On the information about individual utilities used in social choice

This paper investigates the consequences of the requirement tha a social choice method should convey as much information as possible about the individuals' values. It is first shown that if interpersonal comparisons of utilities are excluded, then the only information about individual values that can be used in the context of social choice is ordinal information. It is then argued that Arrow's independence condition demands that only a part of the information about the individual preferences be used in the social choice. Finally, it is shown that the requirement of maximal information gives strong support to Smith's (1973) ‘separability’ condition. In combination with Smith's result this shows that the choice methods that convey the most information about individual preferences are the ‘generalized point systems’ (also called ‘scoring functions’ or ‘ranking systems’).     

Coalitional power structure without the pareto principle

In this paper we have extended Arrow's analysis to a framework where for any given profile of individual preference orderings the decision procedure specifies a non-trivial probability distribution over possible social orderings. We have demonstrated that if the social decision procedure satisfies certain probabilistic versions of weak independence of irrelevant alternatives, then it is characterized by a ‘power’ structure for all possible coalitions of individuals without assuming either the Pareto Principle or its antecedents. A generalised version of Arrow's impossibility theorem follows as a special case of our result. We have weakened Arrow's independence condition, and have shown the existence of a hierarchy of dictators without imposing the Pareto criterion.     

Nash equilibria for non-binary choice rules

We prove the existence of equilibria in games with players who employ abstract (non-binary) choice rules. This framework goes beyond the standard, transitive model and encompasses games where players have non-transitive preferences (e.g., skew-symmetric bilinear preferences).      

Arrow's theorem: unusual domains and extended codomains

In this paper we establish Arrow's theorem in a general ordinal case. When some configurations are allowed in the domain and if this domain is included in the codomain, the only social functions satisfying the independence condition and the weak Pareto Principle are the absolute dictatorships or the absolute oligarchies.     

The categorical approach to social choice theory

A number of different problems in social choice theory are given a common treatment in the framework of category theory. A social choice function is a map μ from a set Hom(A, PX) of profiles - in applications functions from a set A of agents to the set PX of linear orders on a set X of alternatives - to a set GPX the nature of which depends on the application. The main theorem established is applied to social welfare functions and strategy-proof voting schemes.     

Constitutional implementation of social choice correspondences

A game form constitutionally implements a social choice correspondence if it implements it in Nash equilibrium and, moreover, the associated effectivity functions coincide. This paper presents necessary and sufficient conditions for a unanimous social choice correspondence to be constitutionally implementable, and sufficient and almost necessary conditions for an arbitrary (but surjective) social choice correspondence to be constitutionally implementable. It is shown that the results apply to interesting classes of scoring and veto social choice correspondences.     

Condorcet — a man of the avant-garde

In 1785 Condorcet proposed a method to aggregate qualitative data, but until very recently this method was attributed to contemporary authors and its importance completely neglected.     

On a compromise social choice correspondence

This paper analyzes the compromise social choice correspondence derived from the τ-value of digraph games. Monotonicity of this correspondence is shown. A connection between several properties of social choice correspondences based on game theoretical solutions and game theoretical properties of the underlying solutions is given. Applications to several game theoretic solutions are provided.     

Symmetric social choices and collective rationality

A social choice function C defined on the m-element subsets of a set X for n voters assigns a non-empty subset C(A,R) of A to each pair (A,R) in which ¦A¦ = m,A X, and R is an n-tuple of voter preference orders on X. When n > 2, 3 m < ¦X¦, and C satisfies natural conditions of independence, symmetry among voters and alternatives, and collective rationality, it is proved that C is completely indecisive in the sense that C(A,R) = A for all (A,R). This non-binary result complements a binary (m = 2) result proved by Hanson. It suggests that there is a fundamental incompatibility between collective rationality and conditions designed to assure equitable treatment of voters and of alternatives.     

The median procedure in cluster analysis and social choice theory

Classical approachs for fitting and aggregation problems, specially in cluster analysis, social choice theory and paired comparisons methods, consist in the minimization of a remoteness function between relational data and a relational model. The notion of median, with its algebraic, metric, geometrical and statistical aspects, allow a unified treatment of many of base problems. Properties of median procedures are organized according to four directions: stabilities and axiomatic characterizations; Arrow-like properties; combinatorial properties; effective computational possibilities. Finally, interesting mathematical problems, related to the notion of median are surveyed.     

