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.

Let ${\displaystyle X}$ be a set of possible "states of the world" or "alternatives". Society wishes to choose a single state from ${\displaystyle X}$. For example, in a single-winner election, ${\displaystyle X}$ may represent the set of candidates; in a resource allocation setting, ${\displaystyle X}$ may represent all possible allocations of the resource.

Let ${\displaystyle I}$ be a finite set, representing a collection of individuals. For each ${\displaystyle i\in I}$, let ${\displaystyle u_{i}:X\longrightarrow \mathbb {R} }$ 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 ${\displaystyle (u_{i})_{i\in I}}$ to select some element(s) from ${\displaystyle X}$ 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 ${\displaystyle x\in X}$ which maximizes the utilitarian sum

The utilitarian rule is easy to interpret and implement when the functions u_i represent some tangible, measurable form of utility. For example:

When the functions u_i represent some abstract form of "happiness", the utilitarian rule becomes harder to interpret. For the above formula to make sense, it must be assumed that the utility functions ${\displaystyle (u_{i})_{i\in I}}$ are both cardinal and interpersonally comparable at a cardinal level.

The notion that individuals have cardinal utility functions is not that problematic. Cardinal utility has been implicitly assumed in decision theory ever since Daniel Bernoulli's analysis of the St. Petersburg paradox. Rigorous mathematical theories of cardinal utility (with application to risky decision making) were developed by Frank P. Ramsey, Bruno de Finetti, von Neumann and Morgenstern, and Leonard Savage. However, in these theories, a person's utility function is only well-defined up to an "affine rescaling". Thus, if the utility function ${\displaystyle u_{i}:X\longrightarrow \mathbb {R} }$ is valid description of her preferences, and if ${\displaystyle r_{i},s_{i}\in \mathbb {R} }$ are two constants with ${\displaystyle s_{i}>0}$, then the "rescaled" utility function ${\displaystyle v_{i}(x):=s_{i}\,u_{i}(x)+r_{i}}$ is an equally valid description of her preferences. If we define a new package of utility functions ${\displaystyle (v_{i})_{i\in I}}$ using possibly different ${\displaystyle r_{i}\in \mathbb {R} }$ and ${\displaystyle s_{i}>0}$ for all ${\displaystyle i\in I}$, and we then consider the utilitarian sum