What is wrong with Allais' Certainty Effect?

Several choice situations are constructed to explore whether the violation of expected utility theory in an Allais paradox choice situation can be attributed to what Tversky and Kahneman (1986) describe as the Allais certainty effect. Problems are developed where the Allais certainty effect would be expected to occur but results show it does not. Other problems demonstrate that the Allais ‘paradox’ is observed in the absence of the Allais certainty effect. The study concludes that, although expected utility theory is known to be wrong through the Allais paradox, the Allais certainty effect does not appear to be able to rescue it.     

艾勒悖论(Allais Paradox)另释

艾勒悖论违背了期望效用(Expected Utility)理论的独立性(independence)原则,成为欲推翻期望效用理论的杠杆。“齐当别”抉择模型不将风险决策行为看成是追求某种“最大期望值”的抉择反应,而将其看成是“最好可能结果之间的取舍”或者“最坏可能结果之间的取舍”。此项研究设计了一“判断”技术,并用此对艾勒设计的选择问题加以检验。实验表明,判断结果所示的“齐当别”策略能满意地对艾勒选择题作出解释。     

艾勒悖论再探——三结果决策的齐当别式表征:“好-坏”抑或“最好-最坏”?

艾勒悖论的第一项决策任务属于三结果决策任务,这对齐当别模型以及鉴别其不同判断任务间的优势都具有特殊意义。本研究通过3项子研究系统的探讨了不同表征方式("最好-最坏","好-坏")、不同判断任务(经典判断任务与联合判断任务)、不同数量级的结果值(万元级、百万元级以及亿元级)以及不同决策情境(迫选,非迫选)上,齐当别模型对艾勒悖论的解释力。结果显示:无论是迫选情境还是非迫选情境,经典的艾勒悖论均可被齐当别模型解释,但应基于"好-坏"而非"最好-最坏"表征维度;齐当别模型的联合判断任务在三结果决策任务上的预测力也再次被证明优于经典判断任务。     

Money, Kisses, And Electric Shocks: On the Affective Psychology of Risk

Prospect theory's S-shaped weighting function is often said to reflect the psychophysics of chance. We propose an affective rather than psychophysical deconstruction of the weighting function resting on two assumptions. First, preferences depend on the affective reactions associated with potential outcomes of a risky choice. Second, even with monetary values controlled, some outcomes are relatively affect-rich and others relatively affect-poor. Although the psychophysical and affective approaches are complementary, the affective approach has one novel implication: Weighting functions will be more S-shaped for lotteries involving affect-rich than affect-poor outcomes. That is, people will be more sensitive to departures from impossibility and certainty but less sensitive to intermediate probability variations for affect-rich outcomes. We corroborated this prediction by observing probability-outcome interactions: An affect-poor prize was preferred over an affect-rich prize under certainty, but the direction of preference reversed under low probability. We suggest that the assumption of probability-outcome independence, adopted by both expected-utility and prospect theory, may hold across outcomes of different monetary values, but not different affective values.     

Causes of Allais common consequence paradoxes: An experimental dissection

The common consequence paradox of Allais can be decomposed into three simpler principles: transitivity, coalescing, and restricted branch independence. Different theories attribute such paradoxes to violations of restricted branch independence only, to coalescing only, or to both. This study separates tests of these two properties in order to compare these theories. Although rank-dependent utility (RDU) theories, including cumulative prospect theory (CPT), violate branch independence, the empirical pattern of violations is opposite that required by RDU theories to account for Allais paradoxes. Data also show systematic violations of coalescing, which refute RDU theories. The findings contradict both original and CPTs with or without their editing principles of combination and cancellation. Modal choices were well predicted by Birnbaum's RAM and TAX models with parameters estimated from previous data. The effects of event framing on these tests were also assessed and found to be negligible.     

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 Common Ratio Effect in Choice,Pricing, and Happiness Tasks

The Allais common ratio effect is one of the most robust violations of rational decision making under risk. In this paper, we conduct a novel test of the common ratio effect in which we elicit preferences for the common ratio choice alternatives in choice, pricing, and happiness rating tasks. We find large shifts in preference patterns across tasks, both within and between subjects. In particular, we find that both the consistency and distribution of responses differ systematically across tasks, with modal choices replicating the Allais preference pattern, modal happiness ratings exhibiting consistent risk aversion, and modal prices maximizing expected value. We discuss the predictions of various cognitive explanations of the common ratio effect in the context of our experiment. We find that a dual process framework provides the most complete account of our results. Surprisingly, we also find that although the Allais pattern was the modal behavior in the choice task, none of the 158 respondents in our experiment exhibited the Allais pattern simultaneously in choice, happiness, and pricing tasks. Our results constitute a new paradox for the leading theories of choice under risk. Copyright © 2017 John Wiley & Sons, Ltd.     

