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Unrestricted domain is one of the conditions for Arrow's impossibility theorem. Under that theorem, it is impossible to have a social choice function that satisfies unrestricted domain, Pareto efficiency, independence of irrelevant alternatives, and non-dictatorship. However, the conditions of the theorem can be satisfied if unrestricted domain ...
Arrow's theorem assumes as background that any non-degenerate social choice rule will satisfy: [15]. Unrestricted domain – the social choice function is a total function over the domain of all possible orderings of outcomes, not just a partial function.
Independence of irrelevant alternatives (IIA) is an axiom of decision theory which codifies the intuition that a choice between and should not depend on the quality of a third, unrelated outcome . There are several different variations of this axiom, which are generally equivalent under mild conditions.
Tarski's theorem about choice: For every infinite set A, there is a bijective map between the sets A and A×A. Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other. Given two non-empty sets, one has a surjection to the other. Every surjective function has a right inverse.
Social choice theory is a branch of welfare economics that extends the theory of rational choice to collective decision-making. [1] Social choice studies the behavior of different mathematical procedures (social welfare functions) used to combine individual preferences into a coherent whole.
In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle or restricted forms of it. The theorem was discovered in 1975 by Radu Diaconescu [ 1 ] and later by Goodman and Myhill . [ 2 ]
Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem ; Büchi-Elgot-Trakhtenbrot theorem (mathematical logic) Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory, Cantor's diagonal argument) Church–Rosser theorem (lambda calculus)
Theorem: Any matching law selection rule satisfies Luce's choice axiom. Conversely, if P ( a ∣ A ) > 0 {\displaystyle P(a\mid A)>0} for all a ∈ A ⊂ X {\displaystyle a\in A\subset X} , then Luce's choice axiom implies that it is a matching law selection rule.