Is Probability Weighting Sensitive to the Magnitude of Consequences? An Experimental Investigation on Losses

Experimental investigations of the probability weighting function over losses are scarce and all involve small payoffs. The paper aims to give new insight into the probability weighting function for losses, by eliciting it through a simple two-stage semi-parametric procedure over more realistic losses, and by investigating its sensitivity to the magnitude of the payoffs. Current data confirm previous evidence of convex utility functions and inverse-S-shaped weighting functions. Still, at least for small probabilities, probability weighting appears to be affected by the size of consequences: the larger the losses, the more aversive the gambles and the more pessimistic the subjects are.     

Comparing risk preferences over financial and environmental lotteries

This paper investigates whether preferences over environmental risks are best modeled using probability-weighted utility functions or can be reasonably approximated by expected utility (EU) or subjective EU models as is typically assumed. I elicit risk attitudes in the financial and environmental domains using multiple-price list experiment. I examine how subjects?? behavioral, attitudinal, and demographic characteristics affect their probability weighting functions first for financial risks, then for oil-spill risks. I find that most subjects tend to overweight extreme positive outcomes relative to expected utility in both the environmental and financial domains. Subjects are more likely to overemphasize low probability, extreme environmental outcomes than low probability, extreme financial outcomes, leading subjects to offer more support for mitigating environmental gambles than financial gambles with the same odds and equivalent outcomes. I conclude that EU models are likely to underestimate subjects?? willingness to pay for environmental cleanup programs or policies with uncertain outcomes.     

An empirical test of ordinal independence

In this article, we test Green and Jullien's (1988) Ordinal Independence (OI) Axiom, an axiom necessary for any rank-dependent expected utility (RDEU) model, including Cumulative Prospect Theory (Tversky and Kahneman, 1992). We observe systematic violations of OI (some within-subject violation rates of over 50%). These patterns of choice cannot be explained by any RDEU theory alone. We suggest that subjects are employing an editing operation prior to evaluation: if an outcome-probability pair is common to both gambles, it is cancelled when the commonality is transparent; otherwise, it is not cancelled. We interpret the results with respect to both original and cumulative prospect theory and the known empirical properties of the weighting function.     

Anxiety and Decision Making with Delayed Resolution of Uncertainty

In many real-world gambles, a non-trivial amount of time passes before the uncertainty is resolved but after a choice is made. An individual may have a preference between gambles with identical probability distributions over final outcomes if they differ in the timing of resolution of uncertainty. In this domain, utility consists not only of the consumption of outcomes, but also the psychological utility induced by an unresolved gamble. We term this utility anxiety. Since a reflective decision maker may want to include anxiety explicitly in analysis of unresolved lotteries, a multiple-outcome model for evaluating lotteries with delayed resolution of uncertainty is developed. The result is a rank-dependent utility representation (e.g., Quiggin, 1982), in which period weighting functions are related iteratively. Substitution rules are proposed for evaluating compound temporal lotteries. The representation is appealing for a number of reasons. First, probability weights can be interpreted as the cognitive attention allocated to certain outcomes. Second, the model disaggregates strength of preference from temporal risk aversion and thus provides some insight into the old debate about the relationship between von Neumann–Morgenstern utility functions and strength of preference value functions.     

Testing Prospect Theories Using Probability Tradeoff Consistency

The two versions of prospect theory, original prospect theory (OPT; Kahneman and Tversky, 1979) and cumulative prospect theory (CPT; Tversky and Kahneman, 1992), use different composition rules to combine the value function and the probability weighting function and hence value gambles with two or more non-zero outcomes differently. Previous tests of OPT and CPT have yielded mixed results, with some investigations supporting OPT and some supporting CPT. We extend the probability tradeoff consistency axiom used in Abdellaoui (2002) for CPT to OPT, and develop a critical test of the two prospect theories based on their respective probability tradeoff consistency conditions. An empirical investigation of the critical test shows that choices are consistent with OPT, but not CPT, for gambles that do not involve a certainty effect, and consistent with both CPT and OPT for gambles that do involve a certainty effect, provided that an editing operation is invoked for OPT.     

The Utility of Gambling Reconsidered

The utility of gambling, which entails an intrinsic utility or disutility of risk, has been alluded to in the economics literature for over a century. This paper demonstrates that any utility of gambling almost unavoidably implies a violation of fundamental rationality properties, such as transitivity or stochastic dominance, for static choices between gambles. This result may explain why the utility of gambling, a phenomenon so widely discussed, has never been formalized in the economics literature. The model of this paper accommodates well-known deviations from expected utility, such as the Allais paradox and the coexistence of gambling and insurance, while minimally deviating from expected utility.     

