Evidence against Rank-Dependent Utility Theories: Tests of Cumulative Independence,Interval Independence,Stochastic Dominance,and Transitivity

This study tests between two modern theories of decision making. Rank- and sign-dependent utility (RSDU) models, including cumulative prospect theory (CPT), imply stochastic dominance and two cumulative independence conditions. Configural weight models, with parameters estimated in previous research, predict systematic violations of these properties for certain choices. Experimental data systematically violate all three properties, contrary to RSDU but consistent with configural weight models. This study also tests whether violations of stochastic dominance can be explained by violations of transitivity. Violations of transitivity may be evidence of a dominance detecting mechanism. Although some transitivity violations were observed, most choice triads violated stochastic dominance without violating transitivity. Judged differences between gambles were not consistent with the CPT model. Data were not consistent with the editing principles of cancellation and combination. The main findings are interpreted in terms of coalescing, the principle that equal outcomes can be combined in a gamble by adding their probabilities. RSDU models imply coalescing but configural weight models violate it, allowing configural weighting to explain violations of stochastic dominance and cumulative independence.     

Discounting by Probabilistic Waiting

In everyday life, many probabilistic situations may be characterized as probabilistic waiting. A gambler, for example, bets repeatedly at the racetrack, the casino, or the card table. The gambler may not win on the first try, but if a gamble is repeated enough times, a win is almost certain to occur eventually. If repeated gambles are structured as strings of losses ending in a win (probabilistic waiting) and the amount won is discounted by the delay caused by the series of losses, then strings with many losses will be discounted more than those with fewer losses, thereby causing subjective value of the series of gambles as a whole to increase. The current study used the opposite effect that amount has on the degree of delay and probability discounting as a marker to determine whether people evaluate situations involving probabilistic waiting as they evaluate situations involving delayed outcomes or as situations involving probabilistic outcomes. We find that the more likely a probabilistic waiting situation is to end in reward (e.g., a gamble is repeated indefinitely until reward is obtained), the more that situation conforms to delay discounting; the less likely a probabilistic waiting situation is to end in reward (e.g., a fixed, small number of gambles), the more that situation conforms to probability discounting. We argue that the former situation is applicable to pathological gambling, and that people with steep delay discount functions would therefore be more likely to have gambling problems. Copyright © 2015 John Wiley & Sons, Ltd.     

The disjunction effect reexamined: Relevant methodological issues and the fallacy of unspecified percentage comparisons

Leonard J. Savage’s sure-thing principle (1954) is a key assumption of the consequentialist conception of decision making under uncertainty, which more-or-less assumes that decision makers are rational and thorough. The sure-thing principle states that if some option x is preferred given some other event A occurs, and if option x is preferred given this event A does not occur, then x should be preferred even when the outcome of A is unknown. Tversky and Shafir [Tversky, A., &; Shafir, E. (1992). The disjunction effect in choice under uncertainty. Psychological Science, 3(5) 305–309] claim that this basic principle is frequently violated in two-step gambles. They call such violations disjunction effects. Kuhberber, Komunska, and Perner [Kuhberber, A., Komunska, D., &; Perner, J. (2001). The disjunction effect: does it exist for two step gambles? Organizational Behavior and Human Decision Processes, 85(2) 250–264] attempted to replicate Tversky and Shafir’s findings and claim their results show that people do not violate the sure-thing principle in repeated gambles. This article evaluates Kuhberger, Komunska, and Perner’s claims, suggesting they did not appropriately analyze their results, and further provides evidence that people do regularly violate the sure-thing principle in two-step gambles, providing further evidence for the reality of disjunction effects.     

