Probability Weighting Function Research Articles

A 'Certain-Uncertain' Inconsistency of the Main Experimental System of Utility and Prospect Theories and Models

In the main random-lottery incentive system of experiments, the choices of certain outcomes are stimulated by random, uncertain lotteries. This “certain-uncertain” inconsistency is quite evident, but only recently revealed. Because of this inconsistency, conclusions from a random-lottery incentive experiment, which includes a certain outcome, cannot be unquestionably correct. The well-known experiment and purely mathematical theorems support this evidently. So, the main result of the article is that the prevailing experimental system of utility and prospect theories and models may need additional independent analyses in the content of the “certain-uncertain” inconsistency. Because of this inconsistency, one cannot exclude that, at the probability p=1, this system hides such a topological feature as a discontinuity of the probability weighting function. Such a discontinuity is also supported by the theorems and experiment. Owing to this discontinuity, these theories and models can be qualitatively changed. For example, the probability weighting (Prelec) function can transform from the inverse S-shaped curve into a roughly straight approximation of its own middle part.

ARROW–DEBREU EQUILIBRIA FOR RANK‐DEPENDENT UTILITIES

AbstractWe provide conditions on a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents under which Arrow–Debreu equilibria exist. When such an equilibrium exists, we show that the state‐price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.

Assessing diagnostic accuracy improvement for survival or competing‐risk censored outcomes

AbstractDiagnostic accuracy studies have progressed in the past decade to consider survival outcomes beyond the traditional dichotomous outcome. Another recent advance is the appearance of novel measures for diagnostic accuracy improvement by adding new markers. In this paper we attempt to integrate these two evolving areas and contribute a discussion on assessing diagnostic accuracy improvement for censored survival outcomes. More importantly, we consider competing‐risk censoring in addition to independent censoring, and provide inferential procedures. Particularly, we consider fitting regression models based on cumulative incidence functions for the primary event, and propose parallel estimators for the adapted accuracy improvement measures based on inverse probability weighting and bivariate cumulative incidence function estimation. Both estimators perform very well in simulations and in an application to a breast cancer study. The Canadian Journal of Statistics 42: 109–125; 2014 © 2014 Statistical Society of Canada

確率情報の図的表現下での確率荷重関数

確率情報の図的表現下での確率荷重関数

Music-evoked incidental happiness modulates probability weighting during risky lottery choices

We often make decisions with uncertain consequences. The outcomes of the choices we make are usually not perfectly predictable but probabilistic, and the probabilities can be known or unknown. Probability judgments, i.e., the assessment of unknown probabilities, can be influenced by evoked emotional states. This suggests that also the weighting of known probabilities in decision making under risk might be influenced by incidental emotions, i.e., emotions unrelated to the judgments and decisions at issue. Probability weighting describes the transformation of probabilities into subjective decision weights for outcomes and is one of the central components of cumulative prospect theory (CPT) that determine risk attitudes. We hypothesized that music-evoked emotions would modulate risk attitudes in the gain domain and in particular probability weighting. Our experiment featured a within-subject design consisting of four conditions in separate sessions. In each condition, the 41 participants listened to a different kind of music—happy, sad, or no music, or sequences of random tones—and performed a repeated pairwise lottery choice task. We found that participants chose the riskier lotteries significantly more often in the “happy” than in the “sad” and “random tones” conditions. Via structural regressions based on CPT, we found that the observed changes in participants' choices can be attributed to changes in the elevation parameter of the probability weighting function: in the “happy” condition, participants showed significantly higher decision weights associated with the larger payoffs than in the “sad” and “random tones” conditions. Moreover, elevation correlated positively with self-reported music-evoked happiness. Thus, our experimental results provide evidence in favor of a causal effect of incidental happiness on risk attitudes that can be explained by changes in probability weighting.

