Common Ratio Effect Research Articles

Does the Allais paradox survive with non-monetary consequences?

The form of the Allais paradox known as the common ratio effect (CRE) is a violation of deterministic expected utility theory that has been widely replicated with monetary outcomes. Its robustness has stimulated the development of numerous alternative models of risky choice. However, much less is known about the prevalence of the CRE in decisions involving non-monetary outcomes. We conduct a controlled laboratory comparison of the CRE for money versus consumer goods. The CRE is very strong with money, but largely disappears for goods, primarily as a result of differences in risk attitudes between goods and money. We caution against assuming that findings from experiments involving monetary lotteries will reliably generalise to other types of consequences.

Distinguishing Common Ratio Preferences from Common Ratio Effects Using Paired Valuation Tasks

Without strong assumptions about how noise manifests in choices, we can infer little from existing empirical observations of the common ratio effect (CRE) about whether there exists an underlying common ratio preference (CRP). We propose to solve this inferential challenge using paired valuations, which yield valid inference under common assumptions. Using this approach in an online experiment with 900 participants, we find no evidence of a systematic CRP. To reconcile our findings with existing evidence, we present the same participants with paired choice tasks and demonstrate how noise can generate a CRE even for individuals without an associated CRP. (JEL C91, D81, D91)

How common is the common-ratio effect?

The common-ratio effect and the Allais Paradox (common-consequence effect) are the two best‐known violations of Expected Utility Theory. We reexamine data from 39 articles reporting experiments (143 designs/parameterizations, 14,909 observations) and find that the common-ratio effect is systematically affected by experimental design and implementation choices. The common-ratio effect is more likely to be observed in experiments with a low common-ratio factor, a high ratio of middle to highest outcome, when lotteries are presented as simple probability distributions (not in a compound/frequency form), and with high hypothetical incentives.

The normalized expected utility – entropy and variance model for decisions under risk

The normalized expected utility - entropy and variance (NEU-EV) decision model and associated risk measure are proposed for the analysis and modelling of decisions under risk. The model depends on normalized entropy and variance and expected utility, can be applied to decision problems depending on actions, where the states have different numbers of outcomes. Several properties of risk perception and examples of certainty effects, common ratio effects and common consequence effects are analysed using this model, considering actions with non-negative outcomes. The results show that the NEU-EV model is an adequate model for explaining these risk decision problems.

Expected Utility, Independence, and Continuity

In this paper, we examine some characterizing implications of the expected utility model with a particular focus on independence and continuity. Many well documented choice anomalies under risk (e.g., the common consequence and ratio effects) show that even weaker forms of independence can be violated by experimental subjects. We demonstrate that these violations have a close relation when continuity holds, but their connection becomes less pronounced when continuity is dropped. We show that while retaining the independence axiom, replacing continuity with two intuitive conditions, substitution and monotonicity, can characterize the expected utility model. In this case, weakening the independence axiom can still yield an expected utility, but in a weaker sense implying that our axiomatization is tight.

Salience and social choice

The axioms of expected utility and discounted utility theory have been tested extensively. In contrast, the axioms of social welfare functions have only been tested in a few questionnaire studies involving choices between hypothetical income distributions. In a controlled experiment with 100 subjects placed in the role of social planners, we test five fundamental properties of social welfare functions to determine the efficacy of traditional social choice models in predicting social planner allocations when presented with choice sets designed to test the axioms of the theory. We find that three properties of the standard social welfare functions tested are systematically violated, producing an Allais paradox, a common ratio effect, and a framing effect in social choice. We find support for scale invariance and a preference for tail-increasing transfers. Our experiment also enables us to test a model of salience-based social choice which predicts the systematic deviations and highlights the close relationship between these anomalies and the classical paradoxes for risk and time.

