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.
[1]: sub.2.5 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.
be a set of possible "states of the world" or "alternatives".
Society wishes to choose a single state from
may represent all possible allocations of the resource.
be a finite set, representing a collection of individuals.
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
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
which maximizes the utilitarian sum The utilitarian rule is easy to interpret and implement when the functions ui represent some tangible, measurable form of utility.
For example:[1]: 44 When the functions ui 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
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".
is an equally valid description of her preferences.
If we define a new package of utility functions
, and we then consider the utilitarian sum then in general, the maximizer of
Thus, in a sense, classic utilitarian social choice is not well-defined within the standard model of cardinal utility used in decision theory, unless a mechanism is specified to "calibrate" the utility functions of the different individuals.
Relative utilitarianism proposes a natural calibration mechanism.
The Relative Utilitarian social choice rule selects the element in
which maximizes the utilitarian sum As an abstract social choice function, relative utilitarianism has been analyzed by Cao (1982),[2] Dhillon (1998),[3] Karni (1998),[4] Dhillon and Mertens (1999),[5] Segal (2000),[6] Sobel (2001)[7] and Pivato (2008).
[8] (Cao (1982) refers to it as the "modified Thomson solution".)
Every Pareto efficient social choice function is necessarily a utilitarian choice function, a result known as Harsanyi's utilitarian theorem.
Specifically, any Pareto efficient social choice function must be a linear combination of the utility functions of each individual utility function (with strictly positive weights).
In the context of voting, the utilitarian rule leads to several voting methods: In the context of resource allocation, the utilitarian rule leads to: