Editing Social welfare function (section)

{{Short description|Function that ranks states of society according to their desirability}}
{{For|the practical application of social choice to elections|Electoral system}}

{{Electoral systems}}
{{Economics sidebar}}

In [[welfare economics]] and [[social choice theory]], a '''social welfare function'''—also called a '''social''' '''ordering''', '''ranking''', '''utility''', or '''choice''' '''function'''—is a [[function (mathematics)|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.<ref>[[Amartya K. Sen]], 1970 [1984], ''Collective Choice and Social Welfare'', ch. 3, "Collective Rationality." p. 33, and ch. 3*, "Social Welfare Functions." [http://www.citeulike.org/user/rlai/article/681900 Description.]</ref> It can be seen as mathematically formalizing [[Jean-Jacques Rousseau|Rousseau]]'s idea of a [[general will]].

Social choice functions are studied by [[Economist|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 distribution|income distributions]]).<ref>{{Cite book |last=Tresch |first=Richard W. |title=Public Sector Economics |publisher=PALGRAVE MACMILLAN |year=2008 |isbn=978-0-230-52223-7 |location=New York |pages=67}}</ref> They are also used by [[Democracy|democratic]] governments to choose between several options in [[Election|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 utility|'''Ordinal''']] (or [[ranked voting]]) functions only use [[Ordinal utility|ordinal]] information, i.e. whether one choice is better than another.
* [[Cardinal utility|'''Cardinal''']] (or [[rated voting]]) functions also use [[Cardinal utility|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 choice theory|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|second-best]].