Manipulation and the Pareto rule

This paper examines rules that map preference profiles into choice sets. There are no agendas other than the entire set of alternatives. A rule is said to be “manipulable” if there is a person i, and a preference profile, such that i prefers the choice set obtained when he is dishonest to the one obtained when he is honest. It is “nonmanipulable” if this can never happen. The paper indicates how preferences over choice sets might be sensibly derived from preferences over alternatives, and discusses seven different notions of manipulability associated with seven different assumptions about preferences over sets of alternatives. The paper has two sections of results. In the first I show that the Pareto rule, that is, the rule that maps preference profiles into corresponding sets of Pareto optima, is nonmanipulable in four of the seven senses of manipulability, and manipulable in three of them. In the second section, I examine this conjecture: If an arbitrary rule is nonmanipulable and nonimposed, and if indifference is disallowed, then every choice set must be contained in the set of Pareto optima. The conjecture is true under the strongest definition of nonmanipulability.     

Pareto violations of parliamentary voting systems

It has been known for a long time that many binary voting rules can select a Pareto dominated outcome, that is an outcome such that there exists some other alternative which is preferred by every voter. In this paper, we show that some of these rules can select an outcome Pareto dominated in a much stronger sense. Furthermore, our main results are concerned with the evaluation of the likelihood of Pareto dominated outcomes under four social choice rules commonly used in Parliaments or in committees. Given a set of four alternatives and a set of n individuals, we assume anonymous profiles and using analytical methods we compute the proportion of profiles at which the Pareto criterion is violated. Our results show that one should not be especially worried about the existence of the possibility as such. We are grateful to two anonymous referees for very helpful remarks and suggestions.     

Coalitional manipulation and the Pareto rule

This paper examines aggregation procedures that map profiles of individual preferences into choice sets. An aggregation procedure is said to be “manipulable by a coalition” if there is a group of individuals, and a preference profile, such that every member of the group prefers the choice set obtained when they are misrepresenting their preferences, to the one obtained when they are honest. We show that the Pareto rule, which is an aggregation procedure that maps profiles of individual preferences into corresponding sets of Pareto optima, is not manipulable by any coalition of individuals under various behavioural assumptions which relate preferences over choice sets to preferences over alternatives. The non-manipulability of the Pareto rule by a single individual follows as a special case under these behavioural assumptions.     

Nash implementation with partially honest individuals

We investigate the problem of Nash implementation in the presence of “partially honest” individuals. A partially honest player is one who has a strict preference for revealing the true state over lying when truthtelling does not lead to a worse outcome than that which obtains when lying. We show that when there are at least three individuals, all social choice correspondences satisfying No Veto Power can be implemented. If all individuals are partially honest and if the domain is separable, then all social choice functions can be implemented in strictly dominant strategies by a mechanism which does not use “integer/modulo games”. We also provide necessary and sufficient conditions for implementation in the two-person case, and describe some implications of these characterization conditions.     

The (sub-)optimality of the majority rule

We consider collective choice from two alternatives. Ex-ante, each agent is uncertain about which alternative she prefers, and may be uncertain about the intensity of her preferences. An environment is given by a probability distribution over utility vectors that is symmetric across agents and neutral across alternatives. In many environments, the majority voting rule maximizes agents? ex-ante expected utilities among all anonymous and dominant-strategy implementable choice rules. But in some environments where the agents? utilities are stochastically correlated, other dominant-strategy choice rules are better for all agents. If utilities are stochastically independent across agents, majority voting is ex-ante optimal among all anonymous and incentive-compatible rules. We also compare rules from an interim-viewpoint.     

Preference for flexibility in the absence of learning: the risk attitude effect

Individuals’ preferences over opportunity sets may display “preference for flexibility” which prescribes to gradually eliminate alternatives from a given set until a final choice is made. One rationale for this preference for flexibility is individuals’ incentive to postpone the final choice in order to better learn their underlying preferences over basic alternatives. In this paper we show that even in the absence of learning, preference for flexibility arises if individuals are risk-averse or, at least, are not very risk-seeking. Thus, individual’s attitude towards risk provides yet another rationale for preference for flexibility. One of our results is that in the absence of learning, risk-neutral as well as risk-averse individuals display the same, maximal preference for flexibility. We thank Han Bleichrodt, Robert Dur, Chaim Fershtman, Maarten Janssen, Peran van Reeven, Peter Wakker, and Timothy van Zandt for helpful comments to and inspiring discussions. We are very grateful to the anonymous referee for very constructive comments and suggestions.     

