Judgment aggregation: (im)possibility theorems

The aggregation of individual judgments over interrelated propositions is a newly arising field of social choice theory. I introduce several independence conditions on judgment aggregation rules, each of which protects against a specific type of manipulation by agenda setters or voters. I derive impossibility theorems whereby these independence conditions are incompatible with certain minimal requirements. Unlike earlier impossibility results, the main result here holds for any (non-trivial) agenda. However, independence conditions arguably undermine the logical structure of judgment aggregation. I therefore suggest restricting independence to “premises”, which leads to a generalised premise-based procedure. This procedure is proven to be possible if the premises are logically independent.     

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.     

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.     

A characterization of consistent collective choice rules

We characterize a class of collective choice rules such that collective preference relations are consistent. Consistency is a weakening of transitivity and a strengthening of acyclicity requiring that there be no cycles with at least one strict preference, which excludes the possibility of a “money pump.” The properties of collective choice rules used in our characterization are unrestricted domain, strong Pareto, anonymity and neutrality. If there are at most as many individuals as there are alternatives, the axioms provide an alternative characterization of the Pareto rule. If there are more individuals than alternatives, however, further rules become available.     

Judgment aggregation in search for the truth

We analyze the problem of aggregating judgments over multiple issues from the perspective of whether aggregate judgments manage to efficiently use all voters' private information. While new in judgment aggregation theory, this perspective is familiar in a different body of literature about voting between two alternatives where voters' disagreements stem from conflicts of information rather than of interest. Combining the two bodies of literature, we consider a simple judgment aggregation problem and model the private information underlying voters' judgments. Assuming that voters share a preference for true collective judgments, we analyze the resulting strategic incentives and determine which voting rules efficiently use all private information. We find that in certain, but not all cases a quota rule should be used, which decides on each issue according to whether the proportion of ‘yes’ votes exceeds a particular quota.     

Priorities in the location of multiple public facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We characterize the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among “interest groups”. We characterize the subclasses of priority rules that respectively satisfy anonymity, avoid the no-show paradox, strategy-proofness and population-monotonicity. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Any such rule can also be viewed as a collection of generalized peak-selection median rules, that are linked across populations, in a way that we describe.     

Aggregation of binary evaluations

We study a general aggregation problem in which a society has to determine its position (yes/no) on each of several issues, based on the positions of the members of the society on those issues. There is a prescribed set of feasible evaluations, i.e., permissible combinations of positions on the issues. This framework for the theory of aggregation was introduced by Wilson and further developed by Rubinstein and Fishburn. Among other things, it admits the modeling of preference aggregation (where the issues are pairwise comparisons and feasibility reflects rationality), and of judgment aggregation (where the issues are propositions and feasibility reflects logical consistency). We characterize those sets of feasible evaluations for which the natural analogue of Arrow's impossibility theorem holds true in this framework.     

Symmetry and impartial lotteries

A prize is to be awarded, so each candidate designates one of his peers on a ballot. The ballots determine the lottery that selects the winner, and impartiality requires that no candidate's choice of designee impacts his own chance of winning, removing incentives for strategic ballot submission. The primary results are (1) a characterization of all impartial rules that treat agents symmetrically as voters, and (2) a characterization of all impartial rules that treat agents symmetrically as candidates. Each rule in either class may be represented as a randomization over a finite set of simple rules. These results have immediate interpretation in a second context: the division of surplus among team members. Corollaries include the constant rule impossibility of Holzman and Moulin (2013), a new dictatorship impossibility, and the first axiomatic characterization of uniform random dictatorship.     

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].     

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 Afriat Theorem for the collective model of household consumption

We provide a nonparametric ‘revealed preference’ characterization of rational household behavior in terms of the collective consumption model, while accounting for general individual preferences that can be non-convex. Our main result is the Collective Afriat Theorem, which parallels the well-known Afriat Theorem for the unitary model. First, it provides a characterization of collectively rational consumption behavior in terms of collective Afriat inequalities. Next, it implies the Collective Axiom of Revealed Preference (CARP) as a testable necessary and sufficient condition for data consistency with collective rationality. Finally, the theorem has some interesting testability implications. With only a finite set of observations, the nature of consumption externalities (positive or negative) in the intra-household allocation process is non-testable. The same non-testability conclusion holds for privateness (with or without externalities) or publicness of consumption. By contrast, concavity of individual utility functions (representing convex preferences) turns out to be testable. In addition, monotonicity is testable for the model that assumes all household consumption is public.     

