Editing Social welfare function

{{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]].

== {{Anchor|Construction|Construct|Constructing}}Terminology and equivalence ==
Some authors maintain a distinction between three closely-related concepts: 

# A social ''choice'' function selects a single best outcome (a single candidate who wins, or multiple if there happens to be a tie).
# A social ''ordering'' function lists the candidates, from best to worst.
# A social ''scoring'' function maps each candidate to a number representing their quality. 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. Deleting the best outcome, then finding the new winner, results in a runner-up who is assigned second place. Repeating this process gives a full ranking of all candidates.<ref>{{Cite journal |last=Quesada |first=Antonio |date=2002 |title=From social choice functions to dictatorial social welfare functions |url=https://ideas.repec.org//a/ebl/ecbull/eb-02d70006.html |journal=Economics Bulletin |volume=4 |issue=16 |pages=1–7}}</ref>

Because of this close relationship, the three kinds of functions are often conflated by [[abuse of terminology]].

=== Example ===
Consider an [[Instant-runoff voting|instant-runoff]] election between Top, Center, and Bottom. Top has the most first-preference votes; Bottom has the second-most; and Center (positioned [[Political spectrum|between the two]]) has the fewest first preferences. 
{| class="wikitable"
|+
!
!Round 1
!Round 2
|-
!Top
|40
|'''53'''
|-
!Center
|<s>26</s>
|''Eliminated''
|-
!Bottom
|34
|47
|}
Under instant-runoff voting, Top is the winner. Center is eliminated in the first round, and their second-preferences are evenly split between Top and Bottom, allowing Top to win.

To find the second-place finisher, we find the winner if Top had not run. In this case, the election is between Center and Bottom.
{| class="wikitable"
!Runner-up
!Round 1
|-
!—
|''Excluded''
|-
!Center
|'''66'''
|-
!Bottom
|34
|}
(Note that the finishing order is not the same as the elimination order for [[Sequential elimination method|sequential elimination methods]]: despite being eliminated first, Center is the runner-up in this election.)

==Ordinal welfare==
{{Multiple issues|{{technical|section|date=March 2024}}
{{Copy editing|section|date=March 2024}}
{{More citations needed|section|date=March 2024}}
{{rewrite||section|date=March 2024}}|section=y}}
In a 1938 article, [[Abram Bergson]] introduced the term ''social welfare function,'' with the intention "to state in precise form the value judgments required for the derivation of the conditions of maximum economic welfare." The function was real-valued and [[Differentiable function|differentiable]]. It was specified to describe the society as a whole. Arguments of the function included the quantities of different commodities produced and consumed and of [[factors of production|resources]] used in producing different commodities, including labor.

Necessary general conditions are that at the maximum value of the function:
* The marginal "dollar's worth" of welfare is equal for each individual and for each commodity
* The marginal "dis-welfare" of each "dollar's worth" of labor is equal for each commodity produced of each labor supplier
* The marginal "dollar" cost of each unit of resources is equal to the marginal value productivity for each commodity.
Bergson argued that [[welfare economics]] had described a standard of economic efficiency despite dispensing with ''interpersonally-comparable'' [[cardinal utility]], the hypothesization of which may merely conceal value judgments, and purely subjective ones at that.

Earlier neoclassical welfare theory, heir to the classical [[utilitarianism]] of [[Jeremy Bentham|Bentham]], often treated the [[law of diminishing marginal utility]] as implying interpersonally comparable utility. Irrespective of such comparability, income or wealth ''is'' measurable, and it was commonly inferred that redistributing income from a rich person to a poor person tends to increase total utility (however measured) in the society. But Lionel Robbins ([[An Essay on the Nature and Significance of Economic Science|1935]], ch. VI) argued that how or how much utilities, as mental events, change relative to each other is not measurable by any empirical test, making them [[unfalsifiable]]. Robbins therefore rejected such as incompatible with his own philosophical [[behaviorism]].

