Ranking opportunity sets: An approach based on the preference for flexibility

We describe a criterion to evaluate subsets of a finite set of alternatives which are considered as opportunity sets. The axioms for set comparison are motivated within the preference for flexibility framework. We assume the preference over the universal set of alternatives to be made of two disjoint binary relations. The result is the axiomatic characterization of a procedure which is formally similar to the leximax ordering, but in our case it incorporates the presence of some uncertainty about the decision-maker final tastes. Received: 20 January 1999/Accepted: 20 October 1999     

On strategy-proof social choice correspondences

We introduce two new concepts of strategy-proofness for social choice correspondences based on the theory of preferences over sets of alternatives under complete uncertainty. One is based on Pattanaik and Peleg (Soc Choice Welf 1:113–122, 1984) and the other is based on Bossert et al. (Econ Theory 16:295–312, 2000). We prove that there is no social choice correspondence satisfying anonymity, neutrality, a range condition, and either of our concepts of strategy-proofness. An erratum to this article can be found at     

Recoverability of choice functions and binary relations: some duality results

Banerjee and Pattanaik (1996) proved a theorem that the maximal set with respect to a quasi-ordering can be fully recovered by defining the greatest sets with respect to each and every ordering extension thereof and taking their union. Donaldson and Weymark (1998) proved a theorem that a quasi-ordering can be fully recovered by taking the intersection of all the ordering extensions thereof. These recoverability theorems are obviously related, but their exact relationship has never been clarified in the literature. This paper examines the issue of choice-functional recoverability and relational recoverability in a general framework, and establishes several remarkable duality relationships. Thanks are due to Professors Walter Bossert, Prasanta K. Pattanaik, Clemens Puppe, Amartya K. Sen, and John Weymark, with whom we had several opportunities to discuss this and related issues. The first author also would like to express his gratitude to the Andrew Young School of Policy Studies, Georgia State University, and the Scientific Research Grant for Policy Areas (B) Number 603 from the Ministry of Education, Science and Culture of Japan for their financial support which made this collaborative research possible. We are also grateful to the referees whose comments have led to several improvements of the paper.     

Social choice and the indexing dilemma

This paper distinguishes an index ordering and a social ordering function as a simple way to formalize the indexing problem in the social choice framework. Two main conclusions are derived. First, the alleged dilemma between welfarism and perfectionism is shown to involve a third possibility, exemplified by the fairness approach to social choice. Second, the idea that an individual is better off than another whenever he has more (goods, functionings, etc.) in all dimensions, which is known to enter in conflict with the Pareto principle, can be partly salvaged by adopting the fairness approach. This paper has benefited very much from comments by M. Salles and from stimulating interactions with K. Basu, P. Pattanaik, B. Tungodden and the participants at the Philosophical Aspects of Social Choice conference in Caen.     

Freedom of choice

Inspired by the discussion of different functions of freedom of choice (instrumental versus intrinsic value) by Sen and others and an axiomatic characterization of an intrinsic aspect by Pattanaik and Xu, we compare unique axiomatic characterizations of three classes of rankings of opportunity sets in terms of freedom of choice: First, we investigate the simple cardinality-based ranking proposed by Pattanaik and Xu and a generalization of this. Secondly, we propose a new criterion that is based on the comparison of the ranges of sets of options. Thirdly, we solve possibly occuring conflicts between these two criteria.I am indebted to Wulf Gaertner, Yongsheng Xu, Walter Bossert, the participants of the Economic Theory Workshop of the University of Rochester, and anonymous referees for helpful discussions and comments. Financial support by Deutsche Forschungsgemeinschaft is gratefully acknowledged.     

An axiomatic characterization of the lexicographic maximin extension of an ordering over a set to the power set

The lexicographic maximin extension of an ordering is an important and widely used tool in social choice theory. We provide an axiomatization of it by means of five axioms. When the basic ordering is linear the following four (independent) axioms are sufficient: (1) Gärdenfors principle; (2) Neutrality; (3) Strong Fishburn monotonicity; and (4) Extension. Our result may also have applications in the theory of individual choice under uncertainty.     

