Measures of freedom

According to the capability approach and more orthodox theories of individual well-being, freedom is an important ingredient of individual welfare. In operationalising these theories, therefore, it is crucial to have a practical measure of freedom. This paper analyses the measures of freedom proposed by Pattanaik and Xu, Steiner and Carter, Kramer, and Rosenbaum. We provide two interpretations of the first three measures. It is shown that one of these interpretations can be useful in empirical applications, while the other, together with additional arguments, considerably weakens Kramer's measure of freedom. Finally, it is shown that Rosenbaum's measure is not satisfactory as well because it yields counterintuitive results.     

A note on the redundancy of an axiom in the Pattanaik-Peleg characterization of the lexicographic maximin extension

P. K. Pattanaik and B. Peleg have provided an axiomatic characterization of the lexicographic maximin extension of an ordering over a finite set to an ordering over the set of all non-empty subsets of that set. This note shows that one of their axioms, namely the Union axiom, is redundant, thus obtaining a more economical characterization composed of the remaining four axioms (which are independent by virtue of the examples given by Pattanaik and Peleg).     

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.     

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.     

Fuzzy choice functions

The R-greatest and maximal sets of standard choice theory are extended to fuzzy R-greatest and fuzzy maximal sets. Unlike the precise counterparts of these concepts, these two sets do not in general coincide when preferences are reflexive and connected. A stronger than usual version of connectedness under which the two sets are equal is provided. The concept of a fuzzy choice function is introduced and conditions under which a fuzzy choice function may be rationalized as a fuzzy R-greatest or a fuzzy maximal set are discussed. Rationalizability with transitive and weakly transitive fuzzy preference relations is also considered.We are indebted to Professor P. K. Pattanaik for his comments on an earlier version of this paper. We also wish to acknowledge comments made by an anonymous referee from which this paper has benefited greatly. The usual caveat about errors applies.     

Quasi-decisiveness, quasi-ultrafilter, and social quasi-orderings

The aim of the present paper is to provide an axiomatic analysis of incomplete social judgments. In particular, we clarify the underlying power structure of Arrovian collective choice rules when social preferences are allowed to be incomplete. We propose the concept of quasi-decisiveness and investigate the properties of the collection of quasi-decisive sets associated with an Arrovian collective choice rule. In the course of this, we offer a series of applications.     

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     

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     

Stable sets and standards of behaviour

In this paper we present a constructive, behavioural and axiomatic approach to the notion of a stable set as a model of the standard of behaviour of a social organisation. The socially stable set we introduce is a generalisation of the von Neumann-Morgenstern stable set. In contrast with the original version, our stability concept is always solvable. The standard of behaviour, reflecting the established conceptual order of a society or organisation, emerges from a dominance relation on alternative conceptions that are relevant with regard to a certain issue. This common social choice phenomenon, that permeates our societies and organisations, we have tried to clarify. Two axiomatic characterisations as well as a construction algorithm for socially stable sets are presented. These characterisations are based on behavioural postulates regarding the individual or collective strategic behaviour of effective sets. Relations between socially stable sets and other notions of stability are discussed. Received: 4 May 1998/13 March 2000     

Libertarian collective decision-making: A new framework

A model of collective decision-making is presented which enables us to analyze the allocation of individual rights and liberties. The model broadens the traditional social choice framework. In social choice theory it is usually assumed that individuals have preferences over a set of feasible alternatives and that society reaches a decision on the basis of those preferences only. In the model presented in this paper, decision procedures do not only take information about individual preferences as input, but also information about individual freedom. The specific decision procedures we present and analyze can be characterized as libertarian.This paper was written while I was a visitor at the University of California, Riverside. The visit was made possible by financial support of the Netherlands Organization of Scientific Research (NWO). I am extremely grateful to Prasanta Pattanaik for many stimulating discussions on the subject. For helpful comments I also thank Ad van Deemen, Grahame Lock, Huib Pellikaan and Marcel Wissenburg. Finally, I express my thanks to Wulf Gaertner and two anonymous referees.     

Freedom to veto

Several sets of axioms have been proposed to characterize rankings of opportunity sets in terms of freedom of choice. In these models it has been assumed that being in a position to choose from more options is preferred to having fewer options. We tested the empirical validity of that assumption experimentally. Combining a dictator game (a no-choice situation for the receiver) and an ultimatum game (the receiver can choose between two options) we investigated whether receivers prefer to have some freedom of choice (in the ultimatum game) over having no freedom of choice (in the dictator game) even in the presence of monetary incentives to choose otherwise. The experimental results show that a strong majority of players is not willing to give up the option to veto without monetary incentives to do so. However, players are often willing to trade their freedom to veto even for a small bonus. The higher the monetary incentives the more players give up their veto power.     

Contraction consistent stochastic choice correspondence

We model a general choice environment via probabilistic choice correspondences, with (possibly) incomplete domain and infinite universal set of alternatives. We offer a consistency restriction regarding choice when the feasible set contracts. This condition, ‘contraction consistency’, subsumes earlier notions such as Chernoff’s Condition, Sen’s α and β, and regularity. We identify a restriction on the domain of the stochastic choice correspondence (SCC), under which contraction consistency is equivalent to the weak axiom of revealed preference in its most general form. When the universal set of alternatives is finite, this restriction is also necessary for such equivalence. Analogous domain restrictions are also identified for the special case where choice is deterministic but possibly multi-valued. Results due to Sen (Rev Econ Stud 38:307–317, 1971) and Dasgupta and Pattanaik (Econ Theory 31:35–50, 2007) fall out as corollaries. Thus, conditions are established, under which our notion of consistency, articulated only in reference to contractions of the feasible set, suffices as the axiomatic foundation for a general revealed preference theory of choice behaviour.     

