The possibility of judgment aggregation on agendas with subjunctive implications

The new field of judgment aggregation aims to find collective judgments on logically interconnected propositions. Recent impossibility results establish limitations on the possibility to vote independently on the propositions. I show that, fortunately, the impossibility results do not apply to a wide class of realistic agendas once propositions like “if a then b” are adequately modelled, namely as subjunctive implications rather than material implications. For these agendas, consistent and complete collective judgments can be reached through appropriate quota rules (which decide propositions using acceptance thresholds). I characterise the class of these quota rules. I also prove an abstract result that characterises consistent aggregation for arbitrary agendas in a general logic.     

Introduction to judgment aggregation

This introduces the symposium on judgment aggregation. The theory of judgment aggregation asks how several individuals' judgments on some logically connected propositions can be aggregated into consistent collective judgments. The aim of this introduction is to show how ideas from the familiar theory of preference aggregation can be extended to this more general case. We first translate a proof of Arrow's impossibility theorem into the new setting, so as to motivate some of the central concepts and conditions leading to analogous impossibilities, as discussed in the symposium. We then consider each of four possible escape-routes explored in the symposium.     

The premiss-based approach to judgment aggregation

In the framework of judgment aggregation, we assume that some formulas of the agenda are singled out as premisses, and that both Independence (formula-wise aggregation) and Unanimity Preservation hold for them. Whether premiss-based aggregation thus defined is compatible with conclusion-based aggregation, as defined by Unanimity Preservation on the non-premisses, depends on how the premisses are logically connected, both among themselves and with other formulas. We state necessary and sufficient conditions under which the combination of both approaches leads to dictatorship (resp. oligarchy), either just on the premisses or on the whole agenda. Our analysis is inspired by the doctrinal paradox of legal theory and is arguably relevant to this field as well as political science and political economy. When the set of premisses coincides with the whole agenda, a limiting case of our assumptions, we obtain several existing results in judgment aggregation theory.     

Factoring out the impossibility of logical aggregation

According to a theorem recently proved in the theory of logical aggregation, any nonconstant social judgment function that satisfies independence of irrelevant alternatives (IIA) is dictatorial. We show that the strong and not very plausible IIA condition can be replaced with a minimal independence assumption plus a Pareto-like condition. This new version of the impossibility theorem likens it to Arrow's and arguably enhances its paradoxical value.     

Some remarks on ranking opportunity sets and Arrow impossibility theorems: correspondence results

An earlier work by Dutta and Sen provides characterizations of a set of decision rules for the ranking of opportunity sets. This paper begins by demonstrating the redundancy of one of the axioms in the said characterizations and goes on to analyze in detail one of the theorems, the Generalized Utilitarian rules theorem, which is incorrect. Basically, we find that one of the axioms that the authors provide is not in all cases satisfied for the Utilitarian rules. In this paper, we discuss this issue before proposing an alternative characterization which preserves the spirit of the original authors.     

On the universally beneficial manipulation conjecture

The universally beneficial manipulation conjecture of Campbell and Kelly states that for a social choice rule, if everyone gains as a result of any optimal manipulation, then the rule satisfies universally beneficial manipulation, i.e., everyone gains as a result of any manipulation, optimal or not. We prove the conjecture for three alternatives and any number of individuals. We also construct a counterexample in a general form with more than three alternatives and at least two individuals.     

Majority voting on restricted domains

In judgment aggregation, unlike preference aggregation, not much is known about domain restrictions that guarantee consistent majority outcomes. We introduce several conditions on individual judgments sufficient for consistent majority judgments. Some are based on global orders of propositions or individuals, others on local orders, still others not on orders at all. Some generalize classic social-choice-theoretic domain conditions, others have no counterpart. Our most general condition generalizes Sen's triplewise value-restriction, itself the most general classic condition. We also prove a new characterization theorem: for a large class of domains, if there exists any aggregation function satisfying some democratic conditions, then majority voting is the unique such function. Taken together, our results support the robustness of majority rule.     

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.     

Abstract Arrowian aggregation

In a general framework of abstract binary aggregation, we characterize aggregation problems in terms of the monotone Arrowian aggregators they admit. Specifically, we characterize the problems that admit non-dictatorial, locally non-dictatorial, anonymous, and neutral monotone Arrowian aggregation, respectively. As a consequence of these characterizations, we also obtain new results on the possibility of strategy-proof social choice and the “concrete Arrowian” aggregation of preferences into a social ordering on generalized single-peaked domains.     

Representation and aggregation of preferences under uncertainty

We axiomatize in the Anscombe-Aumann setting a wide class of preferences called rank-dependent additive preferences that includes most known models of decision under uncertainty as well as state dependent versions of these models. We prove that aggregation is possible and necessarily linear if and only if (society's) preferences are uncertainty neutral. The latter means that society cannot have a non-neutral attitude toward uncertainty on a subclass of acts. A corollary to our theorem is that it is not possible to aggregate multiple prior agents, even when they all have the same set of priors. A number of ways to restore the possibility of aggregation are then discussed.     

