Essential alternatives and freedom rankings

We study the problem of ranking opportunity sets in terms of freedom of choice. The analysis is based on the notion of essential alternatives introduced in Puppe (J Econ Theory 68:174–199, 1996). An alternative in an opportunity set is called essential if by deleting it, the reduced opportunity set offers less freedom than the original set. We provide an axiomatic characterization of the ranking according to which an opportunity set offers more freedom than another opportunity set if its share of essential elements in their union is larger.     

Ranking opportunity sets on the basis of their freedom of choice and their ability to satisfy preferences: A difficulty

This paper examines a possibility of enlarging the domain of definition of individual preferences suggested by the recent literature on freedom of choice. More specifically, the possibility for an individual to have preferences that depend upon both the opportunity set that she faces and the particular alternative that she chooses from that set is considered. Even more specifically, the possibility for these preferences to value freedom of choice, as defined by the set theoretic relation of inclusion, while being consistent, in a certain sense, with the existence of a preference ordering over the options contained in opportunity sets is investigated. It is shown in the paper that a necessary condition for the existence of any transitive extended preferences of this type is for freedom of choice to be given no intrinsic importance. Received: 22 November 1995 / Accepted: 11 January 1997     

Equitable opportunities in economic environments

 In this paper, we extend the axiomatic analysis of equitable opportunities developed in Kranich [6] from finite to continuous opportunity sets. This extended framework is amenable to economic applications. The main results establish conditions under which an ordinal ranking of profiles of opportunity sets can be represented by a cardinal advantage function which describes both the extent of inequality and the distribution of advantage among the agents. Received: 3 November 1993/Accepted: 18 December 1995     

Ranking opportunity sets in the space of functionings

We develop a ranking of compact, convex and comprehensive opportunity sets defined in the evaluative space of individual functionings. We suppose the existence of a target, that is a multi-dimensional bliss point in terms of functionings. This leads us to define concepts such as essentiality and freedomin a novel way. As a main result, we give an axiomatic characterization of the ranking obtained by minimizing the Euclidean distance between each opportunity set and the target.     

Extended preferences and freedom of choice

The common choice theory in economics is based on the assumption that an individual is defined in terms of a binary preference relation. This preference relation is defined over alternatives without taking into account menu dependence and, in particular, freedom of choice or, more generally, the set that contains the alternatives. In this study we clarify the nature and the significance of freedom of choice which may positively or negatively affect the individual's welfare. Our proposed extended preference relation of the individual takes into account both the particular alternative and the opportunity set that he faces. This extended relation does not induce ranking of opportunity sets. Its restriction to a particular opportunity set is the paradigmatic preference relation and it can capture the dependence of preferences on freedom of choice. Our main result establishes the inconsistency between dependence of extended preferences on freedom of choice and the existence of a utility that represents the paradigmatic preference relation and any of its restrictions. Received: 30 December 1997/Accepted: 6 September 1999     

The measurement of opportunity inequality: a cardinality-based approach

We consider the problem of ranking distributions of opportunity sets on the basis of equality. First, conditional on a given ranking of individual opportunity sets, we define the notion of an equalizing transformation. Then, assuming that the opportunity sets are ranked according to the cardinality ordering, we formulate the analogues of the notions of the Lorenz partial ordering, equalizing (Dalton) transfers, and inequality averse social welfare functions – concepts which play a central role in the literature on income inequality. Our main result is a cardinality-based analogue of the fundamental theorem of inequality measurement: one distribution Lorenz dominates another if and only if the former can be obtained from the latter by a finite sequence of rank preserving equalizations, and if and only if the former is ranked higher than the latter by all inequality averse social welfare functions. In addition, we characterize the smallest monotonic and transitive extension of our cardinality-based Lorenz inequality ordering. Received: 2 May 1995 / Accepted: 11 October 1996     

Equal factor equivalence in economies with multiple public goods

We reconsider the problem of provision and cost-sharing of multiple public goods. The efficient equal factor equivalent allocation rule makes every agent indifferent between what he receives and the opportunity of choosing the bundle of public goods subject to the constraint of paying r times its cost, where r is set as low as possible. We show that this rule is characterized in economies with a continuum of agents by efficiency, a natural upper bound on everyone's welfare, and a property of solidarity with respect to changes in population and preferences. Received: 3 August 1995 / Accepted : 29 April 1997     

Extending an order to the power set: The Leximax Criterion

This work analyzes the issue of ranking the levels of well-being enjoyed by an agent when dealing with her opportunity set in a freedom of choice context. We try to propose a natural extension of a preference relation over a (possibly) infinite set X to its power set, compatible with the finite Leximax criterion. This research has been supported by Comisión Interministerial de Ciencia y Tecnología, SEC2000-0838 (Ballester), and Dirección General de Enseñanza Superior e Investigación Científica, PB98-0551 (De Miguel).     