Behavioral biases and the representative agent

In this article, we show that behavioral features can be obtained at a group level even if they do not appear at the individual level. Starting from a standard model of Pareto optimal allocations, with expected utility maximizers but allowing for heterogeneity among individual beliefs, we show in particular that the representative agent has an inverse S-shaped probability distortion function as in Cumulative prospect theory.     

Probabilistic Insurance

Probabilistic insurance is an insurance policy involving a small probability that the consumer will not be reimbursed. Survey data suggest that people dislike probabilistic insurance and demand more than a 20% reduction in the premium to compensate for a 1% default risk. While these preferences are intuitively appealing they are difficult to reconcile with expected utility theory. Under highly plausible assumptions about the utility function, willingness to pay for probabilistic insurance should be very close to willingness to pay for standard insurance less the default risk. However, the reluctance to buy probabilistic insurance is predicted by the weighting function of prospect theory. This finding highlights the potential role of the weighting function to explain insurance.     

Violations of the betweenness axiom and nonlinearity in probability

Betweenness is a weakened form of the independence axiom, stating that a probability mixture of two gambles should lie between them in preference. Betweenness is used in many generalizations of expected utility and in applications to game theory and macroeconomics. Experimental violations of betweenness are widespread. We rule out intransitivity as a source of violations and find that violations are less systematic when mixtures are presented in compound form (because the compound lottery reduction axiom fails empirically). We also fit data from nine studies using Gul's disappointment-aversion theory and two variants of EU, which weight separate or cumulative probabilities nonlinearly. The three theories add only one parameter to EU and fit much better.     

Probability weighting and the ‘level’ and ‘spacing’ of outcomes: An experimental study over losses

The main goal of the experimental study described in this paper is to investigate the sensitivity of probability weighting to the payoff structure of the gambling situation—namely the level of consequences at stake and the spacing between them—in the loss domain. For that purpose, three kinds of gambles are introduced: two kinds of homogeneous gambles (involving either small or large losses), and heterogeneous gambles involving both large and small losses. The findings suggest that at least for moderate/high probability of loss do both ‘level’ and ‘spacing’ effects reach significance, with the impact of ‘spacing’ being both opposite to and stronger than the impact of ‘level’. As compared to small-loss gambles, large-loss gambles appear to enhance probabilistic optimism, while heterogeneous gambles tend to increase pessimism.

Testing between alternative models of choice under uncertainty: Some initial results

Experiments have identified a number of well-known violations of expected utility theory, giving rise to alternative models of choice under uncertainty, all of which are able to explain these violations. In this article, predictions of several prominent rival formulations are examined. No single alternative consistently organizes choices. Among the more important inconsistencies, we identify conditions generating systematic fanning in of indifference curves in the unit probability triangle, and find risk-loving over a number of gambles with all positive payoffs, in cases where prospect theory predicts risk aversion.     

Is “Generic Utility Theory” a Suitable Theory of Choice Behavior for Gambles with Mixed Gains and Losses?

Miyamoto's (1988, 1992) generic utility theory (GUT) subsumes a broad class of bilinear utility models. Chechile and Cooke (1997) tested the GUT class of models and found model failure due to the systematic variation of a parameter that should be a positive constant across a range of contexts. In the current study, an improved experimental design is employed to evaluate utility theory. The current study provides further evidence against the GUT class of models for mixed gambles. Moreover, evidence is also provided to demonstrate individual behavior that is incompatible with a coherent bilinear utility theory of choice behavior in the context of mixed gambles with gains and losses.     

It is Whether You Win or Lose: The Importance of the Overall Probabilities of Winning or Losing in Risky Choice

Imagine that you own a five-outcome gamble with the following payoffs and probabilities: ($100, .20; $50, .20; Imagine that you own a five-outcome gamble with the following payoffs and probabilities: ($100, .20; $50, .20; $0, .20; –$25, .20; –$50, .20). What happens when the opportunity to improve such a gamble is provided by a manipulation that adds value to one outcome versus another outcome, particularly when the opportunity to add value to one outcome versus another outcome changes the overall probability of a gain or the overall probability of a loss? Such a choice provides a simple test of the expected utility model (EU), original prospect theory (OPT), and cumulative prospect theory (CPT). A study of risky choices involving 375 respondents indicates that respondents were most sensitive to changes in outcome values that either increased the overall probability of a strict gain or decreased the overall probability of a strict loss. These results indicate more support for OPT rather than CPT and EU under various assumptions about the shape of the utility and value and weighting functions. Most importantly, the main difference between the various expectation models of risky choice occurs for outcomes near the reference value. A second study of risky choice involving 151 respondents again demonstrated the sensitivity of subjects to reducing the probability of a strict loss even at the cost of reduced expected value. Consequently, we argue that theories of how people choose among gambles that involve three or more consequences with both gains and losses need to include measures of the overall probabilities of a gain and of a loss.JEL Classification  D81     

