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.     

Characterization of self-selective social choice functions on the tops-only domain

Self-selectivity is a new kind of consistency pertaining to social choice rules. It deals with the problem of whether a social choice rule selects itself from among other rival such rules when a society is also to choose the choice rule that it will employ in making its choice from a given set of alternatives. Koray [3] shows that a neutral and unanimous social choice function is universally self-selective if and only if it is dictatorial. In this paper, we confine the available social choice functions to the tops-only domain and examine whether such restriction allow us to escape the dictatoriality result. A neutral, unanimous, and tops-only social choice function, however, turns out to be self-selective relative to the tops-only domain if and only if it is top-monotonic, and thus again dictatorial. Received: 8 October 2001/Accepted: 4 June 2002     

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.     

The consistency of choice using experimental design

Conjoint designs provide an appealing context for investigating the consistency of consumer choice. This short article demonstrates that in principle experimental designs can be an effective mechanism for studying the consistency of consumer choice, provided one includes some pairs of attribute combinations which are dominant, such that one is worse or better off on every dimension. Using a preference design from a study on attendance at an international exposition in Australia we identify dominance in a subset of all possible pairs, and use this in two test of choice consistency to identify the extent of illogicality in choice. Discrete choice analysis is used to see if inconsistency is linked to population subgroups. The evidence can be used to point to subgroups who may have difficulty in comprehending controlled experimental instruments, and/or are not rational in the sense defined by our models of behaviour, and who may require special consideration in prediction of choice.     

On the relative strengths of consistency conditions on choice functions

The paper compares a variety of well known set theoretic consistency conditions on choice functions with the two that have been introduced by the author in an earlier paper. The two conditions are non-trivial generalisations of choice functional transitivity and quasi-transitivity respectively. The paper shows if the two conditions are individually stronger than or weaker than or logically independent of 14 other conditions including the rationalisability by an ordering, path independence, the conditions and , idemmorphism, absorbance and superset. The relative strengths of all the 16 conditions have been summed up at the end by an implication diagram.I am indebted to Prof. Satish Jain for the guidence he gave me to write the Ph. D. dissertation in which the results of this paper appear and to the referee for his comments on the earlier version of this paper.     

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     

Self-selective social choice functions

It is not uncommon that a society facing a choice problem has also to choose the choice rule itself. In such situations, when information about voters’ preferences is complete, the voters’ preferences on alternatives induce voters’ preferences over the set of available voting rules. Such a setting immediately gives rise to a natural question concerning consistency between these two levels of choice. If a choice rule employed to resolve the society’s original choice problem does not choose itself, when it is also used for choosing the choice rule, then this phenomenon can be regarded as inconsistency of this choice rule as it rejects itself according to its own rationale. Koray (Econometrica 68: 981–995, 2000) proved that the only neutral, unanimous universally self-selective social choice functions are the dictatorial ones. Here we introduce to our society a constitution, which rules out inefficient social choice rules. When inefficient social choice rules become unavailable for comparison, the property of self-selectivity becomes more interesting and we show that some non-trivial self-selective social choice functions do exist. Under certain assumptions on the constitution we describe all of them.     

Characterization of social choice sets in terms of individuals' maximal sets: The fixed agenda framework

In this paper we analyze the explicit representation of fixed agenda social choice correspondences under different rationality assumptions (independence, Pareto optimality, etc.). It is well known that, under some of these assumptions, the existence of dictators, oligarchies or individuals with veto power can be proven ([6] and [10]); but only a partial characterization of the social choice set is obtained. We establish a relationship between the social choice set and the individuals' maximal sets, which explicitly describes a fixed agenda social choice correspondence that satisfies these rationality assumptions. Received: 30 December 1997/Accepted: 20 December 1999     

Internal consistency of choice,Sen and the spirit of revealed preferences: A behaviorist approach

Until his first papers Sen carried on an argument against the revealed preferences theory and internal consistency of choice. First, the paper presents some of the paradoxes used by Sen. Second, it underlines problems which could occur in using intention or personal meaning to derive preferences. Then, as paradoxes frequently arise from misinterpretation of behaviour, it is suggested to read preferences through a redefinition of the act of choice. Finally, a way of overcoming problems and keeping the spirit of revealed preferences theory is suggested. However, this way imposes some boundaries to the extent of the revealed preferences theory.     

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.     

Manipulating an aggregation rule under ordinally fuzzy preferences

It is well known that many aggregation rules are manipulable through strategic behaviour. Typically, the aggregation rules considered in the literature are social choice correspondences. In this paper the aggregation rules of interest are social welfare functions (SWFs). We investigate the problem of constructing a SWF that is non-manipulable. In this context, individuals attempt to manipulate a social ordering as opposed to a social choice. Using techniques from an ordinal version of fuzzy set theory, we introduce a class of ordinally fuzzy binary relations of which exact binary relations are a special case. Operating within this family enables us to prove an impossibility theorem. This theorem states that all non-manipulable SWFs are dictatorial, provided that they are not constant. This theorem uses a weaker transitivity condition than the one in Perote-Peña and Piggins (J Math Econ 43:564–580, 2007), and the ordinal framework we employ is more general than the cardinal setting used there. We conclude by considering several ways of circumventing this impossibility theorem.     

