Thinking styles and risky decision‐making: Further exploration of the affect–probability weighting link

People usually overweight small probabilities and underweight large probabilities leading to the familiar inverse S‐shaped weighting function. This research explores the link between affect and the structure of probability weighting from the perspective of thinking dispositions, a concept central to dual system theories of reasoning. The effects of affective priming and cognitive load on both probability weighting and the value function are also examined. The evidence suggests that thinking styles do have predictive implications for risky decision‐making. Participants with a more affective thinking style tend to be more risk‐seeking in small probability gambles. However, increasing access to the affective system by affective priming or cognitive load manipulations tend to reduce risk‐seeking behavior in small probability gambles as well as reduce risk averse behavior in large probability gambles. Previous research, manipulating the affective nature of lottery outcomes, found evidence for an increase in curvature (more overweighting of small probabilities and more underweighting of large probabilities) of the weighting function for affect‐rich outcomes, lending support to a hope‐and‐fear deconstruction of probability weighting. The present research suggests that increased anticipatory emotions characterized by the elevation of the weighting function (more overweighting at all probabilities) is also important and could sometimes be more significant than hope‐and‐fear in decision‐making under risk. An integrated approach incorporating the impact of affect on all three, the elevation and curvature of probability weighting as well as the curvature of the value function explains the empirical findings. Copyright © 2010 John Wiley & Sons, Ltd.     

Experimental test of the prospect theory value function: A stochastic dominance approach

According to prospect theory, individuals are risk averse regarding gains but risk seeking regarding losses, implying an S-shaped value function. The S-shaped value function hypothesis is based on experiments in which subjects are asked to choose separately between alternatives with either only positive or only negative outcomes, alternatives which rarely exist in the capital market. In addition, the S-shaped findings may be biased by the “certainty effect” and by probability distortion. In this paper we employ the recently developed prospect stochastic dominance criterion to test the prospect theory S-shaped value function hypothesis with mixed outcomes and with no “certainty effect.” Assuming that subjects do not distort moderate probabilities, we strongly reject the prospect theory S-shaped value function, with at least 76–86% of the choices being inconsistent with such preferences. When possible subjective probability distortions are taken into account, we find that at least 50–66% of the choices are inconsistent with an S-shaped value function.     

Modeling the effect of reward amount on probability discounting

The present study with college students examined the effect of amount on the discounting of probabilistic monetary rewards. A hyperboloid function accurately described the discounting of hypothetical rewards ranging in amount from $20 to $10,000,000. The degree of discounting increased continuously with amount of probabilistic reward. This effect of amount was not due to changes in the rate parameter of the discounting function, but rather was due to increases in the exponent. These results stand in contrast to those observed with the discounting of delayed monetary rewards, in which the degree of discounting decreases with reward amount due to amount-dependent decreases in the rate parameter. Taken together, this pattern of results suggests that delay and probability discounting reflect different underlying mechanisms. That is, the fact that the exponent in the delay discounting function is independent of amount is consistent with a psychophysical scaling interpretation, whereas the finding that the exponent of the probability-discounting function is amount-dependent is inconsistent with such an interpretation. Instead, the present results are consistent with the idea that the probability-discounting function is itself the product of a value function and a weighting function. This idea was first suggested by Kahneman and Tversky (1979), although their prospect theory does not predict amount effects like those observed. The effect of amount on probability discounting was parsimoniously incorporated into our hyperboloid discounting function by assuming that the exponent was proportional to the amount raised to a power. The amount-dependent exponent of the probability-discounting function may be viewed as reflecting the effect of amount on the weighting of the probability with which the reward will be received.     

Probability weights in rank-dependent utility with binary even-chance independence

We examine the effects of a weak version of expected utility's independence axiom on the probability weighting function in rank-dependent utility. Our weak independence axiom says that a 50-50 lottery between a two-outcome gamble and its certainty equivalent is indifferent to the certainty equivalent. A variety of nonlinear probability weighting functions satisfy this axiom, but most weighting functions proposed by others do not. Nevertheless, the axiom accommodates weighting functions that are quite similar to the inverse S-shaped concave-convex functions of others that overvalue small probabilities and undervalue large probabilities.     