Rationality on the rise: Why relative risk aversion increases with stake size

How does risk tolerance vary with stake size? This important question cannot be adequately answered if framing effects, nonlinear probability weighting, and heterogeneity of preference types are neglected. We show that the observed increase in relative risk aversion over gains cannot be captured by the curvature of the value function. Rather, it is predominantly driven by a change in probability weighting of a majority group of individuals who weight probabilities of high gains more conservatively. Contrary to gains, no coherent change in relative risk aversion is observed for losses. These results not only challenge expected utility theory, but also prospect theory.     

Rational behaviour: A comparison between the theory stemming from de Finetti's work and some other leading theories

We present some interesting features of de Finetti's decision theory; then we extended the theory. The extended theory has a normative character and is of the expected utility kind, but it is also very adaptable. By comparing it with some leading theories, we find that our theory is compatible with consideration of the whole probability distribution — it can even accommodate Allais' paradox -, while it is not generally compatible with probability weighting. We are mainly interested in the normative point of view.     

Interpersonal comparisons of utility for 2 of 3 types of people

This article argues that there is a natural solution to carry out interpersonal comparisons of utility when the theory of gambles is supplemented with a group operation of joint receipts. If so, three types of people can exist, and the two types having multiplicative representations of joint receipt have, in contrast to most utility theories, absolute scales of utility. This makes possible, at least in principle, meaningful interpersonal comparisons of utility with desirable properties, thus resolving a long standing philosophical problem and having potentially important implications in economics. Two behavioral criteria are given for the three classes of people. At this point the relative class sizes are unknown.     

On probabilities and loss aversion

This paper reviews the most common approaches that have been adopted to analyze and describe loss aversion under prospect theory. Subsequently, it is argued that loss aversion is a property of observable choice behavior and two new definitions of loss averse behavior are advocated. Under prospect theory, the new properties hold if the commonly used utility based measures of loss aversion are corrected by a probability based measure of loss aversion and their product exceeds 1. It is shown that prominent parametric families of weighting functions, while successful in accommodating empirical findings on probabilistic risk attitudes, may not fit well with the theoretical implications of the new loss averse behavior conditions.     

Common Consequence Conditions in Decision Making under Risk

We generalize the Allais common consequence effect by describing three common consequence effect conditions and characterizing their implications for the probability weighting function in rank-dependent expected utility. The three conditions—horizontal, vertical, and diagonal shifts within the probability triangle—are necessary and sufficient for different curvature properties of the probability weighting function. The first two conditions, shifts in probability mass from the lowest to middle outcomes and middle to highest outcomes respectively, are alternative conditions for concavity and convexity of the weighting function. The third condition, decreasing Pratt-Arrow absolute concavity, is consistent with recently proposed weighting functions. The three conditions collectively characterize where indifference curves fan out and where they fan in. The common consequence conditions indicate that for nonlinear weighting functions in the context of rank-dependent expected utility, there must exist a region where indifference curves fan out in one direction and fan in the other direction.     

A parametric analysis of prospect theory’s functionals for the general population

This article presents the results of an experiment that completely measures the utility function and probability weighting function for different positive and negative monetary outcomes, using a representative sample of N = 1,935 from the general public. The results confirm earlier findings in the lab, suggesting that utility is less pronounced than what is found in classical measurements where expected utility is assumed. Utility for losses is found to be convex, consistent with diminishing sensitivity, and the obtained loss-aversion coefficient of 1.6 is moderate but in agreement with contemporary evidence. The estimated probability weighting functions have an inverse-S shape and they imply pessimism in both domains. These results show that probability weighting is also an important phenomenon in the general population. Women and lower educated individuals are found to be more risk averse, in agreement with common findings. In contrast to previous studies that ascribed gender differences in risk attitudes solely to differences in the degree utility curvature, however, our results show that this finding is primarily driven by loss aversion and, for women, also by a more pessimistic psychological response toward the probability of obtaining the best possible outcome.     

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.