Social welfare function

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In welfare economics and social choice theory, a social welfare function—also called a social ordering, ranking, utility, or choice function—is a function that ranks a set of social states by their desirability. Each person's preferences are combined in some way to determine which outcome is considered better by society as a whole.^[1] It can be seen as mathematically formalizing Rousseau's idea of a general will.

Social choice functions are studied by economists as a way to identify socially-optimal decisions, giving a procedure to rigorously define which of two outcomes should be considered better for society as a whole (e.g. to compare two different possible income distributions).^[2] They are also used by democratic governments to choose between several options in elections, based on the preferences of voters; in this context, a social choice function is typically referred to as an electoral system.

The notion of social utility is analogous to the notion of a utility function in consumer choice. However, a social welfare function is different in that it is a mapping of individual utility functions onto a single output, in a way that accounts for the judgments of everyone in a society.

There are two different notions of social welfare used by economists:

• Ordinal (or ranked voting) functions only use ordinal information, i.e. whether one choice is better than another.
• Cardinal (or rated voting) functions also use cardinal information, i.e. how much better one choice is compared to another.

Arrow's impossibility theorem is a key result on social welfare functions, showing an important difference between social and consumer choice: whereas it is possible to construct a rational (non-self-contradictory) decision procedure for consumers based only on ordinal preferences, it is impossible to do the same in the social choice setting, making any such ordinal decision procedure a second-best.