The effects of payout and probability magnitude on the Allais paradox

The Allais paradox decision bias was first offered as a challenge to the expected utility theory over 60 years ago. Although the Allais paradox is a standard challenge for normative theories of risky choice, its causes are not well understood. The present experiment uses two manipulations of the Allais paradox to investigate the commonly proposed probability-weighting explanation of the paradox. Reducing the magnitude of the outcomes did not affect the size of the Allais paradox, contradicting previous literature and supporting the probability weighting hypothesis. Reducing the probability of the nonzero outcomes to eliminate certainty reduced, but did not eliminate, the Allais paradox, a result inconsistent with probability weighting and other theories of the Allais paradox. The results suggest that the certainty effect alone cannot explain the Allais paradox.     

The psychological underpinnings of relative thinking in price comparisons

This article investigates the psychological underpinnings of relative thinking—the tendency of consumers to consider relative savings, and not just absolute savings, in their decisions to search for a deal or purchase an item. We examine how (i) cognitive load, (ii) the affective-richness of the product, and (iii) the consumer's propensity for intuitive decision-making influence relative thinking. As hypothesized, high cognitive load and affect-rich (vs. affect-poor) products, and individual level preference for intuitive decision-making aggravate this behavior. Our results present clear managerial implications along with developing a better understanding of the behavioral foundations of relative thinking.     

How Affect Shapes Risky Choice: Distorted Probability Weighting Versus Probability Neglect

People's choices between prospects with relatively affect‐rich outcomes (e.g., medical side effects) can diverge markedly from their choices between prospects with relatively affect‐poor outcomes (e.g., monetary losses). We investigate the cognitive mechanisms underlying this “affect gap” in risky choice. One possibility is that affect‐rich prospects give rise to more distortion in probability weighting. Another is that they lead to the neglect of probabilities. To pit these two possibilities against each other, we fitted cumulative prospect theory (CPT) to the choices of individual participants, separately for choices between options with affect‐rich outcomes (adverse medical side effects) and options with affect‐poor outcomes (monetary losses); additionally, we tested a simple model of probability neglect, the minimax rule. The results indicated a qualitative difference in cognitive mechanisms between the affect‐rich and affect‐poor problems. Specifically, in affect‐poor problems, the large majority of participants were best described by CPT; in affect‐rich problems, the proportion of participants best described by the minimax rule was substantially higher. The affect gap persisted even when affect‐rich outcomes were supplemented by numerical information, thus providing no support for the thesis that choices in affect‐rich and affect‐poor problems diverge because the information provided in the former is nonnumerical. Our findings suggest that the traditional expectation‐based framework for modeling risky decision making may not readily generalize to affect‐rich choices. Copyright © 2015 John Wiley & Sons, Ltd.     

Probability as a psychological distance: Construal and preferences

We argue that probability, like space and time, instantiates psychological distance. Unlikely outcomes may seem more remote than likely outcomes and may therefore be construed at a relatively high level. Specifically, when the probability of an outcome is low, ends-related primary features should be more salient than means-related secondary features, but as the probability of the outcome increases, means-related features may become no less and even more salient than ends-related features. Thus, increases in probability should increase the weight of means-related features relative to the weight of ends-related features in decisions, thereby decreasing (or even reversing) the preference for a more desirable/less feasible outcome over a less desirable/more feasible outcome. We observed this pattern in two experiments. Analyses of judgments, monetary decisions, and self-reported reasons for decisions showed that the weight of means-related features was more sensitive to changes in probability than the weight of ends-related features in decisions.     

Cognitive strategies for coping with uncertain outcomes

It was proposed that, when faced with highly desirable but uncertain outcomes, people may employ cognitive strategies in an attempt to influence their future affective responses to the outcomes in question. The present study attempted to demonstrate the use of two such strategies. First, it was hypothesized that when people are faced with a low probability of obtaining a highly desirable outcome, they tend to derogate that outcome by perceiving it as less attractive. Second, it was proposed that when people are faced with uncertainty regarding the occurrence of a highly desirable outcome, they tend to underestimate the likelihood of its occurrence, in an attempt to avoid future disappointment. These hypotheses were tested within the context of a lottery in which subjects were given a low, moderate, or high chance of winning a prize that was either high or low in attractiveness. As predicted, subjects viewed the highly attractive prize as less valuable and attractive when they had a low probability of winning than when the probability of winning was moderate or high. Subjects also perceived themselves as less likely to win when the prize was high in attractiveness than when it was low in attractiveness. The relationship of these findings to studies of self-handicapping and attribute ambiguity is discussed.     