Discriminating among probability weighting functions using adaptive design optimization

Probability weighting functions relate objective probabilities and their subjective weights, and play a central role in modeling choices under risk within cumulative prospect theory. While several different parametric forms have been proposed, their qualitative similarities make it challenging to discriminate among them empirically. In this paper, we use both simulation and choice experiments to investigate the extent to which different parametric forms of the probability weighting function can be discriminated using adaptive design optimization, a computer-based methodology that identifies and exploits model differences for the purpose of model discrimination. The simulation experiments show that the correct (data-generating) form can be conclusively discriminated from its competitors. The results of an empirical experiment reveal heterogeneity between participants in terms of the functional form, with two models (Prelec-2, Linear in Log Odds) emerging as the most common best-fitting models. The findings shed light on assumptions underlying these models.

Prospect Theory for joint time and money consequences in risk and ambiguity

Transport users face complex decisions. Not only are the consequences of their choices uncertain, but they generally involve several attributes, such as time and money. Time-money tradeoffs have been studied in depth in transport economics, and research is now paying increasing attention to the role of uncertainty and information in transport decisions. This paper aims to measure the impact of uncertainty and information on multi-attribute decisions using Prospect Theory. In doing so, the study makes two contributions to transportation literature: one methodological and the other empirical. First, we propose a fast and tractable method for measuring Prospect Theory parameters that capture attitudes towards probabilities (probability weighting function) and attitudes towards losses (loss aversion). The elicitation method does not require the elicitation of the utility function. This makes it particularly suitable in complex multi-attribute decisions where the shape of the utility function is unknown. Second, we present the results of an experiment that uses the proposed method to measure, at the individual level, probability weighting in decisions involving joint time and money consequences in two decision contexts: risk (where probabilities are given) and ambiguity (where the probability distribution is unknown). An experimental setup that exposes subjects to real gains and losses for money and time has been built for this purpose. We observe inverse S-shaped probability weighting and loss aversion for risk. Probability weighting is even more pronounced in ambiguity, where subjects do not have precise information about the probability distribution. We explain how these results and the analysis of ambiguity attitudes in general can offer a better understanding of travelers’ route or transport mode choices.

Range-Dependent Utility

This paper introduces the concept of range-dependent utility. Instead of reference dependence which evaluates outcomes relative to some reference point, we postulate dependence on a given lottery (set of lotteries) outcomes range. In this way the decision maker is a fully rational expected utility maximizer only within a certain range. Range-dependent utility enables experimental results to be explained without recourse to the probability weighting function. Experimental data show that range-dependent utilities can be normalized to obtain decision utility -- a single utility function able to describe decisions involving lotteries denied over different ranges. Both the data analysis as well as theoretical considerations concerning monotonicity indicate that the decision utility should be of S-shape.

Probability Weighting of Rare Events and Currency Returns

We show that the probability weighting of rare events, accounting for investors' attitudes toward extreme downside losses versus upside gains in non-expected utility models, provides a unified explanation for both time-series and cross-sectional variations of currency portfolio returns. We use a simple structural model to show the link between the probability weighting function and pricing kernel, and then estimate them by non-parametric methods using currency options data from 1996 to 2012. The estimates show that a domestic investor over-weights the likelihood of a substantial depreciation or appreciation of foreign currencies, consistent with experimental studies. A global probability weighting measure of left (right) tail events is highly significant in positively (negatively) predicting future currency returns over time series at both individual and portfolios levels. Furthermore, asset pricing tests show that differences in exposure to our global tail weighting measures, of high versus low interest rate currencies and of high versus low past return currencies, can explain the cross-sectional variation in average excess returns across both carry and momentum portfolios. Moreover, our global tail weighting measures remain significant after controlling for existing currency risk factors in the literature, and frequently drive their significance out, in both time-series and cross-sectional return predictability regressions.

Using Preferred Outcome Distributions to Estimate Value and Probability Weighting Functions in Decisions under Risk

In this paper we propose the use of preferred outcome distributions as a new method to elicit individuals' value and probability weighting functions in decisions under risk. Extant approaches for the elicitation of these two key ingredients of individuals' risk attitude typically rely on a long, chained sequence of lottery choices. In contrast, preferred outcome distributions can be elicited through an intuitive graphical interface, and, as we show, the information contained in two preferred outcome distributions is sufficient to identify non-parametrically both the value function and the probability weighting function in rank-dependent utility models. To illustrate our method and its advantages, we run an incentive-compatible lab study in which participants use a simple graphical interface - the Distribution Builder (Goldstein et al. 2008) - to construct their preferred outcome distributions, subject to a budget constraint. Results show that estimates of the value function are in line with previous research but that probability weighting biases are diminished, thus favoring our proposed approach based on preferred outcome distributions.

