Arrow's theorem, many agents, and invisible dictators☆

The paper investigates the structure of coalitional strategy-proof social choice functions – CSP scfs, for short – whose range is a subset of cardinality two of an arbitrary set $A$ of alternatives. The study is conducted in the case where the voters/agents are allowed to express indifference among elements of $A$, and the domain of the scfs consists of preference profiles $P={\left(P_v\right)}_{v\in V}$ over a society $V$ of arbitrary cardinality. A representation formula for the two-valued CSP scfs is obtained that provides the structure of such functions.

This paper examines the aggregation of preferences with a finitely additive measure space of agents. We consider three types of non-dictatorship axioms: non-dictatorship, coalitional non-dictatorship, and atomic non-dictatorship. First, we show that the existence of an atom is a necessary and sufficient condition for the existence of a social welfare function that satisfies weak Pareto, independence of irrelevant alternatives, and coalitional non-dictatorship. Second, we simultaneously impose non-dictatorship and coalitional non-dictatorship, and specify a necessary and sufficient condition for the finitely additive measure that guarantees the compatibility among the axioms. Third, we impose all non-dictatorship axioms and show that the corresponding measure is extremely restricted.

A new property of collective choice rules that we refer to as superset robustness is introduced, and we employ it in several characterization results. The axiom requires that if all individual preference orderings expand weakly (in the sense of set inclusion), then the corresponding social preference relation must also expand weakly. In other words, if a given profile is changed by adding instances of weak preference to some individual relations, then the social weak preference relation for the expanded profile must contain the social weak preference relation for the original—that is, the social relation cannot contract in response to the addition of pairs to the individual relations. We begin by examining social welfare functions (that is, collective choice rules such that the resulting social preferences are orderings) and then move on to rules that generate transitive (but not necessarily complete) social rankings. The remaining results of the paper focus on Suzumura-consistent collective choice rules. In all of these cases, it turns out that the property of superset robustness is closely related to classes of agreement-based collective choice rules. These are rules such that the social relation is determined by collecting the pairs on whose relative rankings the members of the decisive sets agree.

We analyze the decisiveness structures associated with acyclical collective choice rules. In particular, we examine the consequences of adding anonymity to weak Pareto, thereby complementing earlier results on acyclical social choice. Both finite and countably infinite populations are considered. As established in contributions by Donald Brown and by Jeffrey Banks, acyclical social choice is closely linked to prefilters in the presence of the weak Pareto principle. We introduce the notion of a conditional prefilter and use it to generalize their results. In the finite-population case, adding anonymity implies that the collection of decisive sets is a special case of a conditional prefilter. We then identify the decisiveness structure that results from adding anonymity to the weak Pareto principle. Moving to infinite populations, we obtain the decisive set that consists of the entire population as a possibility, along with a new class of prefilters that we refer to as symmetric free Fréchet prefilters. The choice of the term Fréchet prefilter is motivated by the observation that they share a defining property with the well-known Fréchet filter—namely, that all sets in the requisite collection are such that their complement is finite.