An extension of Fishburn's theorem on extending orders

P. C. Fishburn [J. Econom. Theory31 (1983), 000-000] has shown that one can extend any well ordering of a set to a linear ordering of the set of all non-empty subsets of that set, while satisfying two axioms denoted (GP¹) and (M¹). By applying the compactness theorem of logic, this note shows that the well ordering assumption can be waived. Definability and well ordering properties of the extension are also discussed and shown to depend on the class of basic orderings considered.     

Median-based extensions of an ordering over a set to the power set: An axiomatic characterization

In this paper we consider the problem of inducing an ordering over the set of all non-empty subsets of a finite set X of alternatives, given an ordering R over X. Assuming R to be antisymmetric and X to have at least six elements, we provide a set of independent, necessary, and sufficient conditions for the induced ordering to be “median-based” (so that every non-empty subset of X is “indifferent” to its own median set defined in terms of R).     

A CHARACTERIZATION OF THE RAWLSIAN SOCIAL ORDERING OVER INFINITE UTILITY STREAMS

In this paper, we characterize a Rawlsian ordering over infinite utility streams using a fairness principle called Hammond Equity, as well as a weak efficiency principle, Sup Continuity, and a condition that links finite and infinite settings of distributive justice called Preference Continuity.     

Comment on the Kannai-Peleg impossibility theorem for extending orders

Recent work by Y. Kannai and B. Peleg shows that two appealing axioms for extending a linear order on a set of six or more elements to a weak order on the family of non-empty subsets of that set are mutually incompatible. This comment shows that a modest restriction of their monotonicity axiom is compatible for extension with a generalization of their other axiom provided that the linear order on the basic set is a well ordering.     

Inequality as differences: A simple characterization

Differences ordering (DO) is introduced and studied as an alternative to Lorenz ordering (LO). We identify the set of transfers consistent with DO and its class of order-preserving functions.     

The measurement of income mobility: A partial ordering approach

Summary. Given a set of longitudinal data pertaining to two populations, a question of interest is the following: Which population has experienced a greater extent of income mobility? The aim of the present paper is to develop a systematic way of answering this question. We first put forth four axioms for income movement-mobility indices, and show that a familiar class of measures is characterized by these axioms. An unambiguous (partial) ordering is then defined as the intersection of the (complete) orderings induced by the mobility measures which belong to the characterized class; a transformation of income distributions is “more mobile” than another if, and only if, the former is ranked higher than the latter for all mobility measures which satisfy our axioms. Unfortunately, our mobility ordering depends on a parameter, and therefore, it is not readily apparent how one can apply it to panel data directly. In the second part of the paper, therefore, we derive several sets of parameter-free necessary and sufficient conditions which allow one to use the proposed mobility ordering in making unambiguous income mobility comparisons in practice. Received: July 12, 1995; revised version: May 13, 1997     

Arrow's Possibility Theorem for one-dimensional single-peaked preferences

In one-dimensional environments with single-peaked preferences we consider social welfare functions satisfying Arrow's requirements, i.e. weak Pareto and independence of irrelevant alternatives. When the policy space is a one-dimensional continuum such a welfare function is determined by a collection of 2^N strictly quasi-concave preferences and a tie-breaking rule. As a corollary we obtain that when the number of voters is odd, simple majority voting is transitive if and only if each voter's preference is strictly quasi-concave.     

Priorities in the location of multiple public facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We characterize the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among “interest groups”. We characterize the subclasses of priority rules that respectively satisfy anonymity, avoid the no-show paradox, strategy-proofness and population-monotonicity. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Any such rule can also be viewed as a collection of generalized peak-selection median rules, that are linked across populations, in a way that we describe.     

Arrovian aggregation in economic environments: how much should we know about indifference surfaces?

