This chapter discusses results about committees that use majority rule, some related results about majority rule as a social choice rule (that is, without referring to a specific context where majority rule is used), and results about electoral competitions where the winner is selected by majority rule.
Economists have used the term “nonbinary” to describe both choice functional nonbinariness (choice functions that cannot be rationalized as the maximizing outcome of a binary preference relation) and structural nonbinariness (the structure of the model dictates that pairs of alternatives do not belong to the domain of the social choice function). Here we have described necessary and sufficient conditions for oligarchy and dictatorship results in social choice models that are nonbinary in both senses.
This chapter surveys the literature on strategy proofness from a historical perspective. While I discuss the connections with other works on incentives in mechanism design, the main emphasis is on social choice models.
It is shown how simple geometry can be used to analyze and discover new properties about pairwise and positional voting rules as well as for those rules (e.g., runoffs and Approval Voting) that rely on these methods. The description starts by providing a geometric way to depict profiles, which simplifies the computation of the election outcomes. This geometry is then used to motivate the development of a "profile coordinate system," which evolves into a tool to analyze voting rules. This tool, for instance, completely explains various longstanding "paradoxes," such as why a Condorcet winner need not be elected with certain voting rules. A different geometry is developed to indicate whether certain voting "oddities" can be dismissed or must be taken seriously, and to explain why other mysteries, such as strategic voting and the no-show paradox (where a voter is rewarded by not voting), arise. Still another use of geometry extends McGarvey's Theorem about possible pairwise election rankings to identify the actual tallies that can arise (a result that is needed to analyze supermajority voting). Geometry is also developed to identify all possible positional and Approval Voting election outcomes that are admitted by a given profile; the converse becomes a geometric tool that can be used to discover new election relationships. Finally, it is shown how lessons learned in social choice, such as the seminal Arrow's and Sen's Theorems and the expanding literature about the properties of positional rules, provide insights into difficulties that are experienced by other disciplines.
We survey the theory of equity in a variety of concretely specified resource allocation models: classical economies with private goods, economies with production, economies with indivisible goods, when monetary compensations are feasible and when they are not, economies with single-peaked preferences, and economies in which the dividend is a non-homogeneous continuum. We present the central fairness punctual notions, no-envy, egalitarian-equivalence, concepts of equal or equivalent opportunities and the relational principles of monotonicity and consistency.
This article surveys the literature that investigates the consistency of Arrow's social choice axioms when his unrestricted domain assumptions are replaced by domain conditions that incorporate the restrictions on agendas and preferences encountered in economic environments. Both social welfare functions and social choice correspondences are considered.
A competitive market mechanism is a prominent example of a nonbinary social choice rule, typically defined for a special class of economic environments in which each social state is an economic allocation of private goods, and individuals' preferences concern only their own personal consumption. This chapter begins by discussing which Pareto efficient allocations can be characterized as competitive equilibria with lump-sum transfers. It also discusses existence and characterization of such equilibria without lump-sum transfers. The second half of the chapter focuses on continuum economies, for which such characterization results are much more natural, given that agents have negligible influence over equilibrium prices.
Fuzzy set theory has been explicitly introduced to deal with vagueness and ambiguity. One can also use probability theory or techniques borrowed from philosophical logic. In this chapter, we consider fuzzy preferences and we survey the literature on aggregation of fuzzy preferences. We restrict ourselves to "pure aggregation" theory and, accordingly, do not cover strategic aspects of social choice. We present Arrovian aggregation problems in a rather standard framework as well as in a very specific economic environment. We also consider a fuzzy treatment of Sen's impossibility of a Paretian liberal. We distinguish two types of fuzziness: quantitative fuzziness, defined via real numbers, and qualitative fuzziness, defined via linguistic data with a suitable order structure. We outline the thin frontier between impossibility and possibility results.
Ever since Sen crystallized the logical conflict between the welfaristic value of the Pareto principle and the nonwelfaristic value of individual libertarian rights into what he christened the impossibility of a Paretian liberal, there have been many attempts in social choice theory to generalize, or to repudiate, the conflict between welfare and rights. This chapter focuses on this logical conflict and tries to find a way of balancing these two important values in human well-being. We will identify three issues in this line of research, viz., the formal articulation of rights, the social realization of rights, and the initial conferment of rights. We will also examine the sustainability of Sen's concept of individual rights and examine the game form articulation of individual rights as a viable alternative to Sen's proposed concept.