Subjective utility,anticipated utility,and the Allais paradox

One of the most notable counterexamples to expected utility theory is the “Allais paradox” (M. Allais, 1953, Econometrica, 31, 503–546). A number of alternative theories have been proposed in an attempt to resolve this paradox, notably including Karmarkar, 1978, Karmarkar, 1979, 24, 67–72). It is shown that SWU theory necessarily involves violations of dominance, but that the theory can be modified to avoid these violations. The result is a special case of J. Quiggin's anticipated utility theory (1982, Journal of Economic Behaviour and Organisation, 3, 323–343).     

The combined effects of risk and time on choice: Does uncertainty eliminate the immediacy effect? Does delay eliminate the certainty effect?

Researchers have noted parallels between decisions under uncertainty and over time. Three experiments evaluated the theory that uncertainty and time affect choice via a common underlying dimension, such that delaying an outcome is equivalent to making it uncertain. To test this account we asked whether adding uncertainty to outcomes would eliminate the immediacy effect bias. We also asked whether adding time delay to outcomes would eliminate the certainty effect bias. The answer to both questions was yes, provided the prospects were presented singly rather than jointly. In single evaluation uncertainly eliminated the immediacy effect and delay eliminated one form of the certainty effect, while in joint evaluation, these effects did not occur. These findings suggest that at least in some contexts decision makers may equate risk and delay. Other explanations are possible, however, demonstrating that the interaction between risk and delay is complex and not easily understood.     

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.     

Sub-Optimal Choice by Pigeons: Failure to Support The Allais Paradox

Pigeons show a preference for an alternative that provides them with discriminative stimuli (sometimes a stimulus that predicts reinforcement and at other times a stimulus that predicts the absence of reinforcement) over an alternative that provides them with nondiscriminative stimuli, even if the nondiscriminative stimulus alternative is associated with 2.5 times as much reinforcement (Stagner & Zentall, 2010). In Experiment 1 we found that the delay to reinforcement associated with the nondiscriminative stimuli could be reduced by almost one half before the pigeons were indifferent between the two alternatives. In Experiment 2 we tested the hypothesis that the preference for the discriminative stimulus alternative resulted from the fact that, like humans, the pigeons were attracted by the stimulus that consistently predicted reinforcement (the Allais paradox). When the probability of reinforcement associated with the discriminative stimulus that predicted reinforcement was reduced from 100% to 80% the pigeons still showed a strong preference for the discriminative stimulus alternative. Thus, under these conditions, the Allais paradox cannot account for the sub-optimal choice behavior shown by pigeons. Instead we propose that sub-optimal choice results from positive contrast between the low expectation of reinforcement associated with the discriminative stimulus alternative and the much higher obtained reinforcement when the stimulus associated with reinforcement appears. We propose that similar processes can account for sub-optimal gambling behavior by humans.     

The priority heuristic: making choices without trade-offs

Bernoulli's framework of expected utility serves as a model for various psychological processes, including motivation, moral sense, attitudes, and decision making. To account for evidence at variance with expected utility, the authors generalize the framework of fast and frugal heuristics from inferences to preferences. The priority heuristic predicts (a) the Allais paradox, (b) risk aversion for gains if probabilities are high, (c) risk seeking for gains if probabilities are low (e.g., lottery tickets), (d) risk aversion for losses if probabilities are low (e.g., buying insurance), (e) risk seeking for losses if probabilities are high, (f) the certainty effect, (g) the possibility effect, and (h) intransitivities. The authors test how accurately the heuristic predicts people's choices, compared with previously proposed heuristics and 3 modifications of expected utility theory: security-potential/aspiration theory, transfer-of-attention-exchange model, and cumulative prospect theory.     

No escape from Allais: reply to Buchak

In Risk and Rationality, Lara Buchak advertised REU theory as able to recover the modal preferences in the Allais paradox. But we pointed out that REU theory only applies in the “grand world” setting, where it actually struggles with the modal Allais preferences. Buchak offers two replies. Here we enumerate technical and philosophical problems they face.

