Continuous representation of Von Neumann–Morgenstern preferences

This paper considers a continuous representation of preference relations satisfying Grandmont's (1972) Expected Utility Hypothesis. We equip the preferences with the topology of closed convergence, then we show the existence of a jointly continuous expected utility function and consider its uniqueness. Furthermore, we construct an embedding map of the preferences into the set of expected utility functions.     

Construction of a continuous utility function for a class of preferences

We construct a continuous utility indicator for a subclass of continuous preference relations, including some with thick indifference classes, using a measure theoretic technique related to that of Neuefeind (1972). This indicator is not continuous on the full class of continuous preferences endowed with the closed convergence topology. It appears that no such indicator can be constructed, although Mas-Colell (1975) has established that one exists. A finer topology for preferences seems appropriate.     

Revealed preference theory on the choice of lotteries

The choice behavior of a decision-maker is said to be consistent with expected utility maximization if there exists a utility function defined on the set of prizes such that the decision-maker chooses lotteries with the highest expected utility. We present a revealed preference characterization of choice behavior that is consistent with expected utility maximization. A necessary and sufficient condition for expected utility maximization is that there does not exist a way to compound lotteries such that the probability distribution over the final prizes generated by the chosen lotteries of each observation is equal to that generated by the rejected lotteries of each observation. Our result is quite general and can be applied to any compact set of prizes and any choice correspondence.     

The geometry of revealed preference

In this paper, we examine how the geometry underlying revealed preference determines the set of preferences that can be revealed by choices. Specifically, given an arbitrary binary relation defined on a finite set, we ask if and when there exists a data set which can generate the given relation through revealed preference. We show that the dimension of the consumption space affects the set of revealed preference relations. If the consumption space has more goods than observations, any revealed preference relation can arise. Conversely, if the consumption space has low dimension relative to the number of observations, then there exist both rational and irrational preference relations that can never be revealed by choices.     

Similarity of information and behavior with a pointwise convergence topology

A topology on the space of information is defined which makes the latter a complete separable metric space. In this metric, set-theoretic convergent sequences of information converge, and the set of finite partitions of the state space is dense. This topology is weaker than the one studied by Allen. In addition, an expected utility maximizing consumer's demand for commodities is a continuous function of his information, and this topology is the weakest one with that property.     

Neighboring information and distributions of agents' characteristics under uncertainty

This paper proposes a topological structure for information. Specifically, information is considered an arbitrary sub-σ-field of the σ-field (of measurable subsets of states of the world) which represents events. A complete metric is defined on the space of all equivalence classes of sub-σ-fields; its uniformity is the same for all probability measures (on the abstract measurable space of states of the world and events) which are uniformly absolutely continuous. With this topology, economic behavior depends continuously on the agent's information; demand functions under uncertainty are jointly continuous in utilities, endowments, and information. Moreover, the value of information is a continuous function of the information sub-σ-field. Technical properties of the topology are also given.     

A note on the space of preference relations

The space of irreflexive, transitive, and continuous binary relations of R^l endowed with the topology of closed convergence, and subsets of defined by various other properties of the relations are investigated. It is shown that the subsets defined by properties, which one often assumes in general equilibrium theory, have nice topological properties such as compactness or G₈-ness.     

‘Stochastically more risk averse:’ A contextual theory of stochastic discrete choice under risk

Microeconometric treatments of discrete choice under risk are typically homoscedastic latent variable models. Specifically, choice probabilities are given by preference functional differences (given by expected utility, rank-dependent utility, etc.) embedded in cumulative distribution functions. This approach has a problem: Estimated utility function parameters meant to represent agents’ degree of risk aversion in the sense of Pratt (1964) do not imply a suggested “stochastically more risk averse” relation within such models. A new heteroscedastic model called “contextual utility” remedies this, and estimates in one data set suggest it explains (and especially predicts) as well as or better than other stochastic models.     

Stochastic dominance with pair-wise risk aversion

A class of stochastic orders is defined on the set of bivariate distribution functions. This class of orders is linearly orderable by inclusion. A family of utility functions, coherent with each of the stochastic orders previously defined, is determined. These utility functions represent pair-wise risk aversion. The relations with univariate stochastic orders are examined.     

Expected utility theory on mixture spaces without the completeness axiom

A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.     

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.     

On aggregation and welfare analysis

In this article, the beginnings of a new approach to the theory of aggregation are developed. The basic idea is that aggregation should involve two things: (a) data over which social preferences are defined should be mapped into a smaller-dimensional space, and (b) there should exist an ordering on that lower-dimensional space such that an improvement in this criterion implies an improvement for any of a class of social preference functions and of a class of individual preference relations defined over the original space. Results are developed which show that this conception of aggregation can yield meaningful results; particularly with respect to comparisons of ‘real national income’ in two situations, for a given economy.     

