In this chapter, we seek to review some of the central concepts and results in the literature on positionalist voting rules, which goes back to Borda (1781). After the introduction of the basic notation and definitions in earlier parts of the chapter, we explore in Section 3 the distinction...

The implementation problem is the problem of designing a mechanism (game form) such that the equilibrium outcomes satisfy a criterion of social optimality embodied in a social choice rule. If a mechanism has the property that, in each possible state of the world, the set of equilibrium outcomes...

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...

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...

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...

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...

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.

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...

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...