Arrow's celebrated theorem of social choice shows that the aggregation of individual preferences into a social ordering cannot make the ranking of any pair of alternatives depend only on individual preferences over that pair, unless the fundamental weak Pareto and non-dictatorship principles are violated. In the standard model of division of commodities, we investigate how much information about indifference surfaces is needed to construct social ordering functions satisfying the weak Pareto principle and anonymity. We show that local information such as marginal rates of substitution or the shapes “within the Edgeworth box” is not enough, and knowledge of substantially non-local information is necessary.     

Rational preferences under ambiguity

This paper analyzes preferences in the presence of ambiguity that are rational in the sense of satisfying the classical ordering condition as well as monotonicity. Under technical conditions that are natural in an Anscombe?CAumann environment, we show that even for such a general preference model, it is possible to identify a set of priors, as first envisioned by Ellsberg (Q J Econ 75:643?C669, 1961). We then discuss ambiguity attitudes, as well as unambiguous acts and events, for the class of rational preferences we consider.     

Edgeworth's conjecture with infinitely many commodities: commodity differentiation

Summary. Convergence of the cores of finite economies to the set of Walrasian allocations as the number of agents grows has long been taken as one of the basic tests of perfect competition. The present paper examines this test in the most natural model of commodity differentiation: the commodity space is the space of nonnegative measures, endowed with the topology of weak convergence. In Anderson and Zame [12], we gave counterexamples to core convergence in L ¹, a space in which core convergence holds for replica economies and core equivalence holds for continuum economies; in addition, we gave a core convergence theorem under the assumption that traders' utility functions exhibit uniformly vanishing marginal utility at infinity. In this paper, we provide two core convergence results for the commodity differentiation model. A key technical virtue of this space is that relatively large sets (in particular, closed norm-bounded sets) are compact. This permits us to invoke a version of the Shapley-Folkman Theorem for compact subsets of an infinite-dimensional space. We show that, for sufficiently large economies in which endowments come from a norm bounded set, preferences satisfy an equidesirability condition, and either (i) preferences exhibit uniformly bounded marginal rates of substitution or (ii) endowments come from an order-bounded set, core allocations can be approximately decentralized by prices. Received: July 29, 1996; revised version: January 14, 1997     

Maxmin under risk

Summary. Let be a continuous and convex weak order on the set of lotteries defined over a set Z of outcomes. Necessary and sufficient conditions are given to guarantee the existence of a set of utility functions defined on Z such that, for any lotteries p and q, The interpretation is simple: a conservative decision maker has an unclear evaluation of the different outcomes when facing lotteries. She then acts as if she were considering many expected utility evaluations and taking the worst one. Received: January 19, 2000; revised version: December 20, 2000     

A Measure Of The Trade‐Off Between Responsiveness and Nondictatorship for Arrovian Social Welfare Functions

If individuals are never indifferent between distinct alternatives then for any transitive‐valued social welfare function satisfying IIA, and any fraction t, either the set of pairs of alternatives that are socially ordered without consulting more than one individual's preferences comprises at least the fraction t of all pairs, or else the fraction of pairs that have their social ordering determined independently of everyone's s preferences exceeds, or is very close to, 1 ?t. The Pareto criterion is not imposed. (There is also a version of this result for the domain of preferences that admit indifference.)     

Cyclic sets in multidimensional voting models

Simple majority rule usually does not yield an unambiguous consistent outcome. Assuming a characterization of the set of potential outcomes as Rⁿ and an odd number of voters with quasiconcave preferences, a unique, nonempty set of majority rule cycles exists. Sufficient conditions are established for the top cycle set to encompass the entire policy space. Generalized quadratic utility functions satisfy these conditions.     

