Social welfare function

• 1. Introduction

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  Social welfare function is the name economists and political philosophers give to one of the oldest puzzles in human governance: how do you add up the desires of many different people and call the result a single decision for society? Abram Bergson introduced the term in a 1938 article, with the stated intention of putting into precise form the value judgments required to derive conditions of maximum economic welfare. That goal sounds technical, but the stakes are anything but abstract. Should a government distribute income evenly, or concentrate it where it generates the most total wealth? Should an election system crown whoever gets the most first-place votes, or should it weigh the deeper preferences beneath the surface? These questions sit at the heart of every policy debate in every democracy on earth.

  The idea itself reaches back further than 1938. Philosophers trace its intellectual ancestry to Jean-Jacques Rousseau's concept of a general will, the notion that a group of people, properly organized, can express a shared judgment that is more than just a headcount. What Bergson and the economists who followed him tried to do was translate that philosophical intuition into a mathematical object you could actually compute.

  Kenneth Arrow's 1963 book then threw a wrench into the entire enterprise. Arrow showed that any ordinal social welfare function, one that merely ranks outcomes without measuring the distances between them, cannot satisfy a basic axiom of rational behavior called independence of irrelevant alternatives. The implications of that finding are still debated today. They run through electoral systems, income policy, and the deepest questions about what fairness even means.

• 2. Bergson and the Welfare Maximum

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  Abram Bergson's 1938 article did not just name a concept. It laid out the necessary general conditions for a welfare maximum, and each condition was precise. At the peak of the function, the marginal dollar's worth of welfare must be equal for each individual and for each commodity. The marginal discomfort of each dollar's worth of labor must be equal for each commodity produced by each supplier of that labor. And the marginal dollar cost of each unit of resources must equal the marginal value it produces for each commodity. These are the conditions economists call efficiency.

  Bergson was reacting against an older tradition rooted in classical utilitarianism. Earlier neoclassical welfare theory, heir to Jeremy Bentham, often treated the law of diminishing marginal utility as proof that you could compare one person's satisfaction directly to another's. If a rich person loses a dollar, the argument went, society loses less welfare than it gains when a poor person receives that dollar. Bergson saw the problem: this assumed you could place every person's inner experience on a single measuring stick.

  Lionel Robbins had already challenged that assumption directly in a 1935 chapter. He argued that how utilities change relative to each other, as mental events, is not measurable by any empirical test. That makes them unfalsifiable. On those grounds, Robbins rejected interpersonal utility comparisons as incompatible with his own philosophical behaviorism.

  Bergson's response was to build a welfare function that could derive Pareto efficiency without requiring those comparisons. Pareto efficiency holds that a situation is optimal when it is impossible to make at least one person better off without making anyone else worse off. Bergson called a specific improvement, where at least one individual moves to a more preferred position while everyone else is at worst indifferent, an economic welfare increase. That framing later became standard vocabulary under the label Pareto improvement.

• 3. Samuelson Sharpens the Framework

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  Paul Samuelson carried Bergson's framework further. Writing in 1947, Samuelson stressed that the social welfare function is flexible enough to characterize any ethical belief, Pareto-bound or not, provided that belief can be expressed as a complete and transitive ranking of all social alternatives, along with one set out of an infinity of welfare indices and cardinal indicators. That flexibility was the point. The function does not smuggle in a hidden value judgment. It makes the judgment explicit, so it can be examined and debated.

  Samuelson introduced a second mathematical object alongside the welfare function: the possibility function. Each takes as its arguments the full set of utility functions for everyone in society. The welfare function ranks hypothetical sets of utility from ethically lowest to highest, making interpersonal comparisons explicit rather than hidden. The possibility function captures the feasible locus of utility combinations, constrained by technology and resources, after Pareto efficiency has been applied. Welfare maximization then means finding the point where the welfare function is tangent to the possibility function.

  Samuelson drew a deliberate geometric parallel to ordinary consumer theory. The two persons' utilities in a two-person society play the role of two commodities in standard indifference-curve analysis. The welfare function maps onto the indifference-curve surface. The possibility function maps onto the budget constraint. The point of tangency is the optimum, just as in the familiar consumer diagram.

  As Samuelson noted in 1983, Bergson had also clarified something often overlooked: the conditions for production and consumption efficiency are logically distinct from the interpersonal ethical values that a social welfare function embodies. Samuelson's contribution was to make that distinction a formal separation in the mathematics, not just a philosophical footnote.

• 4. Arrow's Impossibility Theorem

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  Kenneth Arrow's 1963 book took direct aim at the ordinal approach, and its conclusion unsettled the entire field. Arrow showed that dropping the requirement of real-valued, and therefore cardinal, social orderings does not simplify the problem. It makes rational behavior at the social level impossible.

  Arrow's version of a social welfare function, which he also called a constitution, maps a set of individual orderings for everyone in society onto a single social ordering of alternatives. The theorem shows that no such function can satisfy independence of irrelevant alternatives, a standard axiom of rational behavior. The axiom says that changing the value assigned to one outcome should not affect choices among outcomes that do not involve it. Arrow illustrated the intuition with a consumer example: if someone buys apples because they prefer apples to blueberries, learning that cherries are on sale should not flip that preference toward blueberries.