Risky decision: Attention and path dependency as a function of response mode

It was previously shown and was here replicated that the amount of money required to induce a subject to exchange one gamble for a second and then the second for a third depends on the second (intermediate) gamble. This path dependency cannot be explained by any of the algebraic models (including SEU), nor can it be explained as a simple attention effect. A model was put forward which explains the effect in terms of differential masking depending on a kind of stimulus—response compatibility. The effect of response mode on path dependency was examined in this study; similarity judgements showed no consistent path dependency nor did difference in rated attractiveness of gambles presented in pairs, but rated attractiveness was different for gambles presented singly. Since it might be that monetary response emphasises the monetary aspects, that is, the winning amount of the gamble, it was thought that a probability response would emphasise the probability of winning. However, when subjects were asked to set the probability of winning $ 12 which would induce them to exchange one gamble for another, not only was no path dependency observed, but the subjects' responses seemed to depend only on the difference in expected value of the gambles. This suggests the possibility of developing response-display modes which would eliminate or at least attenuate the inconsistencies observed in risky decision processes.     

The same but different: an empirical investigation of the reducibility principle

According to classical utility theory the valuation of any lottery should depend only on its outcomes and their respective probabilities, and should not be affected by the nature, complexity and structure of the chance mechanism. Previous research has documented systematic violations of this principle. For example, most subjects prefer lotteries in which the payoffs are contingent on the joint occurrence of multiple (high probability) events to simple lotteries, and lotteries in which the earlier stages offer higher probabilities than the later stages. We review the various violations of this principle and suggest a classification into two major types associated with misunderstanding of chance mechanisms and attitude towards the chance mechanism and process. In the present study 40 subjects were presented with 30 pairs of binary gambles. In any given pair the lotteries had identical outcomes and equal ‘reduced’ probabilities (and thus equal expected values). However, the chance mechanisms varied along a variety of factors such as the size of the sample space, the number of stages, temporal ordering, order of probabilities, their transparency and time constraints. Half the subjects saw lotteries involving gains and the other half considered only losses. After choosing one lottery in each pair, the subjects were asked to explain and justify their choices. The findings revealed systematic violations of the reducibility principle: subjects displayed a preference for lotteries with larger sample spaces, and for lotteries that allow quicker resolutions in the earlier stages. A clear distinction between some patterns of preferences in the gains and loss domains was revealed. In gambles involving gains subjects preferred to have the highest probability on the first stage (and the lowest probability on the last stage), but displayed the opposite preferences for losses. A content analysis of the subjects' stated reasons for their choices identified eight major categories. The most frequently invoked were hope, fun, simplicity, stress and time. Copyright © 2001 John Wiley & Sons, Ltd.     

Additivity and expected utility in risky multiattribute preferences

This experiment analyzed human preferences among even chance gambles for commodity bundles. The purpose of the experiment was to test several independence assumptions that distinguish between models for risky multiattribute preferences. In particular, the additivity and the expected utility part of the additive expected utility model were tested symmetrically. The degree and form of model violations were established, and the effects of instructions and of response modes were examined. All independence assumptions were violated by a bias to prefer a gamble or a commodity bundle that was previously matched against a standard. Systematic and strong violations that superseded this bias questioned the validity of the additive expected utility model. In violation of the additivity part of that model subjects consistently preferred the gamble with more balanced outcomes when comparing gambles with identical marginal probability distributions. This trend, called multiattribute risk aversion, was independent of subjects' single attribute risk attitude. Instructions and response modes had no noticeable impacts on these response patterns.     

Risky decision: Attention effects or masking effects?

The Marshak bid procedure shows that more money is required to induce a S to exchange gamble a for gamble e and then e for b if e differs from a only in the winning amount and differs from b only in the probability of winning, rather than if e differs from a only in the probability of winning. This is contrary to most theories of risky decision making which imply that the amount of money necessary to effect a 2-step exchange between a and b should be independent of the intermediary gamble. One might attempt to explain the effect by saying that the S attends to the dimension which is different between gambles. But the explanation is untenable if one assumes that states of attention are defined as weightings of the dimensions. An alternative explanation is put forward which basically assumes that winning amounts mask differences in probability of winning more than vice versa. The formalization of the theory is given in terms of Fechnerian integration over imperfect differentials.     

