In social choice and operations research, the utilitarian rule (also called the max-sum rule) is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the sum of the utilities of all individuals in society. It is a formal mathematical representation of the utilitarian philosophy, and is often justified by reference to Harsanyi's utilitarian theorem or the Von Neumann–Morgenstern theorem.
Definition Let X {\displaystyle X} be a set of possible "states of the world" or "alternatives". Society wishes to choose a single state from X {\displaystyle X} . For example, in a single-winner election, X {\displaystyle X} may represent the set of candidates; in a resource allocation setting, X {\displaystyle X} may represent all possible allocations of the resource. Let I {\displaystyle I} be a finite set, representing a collection of individuals. For each i ∈ I {\displaystyle i\in I} , let u i : X ⟶ R {\displaystyle u_{i}:X\longrightarrow \mathbb {R} } be a utility function, describing the amount of happiness an individual i derives from each possible state. A social choice rule is a mechanism which uses the data ( u i ) i ∈ I {\displaystyle (u_{i})_{i\in I}} to select some element(s) from X {\displaystyle X} which are "best" for society (the question of what "best" means is the basic problem of social choice theory). The utilitarian rule selects an element x ∈ X {\displaystyle x\in X} which maximizes the utilitarian sum
U ( x ) := ∑ i ∈ I u i ( x ) . {\displaystyle U(x):=\sum _{i\in I}u_{i}(x).}
Tangible utility functions The utilitarian rule is easy to interpret and implement when the functions ui represent some tangible, measurable form of utility. For example:
Consider a problem of allocating wood among builders. The utility functions may represent their productive power – u i ( y i ) {\displaystyle u_{i}(y_{i})} is the number of buildings that agent i {\displaystyle i} can build using y i {\displaystyle y_{i}} units of wood. The utilitarian rule then allocates the wood in a way that maximizes the number of buildings. Consider a problem of allocating a rare medication among patients. The utility functions may represent their chance of recovery – u i ( y i ) {\displaystyle u_{i}(y_{i})} is the probability of agent i {\displaystyle i} to recover by getting y i {\displaystyle y_{i}} doses of the medication. The utilitarian rule then allocates the medication in a way that maximizes the expected number of survivors.
… excerpt ends here. Continue reading the full article.