What shapes the probability weighting function? Influence of affect,numeric competencies,and information formats

Research suggests that people are less sensitive to variations in probability in affect‐rich compared with affect‐poor risky choices. This effect is modeled by a more curved probability weighting function (PWF). We investigated the role of different numeric competencies and the effectiveness of several intervention strategies to decrease this affect‐laden probability distortion. In two experiments, we manipulated the affect‐richness of a risky prospect. In Experiment 1 (N = 467), we measured numeracy and symbolic‐number mapping (i.e., the ability to accurately map numbers onto their underlying magnitudes). The affect‐based manipulations showed the expected effects only in participants with more accurate symbolic‐number mapping, who also reported more differentiated emotional reactions to the various probabilities and displayed more linear PWFs. Instructions to focus on the probability information decreased probability distortion and revealed differences in the use of probability information on the basis of symbolic‐number mapping ability. In Experiment 2 (N = 417), we manipulated the format in which the probability information was presented: using visual aids versus no visual aids and a positive frame (e.g., one person wins) versus combined frame (e.g., one person wins and 99 persons do not win). The affect‐based manipulations had no effect but both the visual aids and combined frame decreased probability distortion. Whereas affect‐richness manipulations require further research, results suggest that probability weighting is at least partially driven by the inability to translate numerical information into meaningful and well‐calibrated affective intuitions. Visual aids and simple framing manipulations designed to calibrate these intuitions can help decision makers extract the gist and increase sensitivity to probabilities.     

Likelihood judgment based on previously observed outcomes: the alternative-outcomes effect in a learning paradigm

Previous research has demonstrated that intuitive perceptions of certainty regarding a focal outcome are sensitive to variations in how evidence supporting nonfocal alternatives is distributed, even when such variations have no bearing on objective probability. We investigated this alternative-outcomes effect in a learning paradigm in which participants made likelihood judgments on the basis of their memory for past observations of relevant outcomes. In Experiment 1, a manipulation of evidence (observed frequencies) across alternative outcomes influenced not only intuitive certainty estimates about a focal outcome but also numeric subjective probabilities. Experiment 2 ruled out the possibility that these effects were attributable to the influence of information loss on frequency estimations. The findings were consistent with the heuristic comparison account, which suggests that the judged likelihood of a focal outcome will be disproportionately influenced by the strength (frequency) of the strongest alternative outcome.     

Cognitive and affective matching effects in persuasion: an amplification perspective

Past research suggests that cognitive and affective attitudes are more open to change toward cognitive and affective (i.e., matched) persuasive attacks, respectively. The present research investigates how attitude certainty influences this openness. Although an extensive literature suggests that certainty generally reduces an attitude's openness to change, the authors explore the possibility that certainty might increase an attitude's openness to change in the context of affective or cognitive appeals. Based on the recently proposed amplification hypothesis, the authors posit that high (vs. low) attitude certainty will boost the resistance of attitudes to mismatched attacks (e.g., affective attitudes attacked by cognitive messages) but boost the openness of attitudes to matched attacks (e.g., affective attitudes attacked by affective messages). Two experiments provide support for this hypothesis. Implications for increasing the openness of attitudes to both matched and mismatched attacks are discussed.     

Does probability weighting matter in probability elicitation?

One of the most widely used methods for probability encoding in decision analysis uses binary comparisons (choices) between two lotteries: one that depends on the values of the random variable of interest and another that is contingent on an external reference chance device (typically a probability wheel). This note investigates the degree to which differences in probability weighting functions between the two types of events could affect the practice of subjective probability encoding. We develop a general methodology to investigate this question and illustrate it with two popular probability weighting functions over the range of parameters reported in the literature. We use this methodology to (a) alert decision analysts and researchers to the possibility of reversals, (b) identify the circumstances under which overt preferences for one lottery over the other are not affected by the weighting function, (c) document the magnitude of the differences between choices based on probabilities and their corresponding weighting functions, and (d) offer practical recommendations for probability elicitation.     

