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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.
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.
Less informally, the social choice function is the function mapping each environment S of available social states (at least two) for any given set of orderings (and corresponding social ordering R) to the social choice set, the set of social states each element of which is top-ranked (by R) for that environment and that set of orderings.
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.
May's theorem states that simple majority voting is the unique social welfare function satisfying all three of the following conditions: [1] Anonymity: The social choice function treats all voters the same, i.e. permuting the order of the voters does not change the result.
Every Pareto efficient social choice function is necessarily a utilitarian choice function, a result known as Harsanyi's utilitarian theorem. Specifically, any Pareto efficient social choice function must be a linear combination of the utility functions of each individual utility function (with strictly positive weights).
In social choice theory, the single-crossing condition is a condition on preferences.It is especially useful because utility functions are generally increasing (i.e. the assumption that an agent will prefer or at least consider equivalent two dollars to one dollar is unobjectionable).