The utilitarian–maximin social welfare function and anomalies in social choice

AbstractTwo major criteria of distributive justice are the utilitarian criterion and the maximin criterion. We offer a simple axiomatic characterization of a mixed utilitarian–maximin social welfare function. This social welfare function explains recent empirical violations of the standard cardinal social choice theory such as the social Allais paradox and the social common ratio effect. In addition, it offers a new foundation for the positively skewed wealth distributions in society. It also provides an objective function for mechanism design applications that trades off maximizing surplus and minimizing inequality.

The Relation between Behavior under Risk and over Time

The paper establishes a tight relation between nonstandard behaviors in the domains of risk and time, by considering a decision-maker with non-expected utility preferences who believes that only present consumption is certain while any future consumption is uncertain. We provide the first complete characterizations of the two-way relations between the certainty effect and present bias, and between the common ratio effect and temporal reversals. (JEL D11, D15, D81, D91)

How Common Is the Common-Ratio Effect?

The common-ratio effect and the Allais paradox are the two best‐known violations of expected utility theory. We reexamine data from 38 experimental articles (127 designs/ parameterizations, 12717 revealed choice patterns) and find that the common-ratio effect is systematically affected by experimental design and implementation choices. The common-ratio effect is more likely to be observed in experiments with a low common-ratio factor, a high ratio of the middle to the highest outcome, when lotteries are presented as simple probability distributions (not in a compound/frequency form), and with real incentives. This latter result is not significant with cluster-robust standard errors.

Indistinguishability of small probabilities, subproportionality, and the common ratio effect

We show that, in the framework of Cumulative Prospect Theory, subproportionality as a property of the probability weighting function alone does not automatically imply the common ratio effect. This issue is particularly relevant for equal-mean lotteries because both risk-averse and risk-seeking behavior have to be predicted there. As a solution, we propose three simple properties of the probability weighting function which are sufficient to accommodate the empirical evidence on the common ratio effect for equal-mean lotteries for any S-shaped value function. These are (1) subproportionality, (2) indistinguishability of small probabilities, and (3) an intersection point with the diagonal lower than 0.5. While subproportionality and a fixed point lower than 0.5 are common assumptions in the literature, the property indistinguishability of small probabilities is introduced for the first time. The ratio of decision weights for infinitesimally small probabilities characterizes indistinguishability and is also an informative measure for the curvature of the probability weighting function at zero. The intuition behind indistinguishability is that, even though the ratio of probabilities stays constant, individuals tend to neglect this relative difference when probabilities get smaller.

Correlated responses on growth traits after two-stage selection for ovulation rate and litter size in rabbits

Rabbit commercial maternal lines are usually selected for litter size (LS) and paternal lines for growth rate (GR). Line OR_LS was selected by ovulation rate (OR) and LS to improve LS more efficiently. In this study, growth traits of line OR_LS were evaluated by estimating the correlated response on weaning weight (WW), slaughter weight (SW) and GR during fattening period as well as their variability (DWW, DSW and DGR, respectively). Data were analyzed using Bayesian inference methods. Heritability estimates were low for growth traits (0.09, 0.13 and 0.14 for WW, SW and GR, respectively) and negligible for growth traits variability (0.01, 0.004 and 0.01 for DWW, DSW and DGR, respectively). Moderate common litter effect ratio (c2; 0.35, 0.28 and 0.27) and low maternal effect ratio (m2; 0.11, 0.05 and 0.01) were obtained for WW, SW and GR, respectively. Both c2 and m2 were lower at slaughter than at weaning. In addition, low common litter effect and negligible maternal effect were observed for growth traits variability. Genetic correlations between LS and both growth traits and their variability were close to zero. Positive genetic correlations were observed between OR and growth traits (0.19, 0.38 and 0.36 for WW, SW and GR, respectively) as well as between OR and growth traits variability (0.35, 0.62 and 0.20 for DWW, DSW and DGR, respectively). Positive correlated responses in both periods were obtained for growth traits, WW, SW and GR (0.037, 0.156 and 0.110 kg, respectively). The correlated response found in growth traits might be due to the positive genetic correlations between OR and these traits. However, selection for OR and LS using independent culling levels did not modify the growth traits variability. Therefore, no negative consequences on growth traits can be expected in current commercial maternal lines.