Ranking opportunity sets in the space of functionings

We develop a ranking of compact, convex and comprehensive opportunity sets defined in the evaluative space of individual functionings. We suppose the existence of a target, that is a multi-dimensional bliss point in terms of functionings. This leads us to define concepts such as essentiality and freedomin a novel way. As a main result, we give an axiomatic characterization of the ranking obtained by minimizing the Euclidean distance between each opportunity set and the target.     

Justifiable preferences over opportunity sets

In this paper we say that a preference over opportunity sets is justifiable if there exists a reflexive and complete binary relation on the set of alternatives, such that one opportunity set is at least as good as a second, if and only if the there is at least one alternative from the first set which is no worse than any alternative of the two sets combined together, with respect to the binary relation on the alternatives. In keeping with the revered tradition set by von Neumann and Morgenstern we call a reflexive and complete binary relation, an abstract game (note: strictly speaking von Neumann and Morgenstern refer to the asymmetric part of a reflexive and complete binary relation as an abstract game; hence our terminology though analytically equivalent, leads to a harmless corruption of the original meaning). In this paper we obtain a necessary and sufficient condition for the justifiability of transitive and quasi transitive preferences over opportunity sets. I would like to thank Prasanta Pattanaik for motivation in this research. I would also like to thank Maurice Salles and an anonymous referee of this journal for useful suggestions leading to considerable improvement in the presentation. An earlier version of this paper was presented at the Sixth Annual Conference on Econometric Modeling for Africa, held at Pretoria from July 4–6, 2001. I would like to thank all the conference participants for their observations on my paper. However, the sole responsibility for errors that do remain is my own.     

Lexicographic measurement of the information contained in opportunity sets

An individual who has to choose one item from an opportunity set or menu may lack complete information about the final consequences attached to her choice. This problem has been modeled by Bossert (2000) who introduces an axiomatic approach to the ranking of menus according to their informational content. We propose a different ranking based on the lexicographic ordering on the vectors of numbers of consequences for each item and characterize it axiomatically. Received: 12 June 1998/Accepted: 5 September 2000     

Consistency between tastes and values: A universalization approach

The object of this paper is to propose a consistency test for an individual involved in collective choice process. Collective choice processes considered in the paper are those that transform individuals ‘tastes’– which reflect the self-interested view point of the individuals – into (social) ranking of alternatives. In addition to her tastes, an individual has values about the way by which collective decision should be made. We distinguish two categories of such values. First, there are end-values that restrict the class of social rankings that the individual considers ethically acceptable. Second there are aggregation-values that specify the way by which the social ranking should depend upon the individuals tastes. The consistency test stands on an hypothetical operation of universalization of the individual tastes to everyone. Five illustrations of the potential usefulness of our approach for interpreting social choice theory and welfare economics are proposed. These illustrations deal with utilitarian aggregation in the presence of income inequality aversion, the so-called ‘ethics of responsibility’ and the aggregation of individual ranking of opportunity sets based on their freedom of choice. A discussion of the relevance of the consistency test for addressing the problem of ‘laundering’ individual preferences is also provided. Received: 25 June 1998/Accepted: 16 March 1999     

Hierarchy of players in swap robust voting games

Ordinarily, the process of decision making by a committee through voting is modeled by a monotonic game the range of whose characteristic function is restricted to {0, 1}. The decision rule that governs the collective action of a voting body induces a hierarchy in the set of players in terms of the a-priori influence that the players have over the decision making process. In order to determine this hierarchy in a swap robust game, one has to either evaluate a power index (e.g., the Shapley–Shubik index, the Banzhaf–Coleman index) for each player or conduct a pairwise comparison between players, whereby a player i is ranked higher than another player j if there exists a coalition in which i is more desirable as a coalition partner than j. In this paper, we outline an alternative mechanism to determine the ranking of players in terms of their a-priori power. This simple and elegant method uses only minimal winning coalitions, rather than the entire set of winning coalitions.     

Bildung and intercultural understanding

The concept of Bildung, sometimes translated as self-cultivation, is located at the core of an influential tradition of educational thought. A key question concerns the relationship between Bildung and interculturality. Drawing on Wilhelm von Humboldt and Hans-Georg Gadamer, and on the so-called transformative learning theory, Bildung can be interpreted as a process of transforming one’s meaning perspective in encounters with others. A meaning perspective is a set of largely implicit presuppositions underlying one’s habitual ways of thinking, feeling and acting. Confrontation with alternative perspectives can be an opportunity to become aware of one’s own perspective, to critically assess it and to transform it. Thus conceived, Bildung is closely related to interculturality.     

Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering

“Subset voting” denotes a choice situation where one fixed set of choice alternatives (candidates, products) is offered to a group of decision makers, each of whom is requested to pick a subset containing any number of alternatives. In the context of subset voting we merge three choice paradigms, “approval voting“ from political science, the “weak utility model” from mathematical psychology, and “social welfare orderings” from social choice theory. We use a probabilistic choice model proposed by Falmagne and Regenwetter (1996) built upon the notion that each voter has a personal ranking of the alternatives and chooses a subset at the top of the ranking. Using an extension of Sen's (1966) theorem about value restriction, we provide necessary and sufficient conditions for this empirically testable choice model to yield a social welfare ordering. Furthermore, we develop a method to compute Borda scores and Condorcet winners from subset choice probabilities. The technique is illustrated on an election of the Mathematical Association of America (Brams, 1988). Received: 18 August 1995 / Accepted: 13 February 1997     

Path independent choice and the ranking of opportunity sets

The indirect utility principle provides an instrumentalist basis for ranking opportunity sets, given an underlying preference ranking on alternatives. Opportunity set A is weakly preferred to B if A includes at least one preference-maximising element from $A\cup B$ . We introduce the Plott consistency principle as a natural extension of this logic to decision-makers who choose amongst alternatives according to a path independent choice function. Such choice functions need not be rationalisable by a preference order. Plott consistency requires that A is an acceptable choice from $\left\{ A, B\right\} $ if A includes at least one element from the set of acceptable choices from $A\cup B$ . We explore necessary and sufficient conditions (imposed on a choice function defined on collections of opportunity sets) for Plott consistency.     

On the multi-preference approach to evaluating opportunities

The purpose of the paper is to provide a general framework for analyzing “preference for opportunities.” Based on two simple axioms a fundamental result due to Kreps is used in order to represent rankings of opportunity sets in terms of multiple preferences. The paper provides several refinements of the basic representation theorem. In particular, a condition of “closedness under compromise” is suggested in order to distinguish the flexibility interpretation of the model from normative interpretations which play a crucial role in justifying the intrinsic value of opportunities. Moreover, the paper clarifies the link between the multiple preference approach and the “choice function” approach to evaluating opportunities. In particular, it is shown how the well-known Aizerman/Malishevski result on rationalizability of choice functions can be obtained as a corollary from the more general multiple preference representation of a ranking of opportunity sets. Received: 3 September 1996 / Accepted: 18 August 1997     

Measuring and implementing equality of opportunity for income

Departing from the welfarist tradition, recent theories of justice focus on individual opportunities as the appropriate standard for distributive judgments. To explore how this philosophical conception can be translated into concrete public policy, we select the income as relevant outcome and the income tax as the relevant redistributive policy, and we address the following questions: (i) what is the degree of opportunity inequality in an income distribution? (ii) how to design an opportunity egalitarian income tax policy? Several criteria for ranking income distributions on the basis of equality of opportunities are derived. Moreover, we characterize an opportunity egalitarian income tax and we formulate criteria for choosing among alternative tax schedules. I would like to thank Peter Lambert for his encouragement and guidance, and Walter Bossert, Maurice Salles, and partecipants at the 1998 Social Choice and Welfare Conference, Vancouver, and at the Conference on Non–Welfaristic Issues in Normative Economics in Caen for their helpful comments and suggestions. Financial support of the Ministero dellIstruzione, Università e Ricerca (Italy) is gratefully acknowledged. Responsibility for any remaining deficiency is mine.     

Modeling Repeatable Events Using Discrete‐Time Data: Predicting Marital Dissolution

I join two methodologies by illustrating the application of multilevel modeling principles to hazard‐rate models with an emphasis on procedures for discrete‐time data that contain repeatable events. I demonstrate this application using data taken from the 1995 National Survey of Family Growth (NSFG) to ascertain the relationship between multiple covariates and risk of subsequent marital dissolution. I consider both fixed‐ and random‐effects versions of the multilevel model, as well as a Generalized Estimating Equation alternative to estimating random effects. I compare results obtained from the various estimators, noting why differences occur, and recommend when to choose the various alternatives. I also provide a set of SAS and STATA programs that can be used to analyze the NSFG data.     

Ranking intersecting Lorenz curves

This paper is concerned with the problem of ranking Lorenz curves in situations where the Lorenz curves intersect and no unambiguous ranking can be attained without introducing weaker ranking criteria than first-degree Lorenz dominance. To deal with such situations two alternative sequences of nested dominance criteria between Lorenz curves are introduced. At the limit the systems of dominance criteria appear to depend solely on the income share of either the worst-off or the best-off income recipient. This result suggests two alternative strategies for increasing the number of Lorenz curves that can be strictly ordered; one that places more emphasis on changes that occur in the lower part of the income distribution and the other that places more emphasis on changes that occur in the upper part of the income distribution. Both strategies turn out to depart from the Gini coefficient; one requires higher degree of downside and the other higher degree of upside inequality aversion than what is exhibited by the Gini coefficient. Furthermore, it is demonstrated that the sequences of dominance criteria characterize two separate systems of nested subfamilies of inequality measures and thus provide a method for identifying the least restrictive social preferences required to reach an unambiguous ranking of a given set of Lorenz curves. Moreover, it is demonstrated that the introduction of successively more general transfer principles than the Pigou–Dalton principle of transfers forms a helpful basis for judging the normative significance of higher degrees of Lorenz dominance. The dominance results for Lorenz curves do also apply to generalized Lorenz curves and thus provide convenient characterizations of the corresponding social welfare orderings.