Auxiliary specifications enable comparison of different social states by each member of society in preference satisfaction. These help define ''[[Pareto efficiency]]'', which holds if all alternatives have been exhausted to put at least one person in a more preferred position with no one put in a less preferred position. Bergson described an "economic welfare increase" (later called a ''Pareto improvement'') as at least one individual moving to a more preferred position with everyone else indifferent. The social welfare function could then be specified in a ''substantively'' individualistic sense to derive Pareto efficiency (optimality). [[Paul Samuelson]] (2004, p.&nbsp;26) notes that Bergson's function "could derive Pareto optimality conditions as ''necessary'' but not sufficient for defining interpersonal normative equity." Still, Pareto efficiency could also characterize ''one'' dimension of a particular social welfare function with distribution of commodities among individuals characterizing ''another'' dimension. As Bergson noted, a welfare improvement from the social welfare function could come from the "position of some individuals" improving at the expense of others. That social welfare function could then be described as characterizing an equity dimension.

Samuelson ([[Foundations of Economic Analysis|1947]], p.&nbsp;221) himself stressed the flexibility of the social welfare function to characterize ''any'' one ethical belief, Pareto-bound or not, consistent with:
* a complete and transitive ranking (an ethically "better", "worse", or "indifferent" ranking) of all social alternatives and
* one set out of an infinity of welfare indices and cardinal indicators to characterize the belief.
As Samuelson (1983, p. xxii) notes, Bergson clarified how production and consumption efficiency conditions are distinct from the interpersonal ethical values of the social welfare function.

Samuelson further sharpened that distinction by specifying the ''welfare function'' and the ''possibility function'' (1947, pp.&nbsp;243–49). Each has as [[Function (mathematics)#The vocabulary of functions|arguments]] the set of utility functions for everyone in the society. Each can (and commonly does) incorporate Pareto efficiency. The possibility function also depends on technology and resource restraints. It is written in implicit form, reflecting the ''feasible'' locus of utility combinations imposed by the restraints and allowed by Pareto efficiency. At a given point on the possibility function, if the utility of all but one person is determined, the remaining person's utility is determined. The welfare function ranks different hypothetical ''sets'' of utility for everyone in the society from ethically lowest on up (with ties permitted), that is, it makes interpersonal comparisons of utility. Welfare maximization then consists of maximizing the welfare function subject to the possibility function as a constraint. The same welfare maximization conditions emerge as in Bergson's analysis.
{| style="border:1px solid #999; text-align:leftcenter; margin: auto;" cellspacing="20"
| For a two-person society, there is a graphical depiction of such welfare maximization at the first figure of [https://web.archive.org/web/20060215000915/http://cepa.newschool.edu/het/essays/paretian/paretosocial.htm#swf Bergson–Samuelson social welfare functions]. Relative to [[consumer theory]] for an ''individual'' as to two commodities consumed, there are the following parallels:
* The respective hypothetical utilities of the two persons in two-dimensional utility space is analogous to respective quantities of commodities for the two-dimensional commodity space of the indifference-curve ''surface''
* The Welfare function is analogous to the indifference-curve ''map''
* The Possibility function is analogous to the budget constraint
* Two-person welfare maximization at the tangency of the highest Welfare function curve on the Possibility function is analogous to tangency of the highest indifference curve on the budget constraint.
|}
[[Kenneth Arrow]]'s [[Social Choice and Individual Values|1963 book]] demonstrated the problems with such an approach, though he would not immediately realize this. Along earlier lines, Arrow's version of a social welfare function, also called a 'constitution', maps a set of individual orderings ([[ordinal utility function]]s) for everyone in society to a social ordering, which ranks alternative social states (such as which of several candidates should be elected).

Arrow found that contrary to the assertions of [[Lionel Robbins]] and other [[Behaviorism|behaviorists]], dropping the requirement of real-valued (and thus [[cardinal utility|cardinal]]) social orderings makes [[Decision theory|rational]] or [[Coherence (philosophical gambling strategy)|coherent]] behavior at the social level impossible. This result is now known as [[Arrow's impossibility theorem]]''.'' Arrow's theorem shows that it is impossible for an ordinal social welfare function to satisfy a standard axiom of [[Decision theory|rational behavior]], called [[independence of irrelevant alternatives]]. This axiom says that changing the value of one outcome should not affect choices that do not involve this outcome. For example, if a customer buys apples because he prefers them to blueberries, telling them that cherries are on sale should not make them buy blueberries instead of apples.