Weak Independence and Social Semi‐Orders

This paper provides variants of Arrow's impossibility theorem, which states that there exists no non‐dictatorial aggregation rule satisfying weak Pareto, independence of irrelevant alternatives and collective rationality. In this paper, independence of irrelevant alternatives and collective rationality are simultaneously relaxed. Weak independence is imposed instead of independence of irrelevant alternatives. Social preferences are assumed to satisfy the semi‐order properties of semi‐transitivity and the interval‐order property. We prove that there exists a vetoer when the number of alternatives is greater than or equal to six.     

An efficiency theorem for incompletely known preferences

There are n agents who have von Neumann-Morgenstern utility functions on a finite set of alternatives A. Each agent i's utility function is known to lie in the nonempty, convex, relatively open set Ui. Suppose L is a lottery on A that is undominated, meaning that there is no other lottery that is guaranteed to Pareto dominate L no matter what the true utility functions are. Then, there exist utility functions ui∈Ui for which L is Pareto efficient. This result includes the ordinal efficiency welfare theorem as a special case.     

A leximin characterization of strategy-proof and non-resolute social choice procedures

Summary. We characterize strategy-proof social choice procedures when choice sets need not be singletons. Sets are compared by leximin. For a strategy-proof rule g, there is a positive integer k such that either (i) the choice sets g(r) for all profiles r have the same cardinality k and there is an individual i such that g(r) is the set of alternatives that are the k highest ranking in i's preference ordering, or (ii) all sets of cardinality 1 to k are chosen and there is a coalition L of cardinality k such that g(r) is the union of the tops for the individuals in L. There do not exist any strategy-proof rules such that the choice sets are all of cardinality to k where . Received: November 8, 1999; revised version: September 18, 2001     

Preference revelation with a limited number of indifference classes

Suppose that g is a strategy-proof social choice rule on the domain of all profiles of complete and transitive binary relations that have exactly m indifference classes. If and the range of g has three or more members, then g is dictatorial. If m = 2, then for any set X of feasible alternatives, there exist non-dictatorial and strategy-proof rules that are sensitive to the preferences of every individual and which have X as range.     

A Measure Of The Trade‐Off Between Responsiveness and Nondictatorship for Arrovian Social Welfare Functions

If individuals are never indifferent between distinct alternatives then for any transitive‐valued social welfare function satisfying IIA, and any fraction t, either the set of pairs of alternatives that are socially ordered without consulting more than one individual's preferences comprises at least the fraction t of all pairs, or else the fraction of pairs that have their social ordering determined independently of everyone's s preferences exceeds, or is very close to, 1 ?t. The Pareto criterion is not imposed. (There is also a version of this result for the domain of preferences that admit indifference.)     

Second-best efficiency of allocation rules: strategy-proofness and single-peaked preferences with multiple commodities

We study strategy-proof allocation rules in economies with perfectly divisible multiple commodities and single-peaked preferences. In this setup, it is known that the incompatibility among strategy-proofness, Pareto efficiency and non-dictatorship arises in contrast with the Sprumont (Econometrica 59:509–519, 1991) one commodity model. We first investigate the existence problem of strategy-proof and second-best efficient rules, where a strategy-proof rule is second-best efficient if it is not Pareto-dominated by any other strategy-proof rules. We show that there exists an egalitarian rational (consequently, non-dictatorial) strategy-proof rule satisfying second-best efficiency. Second, we give a new characterization of the generalized uniform rule with the second-best efficiency in two-agent case.     

Self-Selection Consistent Functions

This paper studies collective choice rules whose outcomes consist of a collection of simultaneous decisions, each one of which is the only concern of some group of individuals in society. The need for such rules arises in different contexts, including the establishment of jurisdictions, the location of multiple public facilities, or the election of representative committees. We define a notion of allocation consistency requiring that each partial aspect of the global decision taken by society as a whole should be ratified by the group of agents who are directly concerned with this particular aspect. We investigate the possibility of designing efficient allocation consistent rules. We also explore whether such rules may in addition respect the Condorcet criterion. Journal of Economic Literature Classification Numbers: D7, D71.     

