Arrow’s theorem as a corollary

Arrow’s Impossibility Theorem is derived from a general result on social aggregation in ‘property spaces’ (S3 convex structures) obtained in prior work. In the derivation, the specific structure of Arrowian aggregation as an aggregation of weak orders plays a purely combinatorial role.     

An approximate solution to Arrow's problem

We impose a lattice-theoretic structure on a class of social welfare functions, which properly include Arrowian social welfare functions, by ordering their families of decisive sets by set inclusion. We show that this lattice is compactly generated and characterize the completely meet irreducible elements of this lattice. Certain completely meet irreducible elements correspond to Arrowian social welfare functions. Those social welfare functions corresponding to the other completely meet irreducible families of decisive sets are approximate solutions to the problem of social choice as posed by Arrow.     

How competition affects schools' performances: Does specification matter?

If the number of individuals is odd, majority rule is the only non-dictatorial strategy-proof social choice rule on the domain of linear orders that admit a Condorcet winner (Campbell and Kelly, 2003). This paper shows that the claim is false when the number of individuals is even, and provides a counterpart to the theorem for the even case.     

Social choice and merit goods

Existing formulations of merit good social preferences, which are usually thought to be non-individualistic, are shown to be implied by three assumptions taken from the Arrowian social choice theoretic framework. Thus an individualistic foundation is provided for merit goods.     

Non-dictatorial extensive social choice

Summary. Different social planners may have different opinions on the well-being of individuals under different social options (Roberts, 1980). If utilities are translation- or ratio-scale measurable, or if the social ranking might be incomplete, or if interplanner comparability is allowed; then there exist non-dictatorial aggregation rules. We propose extensions, intersections, and mixtures of the Pareto, utilitarian, leximin, Kolm-Pollak, and iso-elastic rules.Received: 16 May 2001, Revised: 18 November 2003, JEL Classification Numbers: D63, D71. Correspondence to: Erwin OogheErwin Ooghe, Luc Lauwers: We are extremely grateful to the referee who was willing to review this paper many times. Her/his extensive and in-depth comments had a strong impact. Further thanks are due to Bart Capéau, Marc Fleurbaey, Maurice Salles, Erik Schokkaert, and Alain Trannoy. The first author gratefully acknowledges the financial support by the TMR network Living Standards, Inequality and Taxation (ERBFMRXCT 980248) of the European Communities.     

Characterization of the private alternatives domains admitting arrow social welfare functions

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 alternative effecting individual i alone. Assuming that individuals are selfish (i is indifferent between any two alternatives that have the same ith component) we characterize all the families of permissible individual preferences that admit nondictatorial Arrow-type social welfare functions. We also show that the existence of such a function for a given family of preferences is independent of n provided that is greater than one.     

The structure of strategy-proof social choice — Part I: General characterization and possibility results on median spaces

We define a general notion of single-peaked preferences based on abstract betweenness relations. Special cases are the classical example of single-peaked preferences on a line, the separable preferences on the hypercube, the “multi-dimensionally single-peaked” preferences on the product of lines, but also the unrestricted preference domain. Generalizing and unifying the existing literature, we show that a social choice function is strategy-proof on a sufficiently rich domain of generalized single-peaked preferences if and only if it takes the form of voting by issues (“voting by committees”) satisfying a simple condition called the “Intersection Property.”Based on the Intersection Property, we show that the class of preference domains associated with “median spaces” gives rise to the strongest possibility results; in particular, we show that the existence of strategy-proof social choice rules that are non-dictatorial and neutral requires an underlying median space. A space is a median space if, for every triple of elements, there is a fourth element that is between each pair of the triple; numerous examples are given (some well-known, some novel), and the structure of median spaces and the associated preference domains is analysed.     

Non-manipulable division rules in claim problems and generalizations

When resources are divided among agents, the characteristics of the agents are taken into consideration. A simple example is the bankruptcy problem, where the liquidation value of a bankrupt firm is divided among the creditors based on their claims. We characterize division rules under which no group of agents can increase the total amount they receive by transferring their characteristics within the group. By allowing agents’ characteristics to be multi-dimensional and choosing the meaning of variables appropriately, our model can subsume a number of existing and new allocation problems, such as cost sharing, social choice with transferable utilities, income redistribution, bankruptcy with multiple types of assets, probability updating, and probability aggregation. A number of existing and new results in specific problems are obtained as corollaries.     

Extraneous Alternatives And Strategy-Proofness

It is well known that the Gibbard–Satterthwaite theorem cannot be circumvented by adding extraneous alternatives that are included in the individual preference information but are never selected. We generalize this by proving that, for any domain on which every strategy-proof rule is dictatorial, the addition of extraneous alternatives will not permit the construction of a non-dictatorial and strategy-proof rule if the new domain is a product set. We show how this result, and our other theorem, can be applied to seven families of social choice situations, including those in which more than one alternative is selected.     

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 strategy-proofness characterization of majority rule

Summary. A feasible alternative x is a strong Condorcet winner if for every other feasible alternative y there is some majority coalition that prefers x to y. Let (resp., denote the set of all profiles of linear (resp., merely asymmetric) individual preference relations for which a strong Condorcet winner exists. Majority rule is the only non-dictatorial and strategy-proof social choice rule with domain , and majority rule is the only strategy-proof rule with domain . Received: August 29, 2000; revised version: November 13, 2002 RID="*" ID="*"We are grateful to Wulf Gaertner and our two referees for insightful comments on a previous draft. Correspondence to: D. E. Campbell     

Voting by committees under constraints

We consider social choice problems where a society must choose a subset from a set of objects. Specifically, we characterize the families of strategy-proof voting procedures when not all possible subsets of objects are feasible, and voters’ preferences are separable or additively representable.     