Children's judgements of gambles: a disordinal violation of utility

Violations of utility are often attributed to people's differential reactions to risk versus certainty or uncertainty, or more generally to the way that people perceive outcomes and consequences. However, a core feature of utility is additivity, and violations may also occur because of averaging effects. Averaging is pervasive in intuitive riskless judgement throughout many domains, as shown with Anderson's Information Integration approach. The present study extends these findings to judgement under risk. Five‐ to 10‐year old children showed a disordinal violation of utility because they averaged the part worths of duplex gambles rather than add them, as adults do, and as normatively prescribed. Thus adults realized that two prizes are better than one, but children preferred a high chance to win one prize to the same gamble plus an additional small chance to win a second prize. This result suggests that an additive operator may not be a natural component of the intuitive psychological concept of expected value that emerges in childhood. The implications of a developmental perspective for the study of judgement and decision are discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

澳门人的风险知觉与赌博行为

采用匿名问卷法,调查澳门人的风险知觉与赌博行为。结果显示：（1）赌场的劝世文既无劝勉也无劝阻人们赌博的作用;（2）社会关系网的钱财支援不影响实际赌博次数;（3）多次性博弈与一次性博弈所采用的是不同的机制,一次性博弈不是由期望值所决定的;（4）相互监视和制裁系统会影响赌博：受法律制裁约束的职业者比不受法律制裁约束的职业者更不好赌;自估在赌场遇见认识人的机会与赌博次数呈负相关;（5）性别及母语等“本质趋向”差异表明,所谓澳门人不好赌有其渊源并可能是在长期的文化沉淀中形成的;（6）受教育程度是影响赌博的有效预测变量。这些发现的理论含义一并进行了讨论。     

The Disjunction Effect: Does It Exist for Two-Step Gambles?

One of the basic axioms of the rational theory of decision under uncertainty is Savage's (1954) Sure Thing Principle. It states that if Prospect x is preferred to Prospect y knowing that Event A occurred, and if x is preferred to y knowing that A did not occur, then x should also be preferred to y when it is not known whether A occurred. Tversky and Shafir (1992) claim to have demonstrated a violation of this principle in two-step gambles, which is termed a disjunction effect. The present article evaluates the replicability of the disjunction effect for two-step gambles. The findings show that people do not violate the sure thing principle in repeated gambles. The validity of alleged violations in other paradigms is discussed.     

The affect heuristic and the attractiveness of simple gambles

Prior studies have observed that the attractiveness of playing a simple gamble (7/36 to win $9; otherwise win nothing) is greatly enhanced by introducing a small loss (7/36 win $9; otherwise lose 5¢). The present studies tested and confirmed an explanation of this finding based on the concept of evaluability and the affect heuristic. Evaluators of the “no‐loss” gamble lack a precise feeling for how good $9 is, hence give it little weight in their judgment. In the second gamble, comparison with the small loss makes $9 “come alive with feeling” and become weighted in the judgment, thus increasing the attractiveness of the gamble. These results demonstrate the importance of contextual factors in determining affect and preference for simple risk‐taking opportunities. They show that the meaning, utility, and weighting of even a very familiar monetary outcome such as $9 is not fixed, but depends greatly on these contextual factors. Copyright © 2007 John Wiley & Sons, Ltd.     

Information overload or search-amplified risk? Set size and order effects on decisions from experience

How do changes in choice-set size influence information search and subsequent decisions? Moreover, does information overload influence information processing with larger choice sets? We investigated these questions by letting people freely explore sets of gambles before choosing one of them, with the choice sets either increasing or decreasing in number for each participant (from two to 32 gambles). Set size influenced information search, with participants taking more samples overall, but sampling a smaller proportion of gambles and taking fewer samples per gamble, when set sizes were larger. The order of choice sets also influenced search, with participants sampling from more gambles and taking more samples overall if they started with smaller as opposed to larger choice sets. Inconsistent with information overload, information processing appeared consistent across set sizes and choice order conditions, reliably favoring gambles with higher sample means. Despite the lack of evidence for information overload, changes in information search did lead to systematic changes in choice: People who started with smaller choice sets were more likely to choose gambles with the highest expected values, but only for small set sizes. For large set sizes, the increase in total samples increased the likelihood of encountering rare events at the same time that the reduction in samples per gamble amplified the effect of these rare events when they occurred—what we call search-amplified risk. This led to riskier choices for individuals whose choices most closely followed the sample mean.     