     

Resistance can be good or bad: how theories of resistance and dissonance affect attitude certainty

This research investigated how lay theories about resisting persuasion can affect attitude certainty. Specifically, people who believed that resistance was negative (i.e., implies close-mindedness) showed different levels of attitude certainty after resisting persuasive messages than people who believed resistance was positive (i.e., implies intelligence). When people held positive lay theories of resistance and overcame ostensibly strong arguments, they showed increased attitude certainty (compared to those who overcame ostensibly weak arguments). However, individuals who believed that resistance was negative did not show increases in attitude certainty when overcoming strong arguments. Experiment 2 suggests that the effect of lay theories and perceived argument strength on attitude certainty was due to dissonance created by believing that resistance is undesirable but nonetheless resisting persuasion.     

There is a 60% probability,but I am 70% certain: communicative consequences of external and internal expressions of uncertainty

Current theories of probability recognise a distinction between external (un)certainty (frequentistic probabilities) and internal (un)certainty (degrees of belief). The present studies investigated this distinction in lay people's judgements of probability statements formulated to suggest either an internal (“I am X% certain”) or an external (“It is X% certain” or “There is an X% probability”) interpretation. These subtle differences in wording influenced participants' perceptions and endorsements of such statements, and their impressions of the speaker. External expressions were seen to signal more reliable task duration estimates, and a lower degree of external than internal certainty was deemed necessary to advise a course of action. In conversations about football, internal expressions were perceived as signalling more personal interest, and were expected to be on the average 10% higher than corresponding external probabilities. Finally, people who reported their outcome expectations for two major sports events let their degree of interest in these events influence their internal but not their external certainty. These results have implications for the communication of uncertainty and probability.     

Probabilistic reasoning in a classical logic

We offer a view on how probability is related to logic. Specifically, we argue against the widely held belief that standard classical logics have no direct way of modelling the certainty of assumptions in theories and no direct way of stating the certainty of theorems proved from these (uncertain) assumptions. The argument rests on the observation that probability densities, being functions, can be represented and reasoned with naturally and directly in (classical) higher-order logic.     

The Binary Additivity of Subjective Probability Does not Indicate the Binary Complementarity of Perceived Certainty

People's numeric probability estimates for 2 mutually exclusive and exhaustive events commonly sum to 1.0, which seems to indicate the full complementarity of subjective certainty in the 2 events (i.e., increases in certainty for one event are accompanied by decreases in certainty for the other). In this article, however, a distinction is made between the additivity of probability estimates and the complementarity of internal perceptions of certainty. In Experiment 1, responses on a verbal measure of certainty provide evidence of binary noncomplementarity in the perceived likelihoods of possible scenario outcomes, and a comparison of verbal and numeric certainty estimates suggests that numeric probabilities overestimated the complementarity of people's certainty. Experiment 2 used a choice task to detect binary noncomplementarity. Soliciting numeric probability estimates prior to the choice task changed the participants' choices in a direction consistent with complementarity. Possible mechanisms yielding (non)complementarity are discussed.     

Decision making under time pressure, modeled in a prospect theory framework

The current research examines the effects of time pressure on decision behavior based on a prospect theory framework. In Experiments 1 and 2, participants estimated certainty equivalents for binary gains-only bets in the presence or absence of time pressure. In Experiment 3, participants assessed comparable bets that were framed as losses. Data were modeled to establish psychological mechanisms underlying decision behavior. In Experiments 1 and 2, time pressure led to increased risk attractiveness, but no significant differences emerged in either probability discriminability or outcome utility. In Experiment 3, time pressure reduced probability discriminability, which was coupled with severe risk-seeking behavior for both conditions in the domain of losses. No significant effects of control over outcomes were observed. Results provide qualified support for theories that suggest increased risk-seeking for gains under time pressure.     

Criminally Involved Parents Who Misuse Substances and Children’s Odds of Being Arrested as a Young Adult: Do Drug Treatment Courts Mitigate the Risk?

This paper examined (1) the association between parents who are convicted of a substance-related offense and their children’s probability of being arrested as a young adult and (2) whether or not parental participation in an adult drug treatment court program mitigated this risk. The analysis relied on state administrative data from North Carolina courts (2005–2013) and from birth records (1988–2003). The dependent variable was the probability that a child was arrested as a young adult (16–21). Logistic regression was used to compare groups and models accounted for the clustering of multiple children with the same mother. Findings revealed that children whose parents were convicted on either a substance-related charge on a non-substance-related charge had twice the odds of being arrested as young adult, relative to children whose parents had not been observed having a conviction. While a quarter of children whose parents participated in a drug treatment court program were arrested as young adults, parental completion this program did not reduce this risk. In conclusion, children whose parents were convicted had an increased risk of being arrested as young adults, irrespective of whether or not the conviction was on a substance-related charge. However, drug treatment courts did not reduce this risk. Reducing intergenerational links in the probability of arrest remains a societal challenge.