Using Preferred Outcome Distributions to Estimate Value and Probability Weighting Functions in Decisions under Risk

In this paper we propose the use of preferred outcome distributions as a new method to elicit individuals' value and probability weighting functions in decisions under risk. Extant approaches for the elicitation of these two key ingredients of individuals' risk attitude typically rely on a long, chained sequence of lottery choices. In contrast, preferred outcome distributions can be elicited through an intuitive graphical interface, and, as we show, the information contained in two preferred outcome distributions is sufficient to identify non-parametrically both the value function and the probability weighting function in rank-dependent utility models. To illustrate our method and its advantages, we run an incentive-compatible lab study in which participants use a simple graphical interface - the Distribution Builder (Goldstein et al. 2008) - to construct their preferred outcome distributions, subject to a budget constraint. Results show that estimates of the value function are in line with previous research but that probability weighting biases are diminished, thus favoring our proposed approach based on preferred outcome distributions.

Theoretical development of probability weighting function

Theoretical development of probability weighting function

Emotional balance and probability weighting

We find suggestive evidence that emotional balance has an impact on probability weighting incremental to demographic controls. Specifically, low negative affectivity (implying high emotional balance) tends to be a characteristic of those whose probability weighting functions exhibit lower curvature and more neutral elevation. In other words, emotional balance seems to push people in the direction of normative expected utility theory.

Group Representations for Decision Making under Risk and Uncertainty

This paper extends decision making under risk and uncertainty to group theory via representations of invariant behavioural space for prospect theory. First, we predict that canonical specifications for value functions, probability weighting functions, and stochastic choice maps are homomorphic. Second, we derive a continuous singular matrix operator T for affine transformation of a vector space of skewed S-shape value functions V isomorphic to a vector spaceW of inverted S-shaped probability weighting functions. To characterize the transformation, we decompose the operator into shear, scale and translation components. In that milieu, Moore-Penrose psuedoinverse transformation recovers value functions from probability weighting functionals. Removal of 0 from the point spectrum induces nonsingular operators that support group representation of stochastic choice maps in an invariant subspace of the general linear group GL(V). Third, we demonstrate how group theoretic operations on a gamble provide mathematical foundations of probability weighting functions that subsume the Prelec class. Fourth, we predict that a gamble is isomorphic to an invariant cyclic sub-group in weighted probablity space. This result implies that probability weighting functions [and value functions] fluctuate near their extremes, and explain violation of transitivity axioms in decision theory. Moreover, representations include the special unitary group SU(2) and orthogonal group Θ*3. The former includes Pauli’s spin matrices and accounts for skewness. It also provides microfoundations for construction of (1) behavioural stochastic processes from group character in the frequency domain; and (2) Schrodinger-Pauli Hamiltonian to compute, inter alia, time dependent probabilities in decision field theory.

Probability weighting functions implied in options prices

The empirical pricing kernels estimated from index options are non-monotone (Rosenberg and Engle, 2002; Bakshi, Madan, and Panayotov, 2010) and the corresponding risk-aversion functions can be negative (Aït-Sahalia and Lo, 2000; Jackwerth, 2000). We show theoretically that these and several other properties of empirical pricing kernels are consistent with rank-dependent utility model with probability weighting function, which overweights tail events. We also estimate the pricing kernels nonparametrically from the Standard & Poor's 500 index options and construct empirical probability weighting functions. The estimated probability weights typically have the inverse-S shape, which overweights tail events and is widely supported by the experimental decision theory.

Arrow-Debreu Equilibria for Rank-Dependent Utilities

We provide conditions on a one‐period‐two‐date pure exchange economy with rank-dependent utility agents under which Arrow-Debreu equilibria exist. When such an equilibrium exists, we show that the state‐price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.