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.     

Arrovian aggregation in economic environments: how much should we know about indifference surfaces?

Arrow's celebrated theorem of social choice shows that the aggregation of individual preferences into a social ordering cannot make the ranking of any pair of alternatives depend only on individual preferences over that pair, unless the fundamental weak Pareto and non-dictatorship principles are violated. In the standard model of division of commodities, we investigate how much information about indifference surfaces is needed to construct social ordering functions satisfying the weak Pareto principle and anonymity. We show that local information such as marginal rates of substitution or the shapes “within the Edgeworth box” is not enough, and knowledge of substantially non-local information is necessary.     

Linear efficient and symmetric values for TU-games: Sharing the joint gain of cooperation

Recently, Hermandez-Lamoneda et al. (2008) and independently Chameni and Andjiga (2008) gave an analytic formulation for all valued solutions to the n-person TU-games that satisfy linearity, efficiency and symmetry axioms. Our main purpose in this paper is to recast the proposed formulation to a more potentially interpretational one. We are focused on an interpretation based on the idea of marginal contribution, a concept already familiar in the Shapley value and the Solidarity value. A general null player axiom is introduced, and it turns out that any valued solution satisfying the three properties is characterized by a null player model.     

Ordinal aggregation and quantiles

Consider the problem of aggregating a profile of interpersonally comparable utilities into a social utility. We require that the units of measurement of utility used for agents is the same as the units of measurement for society (ordinal covariance) and a mild Pareto condition (monotonicity). We provide several representations of such social aggregation operators: a canonical representation, a Choquet expectation representation, a minimax representation, and a quantile representation (with respect to a possibly non-additive set function on the agents). We also isolate an additional condition that gives us a quantile representation with respect to a probability measure, in both the finite and infinite agents case.     

Voluntary formation of communities for the provision of public projects

In this paper we examine a group formation problem, where heterogeneous individuals partitioned themselves into communities, each choosing its own public project from the given space of feasible projects. The model is that of “horizontal product differentiation” where individuals display distinct preferences over the policy space. We consider the notion of “efficient” configuration that minimizes the total project-related costs and aggregate personalized costs of all individuals, and “sustainable” configurations, those are immune against breakaways by subgroups of individuals. Our main result is that, with a unidimensional project space and single-peaked personalized costs, every efficient partition is sustainable.     

Between liberalism and democracy

We study and characterize axiomatically a class of voting rules, called consent rules, that incorporate aspects of majoritarianism and liberalism. An outcome of the vote specifies who among the voters are eligible to a certain right or qualification. Each outcome serves also as a permissible ballot. Consent rules are parameterized by the weights given to individuals in determining their own qualification. In one of these rules, the liberal rule, each individual's qualification is determined by her. In other rules, an individual's qualification requires social consent in various degrees. We also show the relation between self-determination and the liberal rule.     

Community standards

I introduce a model of community standards relevant to the judicial determination of obscenity. Standards are defined as subjective judgments restricted only by a simple reasonableness condition. Individual standards are aggregated to form the community standard. Several axioms reflect legal concerns. These require that the community standard (a) preserve unanimous agreements, (b) become more permissive when all individuals become more permissive, and not discriminate, ex ante, (c) between individuals and (d) between works. I show that any rule which satisfies these properties must be “similar” to unanimity rule. I also explore the relationship between the model and the doctrinal paradox of Kornhauser and Sager [12].     

Condorcet Jury Theorem: An example in which informative voting is rational but leads to inefficient information aggregation

Recent research on the Condorcet Jury Theorem has proven that informative voting (that is, voting according to one’s signal) is not necessarily rational. With two alternatives, rational voting typically leads to the election of the correct alternative, in spite of the fact that not all voters vote informatively. We prove that with three alternatives, there are cases in which informative voting is rational and yet leads to the election of a wrong alternative.     

Role of linking mechanisms in multitask agency with hidden information

We investigate the adverse selection problem where a principal delegates multiple tasks to an agent. We characterize the virtually implementable social choice functions by using the linking mechanism proposed by Jackson and Sonnenschein (2007) [20] that restricts the message spaces. The principal does not require any incentive wage schemes and can therefore avoid any information rent and welfare loss. We show the resemblance between the functioning of this message space restriction and that of incentive wage schemes. We also extend the results of the single-agent model to the multi-agent model.     

Efficiency and consistency for locating multiple public facilities

In the problem of locating multiple public facilities studied by Barberà and Beviá [Self-selection consistent functions, J. Econ. Theory 105 (2002) 263-277; Locating public facilities by majority: stability, consistency and group formation, Games Econ. Behav. 56 (2006) 185-200], we offer simple necessary and sufficient conditions for efficiency, decentralizability of efficient decisions in a game of community division and local public goods provision, and a constructive algorithm for efficient and consistent decisions.