Social welfare function

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.

Some authors maintain a distinction between three closely-related concepts: Every social ordering can be made into a choice function by considering only the highest-ranked outcome.

Deleting the best outcome, then finding the new winner, results in a runner-up who is assigned second place.

Top has the most first-preference votes; Bottom has the second-most; and Center (positioned between the two) has the fewest first preferences.

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."

Necessary general conditions are that at the maximum value of the function: 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.

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.

Auxiliary specifications enable comparison of different social states by each member of society in preference satisfaction.

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. 26) notes that Bergson's function "could derive Pareto optimality conditions as necessary but not sufficient for defining interpersonal normative equity."

Samuelson (1947, p. 221) himself stressed the flexibility of the social welfare function to characterize any one ethical belief, Pareto-bound or not, consistent with: 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.

It is written in implicit form, reflecting the feasible locus of utility combinations imposed by the restraints and allowed by Pareto efficiency.

Kenneth Arrow's 1963 book demonstrated the problems with such an approach, though he would not immediately realize this.

Arrow found that contrary to the assertions of Lionel Robbins and other behaviorists, dropping the requirement of real-valued (and thus cardinal) social orderings makes rational or coherent behavior at the social level impossible.

Arrow's theorem shows that it is impossible for an ordinal social welfare function to satisfy a standard axiom of rational behavior, called independence of irrelevant alternatives.

John Harsanyi later strengthened this result by showing that if societies must make decisions under uncertainty, the unique social welfare function satisfying coherence and Pareto efficiency is the utilitarian rule.

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

Amartya Sen proposed a welfare function in 1973: The average per capita income of a measured group (e.g. nation) is multiplied with

This welfare function marks the income, which a randomly selected Euro most likely belongs to.

Symmetry: 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.

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.

For example, the utility function should not depend on whether we measure incomes in cents or dollars.

This family has some familiar members: If we require the Pigou–Dalton principle (that inequality is not a positive good) then