Traditional economics identifies a person's well-being with the goods and services the person consumes and the utility that the person gets from such consumption. This, in turn, has led to the widely used approach of welfarism that uses individual utilities as ingredients for evaluating a society's aggregate welfare. This approach has long been contested as being too restrictive in its view of what constitutes human well-being and for its commodity fetish. What has injected new life into this critique is the emergence of an alternative approach, which replaces the traditional concern for commodities and utility with functionings and capabilities. While the origins of this "capabilities approach" go back to the works of John Stuart Mill, Adam Smith, and, in spirit if not in form, to Aristotle, it was the seminal contribution of Amartya Sen in the form of his 1979 Tanner Lectures that gave it shape and structure. Subsequent works by Sen and an enormous outpouring of writing by various authors in economics, philosophy, and sociology have made this a major field of inquiry, which has also led to important practical applications. The present chapter is a survey of this new field of study. In Sen's terminology a "functioning" is what an individual chooses to do or to be, in contrast to a "commodity," which is an instrument which enables her to achieve different functionings. While functioning is central to the notion of human well-being, it is not merely the achieved functionings that matter but the freedom that a person has in choosing from the set of feasible functionings, which is referred to as the person's "capability." Beginning with a discussion of these ideas in history, the present chapter tries to present a comprehensive review of the recent literature, including formalizations and applications. It is important to recognize that a full formalization may not be feasible, since there are important dimensions of life that are germane to the capabilities approach that may be impossible to capture in a single formalization. Nevertheless, the capability approach itself has been immensely useful in the context of studying poverty, gender issues, political freedom, and the standard of living. It has also resulted in the creation of the Human Development Index (HDI), popularized by UNDP's Human Development Reports since 1990. This chapter critically examines the HDI and recent advances in the human development literature.
This chapter presents a simple introduction to the main results in topological social choice theory. Given a continuous social welfare function, these results show the following: (i) Unanimity and Anonymity are incompatible; (ii) Weak Pareto and No Veto are incompatible; and (iii) Weak Pareto implies the existence of a Strategic Manipulator. Given the role of continuity in all these results, its justification is critically discussed. Finally, a remarkable proof of Arrow's theorem using topological methods is presented.
This paper reexamines key results from the measurement of opportunity freedom, or the extent to which a set of options offers a decision maker real opportunities to achieve. Three cases are investigated: no preferences, a single preference, and plural preferences. The three corresponding evaluation methods-the cardinality relation, the indirect utility relation, and the effective freedom relation-and their variations are considered within a common axiomatic framework. Special attention is given to representations of freedom rankings, with the goal of providing practical approaches for measuring opportunity freedom and the extent of people's capabilities.
This chapter provides a survey of utilitarian theories of justice. We review and discuss axiomatizations of utilitarian and generalized-utilitarian social-evaluation functionals in a welfarist framework. Section 2 introduces, along with some basic definitions, social-evaluation functionals. Furthermore, we discuss several information-invariance assumptions. In Section 3, we introduce the welfarism axioms unrestricted domain, binary independence of irrelevant alternatives and Pareto indifference, and use them to characterize welfarist social evaluation. These axioms imply that there exists a single ordering of utility vectors that can be used to rank all alternatives for any profile of individual utility functions. We call such an ordering a social-evaluation ordering, and we introduce several examples of classes of such orderings. In addition, we formulate some further basic axioms. Section 4 provides characterizations of generalized-utilitarian social-evaluation orderings, both in a static and in an intertemporal framework. Section 5 deals with the special case of utilitarianism. We review some known axiomatizations and, in addition, prove a new characterization result that uses an axiom we call incremental equity. In Section 6, we analyze generalizations of utilitarian principles to variable-population environments. We extend the welfarism theorem to a variable-population framework and provide a characterization of critical-level generalized utilitarianism. Section 7 provides an extension to situations in which the alternatives resulting from choices among feasible actions are not known with certainty. In this setting, we discuss characterization as well as impossibility results. Section 8 concludes.
This chapter is concerned with issues arising from the construction of ethical measures of inequality and poverty. The recent literature on measurement of inequality and poverty emphasizes the close connection between social welfare functions and ethical indices of inequality and poverty. This chapter surveys the main issues in this literature. In particular, we discuss how indices of inequality can be constructed from social welfare functions, and vice versa. Other issues include the equivalence theorems which provide the analytical foundations of the approach which declares one distribution to be more equal than another only when all sensible measures agree on the ranking. The chapter also discusses the measurement of mobility. Finally, the chapter describes some of the parallel issues which arise in the measurement of poverty.
Social Choice traditionally employs the preferences of voters or agents as primitives. However, in most situations of constitutional decision-making the beliefs of the members of the electorate determine their secondary preferences or choices. Key choices in US political history, such as the ratification of the Constitution in 1787 and the election of Lincoln in 1860, were conditioned by changing beliefs as regards the truth of propositions about the political universe. Preference-based models of election tend to conclude that candidates, or parties, converge to a vote-maximizing policy position at the center. Empirical work suggests that such a conclusion is invalid. This chapter argues, on the contrary, that parties or candidates adopt positions that optimize, in a Nash equilibrium sense, with respect to both their beliefs over electoral response, and their beliefs over appropriate policy choices. The analysis indicates that political choices will be different depending on whether plurality (first past the post) or proportionality is used as the method of electoral representation.