Axiomatic derivation of scoring rules without the ordering assumption

Earlier derivations of scoring rules, by Smith (1973) and Young (1975), assumed that a voter can express only a rank ordering of the alternatives on his or her ballot. This paper shows that scoring rules can be derived without this ordering assumption. It is shown that a voting rule must be a scoring rule if it satisfies three basic axioms: reinforcement, overwhelming majorities, and neutrality. Other range and nonreversal axioms are also discussed.     

On ranking linear budget sets in terms of freedom of choice

This paper examines how freedom of choice as reflected in an agents opportunity sets can be measured in economic environments where opportunity sets are linear budget sets in the non-negative orthant of the n-dimensional real space. Three axioms, Symmetry, Monotonicity and Invariance of Scaling Effects, are proposed for this purpose and the measure based on volumes of budget sets is characterized. I am grateful to Prasanta K. Pattanaik for his encouragement and fruitful discussions over many years on this and related subjects. The first draft of this paper was written while I was visiting the Institute of Economic Research at Hitotsubashi University in Japan. I would like to thank its hospitality. Helpful comments of a referee are gratefully acknowledged.     

Intergenerational preference orderings

This paper investigates preference orderings on infinite horizon intergenerational consumption streams. The trade-offs inherent in the selection of moral precepts for an intergenerational preference ordering are made explicit by establishing the mutual inconsistency of an appealing set of axioms.I am grateful to Don Campbell for many valuable discussions and comments. I have also benefitted from the comments of Mike Peters, participants in the Theory Workshop at the University of Toronto, and from a referee and editor of this journal.     

Characterizations of two power indices for voting games with r alternatives

In the first three sections of this paper we present a set of axioms which provide a characterization of an extension of the Banzhaf index to voting games with r alternatives, such as the United Nations Security Council where a nation can vote “yes”, “no”, or “abstain”. The fourth section presents a set of axioms which characterizes a power index based on winning sets instead of pivot sets. Received: 4 April 2000/Accepted: 30 April 2001     

Rules for aggregating information

We present a model of information aggregation in which agents’ information is represented through partitions over states of the world. We discuss three axioms, meet separability, upper unanimity, and non-imposition, and show that these three axioms characterize the class of oligarchic rules, which combine all of the information held by a pre-specified set of individuals.     

Twofold optimality of the relative utilitarian bargaining solution

Given a bargaining problem, the relative utilitarian (RU) solution maximizes the sum total of the bargainer’s utilities, after having first renormalized each utility function to range from zero to one. We show that RU is “optimal” in two very different senses. First, RU is the maximal element (over the set of all bargaining solutions) under any partial ordering which satisfies certain axioms of fairness and consistency; this result is closely analogous to the result of Segal (J Polit Econ 108(3):569–589, 2000). Second, RU offers each person the maximum expected utility amongst all rescaling-invariant solutions, when it is applied to a random sequence of future bargaining problems generated using a certain class of distributions; this is recalls the results of Harsanyi (J Polit Econ 61:434–435, 1953) and Karni (Econometrica 66(6):1405–1415, 1998).     

Measuring autonomy freedom

In the measurement of autonomy freedom, the admissible potential preference relations are elicited by means of the concept of ‘reasonableness’. In this paper we argue for an alternative criterion based on information about the decision maker’s ‘awareness’ of his available opportunities. We argue that such an interpretation of autonomy fares better than that based on reasonableness. We then introduce some axioms that capture this intuition and study their logical implications. In the process, a new measure of autonomy freedom is characterized, which generalizes some of the measures so far constructed in the literature.We thank Martin van Hees, Robert Sugden, Prasanta Pattanaik, the participants at workshops at the CPNSS, London School of Economics, and at the University of Caen and an anonymous referee for their suggestions. The support of the CPNSS is gratefully acknowledged. This paper is part of a research project on “The Analysis and Measurement of Freedom” funded by the Ministero dell’Istruzione, Università e Ricerca (Italy). Its financial support is gratefully acknowledged.     

