An experimental investigation of transitivity in set ranking

A decision under ‘complete uncertainty’ is one where the decision maker knows the set of possible outcomes for each decision, but cannot assign probabilities to those outcomes. This way, the problem of ranking decisions is reduced to a problem of ranking sets of outcomes. All rankings that have emerged in the literature in this domain imply transitivity. In the current study, transitivity is subjected to an empirical evaluation in two experiments, where subjects are asked to choose between sets of monetary consequences. After analysis with a Bayes Factor, very few violations of transitivity were found. It can be concluded that transitivity seems a plausible condition for the ranking of sets of monetary consequences.     

The effect of unconditional preferences on Sen’s paradox

Theory and Decision - Sen’s Liberal paradox describes a conflict between weak Pareto, minimal liberalism, and either transitivity or a best element over a domain of individual preferences....     

On the Cycle-Transitivity of the Dice Model

We introduce the notion of a dice model as a framework for describing a class of probabilistic relations. We investigate the transitivity of the probabilistic relation generated by a dice model and prove that it is a special type of cycle-transitivity that is situated between moderate stochastic transitivity or product-transitivity on the one side, and ukasiewicz-transitivity on the other side. Finally, it is shown that any probabilistic relation with rational elements on a three-dimensional space of alternatives which possesses this particular type of cycle-transitivity, can be represented by a dice model. The same does not hold in higher dimensions.     

Social choice, the strong Pareto principle, and conditional decisiveness

This paper examines social choice theory with the strong Pareto principle. The notion of conditional decisiveness is introduced to clarify the underlying power structure behind strongly Paretian aggregation rules satisfying binary independence. We discuss the various degrees of social rationality: transitivity, semi-transitivity, the interval-order property, quasi-transitivity, and acyclicity.     

How vicious are cycles of intransitive choice?

Transitivity is a compelling requirement of rational choice, and a transitivity axiom is included in all classical theories of both individual and group choice. Nonetheless, choice contexts exist in which choice might well be systematically intransitive. Moreover, this can occur even when the context is transparent, and the decision maker is reflective. The present paper catalogues such choice contexts, dividing them roughly into the following classes:

An experimental investigation of violations of transitivity in choice under uncertainty

Several models of decision-making imply systematic violations of transitivity of preference. Our experiments explored whether people show patterns of intransitivity predicted by regret theory and majority rule. To distinguish “true” violations from those produced by “error,” a model was fit in which each choice can have a different error rate and each person can have a different pattern of true preferences that need not be transitive. Error rate for a choice is estimated from preference reversals between repeated presentations of that same choice. Our results showed that very few people repeated intransitive patterns. We can retain the hypothesis that transitivity best describes the data of the vast majority of participants.

What can the regression model of human judgment learn from multi-attribute decision theory?

Zeleny's recent conjecture that multi-attribute decision theory may help to overcome the inadequacies of the linear regression model is incorrect. Recognition of the information processing advantages inherent in multiple -attribute decision situations combined with a requirement of transitivity itself implies linear objective functions. This follows from some recent developments by a psychologist and an economist in the analysis of individual and collective decision processes, developments which do not take as their starting point the paradigm of choice offered in utility theory.     

Rationality with preference discovery costs

Economic theory assumes that preferences are rational. However, experiments have found small violations of transitivity. This paper develops a model of rationality with preference discovery costs. Introspection is costly. Thus, agents may find it optimal to use less than full effort, even though this raises the risk of making a poor choice. This model could potentially explain the intransitivities observed in the data while retaining rationality and optimization.     

Extreme Copulas and the Comparison of Ordered Lists

We introduce two extreme methods to pairwisely compare ordered lists of the same length, viz. the comonotonic and the countermonotonic comparison method, and show that these methods are, respectively, related to the copula T _M (the minimum operator) and the Ł ukasiewicz copula T _L used to join marginal cumulative distribution functions into bivariate cumulative distribution functions. Given a collection of ordered lists of the same length, we generate by means of T _M and T _L two probabilistic relations Q ^M and Q ^L and identify their type of transitivity. Finally, it is shown that any probabilistic relation with rational elements on a 3-dimensional space of alternatives which possesses one of these types of transitivity, can be generated by three ordered lists and at least one of the two extreme comparison methods.     

The Utility of Gambling Reconsidered

The utility of gambling, which entails an intrinsic utility or disutility of risk, has been alluded to in the economics literature for over a century. This paper demonstrates that any utility of gambling almost unavoidably implies a violation of fundamental rationality properties, such as transitivity or stochastic dominance, for static choices between gambles. This result may explain why the utility of gambling, a phenomenon so widely discussed, has never been formalized in the economics literature. The model of this paper accommodates well-known deviations from expected utility, such as the Allais paradox and the coexistence of gambling and insurance, while minimally deviating from expected utility.     