When payoffs look like probabilities: Separating form and content in risky choice.

Paralleling research in perception, behavioral models of risky choice posit "psychophysical" transformations of material outcomes and probabilities. Prospect theory assumes a value function that is concave for gains and convex for losses, and an inverse S-shaped probability weighting function. But in typical experiments, form and content are confounded: Probabilities are represented on a bounded numerical scale, whereas representations of monetary gains (losses) are unbounded above (below). To unconfound form and content, we conducted experiments using a probability-like representation of outcomes and an outcome-like representation of probability. We show that interchanging numerical representations can interchange the resulting psychophysical functions: A proportional (rather than absolute) representation of outcomes leads to an inverse S-shaped value function for gains. This alternative value function generates novel framing effects, a common ratio effect for bounded gains, and a "framing interaction," where gain-loss framing matters less for proportional outcomes. In addition, we show that an absolute (rather than proportional) representation of probability reduces sensitivity to large probabilities. These findings highlight the deeply constructive nature of the psychophysics of risky choice, and suggest that traditional models may reflect subjective reactions to numerical form rather than substantive content. (PsycINFO Database Record

A second-generation disappointment aversion theory of decision making under risk

This paper presents a new decision theory for modelling choice under risk. The new theory is a two-parameter generalization of expected utility theory. The proposed theory assumes that a decision maker: (1) behaves as if maximizing expected utility; but (2) may experience disappointment (elation) when the utility of a lottery’s outcome falls short of (exceeds) the expected utility of the lottery; and (3) may have a preference for gambling (attraction/aversion to positively/negatively skewed lotteries). The proposed theory can rationalize the fourfold pattern of risk attitudes; the common ratio effect and the reverse thereof (in certain types of decision problems); the Allais paradox in classical common consequence problems and the reverse Allais paradox—in common consequence problems with an even split of a probability mass; violations of the betweenness axiom; switching behavior in the Samuelson’s example; violations of ordinal, upper and lower cumulative independence (which falsify rank-dependent utility and cumulative prospect theory); and preference reversals between valuations and choice. In application to insurance, the theory can rationalize full insurance with an actuarially unfair premium and aversion to probabilistic insurance. In application to optimal portfolio investment, the theory can rationalize the equity premium puzzle.

À la recherche de la signification perdue du paradoxe d’Allais

Dans « Le paradoxe d’Allais. Comment lui rendre sa signification perdue ? », Philippe Mongin [2014] propose une image contestable, parce qu’abusivement rassurante, de la rationalité des sujets participant aux expériences de MacCrimmon et Larsson [1979]. Nous montrons cela en restituant les données des auteurs pour l’effet de conséquence commune et l’effet de rapport commun (deux paradoxes d’Allais), que Mongin assimile, bien qu’il distingue ces deux effets. Deux tendances dans les données nous conduisent à rejoindre Mongin pour conclure que ces expériences sont une source d’inspiration fructueuse pour la théorie de la décision, ce que nous illustrons brièvement par le rôle qu’elles ont joué dans l’apparition de l’utilité espérée dépendante du rang et dans le tournant interindividuel en psychologie cognitive de la décision.

The Common Ratio Effect in Choice, Pricing, and Happiness Tasks

AbstractThe 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.

The persistence of common-ratio effects in multiple-play decisions

AbstractPeople often make more rational choices between monetary prospects when their choices will be played out many times rather than just once. For example, previous research has shown that the certainty effect and the possibility effect (two common-ratio effects that violate expected utility theory) are eliminated in multiple-play decisions. This finding is challenged by seven new studies (N= 2391) and two small meta-analyses. Results indicate that, on average, certainty and possibility effects are reduced but not eliminated in multiple-play decisions. Moreover, in our within-participants studies, the certainty and possibility choice patterns almost always remained the modal or majority patterns. Our primary results were not reliably affected by prompts that encouraged a long-run perspective, by participants’ insight into long-run payoffs, or by participants’ numeracy. The persistence of common-ratio effects suggests that the oft-cited benefits of multiple plays for the rationality of decision makers’ choices may be smaller than previously realized.