Unmediated communication in games with complete and incomplete information

We study the effects of adding unmediated communication to static, finite games of complete and incomplete information. We characterize S^U(G), the set of outcomes of a game G, that are induced by sequential equilibria of cheap talk extensions. A cheap talk extension of G is an extensive-form game in which players communicate before playing G. A reliable mediator is not available and players exchange private or public messages that do not affect directly their payoffs. We first show that if G is a game of complete information with five or more players and rational parameters, then S^U(G) coincides with the set of correlated equilibria of G. Next, we demonstrate that if G is a game of incomplete information with at least five players, rational parameters and full support (i.e., all profiles of types have positive probability), then S^U(G) is equal to the set of communication equilibria of G.     

Strategy-proof preference aggregation: Possibilities and characterizations

An aggregation rule maps each profile of individual strict preference orderings over a set of alternatives into a social ordering over that set. We call such a rule strategy-proof if misreporting one's preference never produces a different social ordering that is between the original ordering and one's own preference. After describing two examples of manipulable rules, we study in some detail three classes of strategy-proof rules: (i) rules based on a monotonic alteration of the majority relation generated by the preference profile; (ii) rules improving upon a fixed status-quo; and (iii) rules generalizing the Condorcet–Kemeny aggregation method.     

Bargaining under Uncertainty and the Monotone Path Solutions

Uncertainty with respect to the feasible set of utility vectors is introduced in an axiomatic bargaining model. Given a criterion for nonprobabilistic decision-making under uncertainty, a natural efficiency requirement can be imposed on a bargaining solution. Using the maximin ordering, thestrictly monotone pathsolutions (generalizations of theegalitariansolution) to the bargaining problem are characterized as the only continuous solutions that satisfy this efficiency axiom. If the maximin criterion is replaced by the maximax ranking or a strict convex combination of the maximin and the maximax criterion, imposing our efficiency axiom and continuity leads to thedictatorialsolutions.Journal of Economic LiteratureClassification Number: C78.     

Increasing interdependence of multivariate distributions

Orderings of interdependence are useful in many economic contexts: in assessing ex post inequality under uncertainty; in comparing multidimensional inequality; in valuing portfolios of assets or insurance policies; and in assessing systemic risk. We explore five orderings of interdependence for multivariate distributions: greater weak association, the supermodular ordering, the convex-modular ordering, the dispersion ordering, and the concordance ordering. For two dimensions, all five are equivalent, whereas for three dimensions, the first four are strictly ranked and the last two are equivalent, and for four or more dimensions, all five are strictly ranked. For the special case of binary random variables, we establish some equivalences among the orderings.     

Tops-only domains

In this paper we consider the standard voting model with a finite set of alternatives A and n voters and address the following question: what are the characteristics of domains \({\mathcal D}\) that induce the property that every strategy-proof social choice function \({f: {\mathcal D}^n \rightarrow A}\) satisfying unanimity, has the tops-only property? We first impose a minimal richness condition which ensures that for every alternative a, there exists an admissible ordering where a is maximal. We identify conditions on \({\mathcal D}\) that are sufficient for strategy-proofness and unanimity to imply tops onlyness in the general case of n voters and in the special case, n = 2. We provide an algorithm for constructing tops-only domains from connected graphs with elements of A as nodes. We provide several applications of our results. Finally, we relax the minimal richness assumption and partially extend our results.     

Designer path independent choice functions

Summary. This paper provides an algorithm for the construction of all PICFs on a finite set of alternatives, V, designed by an a priori given set I of initial choices as well as the determination of whether the initial set I is consistent with path independence. The algorithm is based on a new characterization result for path independent choice functions (PICF) on finite domains and uses that characterization as the basis of the algorithm. The characterization result identifies two properties of a partition of the Boolean algebra as necessary and sufficient for a choice function C to be a PICF: (i): For every subset A of V the set is an interval in the Boolean algebra 2 ^V . (ii): If A/B is an interval in the Boolean algebra such that C(A) = C(B) and if M/N is an upper transpose of A/B then C(M) = C(N). The algorithm proceeds by expanding on the implications of these two properties.Received: 5 November 2003, Revised: 20 July 2004, JEL Classification Numbers: D00, D70.