Numerical Representation of Incomplete and Nontransitive Preferences and Indifferences on a Countable Set

This note considers preference structures over countable sets which allow incomparable outcomes and nontransitive preferences and indifferences. Necessary and sufficient conditions are provided under which such a preference structure can be represented by means of utility function and a threshold function.Incomplete preferences; nontransitive preferences; threshold functions; utility theory

Weak continuity of preferences with nontransitive indifference

We characterize weak continuity of an interval order on a topological space by using the concept of a scale in a topological space.Weakly continuous interval order; continuous numerical representation

Do transitive preferences always result in indifferent divisions?

The transitivity of preferences is one of the basic assumptions used in the theory of games and decisions. It is often equated with rationality of choice and is considered useful in building rankings. Intransitive preferences are considered paradoxical and undesirable. This problem is discussed by many social and natural sciences. The paper discusses a simple model of sequential game in which two players in each iteration of the game choose one of the two elements. They make their decisions in different contexts defined by the rules of the game. It appears that the optimal strategy of one of the players can only be intransitive! (the so-called \textsl{relevant intransitive strategies}.) On the other hand, the optimal strategy for the second player can be either transitive or intransitive. A quantum model of the game using pure one-qubit strategies is considered. In this model, an increase in importance of intransitive strategies is observed -- there is a certain course of the game where intransitive strategies are the only optimal strategies for both players. The study of decision-making models using quantum information theory tools may shed some new light on the understanding of mechanisms that drive the formation of types of preferences.Comment: 16 pages, 5 figure

A BAYESIAN MALLOWS APPROACH TO NONTRANSITIVE PAIR COMPARISON DATA : HOW HUMAN ARE SOUNDS?

We are interested in learning how listeners perceive sounds as having human origins. An experiment was performed with a series of electronically synthesized sounds, and listeners were asked to compare them in pairs. We propose a Bayesian probabilistic method to learn individual preferences from nontransitive pairwise comparison data, as happens when one (or more) individual preferences in the data contradicts what is implied by the others. We build a Bayesian Mallows model in order to handle nontransitive data, with a latent layer of uncertainty which captures the generation of preference misreporting. We then develop a mixture extension of the Mallows model, able to learn individual preferences in a heterogeneous population. The results of our analysis of the musicology experiment are of interest to electroacoustic composers and sound designers, and to the audio industry in general, whose aim is to understand how computer generated sounds can be produced in order to sound more human.Peer reviewe

Transitive regret

Preferences may arise from regret, i.e., from comparisons with alternatives forgone by the decision maker. We ask whether regret-based behavior is consistent with non-expected utility theories of transitive choice and show that the answer is no. If choices are governed by ex ante regret and rejoicing then non-expected utility preferences must be intransitive.Regret, transitivity, non-expected utility

Consumer theory with bounded rational preferences

The neoclassical consumer maximizes utility and makes choices by completely preordering the feasible alternatives and weighing when indifferent. The consumer studied in this paper chooses by weighing when indifferent and also when indecisive, without necessarily preordering the alternatives or exhausting her budget. Preferences therefore need not be complete, transitive or non-satiated but are assumed strictly convex and "adaptive". The latter axiom is new and parallels that of ambiguity aversion in choice under uncertainty.preferences: incomplete, intransitive, convex, adaptive; representation; demand.

The transitive core: inference of welfare from nontransitive preference relations

Abstract This paper studies welfare criteria under an environment in which a decision maker is endowed with a nontransitive preference relation. In such an environment, the classical utilitarian welfare criterion may not identify the welfare order, and the problem of maximizing the decision maker's welfare becomes ambiguous. In order to find a criterion that applies to nontransitive preference relations, I propose a set of desirable properties of welfare criteria and uniquely identify a consistent rule that infers welfare orders from nontransitive preference relations. This rule, called the transitive core, is applied to a variety of nontransitive preference models, such as semiorders on a commodity space, relative discounting time preferences, regret preferences on risky prospects, and collective preference relations induced by the majority criterion. These examinations show that the proposed method provides successful inference of welfare in respective contexts. JEL Classification: D11, D60

General Equilibrium without Utility Functions: How far to go?

How far can we go in weakening the assumptions of the general equilibrium model? Existence of equilibrium, structural stability and finiteness of equilibria of regular economies, genericity of regular economies and an index formula for the equilibria of regular economies have been known not to require transitivity and completeness of consumers’ preferences. We show in this paper that if consumers’ non-ordered preferences satisfy a mild version of convexity already considered in the literature, then the following properties are also satisfied: 1) the smooth manifold structure and the diffeomorphism of the equilibrium manifold with a Euclidean space; 2) the diffeomorphism of the set of no-trade equilibria with a Euclidean space; 3) the openness and genericity of the set of regular equilibria as a subset of the equilibrium manifold; 4) for small trade vectors, the uniqueness, regularity and stability of equilibrium for two version of tatonnement; 5) the pathconnectedness of the sets of stable equilibria.general equilibrium; equilibrium manifold; natural projection; demand functions