Editing Social welfare function (section)

=== 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.