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.     

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.     

Opportunity sets and individual well-being

 An opportunity set ranking rule assigns an ordering of opportunity sets to each individual utility function (defined on the universal set of alternatives) within the domain of this rule. Using an axiomatic approach, this paper characterizes a general class of opportunity set ranking rules which are based on the utilities associated with the elements of an opportunity set. It is argued that the addition of an alternative to a given opportunity set is not necessarily desirable in terms of overall well-being, and this position is reflected in replacing a commonly used monotonicity axiom with an alternative condition. Received: 15 May 1995/Accepted: 14 December 1995     

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.     

Fair allocation of indivisible goods: the two-agent case

One must allocate a finite set of indivisible goods among two agents without monetary compensation. We impose Pareto-efficiency, anonymity, a weak notion of no-envy, a welfare lower bound based on each agent’s ranking of the subsets of goods, and a monotonicity property w.r.t. changes in preferences. We prove that there is a rule satisfying these axioms. If there are three goods, it is the only rule, together with one of its subcorrespondences, satisfying each fairness axiom and not discriminating between goods.     

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     

Diversity as width

It is argued that if a finite partially ordered population is given, and incomparability is taken as the relevant type of dissimilarity, then diversity comparisons between subpopulations may be conveniently based on widths namely on the maximum number of pairwise incomparable units they include. We introduce two width-based rankings. The first one is the plain width-ranking of subposets as induced by width computation. The second one is the undominated width-ranking of subposets namely the ranking induced by the sizes of undominated subposets. Simple axiomatic characterizations of the foregoing rankings are provided.     

Mutual dependency between capabilities and functionings in Amartya Sen’s capability approach

Amartya Sen’s capability approach has recently been widely discussed as a theoretical basis for making resource allocation decisions in health care. The purpose of this paper is to analyze the relationship between capabilities and functionings in the capability approach. The paper shows that some functionings are not only the result of capabilities, but also their prerequisite. That is, there is a dual role of some functionings as both ends and instruments, resulting in a mutual dependency between capabilities and functionings. Functionings may be a direct requirement for capabilities, but also an indirect one because they ensure the absence of mental disorders or negative thoughts, both of which are relevant constraints on freedom. This has important implications. It supports a policy that ensures for everyone an initial endowment of (1) mental and physical health, (2) education, and (3) other functionings with a direct or indirect impact on capabilities.     

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     

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     

The β-ranking and the β-measure for directed networks: Axiomatic characterizations

The β-measure has been introduced and characterized by van den Brink and Gilles (2000) for measuring the domination in directed networks. The present paper characterizes the ranking induced by the β-measure. It also provides an alternative characterization of the β-measure.     

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).     

Closeness centrality via the Condorcet principle

We provide a characterization of closeness centrality in the class of distance-based centralities. To this end, we introduce a natural property, called majority comparison, that states that out of two adjacent nodes the one closer to more nodes is more central. We prove that any distance-based centrality that satisfies this property gives the same ranking in every graph as closeness centrality. The axiom is inspired by the interpretation of the graph as an election in which nodes are both voters and candidates and their preferences are determined by the distances to the other nodes.     

The original Borda count and partial voting

In a Borda count, bc, M. de Borda suggested the last preference cast should receive 1 point, the voter’s penultimate ranking should get 2 points, and so on. Today, however, points are often awarded to (first, second,..., last) preferences cast as per (n, n?1, ..., 1) or more frequently, (n ?1, n?2,..., 0). If partial voting is allowed, and if a first preference is to be given n or n ? 1 points regardless of how many preferences the voter casts, he/she will be incentivised to rank only one option/candidate. If everyone acts in this way, the bc metamorphoses into a plurality vote... which de Borda criticized at length. If all the voters submit full ballots, the outcome—social choice or ranking—will be the same under any of the above three counting procedures. In the event of one or more persons submitting a partial vote, however, outcomes may vary considerably. This preliminary paper suggests research should consider partial voting. The author examines the consequences of the various rules so far advocated and then purports that M. de Borda, in using his formula, was perhaps more astute than the science has hitherto recognised.     

On the welfare loss caused by inequality of opportunity

Based on the ethical principle of equality of opportunity, this paper presents a measure of the welfare loss that is caused by an unfair distribution of a particular outcome (income, health, education, etc). The key idea is that a fair society should produce outcomes that depend on individuals’ effort and not on their external circumstances such as gender, socioeconomic background, etc. We propose measuring inequality of opportunity as the welfare loss attributed to the outcome differences among individuals who exert a similar level of effort. Our results are in line with those aspects of fairness literature that give priority to the ex post compensation approach to equality of opportunity. Finally, we present an empirical application for the measurement of the welfare loss in the income distribution in Europe. We have observed a high degree of heterogeneity among European countries. The welfare loss due to inequality of opportunity ranges in those, from basically zero to almost one fifth of their potential welfare.     

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     

Generalized Gini Indices of Equality of Opportunity

This article considers the ranking of profiles of opportunity sets on the basis of their equality. A version of the Pigou–Dalton transfer principle that is appropriate for the measurement of opportunity inequality is introduced and used to axiomatize the class of generalized Gini equality of opportunity orderings. A characterization of the class of generalized Gini social preference orderings for opportunity profiles is also provided.     

Public goods: An ideal concept

I see the concept of ‘a public good’ as an ideal concept defined by a pair of related characteristics: an opportunity for gain for a collectivity because of nonrivalness in consumption, where the opportunity for gain is difficult to finance because of the nonexclusion possibility. Even at that ideal level, it is impossible to present a solution able to fully realize the opportunity for gain.Given the difficulty of finding a satisfactory solution even at the ideal level, it would be surprising if an analysis of concrete instantiations of public goods would be simple. I will show that economists use at least thirteen concepts to point to economic problems related to public goods. This is more than opponents of the concept ‘public good,’ Malkin and Wildavsky, accuse the economic profession of using.I will reject the conclusion of Malkin and Wildavsky that the concept of a public good is a social construct and thus a purely culturally relative concept. I will concede, nevertheless, that in trying to realize some of the gains presented by goods having public goods aspects, social and cultural preferences play an important role. My conclusion, which justifies the ideal concept ‘public good’, implies that there is potentially a legitimate governmental role in dealing with public goods. Recognizing the socio-political dimension of economics means that one recognizes that the economic reality is complex. It does not justify, however, scepticism about concepts.     

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     

Transforming collective memory: mnemonic opportunity structures and the outcomes of racial violence memory movements

While some research on collective memory has addressed the creation of memory projects such as memorials and historical commissions, less attention has been paid to explaining variation in these projects’ success. America’s troubled history of lynchings and race riots is one topic increasingly addressed by commemorative projects. This article evaluates factors shaping efforts to imprint past racial violence into broader collective memory. Building on research on constraints imposed by actual pasts and environmental conditions, I argue that structural influences on collective memory, or mnemonic opportunity structures, powerfully affect the success of commemorative initiatives. I identify three major dimensions of mnemonic opportunity structures: (1) an environment’s present-day commemorative capacity, a past incident’s (2) ascribed significance, and (3) the moral valence of key characters at the time it occurred. Qualitative data from multiple case studies and 90 interviews, along with supplementary quantitative and qualitative comparative analyses (QCAs) of all recent projects marking segregation-era racial violence, illustrate the utility of the framework.