A cognitive‐emotional account of the shape of the probability weighting function

Research has shown a tendency of decision makers to overweight small probabilities and to underweight moderate and large probabilities. In standard treatments this is graphically modeled by an inverse S‐shaped probability weighting function. We suggest that emotions play a significant role in the shaping of the probability weighting function. In particular, the weighting function is proposed to be some function of objective probability, expected elation, and expected disappointment. The overweighting of small probabilities results from the anticipated elation after having won, given that winning was very unlikely. The underweighting of large probabilities results from anticipated disappointment after having failed to win, given that winning was very likely. Hence, probability is assumed to influence utility. Three experiments investigate these hypotheses. Experiments 1 and 2 show that a convex function relates probability to surprise. Experiment 3 elicits choice data and further supports the proposed hypotheses. The model adds to the understanding of the cognitive and emotional processes underlying the shape of the probability weighting function. Copyright © 2002 John Wiley & Sons, Ltd.     

Risky visuomotor choices during rapid reaching in childhood

Many everyday actions are implicit gambles because imprecisions in our visuomotor systems place probabilities on our success or failure. Choosing optimal action strategies involves weighting the costs and gains of potential outcomes by their corresponding probabilities, and requires stable representations of one's own imprecisions. How this ability is acquired during development in childhood when visuomotor skills change drastically is unknown. In a rewarded rapid reaching task, 6‐ to 11‐year‐old children followed ‘risk‐seeking’ strategies leading to overly high point‐loss. Adults' performance, in contrast, was close to optimal. Children's errors were not explained by distorted estimates of value or probability, but may reflect different action selection criteria or immature integration of value and probability information while planning movements. These findings provide a starting point for understanding children's risk‐taking in everyday visuomotor situations when suboptimal choices can be dangerous. Moreover, children's risky visuomotor decisions mirror those reported for non‐motor gambles, raising the possibility that common processes underlie development across decision‐making domains.     

On the shape of the probability weighting function

Empirical studies have shown that decision makers do not usually treat probabilities linearly. Instead, people tend to overweight small probabilities and underweight large probabilities. One way to model such distortions in decision making under risk is through a probability weighting function. We present a nonparametric estimation procedure for assessing the probability weighting function and value function at the level of the individual subject. The evidence in the domain of gains supports a two-parameter weighting function, where each parameter is given a psychological interpretation: one parameter measures how the decision maker discriminates probabilities, and the other parameter measures how attractive the decision maker views gambling. These findings are consistent with a growing body of empirical and theoretical work attempting to establish a psychological rationale for the probability weighting function.     

The Influence of Positive Affect on the Decision Rule in Risk Situations: Focus on Outcome (and Especially Avoidance of Loss) Rather Than Probability

Two experiments examined the influence of positive affect on probability estimation and choice. Participants in whom positive affect had been induced, as well as no-manipulation controls, were asked to make both numerical evaluations of verbal probabilities in three-outcome gambles and actual betting decisions about similar gambles. Results from Experiment 1 showed the phenomenon labeledcautious optimism:Positive affect participants significantly overestimated the probabilities associated with phrases for winning relative to their estimates of probability of losing for the same phrases (optimism), while participants in a control condition did not; yet, in actual gambling situations, affect condition participants were much less likely to gamble than were controls when a real loss was possible (caution). Results of the betting task from Experiment 2 further indicated that affect participants used a betting-decision rule that was different from that of controls: They bet less than controls in gambles where potential losses were large, even though probability of loss was small, and they bet more than controls in gambles where the amount of the potential loss was small, even though the probability of loss was moderate or large. These findings suggest that positive affect can promote an overt shift from a decision rule focusing primarily on probabilities to one focusing on utilities or outcome values, especially for losses. Taken together, the results are compatible with an interpretation of the influence of positive affect in terms of an elaboration of positive cognitive material, and purposive behavior in decisions, rather than in terms of mere response bias.     

小概率权重问题及其市场应用

探讨了决策制定中有关“小概率权重”的理论问题和市场应用问题。若干实验结果表明概率的权重大小不是绝对的,而是背景依赖的,在决策问题中价值对比降低或不存在时,小概率事件权重不太可能被高估。然而,对小概率的权重也会因决策人所处的目前状况有关,个体在财富状况等方面的差异会对小概率事件的赋权产生影响。实验结果还证明有关小概率权重问题的理论发现对彩票价格制定、保险费用确定以及选择有效的有奖促销策略具有重要的启示     

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.     

The Negative Effect of Probability Assessments on Decision Quality

It is commonly assumed that uncertain information can be reduced to numerical probabilities without biasing preferences. It is also implicitly assumed in much research and many applications that people can express these probabilities. In contradiction to these assumptions Experiment 1 shows that the production of probability assessments biases decisions in an n-person game. Experiment 2 shows that the explicit assessment of numerical probabilities renders choices between gambles concerning future basketball events less optimal. These findings seem to be a result of overweighting the probability dimension relative to the payoff dimension given numerical judgments. Experiment 2 also suggests that without explicit numerical probability judgments subjects are less likely to violate the dominance principle. The theoretical and practical implications of the results are discussed.     