Separability and aggregation of equivalence relations

We provide axiomatic characterizations of two natural families of rules for aggregating equivalence relations: the family of join aggregators and the family of meet aggregators. The central conditions in these characterizations are two separability axioms. Disjunctive separability, neutrality, and unanimity characterize the family of join aggregators. On the other hand, conjunctive separability and unanimity characterize the family of meet aggregators. We show another characterization of the family of meet aggregators using conjunctive separability and two Pareto axioms, Pareto⁺ and Pareto^?. If we drop Pareto^?, then conjunctive separability and Pareto⁺ characterize the family of meet aggregators along with a trivial aggregator.     

Choice, preference and rationalizability in pareto’s theory of economic behavior

Pareto initiated the shift of economic theory from utility/preference to choice in order to fill a gap between theoretical and empirical economics. His suggestions for an empirically-oriented theory of choice were to be developed decades later in the literature on revealed preference and on the conditions of equivalence between preference-based and choice-based axiomatic structures. In the process, however, substantial departures from Pareto’s implicit design of the situation of choice were introduced. In this paper, the conditions for the rationalizability of choice are re-defined so that they fit the type of situation that Pareto had in mind. The result is that the main consistency axioms of standard choice theory have to give way to a different combination of axioms that concern decisiveness as well as consistency. (JEL:B13, D01, D11).     

Up/Downward Preference Aggregation

Two strict preference aggregation mechanisms are suggested. One determines the collective preference from the most preferred to the least preferred alternative using as inputs the individuals' most preferred alternatives once the alternatives already assigned a position in the collective preference have been removed. The other mechanism does the same the other way around, constructing the preference starting from the least preferred alternative and using the least preferred remaining alternatives as inputs. It is shown that if both procedures are assumed to generate the same collective preference then the resulting aggregation rule must be dictatorial or constant. Weaker versions of these procedures also yield a dictatorial aggregation rule when combined with mild versions of unanimity or surjectivity.     

The Supporting Line Property and the Additive Choice Function for Two-Dimensional Choice Problems

The paper provides an axiomatic characterization of the additive choice function using the additivity property. It is seen that, along with Pareto optimality, symmetry, and a supporting line property, the additivity axiom uniquely characterizes the additive choice function. The additive choice function as defined in this paper does not satisfy Nash's assumption of the independence of irrelevant alternatives. This raises the question of when a choice function is representable, an issue addressed in an appendix.     

Coalitional Strategy-Proofness in Economies with Single-Dipped Preferences and the Assignment of an Indivisible Object

We study two allocation models. In the first model, we consider the problem of allocating an infinitely divisible commodity among agents with single-dipped preferences. In the second model, a degenerate case of the first one, we study the allocation of an indivisible object to a group of agents. Our main result is the characterization of the class of Pareto optimal and coalitionally strategy-proof allocation rules. Alternatively, this class of rules, which largely consists of serially dictatorial components, can be characterized by Pareto optimality, strategy-proofness, and weak non-bossiness (in terms of welfare). Furthermore, we study properties of fairness such as anonymity and no-envy. Journal of Economic Literature Classification Numbers: D63, D71.     

A simple sufficient condition for strong implementation

In an important step forward Maskin [E. Maskin, Nash equilibrium and welfare optimality, Rev. Econ. Stud. 66 (1999) 23–38] showed that two properties – monotonicity and no veto power – are together sufficient for Nash implementation. In contrast to the vast literature that followed, this characterization has two major advantages: First, it is often easy to verify, and second, it has an elegant and simple interpretation. However, there does not exist a similar condition for social choice correspondences that are implementable in strong equilibrium. All existing characterizations are either hard to verify or apply only to comprehensive preference domains. In this paper we improve the situation by giving one such condition. Moreover, using well-known examples we show that this is a practical tool.     

Utility Representation of an Incomplete Preference Relation

We consider the problem of representing a (possibly) incomplete preference relation by means of a vector-valued utility function. Continuous and semicontinuous representation results are reported in the case of preference relations that are, in a sense, not “too incomplete.” These results generalize some of the classical utility representation theorems of the theory of individual choice and paves the way towards developing a consumer theory that realistically allows individuals to exhibit some “indecisiveness” on occasion. Journal of Economic Literature Classification Number: D11.