Shifted risk preferences in pathological gambling

Pathological gambling (PG) is an impulse control disorder characterized by excessive monetary risk seeking in the face of negative consequences. We used tools from the field of behavioral economics to refine our description of risk-taking behavior in pathological gamblers. This theoretical framework allowed us to confront two hypotheses: (1) pathological gamblers distort winning probabilities more than controls; and (2) pathological gamblers merely overweight the whole probability range. Method Eighteen pathological gamblers and 20 matched healthy participants performed a decision-making task involving choices between safe amounts of money and risky gambles. The online adjustment of safe amounts, depending on participants' decisions, allowed us to compute 'certainty equivalents' reflecting the subjective probability weight associated with each gamble. The behavioral data were then fitted with a mathematical function known as the 'probability weighting function', allowing us to disentangle our two hypotheses. The results favored the second hypothesis, suggesting that pathological gamblers' behavior reflects economic preferences globally shifted towards risk, rather than excessively distorted probability weighting. A mathematical parameter (elevation parameter) estimated by our fitting procedure was found to correlate with gambling severity among pathological gamblers, and with gambling affinity among controls. PG is associated with a specific pattern of economic preferences, characterized by a global (i.e. probability independent) shift towards risky options. The observed correlation with gambling severity suggests that the present 'certainty equivalent' task may be relevant for clinical use.

Disposition Effect Among Contrarian and Momentum Investors

We provide evidence of disposition effect propensity for stock trading simulation participants employing a contrarian versus a momentum strategy. We found that even subjects playing with chips rather than real money remain vulnerable to those effects. Both tendencies were generally evident in our sample, but we also found individual differences. Subjects seemed to be contrarians both on position opening and on position closing. The main hypothesis of this paper states that contrarian investors are more prone to the disposition effect than are momentum traders. The model proposed by Dacey and Zielonka [2008] plays a crucial role in formulating this hypothesis. We consider the disposition effect not only in terms of the value function but also of the probability weighting function. In accordance with our hypothesis, we found that contrarian traders are more prone to the disposition effect.

Comparison of the Distortion of Probability Information in Decision Under Risk and an Equivalent Visual Task

Decision makers typically overweight small probabilities and underweight large probabilities. However, there are recent reports that when probability is presented in the form of relative frequencies, this typical pattern reverses. We tested this hypothesis by comparing decision making in two tasks: In one task, probability was stated numerically, and in the other task, it was conveyed through a visual representation. In the visual task, participants chose whether a “stochastic bullet” should be fired at either a large target for a small reward or a small target for a large reward. Participants’ knowledge of probability in the visual task was the result of extensive practice firing bullets at targets. In the classical numerical task, participants chose between pairs of lotteries with probabilities and rewards matched to the probabilities and rewards in the visual task. We found that participants’ probability-weighting functions were significantly different in the two tasks, but the pattern for the visual task was the typical, not the reversed, pattern.

Uncertainty plus prior equals rational bias: An intuitive Bayesian probability weighting function.

Empirical research has shown that when making choices based on probabilistic options, people behave as if they overestimate small probabilities, underestimate large probabilities, and treat positive and negative outcomes differently. These distortions have been modeled using a nonlinear probability weighting function, which is found in several nonexpected utility theories, including rank-dependent models and prospect theory; here, we propose a Bayesian approach to the probability weighting function and, with it, a psychological rationale. In the real world, uncertainty is ubiquitous and, accordingly, the optimal strategy is to combine probability statements with prior information using Bayes' rule. First, we show that any reasonable prior on probabilities leads to 2 of the observed effects; overweighting of low probabilities and underweighting of high probabilities. We then investigate 2 plausible kinds of priors: informative priors based on previous experience and uninformative priors of ignorance. Individually, these priors potentially lead to large problems of bias and inefficiency, respectively; however, when combined using Bayesian model comparison methods, both forms of prior can be applied adaptively, gaining the efficiency of empirical priors and the robustness of ignorance priors. We illustrate this for the simple case of generic good and bad options, using Internet blogs to estimate the relevant priors of inference. Given this combined ignorant/informative prior, the Bayesian probability weighting function is not only robust and efficient but also matches all of the major characteristics of the distortions found in empirical research.