经验决策：概念、研究和展望

传统风险决策研究范式中, 决策信息是事先限定的, 即在决策之前呈现各个决策选项的概率和收益, 被试基于这些信息进行决策。已有研究表明, 在传统风险决策任务中人们会高估小概率事件(rare event)。然而最近出现的一种基于不完整信息的决策形式, 即经验决策却对这一发现提出了挑战。研究发现, 人们在进行经验决策时会表现出对小概率事件的低估, 经验决策和传统决策形式之间存在差异。本文主要介绍经验决策的概念及其研究, 并从学习和不确定程度等角度对两种决策形式进行再认识, 以进一步理解经验决策并提出展望。     

Cognitive processes underlying lottery and sports gambling decisions: The role of stated probabilities and background knowledge

Two studies compared choice and underlying cognitive processes in equivalent decision tasks involving risk and uncertainty (lotteries versus sports gambles including displayed expert probability judgements). In sports gambles, background knowledge was triggered via information on team location, home or away. Otherwise, displayed risk information (stake, winnings, odds, and outcome probabilities) was controlled across gamble type. In a choice study, home win bets were chosen significantly more frequently than draws or away wins, compared to lottery equivalents. In a parallel study eliciting concurrent verbal protocols, participants made fewer evaluations of odds and probabilities, and more statements involving background knowledge in sports gambles. Furthermore, some sports gamble protocols indicated modifications of stated probabilities and decision strategies contingent on domain knowledge. It was concluded that stated probability revision and knowledge-based reasoning are key cognitive processes in sports gambling not normally applied in the lottery paradigms often employed in decision research.     

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.     

Too good to be true? Psychological responses to uncommon options in risk–reward environments

Risks and rewards, or payoffs and probabilities, are inversely related in many choice environments. We investigated people's psychological responses to uncommon combinations of risk and reward that deviate from learned regularities (e.g., options that offer a high payoff with an unusually high probability) as they evaluated risky options. In two experiments (N = 183), participants first priced monetary gambles drawn from environments in which risks and rewards were negatively correlated, positively correlated, or uncorrelated. In later trials, they evaluated gambles with uncommon combinations of risk and reward—that is, options that deviated from the respective environment's risk–reward structure. Pricing, response times, and (in Experiment 2) pupil dilation were recorded. In both experiments, participants took more time when responding to uncommon compared to foreseeable options or when the same options were presented in an uncorrelated risk–reward environment. This result was most pronounced when the uncommon gambles offered higher expected values compared to the other gambles in the set. Moreover, these uncommon, high‐value options were associated with an increase in pupil size. These results suggest that people's evaluations of risky options are based not only on the options' payoffs and probabilities but also on the extent to which they fit the risk–reward structure of the environment.     

The impact of hope, self-esteem, and attributional style on adolescents’ school grades and emotional well-being: A longitudinal study

We examined the distinctiveness of three “positive thinking” variables (self-esteem, trait hope, and positive attributional style) in predicting future high school grades, teacher-rated adjustment, and students’ reports of their affective states. Seven hundred eighty-four high school students (382 males and 394 females; 8 did not indicate their gender) completed Time 1 measures of verbal and numerical ability, positive thinking, and indices of emotional well-being (positive affect, sadness, fear, and hostility), and Time 2 measures of hope, self-esteem, and emotional well-being. Multi-level random coefficient modelling revealed that each positive thinking variable was distinctive in some contexts but not others. Hope was a predictor of positive affect and the best predictor of grades, negative attributional style was the best predictor of increases in hostility and fear, and low self-esteem was the best predictor of increases in sadness. We also found that sadness at Time 1 predicted decreases in self-esteem at Time 2. The results are discussed with reference to the importance of positive thinking for building resilience.     

Choice preferences without inferences: subconscious priming of risk attitudes

We present a procedure for subconscious priming of risk attitudes. In Experiment 1, we were reliably able to induce risk‐seeking or risk‐averse preferences across a range of decision scenarios using this priming procedure. In Experiment 2, we showed that these priming effects can be reversed by drawing participants' attention to the priming event. Our results support claims that the formation of risk preferences can be based on preconscious processing, as for example postulated by the affective primacy hypothesis, rather than rely on deliberative mental operations, as posited by several current models of judgment and decision making. Copyright © 2002 John Wiley & Sons, Ltd.     

Dominance violations in judged prices of two- and three-outcome gambles

The dominance principle states that one should prefer the option with consequences that are at least as good as those of other options for any state of the world. When applied to judged prices of gambles, the dominance principle requires that increasing one or more outcomes of a gamble should increase the judged price of the gamble, with everything else held constant. Previous research has uncovered systematic violations of the dominance principle: people assign higher prices to a gamble with a large probability of winning an amount, Y, otherwise zero, than they do to a superior gamble with the same chance of winning Y, otherwise winning a small amount, X! These violations can be explained by a configural-weight theory in which two-outcome gambles are represented with two sets of decision weights; one set for outcomes having values of zero and another set for lower-valued outcomes that have nonzero values. The present paper investigates whether dominance violations are limited to two-outcome gambles. Results show that people violate the dominance principle with three-outcome gambles even with financial incentives. Furthermore, results could be predicted from the configural-weight theory. The data do not support the view that configural weighting is caused by a shift in strategy that would apply only to two-outcome gambles.     

Differential priming effect for subliminal fear and disgust facial expressions

Compared to neutral or happy stimuli, subliminal fear stimuli are known to be well processed through the automatic pathway. We tried to examine whether fear stimuli could be processed more strongly than other negative emotional stimuli using a modified subliminal affective priming paradigm. Twenty-six healthy subjects participated in two separated sessions. Fear, disgust and neutral facial expressions were adopted as primes, and 50% happy facial stimuli were adopted as a target to let only stronger negative primes reveal a priming effect. Participants were asked to appraise the affect of target faces in the affect appraisal session and to appraise the genuineness of target faces in the genuineness appraisal session. The genuineness instruction was developed to help participants be sensitive to potential threats. In the affect appraisal, participants judged 50% happy target faces significantly more 'unpleasant' when they were primed by fear faces than primed by 50% happy control faces. In the genuineness appraisal, participants judged targets significantly more 'not genuine' when they were primed by fear and disgust faces than primed by controls. These findings suggest that there may be differential priming effects between subliminal fear and disgust expressions, which could be modulated by a sensitive context of potential threat.