferences of decision makers. The risk preferences are expressed by an S-shaped value function with gain and loss parts. This study then illustrates an environmental evaluation of waste treatment plants for waste electrical and electronic equipment (WEEE) in Taiwan. Sensitivity analysis and the rank test demonstrate that the proposed model is rather stable. The PROMETHEE methods have been involved in various applications, especially in environmental management. One core process of PROMETHEE is to establish a preference difference function with two types of thresholds. The range of the slope lines of the linear preference is within the interval of [0, 1]. Working from the concept of the prospect theory, we extend its S-shaped function to the interval range of [-1, 1] so as to express risk preferences that occur in two quadrants. This research assesses a project on 15 local WEEE treatment plants to promote their recycling capability and technology competitiveness. According to the five aspects, the performance measures of the plants are obtained from a field study. The proposed model has an advantage on rank invariance by changing the thresholds in our case with sensitivity analysis demonstrating the robustness of the model. The generalized PROMETHEE III with risk preferences indeed provides an extension for making a decision in an uncertain environment.

Equal Tails: A Simple Method to Elicit Utility Under Violations of Expected Utility

Classical methods to elicit utility are biased because most individuals do not treat probabilities linearly. We propose a simple modification of the classical methods that equates, for all prospects being compared, the range of outcomes. We argue that the modification should work in theory, and test the modification experimentally. Our first experiment confirms that the modified certainty equivalent method reduces the curvature of the S-shaped value function. The second experiment is a novel design that compares the trade-off method with the three classical methods in their original and modified forms. Our equal-tails modification of both the certainty equivalent and the lottery equivalent method produces results consistent with the trade-off method. The lottery equivalent modification is particularly useful to elicit utility points when outcomes are nonquantifiable.

Incremental analysis for generalized TODIM

This study aims to generalize TODIM to eliminate two types of scaling effects by incremental analysis. One effect is due to an inappropriate presentation of the losses part in S-shaped value function in traditional TODIM, in which the criteria with smaller weights will contribute larger dominance values in its computing process. The other effect is derived from aggregating partial dominance measurements among heterogeneous criteria in traditional TODIM (a Portuguese acronym of Interactive and Multi-criteria Decision Making). The proposed approach firstly divides the criteria into two categories, benefits and costs, each of which is manipulated separately so as to facilitate incremental analysis. For each category, the pairwise dominance of one alternative is accumulated and normalized to represent the global dominance of each alternative. The global dominance measurements of alternatives in both categories are then combined according to benefit-cost ratio in order to rank alternatives. In addition, sensitivity analyses on the parameters of the gains/losses value function and the cutoff benefit-cost ratio demonstrate the robustness of the proposed model. An example of the selection of fuel buses is illustrated, and shows that our model is feasible and effective for MCDM problems with risk preference.

Reference-dependent risk sensitivity as rational inference.

Existing explanations of reference-dependent risk sensitivity attribute it to cognitive imperfections and heuristic choice processes. This article shows that behavior consistent with an S-shaped value function could be an implication of rational inferences about the expected values of alternatives. Theoretically, I demonstrate that even a risk-neutral Bayesian decision maker, who is uncertain about the reliability of observations, should use variability in observed outcomes as a predictor of low expected value for outcomes above a reference level, and as a predictor of high expected value for outcomes below a reference level. Empirically, I show that combining past outcomes using an S-shaped value function leads to accurate predictions about future values. The theory also offers a rationale for why risk sensitivity consistent with an inverse S-shaped value function should occur in experiments on decisions from experience with binary payoff distributions.

A Unified Theory: Prospect Theory and Market Dynamics

We measure crowd’s decision weights in trading by cumulative trading volume distribution and determine a reference price about crowd’s assessment value of an individual stock by the maximal volume price in stock market. We assume that prospect theory traders derive behavioral utility in trading not only from the liquidity utility expressed in terms of final trading wealth but also from a trading weight utility and a gain-loss utility relative to the reference price. We find a mathematical relation between them, derive a price-volume differential equation, and obtain two sets of explicit trading volume distribution models. One is the models in crowd’s independent trading, and the other is the behavioral models of reference-dependent preferences in crowd’s interactive trading. Finally, we propose an explicit S-shaped value function in prospect theory from the models in crowd’s interactive trading. The unified theory is successful, consistent with empirical tests in high frequency data.

Prospect Theory and Measurement on Crowdds Subjective Behaviors in Trading

We measure crowd’s decision weights in trading by trading volume distribution and determine a reference price about crowd’s assessment value of an individual stock by the maximum volume price in stock market. We examine a reference-dependent preferences hypothesis in trading by two sets of explicit trading volume distribution models, which are connected to an explicit S-shaped value function in prospect theory. It is true with 82.42% in our tests, using high frequency data in China stock market. We explain the patterns of volume distribution by four behavioral features: mental accounting, disposition, decision weight, and coherence or agreement. Moreover, crowd’s traders update a reference price about the assessment value of an individual stock in jump from time to time. It takes place about 11.92% on a trading day. The measure of subjective behaviors by a volume dimension suggests the new openings of asset pricing models and test methodologies in financial economics.