Aggregation of preferences with a variable set of alternatives

A social choice correspondence called the Essential set is studied with the help of an axiom called Cloning Consistency. Cloning consistency is the requirement that the formal choice rule be insensitive to the replication of alternatives. The Essential set is the support of the optimal mixed strategies in a symmetric two-party electoral competition game. Received: 24 March 1998/Accepted: 3 March 1999     

Rational budgeters in the theory of social choice

The purpose of this paper is to explore duality in the theory of social choice. As application Arrow's Impossibility Theorem and another impossibility theorem using the notion of positive responsiveness are chosen. It will be seen that we can establish notions and theorems which are symmetric to the original ones. However, if we establish impossibility theorems when rational behaviour is described by budget correspondences and not by choice correspondences, we need not assume that every subset of X (a family of alternatives) with cardinality 2 is a budget set. Therefore the dual theorems also may hold for families of competitive budget sets. It will also be shown that although the underlying preferences on X need not be acyclic, local decisiveness on budget sets may lead to global decisiveness on these sets.     

Preferences,rational choices and economic valuation: Some empirical tests

This study focuses on the respondent rationality hypothesis, usually assumed to be true in Discrete Choice Experiments. We examine lexicographic preferences, the influence of consistency, and the role of task complexity in the individual choice process. To this end, we carry out rationality tests in a survey on forest recreation. Results show that choice set orderings do not impact on choice probability. If a violation of continuity or consistency axioms does impact on choice probability, the Willingness-To-Pay estimators calculated using the total sample and the sub-samples of “irrational” respondents are not significantly different. This serves as a basis for discussing the traditional concept of rationality.     

Single-Crossing, Strategic Voting and the Median Choice Rule

This paper studies the strategic foundation of the Representative Voter Theorem (Rothstein in: Pub Choice 72:193–212, 1991), also called the “second version” of the Median Voter Theorem. As a by-product, it also considers the existence of strategy-proof social choice functions over the domain of single-crossing preferences. The main result shows that single-crossing constitutes a domain restriction over the real line that allows not only majority voting equilibria, but also non-manipulable choice rules. In particular, this is true for the median rule, which is found to be group strategic-proof over the full set of alternatives and over every nonempty subset. In addition, the paper also examines the relation between single-crossing and order-restriction. And it uses this relation together with the strategy-proofness of the median rule to prove that the outcome predicted by the Representative Voter Theorem can be implemented in dominant strategies through a simple mechanism. This mechanism is a two-stage voting procedure in which, first, individuals select a representative among themselves, and then the winner chooses a policy to be implemented by the planner.     

The core of social choice problems with monotone preferences and a feasibility constraint

We consider collective choice with agents possessing strictly monotone, strictly convex and continuous preferences over a compact and convex constraint set contained in ₊^k . If it is non-empty the core will lie on the efficient boundary of the constraint set and any policy not in the core is beaten by some policy on the efficient boundary. It is possible to translate the collective choice problem on this efficient boundary to another social choice problem on a compact and convex subset of ₊^c (c<k) with strictly convex and continuous preferences. In this setting the dimensionality results in Banks (1995) and Saari (1997) apply to the dimensionality of the boundary of the constraint set (which is lower than the dimensionality of the choice space by at least one). If the constraint set is not convex then the translated lower dimensional problem does not necessarily involve strict convexity of preferences but the dimensionality of the problem is still lower. Broadly, the results show that the homogeneity afforded by strict monotonicity of preferences and a compact constraint set makes generic core emptyness slightly less common. One example of the results is that if preferences are strictly monotone and convex on ² then choice on a compact and convex constraint exhibits a version of the median voter theorem.I thank Donald Saari for helpful comments on an earlier version of this paper.     

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     

Voting games,indifference, and consistent sequential choice rules

We present four choice functions which characterize the stationary points of sequential search rules derived from a preference relation over outcomes. These functions are contrasted with others in the literature in terms of narrowness of choice as well as their ability to satisfy certain normative and consistency conditions, and it is shown how two of these sets arise as the set of equilibrium outcomes of a voting game under different tie-breaking assumptions.We would like to thank Nicholas Miller for introducing us and two anonymous referees for valuable comments and suggestions.     

On probabilistic rationalizability

This paper examines the problem of rationalizability of a choice system without imposing restrictions on the universal set of alternatives and the family of feasible sets. A choice system is the probabilistic counterpart of a choice function which is basic to much of the theory of individual choice behaviour in economics. We provide a necessary and sufficient condition for a choice system to be induced by a probability measure on a set of utility functions. I would like to thank the editor and two anonymous referees for helpful comments.