Domain Conditions in Social Choice Theory

Domain Conditions in Social Choice Theory

Domain Conditions in Social Choice Theory

Domain Conditions in Social Choice Theory

Synopsis

Offering the most comprehensive and up-to-date survey of current research in an important area of social choice theory, Wulf Gaertner's monograph provides an essential reference for economists and political scientists. In the interests of accessibilty and readability, extensive formal proofs to theorems are not included in the text but are carefully referenced, allowing interested readers to pursue them independently. Though written in a formal style, the mathematical level of the book is designed to be appropriate for graduate students with a basic training in mathematics.

Excerpt

The notation and definitions which we are introducing in this chapter will remain valid throughout chapters 2–5. Chapter 6 will use a somewhat different terminology.

Let X = {x, y, z, …}. denote the set of all conceivable social states and let N = {1, …, ng denote a finite set of individuals or voters (n > 2). Let R stand for a binary relation on X; R is a subset of ordered pairs in the product X × X. We interpret R as a preference relation on X. Without any index, R refers to the social preference relation. When we speak of individual i's preference relation we simply write R . the fact that a pair (x y) is an element of R will be denoted xRy the negation of this fact will be denoted by ¬xRy. R is reflexive if for all x ∈ X : xRx. R is complete if for all x, yX, x ≠ y : xRy or yRx. R is said to be transitive if for all x, y, z ∈ (X : xRy ∨ yRz) → xRz. the strict preference relation (the asymmetric part of R) will be denoted by P : xPy ↔ [xRy ∨ ¬ yRx]. the indifference relation (the symmetric part of R) will be denoted by I: xIy ↔[ xRy ∨ yRx]. We shall call R a preference ordering (or an ordering or a complete preordering) on X if R is reflexive, complete and transitive. in this case, one obviously obtains for all x, yX : xPy ↔: ¬yRx (reflexivity and completeness of R are sufficient for this result to hold), P is transitive and I is an equivalence relation; furthermore for all x, y, z ∈ (X : xPy ∨ yRz)→ xPz. R is said to be quasitransitive if P is transitive. R is said to be acyclical if for all finite sequences {x , …, x } from X it is not the case that x Px ∨ x Px ∨ … ∨x −iPx and x Px . the following implications clearly hold: R transitive → R quasitransitive → R acyclical.

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