[[John Harsanyi]] later strengthened this result by [[Harsanyi's utilitarian theorem|showing]] that if societies must make decisions under [[uncertainty]], the unique social welfare function satisfying [[Coherence (philosophical gambling strategy)|coherence]] and [[Pareto efficiency]] is the [[utilitarian rule]].

==Cardinal welfare==
A '''cardinal social welfare function''' is a function that takes as input numeric representations of individual utilities (also known as [[cardinal utility]]), and returns as output a numeric representation of the collective welfare. The underlying assumption is that individuals utilities can be put on a common scale and compared. Examples of such measures include [[life expectancy]] or per capita income.

For the purposes of this section, income is adopted as the measurement of utility.

The form of the social welfare function is intended to express a statement of objectives of a society.

The [[utilitarian]] or [[Benthamite]] social welfare function measures social welfare as the total or sum of individual utilities:
:<math>W = \sum_{i=1}^n Y_i</math>
where <math>W</math> is social welfare and <math>Y_i</math> is the income of individual <math>i</math> among <math>n</math> individuals in society. In this case, maximizing the social welfare means maximizing the total income of the people in the society, without regard to how incomes are distributed in society. It does not distinguish between an income transfer from rich to poor and vice versa. If an income transfer from the poor to the rich results in a bigger increase in the utility of the rich than the decrease in the utility of the poor, the society is expected to accept such a transfer, because the total utility of the society has increased as a whole. Alternatively, society's welfare can also be measured under this function by taking the average of individual incomes:
:<math>W = \frac{1}{n}\sum_{i=1}^n Y_i = \overline{Y}</math>

In contrast, the max-min or Rawlsian social welfare function (based on the philosophical work of [[John Rawls]]) measures the social welfare of society on the basis of the welfare of the least well-off individual member of society:
:<math>W = \min(Y_1, Y_2, \cdots, Y_n)</math>
Here maximizing societal welfare would mean maximizing the income of the poorest person in society without regard for the income of other individuals.

These two social welfare functions express very different views about how a society would need to be organised in order to maximize welfare, with the first emphasizing total incomes and the second emphasizing the needs of the worst-off. The max-min welfare function can be seen as reflecting an extreme form of [[uncertainty aversion]] on the part of society as a whole, since it is concerned only with the worst conditions that a member of society could face.

[[Amartya Sen]] proposed a welfare function in 1973:
:<math>W_\mathrm{Gini} = \overline{Y} (1-G)</math>
 
The average per capita income of a measured group (e.g. nation) is multiplied with <math>(1-G)</math> where <math>G</math> is the [[Gini index]], a relative inequality measure. James E. Foster (1996) proposed to use one of [[Anthony Barnes Atkinson|Atkinson]]'s Indexes, which is an entropy measure. Due to the relation between Atkinsons entropy measure and the [[Theil index]], Foster's welfare function also can be computed directly using the Theil-L Index.

:<math>W_\mathrm{Theil-L} = \overline{Y} \mathrm{e}^{-T_L}</math>

The value yielded by this function has a concrete meaning. There are several possible incomes which could be earned by a ''person'', who randomly is selected from a population with an unequal distribution of incomes. This welfare function marks the income, which a randomly selected person is most likely to have. Similar to the [[median]], this income will be smaller than the average per capita income.

:<math>W_\mathrm{Theil-T} = \overline{Y} \mathrm{e}^{-T_T}</math>

Here the Theil-T index is applied. The inverse value yielded by this function has a concrete meaning as well. There are several possible incomes to which a ''Euro'' may belong, which is randomly picked from the sum of all unequally distributed incomes. This welfare function marks the income, which a randomly selected Euro most likely belongs to. The inverse value of that function will be larger than the average per capita income.