  The consequences for electoral systems are immediate. Consider an instant-runoff election between three candidates: Top, Center, and Bottom. Top holds the most first-preference votes; Bottom holds the second-most; Center, positioned between the two, has the fewest first preferences. In round one, Center is eliminated, and their second preferences split evenly between Top and Bottom, letting Top win. But if you want to find the second-place finisher by removing Top and re-running the election, Center now defeats Bottom. Despite being eliminated first, Center is the runner-up. The finishing order is not the same as the elimination order, and that mismatch illustrates exactly the kind of inconsistency Arrow's theorem identifies.

  John Harsanyi later strengthened Arrow's result on the cardinal side, showing that if societies must make decisions under uncertainty, the only social welfare function satisfying both coherence and Pareto efficiency is the utilitarian rule.

• 5. Ordinal versus Cardinal Measurement

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  The divide between ordinal and cardinal welfare functions is not a technical footnote. It reflects a deep disagreement about what information a society can legitimately use when making collective decisions.

  Ordinal functions use only rank information: outcome A is better than outcome B, full stop. They do not ask by how much. Ranked voting systems in elections are ordinal in exactly this sense. Cardinal functions go further. They require a numeric representation of how much better one option is than another. Life expectancy and per capita income are examples the literature offers as candidate measures that could put individuals on a common scale.

  The ordinal approach seems more modest and harder to manipulate, but Arrow's theorem shows it is also more constrained. Dropping cardinal measurement does not remove the value judgment from the process. It just hides it inside the choice of voting procedure. The cardinal approach makes the value judgment explicit: you must decide whose utilities count more and by how much, and then you can be held to account for that decision.

  The distinction also maps onto two different practical traditions. Democratic governments choosing among candidates in elections are using ordinal social choice functions, which in that context are called electoral systems. Economists evaluating income distributions or policy outcomes are more likely to reach for cardinal measures. Both traditions are studying the same underlying mathematical object; they simply disagree about what inputs that object should accept.

• 6. Four Welfare Functions Compared

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  Economists have proposed several specific cardinal social welfare functions, and each encodes a different vision of fairness.

  The utilitarian or Benthamite function measures social welfare as the simple sum of all individual utilities, or equivalently as the average. Under this rule, maximizing welfare means maximizing total income across the population, with no regard for how that income is distributed. A transfer from a poor person to a rich person is acceptable if the rich person's utility gain exceeds the poor person's utility loss. The function does not distinguish direction.

  John Rawls supplied the philosophical foundation for a very different rule. The max-min or Rawlsian function measures social welfare solely on the basis of the least well-off individual in society. Maximizing welfare under this rule means maximizing the income of the poorest person, regardless of what happens to everyone else. The function can be read as expressing extreme uncertainty aversion on the part of society: it focuses entirely on the worst outcome any member could face.

  Amartya Sen proposed a third approach in 1973. His function multiplies the average per capita income of a measured group by a factor derived from the Gini index, which is a standard relative measure of inequality. A more unequal distribution therefore reduces the welfare value even if average income stays constant.

  James E. Foster proposed in 1996 to use one of Atkinson's indexes, an entropy measure related to the Theil index. The value that function produces has a concrete interpretation: it identifies the income that a randomly selected person is most likely to have in a population with an unequal income distribution. That income will always be smaller than the average per capita income, because inequality concentrates people at the lower end of the scale. Foster's companion function, built on the Theil-T index, identifies the income that a randomly selected euro most likely belongs to, and that value runs larger than the average.

• 7. Axioms, Harsanyi, and Coherence

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  Behind the specific formulas lies a set of axioms that any reasonable preference ordering over utility profiles should satisfy. Monotonicity says that if one person's utility rises while all others stay fixed, the new profile must be strictly preferred. The example in the literature compares the profile (1, 4, 4, 5) to (1, 2, 4, 5): the second profile should win because one person is simply better off.

  Symmetry requires that relabeling or reordering the values in a profile should not change the ranking. A society that values symmetry treats (1, 4, 4, 5) as equivalent to (5, 4, 4, 1), because the two profiles are reorderings of each other and every person should be treated equally.

  Continuity and independence of unconcerned agents round out the standard four. Independence of unconcerned agents, also called locality or separability, says that if a preference ranking prefers (2, 2, 4) to (1, 3, 4), it must also prefer (2, 2, 9) to (1, 3, 9). The utility of the third agent, who is identical in both comparisons, should carry no weight in the decision between the first two agents.

  Any preference relation satisfying all four of these axioms can be represented as a sum of a continuous increasing function applied separately to each individual's utility. Adding one further axiom, that social choice must be consistent with the axioms of rational choice under uncertainty and thus rule out Dutch Books, forces the weighting function to be exactly the utility functions of each individual. That is Harsanyi's utilitarian theorem: any non-utilitarian social choice function will agree to bets that every single member of society opposes. The theorem leaves open a weaker condition, independence of common scale, which holds that the relation between two utility profiles does not change if both are multiplied by the same constant. That weaker requirement points to the isoelastic family of functions, which includes the utilitarian rule, the Nash bargaining solution, and the leximin ordering as special cases, depending on the value of a single parameter.