Intransitive Indifference Research Articles

Revealed preferences under uncertainty: Incomplete preferences and preferences for randomization

We present a set of experiments testing for incomplete preferences due to uncertainty. In a first experiment, we observe that approximately half of the participants exhibit a choice pattern inconsistent with models assuming complete preferences and Certainty Independence (CI). To understand these participants' behavior, in a second experiment, we design a decision task that distinguishes between models assuming complete preferences and relaxing CI and models of incomplete preferences under uncertainty. We find that about half of the participants in question exhibit behavior consistent with incomplete preferences, about one third shows behavior consistent with a preference for randomization between risky and ambiguous prospects, and the remaining participants' behavior is consistent with both types of preferences. In further experiments we find that the observed choice pattern cannot be attributed to probability weighting, choice mistakes, regret aversion or intransitive indifference under risk and certainty. We also show that the observed behavior is robust to a prize variation in the ambiguous prospect.

Expected Scott–Suppes utility representation

We provide an axiomatic characterization for an expected Scott–Suppes utility representation. Such a characterization is the natural analog of the von Neumann–Morgenstern expected utility theorem for semiorders and it is noted as an open problem by Fishburn (1968). Expected Scott–Suppes utility representation is analytically tractable and can be used in applications that study preferences with intransitive indifference under uncertainty. Our representation offers a decision-theoretical interpretation for epsilon equilibrium as well.

Revealed Incomplete Preferences Under Uncertainty: Evidence for Bewley Preferences

The completeness axiom of choice has been questioned for long and theoretical models of decision making allowing for incomplete preferences have been developed. So far the theoretical accomplishments have not been paired with empirical evidence on the actual existence of incomplete preferences under uncertainty. We provide empirical evidence in support of the existence of incomplete preferences due to multiple priors over an ambiguous event, i.e. Bewley preferences. We design experimental decision tasks where specific choice patterns are consistent with Bewley preferences but inconsistent with models assuming completeness. We find that approximately half of the subjects behave consistent with variational Bewley preferences and that the observed behavioral pattern cannot be attributed to probability weighting, choice mistakes, or intransitive indifference. In a robustness test we show that the observed behavior is robust to a prize variation in the ambiguous prospect and consistent with comparative statics predictions based on variational Bewley preferences.

Formula omitted]-rationalizable choices

Rationalizability has been a main topic in individual choice theory since the seminal paper of Samuelson (1938). The rationalization of a multi-valued choice is classically obtained by maximizing the binary relation of revealed preference, which is fully informative of the primitive choice as long as suitable axioms of choice consistency hold. In line with this tradition, we give a purely axiomatic treatment of the topic of choice rationalization. In fact, we introduce a new class of properties of choice coherence, called axioms of replacement consistency, which examine how the addition of an item to a menu may cause a substitution in the selected set. These axioms are used to uniformly characterize rationalizable choices such that their revealed preferences are quasi-transitive, Ferrers, semitransitive, and transitive. Further, regardless of rationalizability, we study the relationship of these new axioms with some classical properties of choice consistency, such as standard contraction, standard expansion, and WARP. To complete our analysis of the transitive structure of rationalizable choices, we examine the case of revealed preferences satisfying weak (m,n)-Ferrers properties in the sense of Giarlotta and Watson (2014). Originally introduced with the purpose of extending the notions of interval orders and semiorders, these Ferrers properties give a descriptive taxonomy of binary relations displaying a transitive strict preference but an intransitive indifference. Here we suggest a possible economic interpretation of weak (m,n)-Ferrers properties, showing that, in a suitable model of transactions, they provide a way of controlling phenomena of money-pump due to the presence of mixed cycles of preference/indifference. Finally, we define (m,n)-rationalizable choices as those having a weakly (m,n)-Ferrers revealed preference, and characterize these choices by means of additional axioms of replacement consistency.

Axiomatizing bounded rationality: the priority heuristic

This paper presents an axiomatic framework for the priority heuristic, a model of bounded rationality in Selten’s (in: Gigerenzer and Selten (eds.) Bounded rationality: the adaptive toolbox, 2001) spirit of using empirical evidence on heuristics. The priority heuristic predicts actual human choices between risky gambles well. It implies violations of expected utility theory such as common consequence effects, common ratio effects, the fourfold pattern of risk taking and the reflection effect. We present an axiomatization of a parameterized version of the heuristic which generalizes the heuristic in order to account for individual differences and inconsistencies. The axiomatization uses semiorders (Luce, Econometrica 24:178–191, 1956), which have an intransitive indifference part and a transitive strict preference component. The axiomatization suggests new testable predictions of the priority heuristic and makes it easier for theorists to study the relation between heuristics and other axiomatic theories such as cumulative prospect theory.

Comparing financial investments by their state dependent returns: A one-way log utility representation

Abstract In a standard single-period model under risk, we formalize and discuss an intuitive criterion for the binary comparison of financial investments. Two investments – x and y – are compared by calculating the present value of x ’s payoffs using the state dependent returns of y as discount factors. The induced preference is asymmetric but exhibits intransitive indifference. If the feasible set is convex, then the criterion selects a unique maximum element. Interestingly, it can be shown that the induced preference can be represented by a one-way expected utility representation employing logarithmic utility. Besides giving a relevant and illustrative example for a one-way utility representation, this result provides a new interpretation of using logarithmic utility for expected utility based decision-making.