Reference-dependent sympathy

Natural disasters and other traumatic events often draw a greater charitable response than do ongoing misfortunes, even those that may cause even more widespread misery, such as famine or malaria. Why is the response disproportionate to need? The notion of reference dependence critical to Prospect Theory (Kahneman & Tversky, 1979) maintains that self-utility, or benefit to self, is not absolute level of wealth but rather gain or loss relative to a reference point. Four studies show that sympathy (Study 1), dictator offers (Study 2), and judgments of deservingness (Study 3a) are reference-dependent: people respond greater to victims of loss than to victims of chronic conditions. This tendency goes away when people evaluate victims in comparison (Study 3b) and when evaluating affect-poor “statistical victims”, as compared to affect-rich “identifiable victims” (Study 4). Together, these results shed light on seemingly irrational patterns of humanitarian aid.     

Decision biases in intertemporal choice and choice under uncertainty: Testing a common account

Two experiments were performed to test a psychophysical account of parallels between biases in risky choice and intertemporal choice. Experiment 1 demonstrated the common difference effect in intertemporal choice and the common ratio effect in risky choice. As was predicted, these two biases were uncorrelated with each other, although each was correlated across monetary/health domains. This result is consistent with the supposition that these two biases result from psychophysical properties of two different dimensions (time and probability, respectively). Experiment 2 examined the magnitude effect in intertemporal choice and the peanuts effect in risky choice. These two biases were correlated with each other but were uncorrelated across monetary/health domains. This result is consistent with the supposition that these two biases result from psychophysical properties of the same dimension (utility of money or health).     

A theory of perceived risk and attractiveness

People judged both the attractiveness and risk of lotteries to win or lose money. The lotteries were designed to test whether risk and attractiveness judgments show systematic deviations from the simple sum of probability-by-utility-products analogous to (S)EU theory. Our results led to an alternative combination rule for probability and outcome information, with a relative weight averaging component and a configural (i.e., sign- or rank-dependent) probability weighting component. Ratings of risk and attractiveness were negatively correlated, but the two tasks showed systematic differences in the rank order of judgments. Both judgments could be fit by the same configural relative weight averaging model, but with different parameters (especially the sign-dependent probability weighting functions). Risk judgments were more sensitive to the probability of losses and zero outcomes compared to attractiveness judgments, which were more sensitive to the probability of gains. There were individual differences on the extent of this difference in probability weights between risk and attractiveness judgments.     

Music, pandas, and muggers: on the affective psychology of value

This research investigated the relationship between the magnitude or scope of a stimulus and its subjective value by contrasting 2 psychological processes that may be used to construct preferences: valuation by feeling and valuation by calculation. The results show that when people rely on feeling, they are sensitive to the presence or absence of a stimulus (i.e., the difference between 0 and some scope) but are largely insensitive to further variations of scope. In contrast, when people rely on calculation, they reveal relatively more constant sensitivity to scope. Thus, value is nearly a step function of scope when feeling predominates and is closer to a linear function when calculation predominates. These findings may allow for a novel interpretation of why most real-world value functions are concave and how the processes responsible for nonlinearity of value may also contribute to nonlinear probability weighting.     

Violations of Branch Independence in Choices between Gambles

Branch Independence is weaker than Savage's independence axiom; it holds that if two gambles have a common outcome for an event of known probability, the value of that common outcome should have no effect on the preference order induced by other probability-outcome branches. Systematic violations of branch independence were obtained in two experiments with choices between gambles composed of three equally likely, positive outcomes. Most people prefer ($2, $40, $44) over ($2, $10, $98); however, most people prefer ($10, $98, $136) over ($40, $44, $136). These results refute Expected Utility theories. They also refute the theory that people edit and cancel common components in choice. The pattern is opposite that predicted by the weighting function of cumulative prospect theory. Results are consistent with rank dependent, configural weight theory, with w_L> w_M> w_H, where w_L, w_M, and w_Hare the weights of the lowest, medium, and highest outcomes, respectively. In this theory, violations of branch independence depend on relations among weights: results indicate that w_L/w_M< w_M/w_H.