Strategy-proof division with single-peaked preferences and individual endowments

We consider the problem of (re)allocating the total endowment of an infinitely divisible commodity among agents with single-peaked preferences and individual endowments. We propose an extension of the so-called uniform rule and show that it is the unique rule satisfying Pareto optimality, strategy-proofness, reversibility, and an equal-treatment condition. The resulting rule turns out to be peaks-only and individually rational: the allocation assigned by the rule depends only on the peaks of the preferences, and no agent is worse off than at his individual endowment. Received: 8 September 1995/Accepted: 30 October 1996     

Incentive compatibility and feasibility constraints in housing markets

We analyze centralized housing markets under the existence of feasibility constraints on the number of agents and objects involved in the exchanges. We focus on an incomplete information setting where only the information about how each agent ranks her endowment is private. We show that under non-degenerate ex-ante probability distributions over preference profiles, no rule satisfies the joint requirements of individual rationality, (constrained) efficiency, and ordinally Bayesian incentive compatibility.     

An impossibility theorem for matching problems

This paper studies the possibility of strategy-proof rules yielding satisfactory solutions to matching problems. Alcalde and Barberá (Econ Theory 4:417–435, 1994) and Sönmez (Econ Des 1:365–380, 1994) show that efficient and individually rational matching rules are manipulable. We pursue the possibility of strategy-proof matching rules by relaxing efficiency to the weaker condition of respect for unanimity. First, we prove that a strategy-proof rule exists that is individually rational and respects unanimity. However, this rule is unreasonable in the sense that mutually best pairs of agents are matched on only rare occasions. In order to explore the possibility of better matching rules, we introduce a natural condition of “respect for 2-unanimity.” Respect for 2-unanimity states that a mutually best pair of agents should be matched, and an agent wishing to being unmatched should be unmatched. Our second result is negative. Secondly, we prove that no strategy-proof rule exists that respects 2-unanimity. This result implies Roth (Math Oper Res 7:617–628, 1962; J Econ Theory 36:277–288, 1985) showing that stable rules are manipulable.     

Allocating multiple estates among agents with single-peaked preferences

We consider the problem of allocating multiple social endowments (estates) of a perfectly divisible commodity among a group of agents with single-peaked preferences when each agent’s share can come from at most one estate. We inquire if well-known single-estate rules, such as the Uniform rule, the Proportional rule or the fixed-path rules can be coupled with a matching rule so as to achieve efficiency in the multi-estate level. On the class of problems where all agents have symmetric preferences, any efficient single-estate rule can be extended to an efficient multi-estate rule. If we allow asymmetric preferences however, this is no more the case. For nondictatorial single-estate rules that satisfy efficiency, strategy proofness, consistency, and resource monotonicity, an efficient extension to multiple estates is impossible. A similar impossibility also holds for single-estate rules that satisfy efficiency, peak-only, and a weak fairness property. We would like to express our gratitude to Bhaskar Dutta, Semih Koray, Hervé Moulin, and Yuntong Wang as well as an associate editor and two anonymous referees of this journal for detailed comments and suggestions. We also thank the seminar participants at Bilkent University, Indian Statistical Institute, Bilgi University, University of Warwick, ASSET 2003, and BWED XXVI.     

Markovian assignment rules

We analyze dynamic assignment problems where agents successively receive different objects (positions, offices, etc.). A finite set of n vertically differentiated indivisible objects are assigned to n agents who live n periods. At each period, a new agent enters society, and the oldest agent retires, leaving his object to be reassigned. We define independent assignment rules (where the assignment of an object to an agent is independent of the way other objects are allocated to other agents), efficient assignment rules (where there does not exist another assignment rule with larger expected surplus), and fair assignment rules (where agents experiencing the same circumstances have identical histories in the long run). When agents are homogenous, we characterize efficient, independent and fair rules as generalizations of the seniority rule. When agents draw their types at random, we prove that independence and efficiency are incompatible, and that efficient and fair rules only exist when there are two types of agents. We characterize two simple rules (type-rank and type-seniority) which satisfy both efficiency and fairness criteria in dichotomous settings.     

A characterization of the uniform rule with several commodities and agents

We consider a problem of allocating infinitely divisible commodities among a group of agents. More specifically, there are several commodities to be allocated and agents have continuous, strictly convex, and separable preferences. We establish that a rule satisfies strategy-proofness, unanimity, weak symmetry, and nonbossiness if and only if it is the uniform rule. This result extends to the class of continuous, strictly convex, and multidimensional single-peaked preferences.     