Sophisticated preference aggregation

A Sophisticated Social Welfare Function (SSWF) is a mapping from profiles of individual preferences into a sophisticated preference which is a pairwise weighted comparison of alternatives. We characterize Pareto optimal and pairwise independent SSWFs in terms of oligarchies that are induced by some power distribution in the society. This is a fairly large class ranging from dictatoriality to anonymous aggregation rules. Our results generalize the impossibility theorem of Arrow (Social choice and individual values. Wiley, New York, 1951) and the oligarchy theorem of Gibbard (Intransitive social indifference and the Arrow dilemma, University of Chicago, unpublished manuscript, 1969).

Intransitivity and Vague Preferences

This paper is concerned with intransitivity in normative rational choice. It focuses on a class of intransitivities which have received little attention, those involving vague preferences. “Vague preferences” are defined in terms of vague predicates such as “red” or “bald.” Such preferences appear common, and intransitive indifference is argued to be an unavoidable feature of them. Such preferences are argued to undermine intransitive strict preference also. Various formal theories of vagueness are applied to an example of vague preferences, but none of them provide a justification for the transitivity axiom.

A representation for intransitive indifference relations

Binary relations representable by utility functions with multiplicative error are considered. It is proved that if the error is a power of utility then the underlying binary relation is either an interval order, or a semiorder. Moreover, semiorders can be characterized among interval orders by the magnitude of the power of utility that is used in the form of error function.

Intransitive indifference and rationalizability of choice functions on general domains

This paper examines various intransitive preferences in terms of a behavior of a chooser in the general-domain framework, where a choice function may not be defined over all possible subsets of the fixed universe of alternatives. In characterizing the quasi-transitive rationalization of a choice function, the results of this paper show the trade off between the assumption on the domain of a choice function and the requirement that the choice set is single-valued or multi-valued.

Additive Utility with Intransitive Indifference and without Independence: A Homogeneous Case

In the homogeneous case of one type of goods or objects, we prove the existence of an additive utility function without assuming transitivity of indifference and independence. The representation reveals a positive factor smaller than 1 that infuences rational choice beyond the utility function and explains departures from these standard axioms of utility theory (factor equals to 1).

Fuzzy interval and semi-orders

The seminal paper by Luce [Econometrica 24 (1956) 178] on intransitive indifference provoked a rich body of research on how to model such a type of indifference. Interval orders and semi-orders play a central role in the modelization of intransitive indifference. This paper is an attempt to fuzzify interval orders and semi-orders. We use an additive fuzzy preference structure to argue for a fuzzification of interval and semi-orders. We show that fuzzy interval orders are to a degree semi-orders, however, against expectations, we cannot show that fuzzy interval graphs are to a degree fuzzy indifference graphs.

Order isomorphisms for preferences with intransitive indifference

An order isomorphism from one preference relation to another is a function between their underlying sets that preserves both preference and indifference. A utility function is an order isomorphism, as is a well-known two-function representation for preference relations with intransitive indifference. Necessary and sufficient conditions are given for the existence of an order isomorphism from a given preference relation to the set of sequences of zeros and ones ordered by Pareto dominance. This means of preference representation is shown via examples to be quite general.

Aggregation of semiorders: intransitive indifference makes a difference

A semiorder can be thought of as a binary relationP for which there is a utilityu representing it in the following sense:xPy iffu(x) −u(y) > 1. We argue that weak orders (for which indifference is transitive) can not be considered a successful approximation of semiorders; for instance, a utility function representing a semiorder in the manner mentioned above is almost unique, i.e. cardinal and not only ordinal. In this paper we deal with semiorders on a product space and their relation to given semiorders on the original spaces. Following the intuition of Rubinstein we find surprising results: with the appropriate framework, it turns out that a Savage-type expected utility requires significantly weaker axioms than it does in the context of weak orders.

An algorithmic proof that semiorders are representable

For preference patterns with intransitive indifference, Scott and Suppes showed that an asymmetric digraph is representable as a consistent preference pattern if and only if it is a semiorder. Rabinovitch gave a constructive proof that every finite semiorder is representable. We give a simpler algorithmic proof of this result.

Revealed preference and intransitive indifference

This paper approaches revealed preference from an assumption of a type of bounded rationality on the part of the consumer. We assume that the individual may have discrimination problems with regard to alternative choices (specifically, we assume that the underlying preferences of the consumer are described by a semiorder). The main result of this paper is the description of a revealed preference condition which is satisfied by a set of partial choice data if and only if the partial choice function can be rationalized by semiorder preferences.

Revealed preference and the axiomatic foundations of intransitive indifference: The case of asymmetric subrelations

This paper provides an axiomatic structure for various binary preference relations that are reflexive, connected, and transitive in their asymmetric subrelations known as quasi-transitive orderings. A choice function generates various (distinct) binary preference relations, such as the base relation, the revealed preference relation, and the wide revealed preference relation. For each interpretation of a binary preference relation, we present an axiom that ensures the existence of a quasi-transitive preference ordering. A description of a choice procedure is considered which one often observes in real life and it is shown that the path independence of such a procedure is a sufficient condition for a quasi-transitive ordering of various binary preference relations.

Intransitive indifference and incomparability

This paper presents the concept of an incomplete semiorder and the result that such a structure can be extended to a semiorder. Corresponding results exist for interval orders. An economic application is provided.

Intransitive Indifference and Revealed Preference

Intransitive Indifference and Revealed Preference

Intransitive indifference: The semi-order problem

Intransitive indifference: The semi-order problem