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Social choice theory is the study of theoretical and practical methods to aggregate or combine individual preferences into a collective social welfare function. The field generally assumes that individuals have preferences , and it follows that they can be modeled using utility functions , by the VNM theorem .
The Introduction contrasts voting and markets with dictatorship and social convention (such as those in a religious code). Both exemplify social decisions. Voting and markets facilitate social choice in a sense, whereas dictatorship and convention limit it. The former amalgamate possibly differing tastes to make a social choice.
In economics and social choice, a function satisfies anonymity, neutrality, or symmetry if the rule does not discriminate between different participants ahead of time. For example, in an election, a voter-anonymous function is one where it does not matter who casts which vote, i.e. all voters' ballots are equal ahead of time.
For example, the standard social scoring function for first-preference plurality is the total number of voters who rank a candidate first. Every social ordering can be made into a choice function by considering only the highest-ranked outcome. Less obviously, though, every social choice function is also an ordering function.
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.
In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed. Intuitively, unrestricted domain is a common requirement for social choice functions, and is a condition for Arrow's impossibility theorem.
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.
Formally, a social choice function is a map : () from the set of functions between N→Rel(X), to the power set of X. (Intuitively, the social choice function represents a societal principle for choosing one or more social outcomes based on individuals' preferences.