A note on the non-emptiness of the stability set

Peleg (1978) presents a necessary and sufficient condition for the non-emptiness of the core, by considering quota games. The purpose of this paper is to present a similar result for the non-emptiness of another solution concept introduced by Rubinstein (1980) and called the stability set. Received: 25 January 1999/Accepted: 6 July 1999     

Top-Pair and Top-Triple Monotonicity

We say that a social choice function (SCF) satisfies Top-k Monotonicity if the following holds. Suppose the outcome of the SCF at a preference profile is one of the top k-ranked alternatives for voter i. Let the set of these k alternatives be denoted by B. Suppose that i’s preference ordering changes in such a way that the set of first k-ranked alternatives remains the set B. Then the outcome at the new profile must belong to B. This definition of monotonicity arises naturally from considerations of set “improvements” and is weaker than the axioms of strong positive association and Maskin Monotonicity. Our main results are that if there are two voters then a SCF satisfies unanimity and Top-2 or Top-pair Monotonicity if and only if it is dictatorial. If there are more than two voters, then Top-pair Monotonicity must be replaced by Top-3 Monotonicity (or Top-triple Monotonicity) for the analogous result. Our results demonstrate that connection between dictatorship and “improvement” axioms is stronger than that suggested by the Muller–Satterthwaite result (Muller and Satterthwaite in J Econ Theory 14:412–418, 1977) and the Gibbard–Sattherthwaite theorem.     

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.     

Equality of Opportunity versus Equality of Opportunity Sets

We characterize two different approaches to the idea of equality of opportunity. Roemer’s social ordering is motivated by a concern to compensate for the effects of certain (non-responsibility) factors on outcomes. Van de gaer’s social ordering is concerned with the equalization of the opportunity sets to which people have access. We show how different invariance axioms open the possibility to go beyond the simple additive specification implied by both rules. This offers scope for a broader interpretation of responsibility-sensitive egalitarianism.     

The solidarity axiom for quasi-linear social choice problems

Recently, Moulin gave various axiomatic characterizations of solutions to quasi-linear social choice problems. He used a consistency axiom, which relates solutions for societies of different sizes, in addition to some basic axioms. In this paper, we introduce another axiom relating solutions for societies of different sizes, called the Solidarity Axiom. This axiom demands that when additional agents enter the scene, all of the original agents be affected in the same direction, i.e., all of them gain or all of them lose. Our main result is a complete characterization of solutions satisfying the solidarity axiom, in addition to Pareto optimality, anonymity and two normalization axioms. All solutions satisfying these five axioms are in the egalitarian spirit; each solution assigns to every agent an equal share of the surplus over some reference level, but uses a different method to compute the reference level. Then, using additional milder axioms, we give further characterization results concerning various subfamilies.I am extremely grateful to Professor William Thomson for his numerous comments which have considerably improved the exposition of the paper. I would like to thank Professor Hervé Moulin for helpful discussions and insightful comments. The comments from an editor and a referee of this journal are also gratefully acknowledged. I retain, however, full responsibility for any shortcommings     

Unemployment duration and the measurement of unemployment

We propose a framework in which the welfare bases of various unemployment measures available in the literature can be analyzed and a decomposition of these measures can be made into aggregate unemployment and the concentration of its distribution. A set of axioms for an unemployment measure is then introduced, which relate to the sensitivity of an unemployment measure to the inequality in the distribution of the unemployment. An unemployment measure is derived on the basis of these axioms that captures the distributional considerations in the measurement of unemployment.      

Extended Pareto rules and relative utilitarianism

This paper introduces the “Extended Pareto” axiom on Social Welfare Functions and gives a characterization of the axiom when it is assumed that the Social Welfare Functions that satisfy it in a framework of preferences over lotteries also satisfy the restrictions (on the domain and range of preferences) implied by the von-Neumann Morgenstern axioms. With the addition of 2 other axioms: “Anonymity” and a weak version of Arrow's Independence of Irrelevant Alternatives axiom: “Weak IIA” it is shown that there is a unique Social Welfare Function called “Relative Utilitarianism” that consists of normalising individual utilities between 0 and 1 and adding them. Received: 7 June 1994 / Accepted: 28 April 1997