Tri-Reference Point Theory of Decision Making: From Principles to Applications

Tri-reference point (TRP) theory (Wang, 2008a; Wang Johnson, 2012) makes use of three reference points, minimum requirement (MR), status quo (SQ), and goal (G) to demarcate choice outcome space into four functional regions: failure, loss, gain, and success. Based on the priority order of the reference points: MR G SQ, the model derives from the four regions a double S-shaped value function, connected at the point of SQ. Risk preferences switched between risk-seeking and risk-aversion when the distribution of a gamble straddles a different reference point and resulted in gain-loss and success-failure asymmetries. In sum, the basic task in making risky choices is to maximize the likelihood of reaching a goal and minimize the likelihood of falling below the MR at the same time. The TRP theory synthetically combines the powerful concept of mean-variance used in statistics and finance with the concept of reference points in the behavioral decision making literature takes into consideration the mean-variance distribution of a choice option and its relationship with the three reference points in order to reach adaptive decisions. In this paper, we introduce the basic assumptions, operational principles, experimental tests of the TRP theory, and compare the TRP theory against expected utility theory, and prospect theory. We also discuss practical guidance and implications of the TRP theory for managerial decision making.

Non-additive multi-attribute fuzzy target-oriented decision analysis

As an emerging area considering behavioral aspects of decision making, target-oriented decision model lies in the philosophical root of bounded rationality as well as represents the S-shaped value function. This paper deals with multi-attribute decision analysis from target-oriented viewpoint. First, the basic (random) target-oriented decision model is extended to involve three types of target preferences: benefit target, cost target, and equal target. Next, since applying fuzzy set theory in decision analysis allows the decision maker to specify imprecise aspiration levels, fuzzy target-oriented decision analysis is formulated to model three typical types of fuzzy targets: fuzzy min, fuzzy max, and fuzzy equal. Also, different attitudes are used to derive target achievement functions, which can be viewed as a support for “probability as psychological distance”. Furthermore, we have proved that multi-attribute target-oriented decision analysis has a similar structure with discrete fuzzy measure and Choquet integral. Hence, we propose using discrete fuzzy measure and Choquet integral to model non-additive multi-attribute target-oriented decision analysis. In particular, the λ-measure is applied to reduce the difficulty of collecting information via a designed bisection search algorithm. Finally, a new product development example is used to illustrate the effectiveness and advantages of our model. The main advantages of our target-oriented decision model are its abilities to model the fuzzy uncertainty of targets as well as capture the non-additive behaviors among targets by means of discrete fuzzy measure and Choquet integral.

Prospect theory and utility theory: Temporary versus permanent attitude toward risk

Prospect theory (PT), which relies on subjects’ behavior as observed in laboratory experiments, contradicts the behavior predicted by the Expected Utility (EU) paradigm. Having wealth of $100,000 or having wealth of $90,000 and winning $10,000 in a lottery is the same by EU paradigm but not the same by Markowitz (1952) and by PT (1979) which emphasizes the importance of change of wealth rather than total wealth on welfare. In this study, we resolve this contradiction by introducing the concept of temporary attitude toward risk (TATR) and permanent attitude toward risk (PATR). Using these concepts, we build a model that merges both the PT and the EU paradigms. The TATR and PATR concepts explain recent experimental findings and the observed stock price overreaction. We show that a positive risk premium with decreasing absolute risk aversion (DARA) can be consistent with the S-shaped value function used in PT.

Prospect Theory and Utility Theory: Temporary Versus Permanent Attitude towards Risk

Prospect Theory (PT), which relies on subjects' behavior as observed in laboratory experiments, contradicts the behavior predicted by the Expected Utility (EU) paradigm. Having wealth of $100,000 or having wealth of $90,000 and wining $10,000 in a lottery is the same by EU paradigm but not the same by Markowitz (1952) and by PT (1979) which emphasizes the importance of change of wealth rather total wealth on welfare. In this study, we resolve this contradiction by introducing the concept of Temporary Attitude Towards Risk (TATR) and Permanent Attitude Towards Risk (PATR). Using these concepts, we build a model that merges both the PT and the EU paradigms. The TATR and PATR concepts explain recent experimental findings and the observed stock price overreaction. We show that a positive risk premium with decreasing absolute risk aversion (DARA) can be consistent with the S-shaped value function used in PT.

Sunk costs and sunk benefits: A re-examination of re-investment decisions

Prior experimental studies supporting the prospect theory explanation of the sunk-cost effect manipulate the framing of the initial investment, describing it either in neutral terms or as a prior loss. This paper subjects the prospect theory explanation to further examination, but takes an alternative experimental approach based on the differential risk taking behaviour predicted by prospect theory’s S-shaped value function. The experiments manipulate whether an initial investment produces a sunk cost (prior loss) or a sunk benefit (prior gain) and investigate the impact of this on the likelihood of authorising an incremental investment held constant across treatment conditions. To ensure the results are robust to the type of incremental investment, two experiments are conducted across which the outcomes of the incremental investment are manipulated to produce poor or good investment opportunities. In all cases the results fail to support a higher likelihood of authorising the incremental investment following a sunk cost than a sunk benefit. In isolation, therefore, prospect theory is unable to explain fully the sunk-cost effect.