=== Axioms of cardinal welfarism ===
Suppose we are given a [[preference relation]] ''R'' on utility profiles. ''R'' is a weak [[total order]] on utility profiles—it can tell us, given any two utility profiles, if they are indifferent or one of them is better than the other. A reasonable preference ordering should satisfy several axioms:<ref name=moulin2004>{{Cite Moulin 2004}}</ref>{{rp|66–69}}

1. [[Monotonicity criterion|'''Monotonicity''']]: if the utility of one individual increases, while all other utilities remain equal, ''R'' should strictly prefer the second profile. For example, it should prefer the profile (1, 4, 4, 5) to (1, 2, 4, 5). Such a change is called a [[Pareto optimality|Pareto improvement]].

2. [[Anonymity (social choice)|'''Symmetry''']]: [[Permutation|reordering or relabeling]] the values in the utility profile should not change the output of ''R''. This axiom formalizes the idea that every person should be treated equally in society. For example, ''R'' should be indifferent between (1, 4, 4, 5) and (5, 4, 4, 1), since these profiles are reorders of each other.  

3. '''Continuity''': for every profile ''v'', the set of profiles weakly better than ''v'' and the set of profiles weakly worse than ''v'' are [[closed set]]s.{{Technical inline|date=March 2024}}

4. '''Independence of unconcerned agents:''' ''R'' should be independent of individuals whose utilities have not changed. For example, if ''R'' prefers (2, 2, 4) to (1, 3, 4), it also prefers (2, 2, 9) to (1, 3, 9); the utility of agent 3 should not affect the comparison between two utility profiles of agents 1 and 2. This property can also be called '''locality''' or '''separability'''. It allows us to treat allocation problems in a local way, and separate them from the allocation in the rest of society.

Every preference relation with properties 1–4 can be represented as by a function ''W'' which is a sum of the form:
:<math>W(u_1,\dots,u_n) = \sum_{i=1}^n w(u_i)</math>
where ''w'' is a continuous increasing function.

=== Harsanyi's theorem ===
Introducing one additional axiom—the nonexistence of [[Coherence (philosophical gambling strategy)|Dutch Books]], or equivalently that social choice behaves according to the [[Von Neumann–Morgenstern utility theorem|axioms of rational choice]]—implies that the social choice function must be the [[utilitarian rule]], i.e. the weighting function <math>w(u)</math> must be equal to the utility functions of each individual. This result is known as [[Harsanyi's utilitarian theorem]].

=== Non-utilitarian ===
By Harsanyi's theorem, any non-utilitarian social choice function will be incoherent; in other words, it will agree to some bets that are unanimously opposed by every member of society. However, it is still possible to establish properties of such functions. 

Instead of imposing rational behavior on the social utility function, we can impose a weaker criterion called '''independence of common scale''': the relation between two utility profiles does not change if both of them are multiplied by the same constant. For example, the utility function should not depend on whether we measure incomes in cents or dollars.

If the preference relation has properties 1–5, then the function ''w'' must be the [[Isoelastic utility|isoelastic function]]:

<math>\frac{c^{1-\eta} - 1}{1-\eta}</math>

This family has some familiar members:
* The limit when <math>\eta \to -\infty</math> is the ''leximin'' ordering.
* For <math>\eta = 0</math> we get the [[Nash bargaining solution]]—maximizing the product of utilities.
* For <math>\eta = 1</math> we get the [[utilitarian]] welfare function—maximizing the sum of utilities.
* The limit when <math>\eta \to \infty</math> is the ''leximax'' ordering.

If we require the '''[[Pigou–Dalton principle]]''' (that inequality is not a positive good) then <math>\eta</math> in the above family must be at most 1.

== See also ==
{{div col|colwidth=22em}}
* [[Aggregation problem]]
* [[Arrow's impossibility theorem]]
* [[Community indifference curve]]
* [[Distribution (economics)]]
* [[Economic welfare]]
* [[Extended sympathy]]
* [[Gorman polar form]]
* [[Justice (economics)]]
* [[Liberal paradox]]
* [[Production-possibility frontier]]
* [[Social choice theory]]
* [[Welfare economics]]
{{div col end}}

