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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 ...
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.
Arrow's Theorem [1]: The 3 conditions of the constitution imply a dictator who prevails as to the social choice whatever that individual's preference and those of all else. An alternate statement of the theorem adds the following condition to the above:
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.
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.
ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice.. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension V[G] such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property.
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.
The revelation principle is a fundamental result in mechanism design, social choice theory, and game theory which shows it is always possible to design a strategy-resistant implementation of a social decision-making mechanism (such as an electoral system or market). [1] It can be seen as a kind of mirror image to Gibbard's theorem.