Rationality from a computational standpoint

A computationally viable choice function possesses, in addition to other important properties, adaptability - as new alternatives become feasible the set of best alternatives corresponding to the new feasible set can be located without having to reconsider previously rejected alternatives. Adaptability is shown to be closely related to, in fact characteristic of, the standard notions of rationality (expressed in terms of transitivity properties). Adaptability provides a generalization of conventional rationality and in addition leads to the choice function proposed by Schwartz, albeit from an entirely different route, that of computational viability.     

Generalized similarity judgments: An alternative explanation for choice anomalies

Rubinstein's (1988) procedure for choosing between risky prospects, based, in part, upon similarities between prizes and probabilities across lotteries, is modified and extended to apply to a more general class of binary choices. This modified procedure is shown to imply behaviors following from Loomes and Sugden's (1982) Regret Theory, although under more general conditions, and provides an alternative explanation for much of the data which led to the specification of Prospect Theory's value and decision weighing functions. The procedure also explains observed violations of stochastic dominance, transitivity, and invariance not accounted for in available alternatives to expected utility.     

The expected likelihood of transitivity: A survey

May developed an algebraic choice model to describe pairwise comparisons from an empirical study. A probabilistic choice variation of May's model has also been developed. This study presents a survey of work that considers the expected likelihood that a subject using the probabilistic model will have transitive responses for pairwise choices on a set of three alternatives. Of particular interest is the impact that various factors that influence the probabilistic choice model have on the expected likelihood of transitivity. These factors include the degree of accuracy with which the subject perceives the attributes of the alternatives, the number of attributes of comparison, and the consistency with which alternatives are ranked across attributes.This research was supported through a fellowship from the Center for Advanced Study of the University of Delaware.     

Measuring credibility of compensatory preference statements when trade-offs are interval determined

This paper studies how an overall fuzzy preference relation can be constructed in the compensatory context of the simple additive difference model, when imprecision on the trade-offs has to be taken into account. Three credibility indices of preferences are analysed and illustrated by a numerical example. Arguments are presented supporting the use of the third index, for which an interesting transitivity property (which was an open problem) is proved.     

Testing transitivity in choice under risk

Recently proposed models of risky choice imply systematic violations of transitivity of preference. This study explored whether people show the predicted intransitivity of the two models proposed to account for the certainty effect in Allais paradoxes. In order to distinguish “true” violations from those produced by “error,” a model was fit in which each choice can have a different error rate and each person can have a different pattern of preferences that need not be transitive. Error rate for a choice is estimated from preference reversals between repeated presentations of the same choice. Results showed that few people repeated intransitive patterns. We can retain the hypothesis that all participants were transitive.     

Characterization of the existence of semicontinuous weak utilities for binary relations

We characterize the existence of semicontinuous weak utilities in a general framework, where the axioms of transitivity and acyclicity are relaxed to that of consistency in the sense of Suzumura (Economica 43:381?C390, 1976). This kind of representations allow us to transfer the problem of the existence of the ${{\mathcal{G}}{\mathcal{O}}{\mathcal{C}}{\mathcal{H}}{\mathcal{A}}}$ set of a binary relation to the easier problem of getting maxima of a real function. Finally, we show that the maxima of these representations correspond to the different levels of satiation that each of individual has (an individual reaches his or her level of satiation when an increase of consuming an alternative product/service brings no increase in utility).     

Nontransitive preferences in decision theory

Intransitive preferences have been a topic of curiosity, study, and debate over the past 40 years. Many economists and decision theorists insist on transitivity as the cornerstone of rational choice, and even in behavioral decision theory intransitivities are often attributed to faulty experiments, random or sloppy choices, poor judgment, or unexamined biases. But others see intransitive preferences as potential truths of reasoned comparisons and propose representations of preferences that accommodate intransitivities. This article offers a partial survey of models for intransitive preferences in a variety of decisional contexts. These include economic consumer theory, multiattribute utility theory, game theory, preference between time streams, and decision making under risk and uncertainty. The survey is preceded by a discussion of issues that bear on the relevance and reasonableness of intransitivity.     

Risk Premiums and Benefit Measures for Generalized-Expected-Utility Theories

Standard tools for the analysis of economic problems involving uncertainty, including risk premiums, certainty equivalents and the notions of absolute and relative risk aversion, are developed without making specific assumptions on functional form beyond the basic requirements of monotonicity, transitivity, continuity, and the presumption that individuals prefer certainty to risk. Individuals are not required to display probabilistic sophistication. The approach relies on the distance and benefit functions to characterize preferences relative to a given state-contingent vector of outcomes. The distance and benefit functions are used to derive absolute and relative risk premiums and to characterize preferences exhibiting constant absolute risk aversion (CARA) and constant relative risk aversion (CRRA). A generalization of the notion of Schur-concavity is presented. If preferences are generalized Schur concave, the absolute and relative risk premiums are generalized Schur convex, and the certainty equivalents are generalized Schur concave.