Distribution of coalitional power in randomized multi-valued social choice

This paper considers the distribution of coalitional influence under probabilistic social choice functions which are randomized social choice rules that allow social indifference by mapping each combination of a preference profile and a feasible set to a social choice lottery over all possible choice sets from the feasible set. When there are at least four alternatives in the universal set and ex-post Pareto optimality, independence of irrelevant alternatives and regularity are imposed, we show that: (i) there is a system of additive coalitional weights such that the weight of each coalition is its power to be decisive in every two-alternative feasble set; and (ii) for each combination of a feasible proper subset of the universal set and a preference profile, the society can be partioned in such a way that for each coalition in this partition, the probability of society's choice set being contained in the union of the best sets of its members is equal to the coalition's power or weight. It is further shown that, for feasible proper subsets of the universal set, the probability of society's choice set containing a pair of alternatives that are not jointly present in anyone's best set is zero. Our results remain valid even when the universal set itself becomes feasible provided some additional conditions hold. Received: 10 May 1999/Accepted: 18 June 2000 I would like to thank Professor Prasanta Pattanaik for suggesting to me the line of investigation carried out in this paper. I am solely responsible for any remaining errors and omissions.     

Reference functions and possibility theorems for cardinal social choice problems

 In this paper, we provide axiomatic foundations for social choice rules on a domain of convex and comprehensive social choice problems when agents have cardinal utility functions. We translate the axioms of three well known approaches in bargaining theory (Nash 1950; Kalai and Smorodinsky 1975; Kalai 1977) to the domain of social choice problems and provide an impossibility result for each. We then introduce the concept of a reference function which, for each social choice set, selects a point from which relative gains are measured. By restricting the invariance and comparison axioms so that they only apply to sets with the same reference point, we obtain characterizations of social choice rules that are natural analogues of the bargaining theory solutions. Received: 8 August 1994/Accepted: 12 February 1996     

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     

Choice,freedom, and freedom of choice

This paper argues in favour of a distinction between freedom and freedom of choice – a distinction that economists and political philosophers have so far either ignored or drawn wrongly. Drawing the distinction correctly may help to resolve a number of disputes in contemporary political philosophy and non-welfarist normative economics regarding the so-called preference-based account of freedom and the relevance, to judgements about freedom, of degrees of similarity between agents options. The paper begins by setting out three much discussed axioms for the measurement of freedom (of choice?) originally put forward by Pattanaik and Xu. It is suggested that the problems these axioms give rise to can be solved by distinguishing correctly between freedom and freedom of choice. The paper then sets out definitions of freedom, choice and freedom of choice, justifying these in philosophical terms and arguing their superiority to alternative definitions. Finally, on the basis of these definitions and with reference to Pattanaik and Xus axioms, it is shown that an agent can enjoy freedom without enjoying freedom of choice, and that she can enjoy an increase in one of these without enjoying an increase in the other. For their helpful comments on earlier drafts of this paper, I should like to thank Keith Dowding, Martin van Hees, Matthew Kramer, Marco Negri, Serena Olsaretti, Olof Page, Mario Ricciardi, Alan Ritter, Hillel Steiner, Kotaro Suzumura and audiences at seminars in Manchester, Palermo, Pavia and Oxford. I am grateful to the Italian Ministry for Higher Education (MURST) and the Italian National Research Council (CNR) for funding that facilitated the completion of the paper.     

On Preference, Freedom and Diversity

We study the problem of ranking sets of options in terms of freedom of choice. We propose a framework in which both the diversity of the options and the preferences of the agent over the options do play a role. We formulate some axioms that reflect these two aspects of freedom and we study their logical implications. Two different criteria for ranking sets are characterized, which generalize some of the rankings proposed so far in the literature.     

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.     

On real economic freedom

Among the various concepts of freedom important for economics, ranking or measuring the freedom of choice provided by budget sets has an important place. The volume ranking has strange properties and cannot be justified by unit invariance and symmetry. The pointed distance (of the budget hyperplane from the origin along some line) provides a measure or ranking that coincides with the standard “purchasing power” or real income. The linear price index is practically unavoidable for measuring or ranking freedom. This is applied to the determination of income distribution and taxation implied by the equal freedom of choice of different domains. Concepts of equal or compared potential freedoms and utility-freedom relate freedom analysis to the basic classical concepts of fair allocation (equity-no-envy, egalitarian equivalence, etc.). The crucial difference between the two opposite concepts of invariance in comparisons is emphasized.     

Measuring freedom of choice: An alternative view of a recent literature

A recent literature has emerged in social choice theory which attributes intrinsic importance to freedom in the evaluation of states of affairs. The literatures philosophical basis lies in Berlins notion of positive liberty. Accordingly, axiomatic measures of availability of choices are developed and the information they convey used for ranking states on the basis of the extent of liberty they offer to individuals. I am grateful to Nancy Cartwright, Dan Hausman and Marco Del Seta for their suggestions to previous drafts of this paper. The usual caveat applies. The support of the Centre for Philosophy of Natural and Social Science at the London School of Economics is also acknowledged.