Probabilistic allocation rules and single-dipped preferences

We consider the problem of allocating an infinitely divisible endowment among a group of agents with single-dipped preferences. A probabilistic allocation rule assigns a probability distribution over the set of possible allocations to every preference profile. We discuss characterizations of the classes of Pareto-optimal and strategy-proof probabilistic rules which satisfy in addition replacement-domination or no-envy. Interestingly, these results also apply to problems of allocating finitely many identical indivisible objects – to probabilistic and to deterministic allocation. Received: 23 November 1998/Accepted: 20 October 2000     

Topological aggregation of preferences: the case of a continuum of agents

 This paper studies the topological approach to social choice theory initiated by G. Chichilnisky (1980), extending it to the case of a continuum of agents. The social choice rules are continuous anonymous maps defined on preference spaces which respect unanimity. We establish that a social choice rule exists for a continuum of agents if and only if the space of preferences is contractible. We provide also a topological characterization of such rules as generalized means or mathematical expectations of individual preferences. Received: 30 November 1994/Accepted: 22 April 1996     

Resource-monotonic solutions to the problem of fair division when preferences are single-peaked

We consider the problem of fairly allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. We search for solutions satisfying resource-monotonicity, the requirement that all agents be affected in the same direction when the amount to divide changes. Although there are resource-monotonic selections from the Pareto solution, there are none satisfying the distributional requirements of no-envy or individual rationality from equal division. We then consider the weakening of resource-monotonicity obtained by allowing only changes in the amount to divide that do not reverse the direction of the inequality between the amount to divide and the sum of the preferred amounts. We show that there is essentially a unique selection from the solution that associates with each economy its set of envy-free and efficient allocations satisfying this property of one-sided resource-monotonicity: it is the uniform rule, a solution that has played a central role in previous analyses of the problem.     

The division problem with voluntary participation

The division problem consists of allocating a given amount of a homogeneous and perfectly divisible good among a group of agents with single-peaked preferences on the set of their potential shares. A rule proposes a vector of shares for each division problem. The literature has implicitly assumed that agents will find acceptable any share they are assigned to. In this article we consider the division problem when agents’ participation is voluntary. Each agent has an idiosyncratic interval of acceptable shares where his preferences are single-peaked. A rule has to propose to each agent either to not participate or an acceptable share because otherwise he would opt out and this would require to reassign some of the remaining agents’ shares. We study a subclass of efficient and consistent rules and characterize extensions of the uniform rule that deal explicitly with agents’ voluntary participation.     

The replacement principle for the provision of multiple public goods on tree networks

This article considers the provision of two public goods on tree networks where each agent has a single-peaked preference. We show that if there are at least four agents, then no social choice rule exists that satisfies efficiency and replacement-domination. In fact, these properties are incompatible, even if agents’ preferences are restricted to a smaller domain of symmetric single-peaked preferences. However, for rules on an interval, we prove that Miyagawa’s (Soc Choice Welf 18:527–541, 2001) characterization that only the left-peaks rule and the right-peaks rule satisfy both of these properties also holds on the domain of symmetric single-peaked preferences. Moreover, if agents’ peak locations are restricted to either the nodes or the endpoints of trees, rules exist on a subclass of trees. We provide a characterization of a family of such rules for this tree subclass.     

Feasible Mechanisms in Economies with Type-Dependent Endowments

We propose two classes of allocation games for N.T.U. and T.U. exchange economies in which initial endowments and preferences depend on the agents’ private information. In both models, agents make non-verifiable claims about their types and effective deposits of consumption goods, which are redistributed by the planner. In a W-allocation game, the agents can withhold part of their endowment, namely consume whatever they do not deposit. In a D-allocation game, the agents can just destroypart of their endowment. W- and D- incentive compatible (I.C.) direct allocation mechanisms ask every agent to reveal his type and to make a deposit consistent with his reported type. The revelation principle holds in full generality for D-I.C. mechanisms but some care is needed for W-I.C. mechanisms. We further investigate the properties of both classes of mechanisms under common assumptions like non-exclusive information and/or constant aggregate endowment. In T.U. economies, W-I.C. and D-I.C. mechanisms are ex ante equivalent.     

Inefficient committees: small elections with three alternatives

We consider small committees which have to elect one of three alternatives using the simple plurality rule. Committee members have common, state-dependent preferences and receive imprecise private signals about the state of nature prior to the election. We are interested in whether the committee decision is efficient, that is whether the probability with which the committee elects the correct alternative is higher than the probability with which one single individual alone—on behalf of the others—would. It has been shown that there exists a unique efficient equilibrium in elections with two alternatives. We show that this result does not extend to elections with more alternatives. Multiple equilibria may exist for the same committee, and there may be both efficient and inefficient ones. Informative voting may or may not be an equilibrium. Also contrary to two-alternative elections, there exist responsive equilibria in which voters vote ‘against’ their signal. As a consequence, only two alternatives receive positive expected vote shares and the outcome is inefficient.     

Solidarity in choosing a location on a cycle

We study the implications of two solidarity conditions on the efficient location of a public good on a cycle, when agents have single-peaked, symmetric preferences. Both conditions require that when circumstances change, the agents not responsible for the change should all be affected in the same direction: either they all gain or they all loose. The first condition, population-monotonicity, applies to arrival or departure of one agent. The second, replacement-domination, applies to changes in the preferences of one agent. Unfortunately, no Pareto-efficient solution satisfies any of these properties. However, if agents’ preferred points are restricted to the vertices of a small regular polygon inscribed in the circle, solutions exist. We characterize them as a class of efficient priority rules.     