Explaining the Paradox of Allais: A Hyper-Bayesian Account

In this paper we show that decision makers who maximize expected utility on a two-dimensional set of the real numbers (i.e., who maximize expected utility on R×R) rather than on the real numbers alone, can act in ways predicted by the Allais paradox. In particular, we show that the ‘common consequence effect’ and the ‘common ratio effect’ as described by Allais in his original 1953 paper can be explained by the extended expected utility model that is described in this paper. The main part of the argument is presented in cognitive-functional terms, whilst a behavioural-axiomatic foundation is provided in an appendix.

The beta stochastic utility (β-SU)

ABSTRACTIn this article, a new model for decision making under uncertainty is presented. Here, we model human attitude toward risks to show that an individual estimate of the expected utility of a lottery follows a generalized Beta distribution with a random error that follows a similar distribution. An individual is said to maximize his stochastic utility when requested to present his preference between risky lotteries. Hypothetically, risky lotteries are those exhibiting wider ranges of rewards where human estimate will not be below the utility of the lowest reward nor above the highest of the lottery. The Beta distribution is bounded and complies to such intuitive preconditions with a variance depending on such bounds. The proposed model will overestimate/underestimate the expected utility of a lottery according to the lottery probability mass and individuals' risk attitudes. By such estimation, our model conforms to the fourfold choice pattern. The model also explains the violations present as inconsistencies in the expected utility theory, such as Allais paradox, common consequence effect, common ratio effect, and the violation of betweenness that can be found in the fourfold choice pattern. For the validation purposes, 13 datasets from literature were collected and tested. The β-SU model fits the data at least as good as other approaches such as the CPT/StEUT and presents higher prediction log-likelihoods and less sum of squared errors in most of the cases, a matter that supports the proposition that human estimates of the expected utility may be drawn out of a generalized Beta distribution.

Is it All Connected? A Testing Ground for Unified Theories of Behavioral Economics Phenomena

We estimate 11 well-studied behavioral phenomena in a group of 190 laboratory subjects (short-term discount rates, small stakes risk aversion, present bias, loss aversion, the endowment effect, aversion to ambiguity and compound lotteries, the common ratio and common consequence effects and sender/receiver behavior in trust games). We study the joint distribution of these behaviors and compare it to the predictions of existing models as a step in the development of a parsimonious, general model of economic choice. We find strong correlations between loss aversion and the endowment effect, and between probability weighting (as measured by the common ratio and common consequence effects) and risk aversion, in line with Cumulative Prospect Theory (CPT). We also find risk aversion to be related to ambiguity aversion, compound lottery aversion and discounting, consistent with the curvature of the utility function being an important determinant of all three behaviors. However, we do not find evidence that probability weighting in the risk domain is related to ambiguity and compound lottery aversion or present bias (as implied by recent extensions to CPT). Behavior in the trust game is unrelated to attitudes to risk or uncertainty. We find little relation between intelligence or personality measures and economic behavior, although we do find overconfidence to be negatively related to many behaviors (particularly ambiguity aversion) and women to be much more loss averse than men.

Modelling decision making under risk and uncertainty by novel utility measures

In this paper, we will address the classical decision theories (the expected value, EV, and expected utility theories, EUTs) along with their violations, such as the common consequence, common ratio effect, violation of betweenness and the fourfold risk pattern. In particular, a numerical method is proposed to determine the utility function of an individual or group of individuals. The approach depends on the individual's evaluation of the certainty equivalent (CE) of a decision problem under uncertainty. Later, we propose an optimisation model to predict human preference between pairs of reward scenarios in which uncertainty is involved. The optimisation model implements binary logistic regression (BLR). Both SPSS and Excel Solver were used in the optimisation and parameter fitting. The presented model is verified via collected survey and literature studies. It is found that the model is able to explain the violations and serve as a new replica to predict human preferences.