Random utility models with all choice probabilities expressible as ‘functions’ of the binary choice probabilities

Given the absence of non-trivial decision rules which are strategy-proof in the Gibbard-Satterthwaite sense, Peleg and Sen have suggested that one might search for decision rules under which there will be always some Nash equilibrium yielding the same outcome as would be yielded by honest voting. This paper shows that most decision rules based on pairwise comparison will fail to satisfy the requirement of Peleg and Sen if one imposes the additional condition that the relevant Nash equilibrium should consist of undominated voting strategies only (a most reasonable condition in the absence of cooperation). It is also shown that under most decision rules based on pairwise comparisons, it will not be possible either to ensure that the outcome of honest voting will be always an outcome of sophisticated voting (in Farquharson's sense) or to ensure that the outcome of honest voting will be always a core outcome.     

Modelling a social dilemma of mode choice based on commuters’ expectations and social learning

This study attempts to apply an agent-based approach to modelling a social dilemma of travel mode choice considering psychological and sociological aspects. A traveller is modelled to have expectations, which shows the traveller’s beliefs about the influence of other group members on his action, as decision-making rules. Social interaction using a group-based interaction is hypothesized to be important. We apply an imitation game based on social learning mechanisms to the model. Two kinds of mechanism are used: payoff-biased and conformist transmission. The model reveals the conditions that make cooperation as a possible outcome are optimistic bandwagon expectations, group-based interactions, and strong conformist transmissions.     

对非二元性选择函数和独立决定性进一步的研究

进一步分析了非二元性选择函数的相关条件与Quesada提出的t元相关独立决定性条件,在非二元性选择环境下,用Quesada提出的条件弱化了Arrow不可能定中无关方案独立性条件,所得结论表明Arrow不可能定中无关方案独立性条件和连同性可以同时弱化.     

On the Deductive Strength of Various Distributivity Axioms for Boolean Algebras in Set Theory

In this article, we shall show the generalized notions of distributivity of Boolean algebras have essential relations with several axioms and properties of set theory, say the Axiom of Choice, the Axiom of Dependence Choice, the Prime Ideal Theorems, Martin's axioms, Lebesgue measurability and so on.     

Multi-utilitarianism in two-agent quasilinear social choice

We introduce a new class of rules for resolving quasilinear social choice problems. These rules extend those of Green [9]. We call such rules multi-utilitarian rules. Each multi-utilitarian rule is associated with a probability measure over the set of weighted utilitarian rules, and is derived as the expectation of this probability. These rules are characterized by the axioms efficiency, translation invariance, transfer monotonicity, continuity, and additivity. By adding recursive invariance, we obtain a class of asymmetric rules generalizing those Green characterizes. A multi-utilitarian rule satisfying strict monotonicity has an associated probability measure with full support.I would like to thank Youngsub Chun, Federico Echenique, Jerry Green, Biung-Ghi Ju, William Thomson, and Walter Trockel for comments and discussions. Two anonymous referees also provided comments that proved very useful. All errors are my own.     

Restricted domains,arrow-social welfare functions and noncorruptible and non-manipulable social choice correspondences: The case of private alternatives

An n-person social choice problem is considered in which the alternatives are n dimensional vectors, with the ith component of such a vector being the part of the alternatives affecting individual i alone. Assuming that individuals are selfish (individual i must be indifferent between any two alternatives with the same components), that they may be indifferent among alternatives and that each individual may choose his preferences out of a different set of permissible preferences, we prove that any set of restricted domains of preferences admits an n person non-dictatorial Arrow-type social welfare function if and only if it admits a two-person Arrow-type social welfare function: we characterize all the sets of restricted domains of preferences which admit two-person Arrow-type social welfare functions (and therefore also admit n-person Arrow-type social welfare functions) and then we prove that we also characterized all the sets of restricted domains of preferences which admit nondictatorial, nonmanipulable, noncorruptible and rational social choice correspondences.     

On uncertainty bounds and growth estimates for fractional fourier transforms

Gelfand–Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand–Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space.