Mixed equilibria in runoff elections

We demonstrate that the Hotelling–Downs model with runoff voting always admits symmetric mixed strategy equilibria for any (even or odd) number of office-motivated candidates (provided they are at least four). In specific, (a) we show that the game does not admit any symmetric atomless equilibrium, (b) we fully characterize a class of symmetric atomic mixed equilibria which exist for any distribution of the voters' ideal policies and (c) we argue that these equilibria are more robust than pure strategy equilibria to introduction of uncertainty about the voters' preferences.     

Justifiable group choice

We study the judgment aggregation problem from the perspective of justifying a particular collective decision by a corresponding aggregation on the criteria. In particular, we characterize the logical relations between the decision and the criteria that enable justification of a majority decision through a proposition-wise aggregation rule with no veto power on the criteria. While the well-studied “doctrinal paradox” provides a negative example in which no such justification exists, we show that genuine possibility results emerge if there is a gap between the necessary and the sufficient conditions for the decision. This happens, for instance, if there is only a partial consensus about the appropriate criteria for the decision, if only a subset of these criteria can be elicited, or if the judgment on criteria is based on probabilistic acceptance thresholds.     

Multidimensional generalized Gini indices

Summary. The axioms that characterize the generalized Gini social evaluation orderings for one-dimensional distributions are extended to the multidimensional attributes case. A social evaluation ordering is shown to have a two-stage aggregation representation if these axioms and a separability assumption are satisfied. In the first stage, the distributions of each attribute are aggregated using generalized Gini social evaluation functions. The functional form of the second-stage aggregator depends on the number of attributes and on which version of a comonotonic additivity axiom is used. The implications of these results for the corresponding multidimensional indices of relative and absolute inequality are also considered.Received: 20 August 2003, Revised: 26 May 2004, JEL Classification Numbers: D63. Correspondence to: John A. WeymarkWe are grateful to our referee for his or her comments.     

Aggregative games and best-reply potentials

This paper introduces quasi-aggregative games and establishes conditions under which such games admit a best-reply potential. This implies existence of a pure strategy Nash equilibrium without any convexity or quasi-concavity assumptions. It also implies convergence of best-reply dynamics under some additional assumptions. Most of the existing literature’s aggregation concepts are special cases of quasi-aggregative games, and many new situations are allowed for. An example is payoff functions that depend on own strategies as well as a linear combination of the mean and the variance of players’ strategies.     

Fair rights

This note argues that analysis of constraints on rights assignments per se is important for social choice theory. By way of illustration, the equity concept of ‘fairness’, when applied to rights, is shown to admit the existence of libertarian social decision functions.     

Computing minimal state space recursive equilibrium in OLG models with stochastic production

Using order-theoretic methods, we derive sufficient conditions for the existence, characterization, and computation of minimal state space recursive equilibrium (RE), as well as Stationary Markov equilibrium (SME) for various classes of stochastic overlapping generations models. In contrast to previous work, our methods focus on constructive methods. Our existence results are obtained for models that include public policy (e.g., social security policies, transfers, taxes, etc), production nonconvexities, elastic labor supply, non-monotone income processes, and long-lived agents. We distinguish conditions under which there exist various subclasses of minimal state space RE, including bounded, monotone, non-monotone, semicontinuous, Lipschitz continuous RE. Finally, we provide monotone equilibrium comparative statics results on the space of economies for some RE.     

Strategy-proof preference aggregation: Possibilities and characterizations

An aggregation rule maps each profile of individual strict preference orderings over a set of alternatives into a social ordering over that set. We call such a rule strategy-proof if misreporting one's preference never produces a different social ordering that is between the original ordering and one's own preference. After describing two examples of manipulable rules, we study in some detail three classes of strategy-proof rules: (i) rules based on a monotonic alteration of the majority relation generated by the preference profile; (ii) rules improving upon a fixed status-quo; and (iii) rules generalizing the Condorcet–Kemeny aggregation method.     

Sequential equilibrium in monotone games: A theory-based analysis of experimental data

A monotone game is an extensive-form game with complete information, simultaneous moves and an irreversibility structure on strategies. It captures a variety of situations in which players make partial commitments and allows us to characterize conditions under which equilibria result in socially desirable outcomes. However, since the game has many equilibrium outcomes, the theory lacks predictive power. To produce stronger predictions, one can restrict attention to the set of sequential equilibria, or Markov equilibria, or symmetric equilibria, or pure-strategy equilibria. This paper explores the relationship between equilibrium behavior in a class of monotone games, namely voluntary contribution games, and the behavior of human subjects in an experimental setting. Several key features of the symmetric Markov perfect equilibrium (SMPE) are consistent with the data. To judge how well the SMPE fits the data, we estimate a model of Quantal Response Equilibrium (QRE) [R. McKelvey, T. Palfrey, Quantal response equilibria for normal form games, Games Econ. Behav. 10 (1995) 6-38; R. McKelvey, T. Palfrey, Quantal response equilibria for extensive form games, Exp. Econ. 1 (1998) 9-41] and find that the decision rules of the QRE model are qualitatively very similar to the empirical choice probabilities.