CONTINUOUS PREFERENCE ORDERINGS REPRESENTABLE BY UTILITY FUNCTIONS

This paper surveys the conditions under which it is possible to represent a continuous preference ordering using utility functions. We start with a historical perspective on the notions of utility and preferences, continue by defining the mathematical concepts employed in this literature, and then list several key contributions to the topic of representability. These contributions concern both the preference orderings and the spaces where they are defined. For any continuous preference ordering, we show the need for separability and the sufficiency of connectedness and separability, or second countability, of the space where it is defined. We emphasize the need for separability by showing that in any nonseparable metric space, there are continuous preference orderings without utility representation. However, by reinforcing connectedness, we show that countably boundedness of the preference ordering is a necessary and sufficient condition for the existence of a (continuous) utility representation. Finally, we discuss the special case of strictly monotonic preferences.     

From revealed preference to preference revelation

Utility functions are regarded as elements of a linear space that is paired with a dual representation of choices to demonstrate the similarity between preference revelation and the duality of prices and quantities in revealed preference. With respect to preference revelation, quasilinear versus ordinal utility and choices in an abstract set versus choices in a linear space are distinguished and their separate and common features are explored. The central thread uniting the various strands is the subdifferentiability of convex functions.     

Perturbed utility and general equilibrium analysis

We study general equilibrium theory of complete markets in an otherwise standard economy with each household having an additive perturbed utility function. Since this function represents a type of stochastic choice theory, the equilibrium of the corresponding economy is defined to be a price vector that makes its mean expected demand equal its mean endowment. We begin with a study of the economic meaning of this notion, by showing that at any given price vector, there always exists an economy with deterministic utilities whose mean demand is just the mean expected demand of our economy with additive perturbed utilities. We then show the existence of equilibrium, its Pareto inefficiency, and the upper hemi-continuity of the equilibrium set correspondence. Specializing to the case of regular economies, we finally demonstrate that almost every economy is regular and the equilibrium set correspondence in this regular case is continuous and locally constant.     

Decision-making under risk: Editing procedures based on correlated similarities, and preference overdetermination

This paper studies editing procedures based on similarity relations in an expected utility maximization context. It shows that these procedures are compatible both with a family of difference-correlated similarities on the prize space and with a set of families (one for each probability) of ratio-correlated similarity relations on the probability space. In view of the properties satisfied by these families of correlated similarities, it is suggested that Rubinstein's preference overdetermination problem can be avoided. Received: 17 April 1996 / Accepted: 24 September 1998     

Recoverability revisited

This study considers the uniqueness problem of the preference relation corresponding to a demand function, which is called the “recoverability problem”. We show that if a demand function has sufficiently wide range and is income-Lipschitzian, then there exists a unique corresponding upper semi-continuous preference relation. Moreover, we explicitly construct a utility function that represents this preference relation. Compared with related research, a feature of our result is that it ensures not only the uniqueness, but also the existence of the corresponding upper semi-continuous preference relation. Further, we introduce two axioms related to demand functions, and show that these axioms are equivalent to the continuity of our preference relation in the interior of the consumption set. In addition to these results, we present three examples that explain why our requirements (including the upper semi-continuity of preference relations and the wide range requirement and income-Lipschitzian property of demand functions) are necessary, and a further two examples in which there is no continuous preference relation corresponding to the given demand function.     

Continuous extension of preferences

We consider the problem of extending preferences from a subset of a commodity space to the entire space. It is a simple consequence of the Tietze extension theorem that continuous preferences can be extended if they are defined on closed subsets of a normal space and are representable by utility functions. We show the following: If the space is a non-separable metric space, then extension of preferences is not always possible. In fact for (path-connected) metric spaces, extension property, utility representation property, and separability are equivalent to each other.     

A von Neumann–Morgenstern representation result without weak continuity assumption

In the paradigm of von Neumann and Morgenstern (1947), a representation of affine preferences in terms of an expected utility can be obtained under the assumption of weak continuity. Since the weak topology is coarse, this requirement is a priori far from being negligible. In this work, we replace the assumption of weak continuity by monotonicity. More precisely, on the space of lotteries on an interval of the real line, it is shown that any affine preference order which is monotone with respect to the first stochastic order admits a representation in terms of an expected utility for some nondecreasing utility function. As a consequence, any affine preference order on the subset of lotteries with compact support, which is monotone with respect to the second stochastic order, can be represented in terms of an expected utility for some nondecreasing concave utility function. We also provide such representations for affine preference orders on the subset of those lotteries which fulfill some integrability conditions. The subtleties of the weak topology are illustrated by some examples.     

Some results on the existence of utility functions on path connected spaces

This paper concerns the existence of utility representations for preferences defined on a path connected space X. This includes any convex set. A classical result of Eilenberg (1941) proves the existence of utility representations when the consumption set is connected and separable. In an infinite dimensional space the above result may not be useful, because we lack, in general, the separability of the space. The non-separable spaces L^∞ and ca(K) are typical examples in mathematical economics. In this paper we show that a continuous preference relation ≽, on X has a continuous utility representation if and only if it is countably bounded, i.e., there is some countable subset F of X such that for all x in X there exist y and z in F with y≽x≽z. An easy corollary states that any continuous preference which has a best and a worst point has a continuous representation. We also obtain a convex continuous preference on a Banach lattice that has not a utility representation, because it is not countably bounded.