Positional rules of collective decision-making

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 between positionalist and non-positionalist social ranking rules and social decision rules. We define alternative notions of positionality and give several examples. Section 4 discusses the positionalist decision procedures in the context of the conditions figuring in Arrow's celebrated impossibility theorem and also in the context of the simple majority rule. In Section 5, we review the literature on the structure of the Borda ranking rule and the Borda decision rule. Some well-known axiomatic characterizations of the Borda ranking rule and the Borda decision rule are discussed here. In Section 6, we discuss some results on the structure of score-based ranking rules and score-based decision rules, which include, as special cases, the Borda ranking rule and the Borda decision rule, respectively. This section also contains a discussion of dominance-based scoring rules and social decision rules known as runoff systems. Section 7 concludes the chapter.