Uncovered Sets

This paper covers the theory of the uncovered set used in the literatures on tournaments and spatial voting. I discern three main extant definitions, and I introduce two new concepts that bound exist- ing sets from above and below: the deep uncovered set and the shallow uncovered set. In a general topological setting, I provide relationships to other solutions and give results on existence and external stability for all of the covering concepts, and I establish continuity properties of the two new uncovered sets. Of note, I characterize each of the uncovered sets in terms of a decomposition into choices from externally stable sets; I define the minimal generalized covering solution, a nonempty refinement of the deep uncovered set that employs both of the new relations; and I define the acyclic Banks set, a nonempty generalization of the Banks set.

Equilibrium Agenda Formation

We develop a definition of equilibrium for agenda formation in general voting settings. The definition is independent of any protocol. We show that the set of equilibrium outcomes for any Pareto efficient voting rule is uniquely determined, and in fact coincides with that of the outcomes generated by considering all full agendas. Under voting by successive elimination (or amendment), the set of equilibrium outcomes corresponds with the Banks set. We also examine the implications in several specific settings and show that studying equilibrium agendas can lead to sharp predictions, in contrast with well-known ``chaos'' theorems.agenda, equilibrium, voting, agenda formation

Set-Monotonicity Implies Kelly-Strategyproofness

This paper studies the strategic manipulation of set-valued social choice functions according to Kelly's preference extension, which prescribes that one set of alternatives is preferred to another if and only if all elements of the former are preferred to all elements of the latter. It is shown that set-monotonicity---a new variant of Maskin-monotonicity---implies Kelly-strategyproofness in comprehensive subdomains of the linear domain. Interestingly, there are a handful of appealing Condorcet extensions---such as the top cycle, the minimal covering set, and the bipartisan set---that satisfy set-monotonicity even in the unrestricted linear domain, thereby answering questions raised independently by Barber\`a (1977) and Kelly (1977).Comment: 14 page

EQUILIBRIUM AGENDA FORMATION

We develop a definition of equilibrium for agenda formation in general voting settings. The definition is independent of any protocol. We show that the set of equilibrium outcomes for any Pareto efficient voting rule is uniquely determined. We also show that for such voting rules, if preferences are strict then the set of equilibrium outcomes coincides with that of the outcomes generated by considering all full agendas for voting by successive elimination and show that the set of equilibrium outcomes corresponds with the Banks set. We also examine the implications in several other settings.Agenda ; Equilibrium ; Voting

The Banks set and the Uncovered Set in budget allocation problems

We examine how a society chooses to divide a given budget among various regions, projects or individuals. In particular, we characterize the Banks set and the Uncovered Set in such problems. We show that the two sets can be proper subsets of the set of all alternatives, and at times are very pointed in their predictions. This contrasts with well-known "chaos theorems," which suggest that majority voting does not lead to any meaningful predictions when the policy space is multidimensional

Two-stage majoritarian choice

We propose a class of decisive collective choice rules that rely on an exogenous linear ordering to partition the majority relation into two acyclic relations. The first relation is used to obtain a shortlist of the feasible alternatives while the second is used to make a final choice. In combination with faithfulness to the underlying majority relation, rules in this class are characterized by two desirable rationality properties: Sen’s expansion consistency and a version of Manzini and Mariotti’s weak WARP. The rules also satisfy natural adaptations of Arrow’s independence of irrelevant alternatives and May’s positive responsiveness

Implementation in adaptive better-response dynamics: Towards a general theory of bounded rationality in mechanisms.

We study the classic implementation problem under the behavioral assumption that agents myopically adjust their actions in the direction of better-responses or bestresponses. First, we show that a necessary condition for recurrent implementation in better-response dynamics (BRD) is a small variation of Maskin monotonicity, which we call quasimonotonicity. We also provide a mechanism for implementation in BRD if the rule is quasimonotonic and excludes worst alternatives – no worst alternative (NWA). Quasimonotonicity and NWA are both necessary and sufficient for absorbing implementation in BRD. Moreover, they characterize implementation in strict Nash equilibria. Under incomplete information, incentive compatibility is necessary for any kind of stable implementation in our sense, while Bayesian quasimonotonicity is necessary for recurrent implementation in interim BRD. Both conditions are also essentially sufficient for recurrent implementation, together with a Bayesian NWA. A characterization of implementation in strict Bayesian equilibria is also provided. Partial implementation results are also obtained.Robust implementation; Bounded rationality; Evolutionary dynamics; Mechanisms;

The Reintegration of Political Science and Economics and the Presumed Imperialism of Economic Theory

No discipline can claim a greater impact on contemporary political theorizing than that of economics, whether that theorizing concerns the study of legislatures, elections, international affairs, or judicial processes. This essay questions, however, whether this impact is a form of "economic imperialism," or the logical development of two disciplines whose artificial separation in the first part of this century merely allowed the development and refinement of the rational choice paradigm, unencumbered by the necessity for considering all of reality. Indeed, applications to specific substantive political matters -- most notably collective and cooperative processes where game theory proves most relevant -- reveal the paradigm's incompleteness. These applications, however, illuminate the necessary theoretical extensions, which is no longer the sole domain of the economist

The Reintegration of Political Science and Economics and the Presumed Imperialism of Economic Theory

No discipline can claim a greater impact on contemporary political theorizing than that of economics, whether that theorizing concerns the study of legislatures, elections, international affairs, or judicial processes. This essay questions, however, whether this impact is a form of "economic imperialism," or the logical development of two disciplines whose artificial separation in the first part of this century merely allowed the development and refinement of the rational choice paradigm, unencumbered by the necessity for considering all of reality. Indeed, applications to specific substantive political matters -- most notably collective and cooperative processes where game theory proves most relevant -- reveal the paradigm's incompleteness. These applications, however, illuminate the necessary theoretical extensions, which is no longer the sole domain of the economist

Why does voting get so complicated? : a review of theories for analyzing democratic participation.

The purpose of this article is to present a sample from the panoply of formal theories on voting and elections to Statistical Science readers who have had limited exposure to such work. These abstract ideas provide a framework for understanding the context of the empirical articles that follow in this volume. The primary focus of this theoretical literature is on the use of mathematical formalism to describe electoral systems and outcomes by modeling both voting rules and human behavior. As with empirical models, these constructs are never perfect descriptors of reality, but instead form the basis for understanding fundamental characteristics of the studied system. Our focus is on providing a general, but not overly simplified, review of these theories with practical examples. We end the article with a thought experiment that applies different vote aggregation schemes to the 2000 presidential election count in Florida, and we find that alternative methods provide different results