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. A social welfare function takes two possible outcomes, then combines every person's preferences to determine which outcome is considered better by society as a whole. Inputs to the function can include any variables that affect the well-being of a society.^[1]

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 more desirable 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.

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 use only ordinal information.
• Cardinal (or rated voting) functions additionally incorporate cardinal information, with agents stating 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.