Competitive manufacturing with fluctuating demand and diverse technology: Mathematical proofs and illuminations on industry output-flexibility

I present an original model of competitive manufacturing with fluctuating demand and diverse technology with mathematical proofs. I discuss Aranoff's output-flexibility indicator, E(AC)-LRMC. I use the model to compute Aranoff's output-flexibility indicator to measure industry output-flexibility. I argue that a measure of industry output-flexibility is β_L(Q₂ − Q₁)/E(Q) I tie the demand-side discussion with the cost-side and show the degree of industry output-flexibility that will emerge under welfare and profit-maximizing pricing rules. I perform comparative statics of changes in technology, of demand, and of frequency of the high-peak state.     

Dynamic stability and reform of political institutions

This paper examines endogenous institutional change in a class of dynamic political games. The political aggregation rules used at date t+1 are instrumental choices under rules at date t. Effectively, rules are “players” who can strategically delegate future policy-making authority to different rules. A political rule is stable if it selects itself. A reform occurs when an alternative rule is selected. The stability of a political rule is shown to depend on whether its choices are dynamically consistent. For instance, simple majority rules can be shown to be dynamically consistent in many common environments where wealth-weighted voting rules are not. The result extends to political rules that incorporate private activities such as extra-legal protests, threats, or private investment. The approach is one way of understanding various explanations of institutional change proposed in the literature. A parametric model of public goods provision gives an illustration.     

Measures of inequality as an aggregation of individual preferences about income distribution: The arrowian case

An inequality preorder is a preorder on a simplex which satisfies symmetry and strict Schur-convexity (the mathematical equivalent of the principle of transfers of Pigou and Dalton). It is shown that we cannot aggregate individual inequality preorders to a collective one if we are interested in Arrow's aggregation rules. The proof uses an interesting result of Kalai, Muller and Satterthwaite (Public Choice 34 (1979), 87–97). Moreover, we prove further results for the aggregation of individual inequality indices when we allow cardinality and interpersonal comparibility of utility.     

Group identification

I study a model of group identification in which individuals' opinions as to the membership of a group are aggregated to form a list of group members. Potential aggregation rules are studied through the axiomatic approach. I introduce two axioms, meet separability and join separability, each of which requires the list of members generated by the aggregation rule to be independent of whether the question of membership in a group is separated into questions of membership in two other groups. I use these axioms to characterize a class of one-vote rules, in which one opinion determines whether an individual is considered to be a member of a group. I then show that the only anonymous one-vote rule is self-identification, in which each individual determines for himself whether he is a member of the group.     

Exchange in a general market with indivisible goods

We show how to restrict trades in exchange markets with heterogeneous indivisible goods so that the resulting restricted exchange markets, the fixed deal exchange markets, have a unique core allocation. Our results on fixed deal exchange markets generalize classical results on the Shapley-Scarf housing market, in which each agent owns one good only. Furthermore, we define the class of fixed deal exchange rules for general exchange markets, and prove that these are the only exchange rules that satisfy strategyproofness, individual rationality, and a weak form of efficiency.     

The communication requirements of social choice rules and supporting budget sets

The paper examines the communication requirements of social choice rules when the (sincere) agents privately know their preferences. It shows that for a large class of choice rules, any minimally informative way to verify that a given alternative is in the choice rule is by verifying a “budget equilibrium”, i.e., that the alternative is optimal to each agent within a “budget set” given to him. Therefore, any communication mechanism realizing the choice rule must find a supporting budget equilibrium. We characterize the class of choice rules that have this property. Furthermore, for any rule from the class, we characterize the minimally informative messages (budget equilibria) verifying it. This characterization is used to identify the amount of communication needed to realize a choice rule, measured with the number of transmitted bits or real variables. Applications include efficiency in convex economies, exact or approximate surplus maximization in combinatorial auctions, the core in indivisible-good economies, and stable many-to-one matchings.     

Fair imposition

We introduce a new mechanism-design problem called fair imposition. In this setting a center wishes to fairly allocate tasks among a set of agents whose cost structures are known only to them, and thus will not reveal their true costs without appropriate incentives. The center, with the power to impose arbitrary tasks and payments on the agents, has the additional goal that his net payment to these agents is never positive (or, that it is tightly bounded if a loss is unavoidable). We consider two different notions of fairness that the center may wish to achieve. The central notion, which we call k-fairness, is in the spirit of max-min fairness. We present both positive results (in the form of concrete mechanisms) and negative results (in the form of impossibility theorems) concerning these criteria. We also briefly discuss an alternative, more traditional interpretation of our setting and results, in the context of auctions.     

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.