The replacement principle in economies with indivisible goods

 We consider the problem of allocating a list of indivisible goods and some amount of an infinitely divisible good among agents with equal rights on these resources, and investigate the implications of the following requirement on allocation rules: when the preferences of some of the agents change, all agents whose preferences are fixed should (weakly) gain, or they should all (weakly) lose. This condition is an application of a general principle of solidarity discussed in Thomson (1990b) under the name “replacement principle”. We look for selections from the no-envy solution satisfying this property. We show that in the general case, when the number of objects is arbitrary, there is no such selection. However, in the one-object case (a single prize), up to Pareto-indifference, there is only one selection from the no-envy solution satisfying the property. Such a solution always selects an envy-free allocation at which the winner of the prize is indifferent between his bundle and the losers’ common bundle. Received: 15 May 1995 / Accepted: 5 June 1996     

A maximal domain of preferences for strategy-proof,efficient, and simple rules in the division problem

The division problem consists of allocating an amount M of a perfectly divisible good among a group of n agents. Sprumont (1991) showed that if agents have single-peaked preferences over their shares, the uniform rule is the unique strategy-proof, efficient, and anonymous rule. Ching and Serizawa (1998) extended this result by showing that the set of single-plateaued preferences is the largest domain, for all possible values of M, admitting a rule (the extended uniform rule) satisfying strategy-proofness, efficiency and symmetry. We identify, for each M and n, a maximal domain of preferences under which the extended uniform rule also satisfies the properties of strategy-proofness, efficiency, tops-onlyness, and continuity. These domains (called partially single-plateaued) are strictly larger than the set of single-plateaued preferences. However, their intersection, when M varies from zero to infinity, coincides with the set of single-plateaued preferences.An earlier version of this paper circulated under the title A maximal domain of preferences for tops-only rules in the division problem. We are grateful to an associate editor of this journal for comments that helped to improve the presentation of the paper and to Matt Jackson for suggesting us the interest of identifying a maximal domain of preferences for tops-only rules. We are also grateful to Dolors Berga, Flip Klijn, Howard Petith, and a referee for helpful comments. The work of Alejandro Neme is partially supported by Research Grant 319502 from the Universidad Nacional de San Luis. The work of Jordi Massó is partially supported by Research Grants BEC2002-02130 from the Spanish Ministerio de Ciencia y Tecnología and 2001SGR-00162 from the Generalitat de Catalunya, and by the Barcelona Economics Program of CREA from the Generalitat de Catalunya. The paper was partially written while Alejandro Neme was visiting the UAB unde r a sabbatical fellowship from the Generalitat de Catalunya.     

On the manipulability of approval voting and related scoring rules

We characterize all preference profiles at which the approval (voting) rule is manipulable, under three extensions of preferences to sets of candidates: by comparison of worst candidates, best candidates, or by comparison based on stochastic dominance. We perform a similar exercise for k-approval rules, where voters approve of a fixed number k of candidates. These results can be used to compare (k-)approval rules with respect to their manipulability. Analytical results are obtained for the case of two voters, specifically, the values of k for which the k-approval rule is minimally manipulable—has the smallest number of manipulable preference profiles—under the various preference extensions are determined. For the number of voters going to infinity, an asymptotic result is that the k-approval rule with k around half the number of candidates is minimally manipulable among all scoring rules. Further results are obtained by simulation and indicate that k-approval rules may improve on the approval rule as far as manipulability is concerned.     

Voting rules and efficiency in one-dimensional bargaining games with endogenous protocol

We analyze a rent-seeking contest that determines the bargaining protocol in a one-dimensional bargaining game, where agents preferences over social outcomes are single-peaked. We relate the incentives of agents to make unproductive and costly efforts/investments to the quota rules that are required to implement agreements. When the contest assigns persistent recognition probabilities, we find that simple majority minimizes the total investments and, hence, inefficiency. In case that the contest recurs each period, multiple equilibria exist with the particularity that in each equilibrium only one agent controls the agenda of the bargaining process.     

Identification of domain restrictions over which acyclic, continuous-valued, and positive responsive social choice rules operate

We consider a social choice problem in various economic environments consisting of n individuals, 4≤n<+∞, each of which is supposed to have classical preferences. A social choice rule is a function associating with each profile of individual preferences a social preference that is assumed to be complete, continuous and acyclic over the alternatives set. The class of social choice rules we deal with is supposed to satisfy the two conditions; binary independence and positive responsiveness. A new domain restriction for the social choice rules is proposed and called the classical domain that is weaker than the free triple domain and holds for almost all economic environments such as economies with private and/or public goods. In this paper we explore what type of classical domain that admits at least one social choice rule satisfying the mentioned conditions to well operate over the domain. The results we obtained are very negative: For any classical domain admitting at least one social choice rule to well operate, the domain consists only of just one profile.     

Equity and the Vickrey allocation rule on general preference domains

We consider the problem of allocating multiple units of an indivisible good among a group of agents in which each agent demands at most one unit of the good and money payment or receipt is required. Under general preference domains that may contain non quasi-linear preferences, the Vickrey allocation rule is characterized by axioms for equity and continuity without use of efficiency: namely, the Vickrey rule is the only rule that satisfies strategy-proofness, weak envy-freeness for equals, non-imposition, and continuity of welfare.