==Notes==
{{Reflist|1}}

== References ==
*[[Kenneth Arrow|Kenneth J. Arrow]], 1951, 2nd ed., 1963, ''[[Social Choice and Individual Values]]'' {{ISBN|0-300-01364-7}} 
* [[Abram Bergson]] (Burk),"A Reformulation of Certain Aspects of Welfare Economics," ''Quarterly Journal of Economics'', 52(2), February 1938, 310–34
* [https://web.archive.org/web/20060215000915/http://cepa.newschool.edu/het/essays/paretian/paretosocial.htm#swf Bergson–Samuelson social welfare functions] in Paretian welfare economics from the New School.
* James E. Foster and [[Amartya Sen]], 1996, ''On Economic Inequality'', expanded edition with annexe, {{ISBN|0-19-828193-5}}.
* [[John C. Harsanyi]], 1987, “interpersonal utility comparisons," ''[[The New Palgrave: A Dictionary of Economics]]'', v. 2, 955–58
* {{Citation | last = Pattanaik | first = Prasanta K. | author-link = Prasanta Pattanaik | contribution = social welfare function ''(definition)'' | editor-last1 = Durlauf | editor-first1 = Steven N. | editor-last2 = Blume | editor-first2 = Lawrence E. | editor-link1 = Steven N. Durlauf | editor-link2 = Lawrence E. Blume | title = The new Palgrave dictionary of economics | publisher = Palgrave Macmillan | edition = 2nd | location = Basingstoke, Hampshire New York | year = 2008 | isbn = 9780333786765  | postscript = .}}
::Also available as: [https://dx.doi.org/10.1057/9780230226203.1567 a journal article.]
* [[Jan de Van Graaff]], 1957, "Theoretical Welfare Economics", 1957, Cambridge, UK: Cambridge University Press.
* [[Lionel Robbins]], 1935, 2nd ed.. ''[[An Essay on the Nature and Significance of Economic Science]]'', ch. VI
* ____, 1938, "Interpersonal Comparisons of Utility: A Comment," ''Economic Journal'', 43(4), 635–41
* [[Paul A. Samuelson]], 1947, Enlarged ed. 1983, ''[[Foundations of Economic Analysis]]'', pp. xxi–xxiv & ch. VIII, "Welfare Economics," {{ISBN|0-674-31301-1}}
* _____, 1977. "Reaffirming the Existence of 'Reasonable' Bergson–Samuelson Social Welfare Functions," ''Economica'', N.S., 44(173), p [https://www.jstor.org/pss/2553553 pp. 81]–88. Reprinted in (1986) ''The Collected Scientific Papers of Paul A. Samuelson'', pp. [https://books.google.com/books?id=UKeJEc46R9AC&pg=PA47=gbs_atb 47–54.] 
* _____, 1981. "Bergsonian Welfare Economics", in S. Rosefielde (ed.), ''Economic Welfare and the Economics of Soviet Socialism: Essays in Honor of Abram Bergson'', [[Cambridge University Press]], Cambridge, pp.&nbsp;223–66. Reprinted in (1986) ''The Collected Scientific Papers of Paul A. Samuelson'', pp.&nbsp;3 [https://books.google.com/books?id=UKeJEc46R9AC&pg=PA225=gbs_atb –46.]
* [[Amartya Sen|Sen, Amartya K.]] (1963). "Distribution, Transitivity and Little's Welfare Criteria," ''Economic Journal'', 73(292), [https://www.jstor.org/stable/2228209 pp. 771]–78.
* _____, 1970 [1984], ''Collective Choice and Social Welfare'' [http://www.citeulike.org/user/rlai/article/681900 (description)], ch. 3, "Collective Rationality." {{ISBN|0-444-85127-5}}
* _____ (1982). ''Choice, Welfare and Measurement'', MIT Press. [https://scholar.harvard.edu/sen/publications/choice-welfare-and-measurement Description] and scroll to chapter-preview [https://books.google.com/books?id=u8GPYeT1qAUC&q=false links.]
* [[Kotaro Suzumura]] (1980). "On Distributional Value Judgments and Piecemeal Welfare Criteria," ''Economica'', 47(186), p [https://www.jstor.org/stable/2553231 pp. 125]–39.
* _____, 1987, “social welfare function," ''The New Palgrave: A Dictionary of Economics'', v. 4, 418–20

{{Authority control}}{{Voting systems}}{{Economics}}

[[Category:Welfare economics]]
[[Category:Social choice theory]]
[[Category:Mathematical economics]]