Changing plans: Dynamic inconsistency and the effect of experience on the reference point

This article presents a new test of a principle of decision making calleddynamic consistency. This principle was tested in an experiment in which participants were asked to make decisions about a second gamble within a sequence of two gambles. Participants were first asked to make a planned choice about the second gamble. The planned choice was madebefore the first gamble was played and was conditioned on the anticipated outcomes of the first gamble.After the first gamble was played, the same participants were asked to make a final choice about the second gamble, conditioned on the experienced outcome of the first gamble. The results showed that participants’ final choices were frequently inconsistent with their plans, even when the anticipated and experienced outcomes were identical. These inconsistencies occurred in a systematic direction. Experiencing an anticipated gain resulted with a change toward risk aversion, and experiencing an anticipated loss resulted in a change toward risk seeking. These results are explained in terms of the effect of actual experience on the reference point used for the evaluation of the decision problem.     

When Time Is of the Essence: Averaging, Aspiration, and the Short Run

A variety of experiments have demonstrated that people's choices among gambles differ according to whether the gamble is to be played just once (the unique case) or multiple times (the repeated case). The reason for this difference appears to be that people are acting in each case so as to increase the likelihood that the chosen alternative will leave them better off than the nonchosen alternative. This paper explores four themes that bear on the difference between unique and repeated gambles. The first traces the historical and theoretical role of weighted averaging in risky choice. The second defends the usefulness of probability-based rules for choices involving aspiration levels. The third presents the idea that choices under risk may reflect dual criteria, one based in weighted averaging and the other based in stochastic control. The fourth challenges the conventional criteria by which choices are judged to be rational or irrational.     

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.     

Are preference reversals reduced when made explicit?

Previous research has shown that preferences for options, such as gambles, can reverse depending on the response mode. These preference reversals have been demonstrated when tasks were performed sequentially. That is, subjects completed one task before beginning another. In an attempt to eliminate preference reversals, we asked subjects to perform tasks simultaneously. That is, subjects made two types of responses for each pair of gambles before evaluating the next pair. In the condition with no financial incentives, preference reversal rates were slightly reduced. In another condition, subjects were paid for their participation and they were allowed to play a gamble with real monetary compensation. A gamble pair was randomly selected, and if a subject's responses in the two tasks were consistent for that pair, he or she was allowed to play the ‘preferred’ gamble. Otherwise, the experimenter selected the gamble from the pair. With these financial incentives, systematic preference reversals were eliminated for two of the three task combinations. Preference reversals continued to occur for attractiveness ratings versus selling prices, although, even for that pair of tasks, the reversal rate was significantly reduced. For all three task pairs, preference orders from the two tasks appeared to merge into more consistent orders.     

When good evidence goes bad: the weak evidence effect in judgment and decision-making

An indispensable principle of rational thought is that positive evidence should increase belief. In this paper, we demonstrate that people routinely violate this principle when predicting an outcome from a weak cause. In Experiment 1 participants given weak positive evidence judged outcomes of public policy initiatives to be less likely than participants given no evidence, even though the evidence was separately judged to be supportive. Experiment 2 ruled out a pragmatic explanation of the result, that the weak evidence implies the absence of stronger evidence. In Experiment 3, weak positive evidence made people less likely to gamble on the outcome of the 2010 United States mid-term Congressional election. Experiments 4 and 5 replicated these findings with everyday causal scenarios. We argue that this “weak evidence effect” arises because people focus disproportionately on the mentioned weak cause and fail to think about alternative causes.