Utilitarian rule

Tangible utility functions

The utilitarian rule is easy to interpret and implement when the functions u_i 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 – ${\displaystyle u_i(y_i)}$ is the number of buildings that agent ${\displaystyle i}$ can build using ${\displaystyle y_i}$ 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 – ${\displaystyle u_i(y_i)}$ is the probability of agent ${\displaystyle i}$ to recover by getting ${\displaystyle y_i}$ doses of the medication. The utilitarian rule then allocates the medication in a way that maximizes the expected number of survivors.

Abstract utility functions

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 ${\displaystyle u_i:X⟶\mathbb{R}}$ 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

${\displaystyle V(x):=\sum _{i\in I}{v}_i(x),}$

then in general, the maximizer of ${\displaystyle V}$ will not be the same as the maximizer of ${\displaystyle U}$. 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

Relative utilitarianism proposes a natural calibration mechanism. For every ${\displaystyle i\in I}$, suppose that the values

${\displaystyle m_i:=\underset{x\in X}{min}{u}_i(x)\text{and}{M}_i:=\underset{x\in X}{max}{u}_i(x)}$

are well-defined. (For example, this will always be true if ${\displaystyle X}$ is finite, or if ${\displaystyle X}$ is a compact space and ${\displaystyle u_i}$ is a continuous function.) Then define

${\displaystyle w_i(x):=\frac{u_i(x)-m_i}{M_i-m_i}}$

for all ${\displaystyle x\in X}$. Thus, ${\displaystyle w_i:X⟶\mathbb{R}}$ is a "rescaled" utility function which has a minimum value of 0 and a maximum value of 1. The Relative Utilitarian social choice rule selects the element in ${\displaystyle X}$ which maximizes the utilitarian sum

${\displaystyle W(x):=\sum _{i\in I}{w}_i(x).}$

As an abstract social choice function, relative utilitarianism has been analyzed by Cao (1982), Dhillon (1998), Karni (1998), Dhillon and Mertens (1999), Segal (2000), Sobel (2001) and Pivato (2008). (Cao (1982) refers to it as the "modified Thomson solution".)

The utilitarian rule and Pareto-efficiency

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).

The utilitarian rule in specific contexts

In the context of voting, the utilitarian rule leads to several voting methods:

• Range voting (also called score voting or utilitarian voting) implements the relative-utilitarian rule by letting voters explicitly express their utilities to each alternative on a common normalized scale.
• Implicit utilitarian voting tries to approximate the utilitarian rule while letting the voters express only ordinal rankings over candidates.

In the context of resource allocation, the utilitarian rule leads to:

• A particular rule for division of a single homogeneous resource;
• Several rules and algorithms for utilitarian cake-cutting – dividing a heterogeneous resource;
• A particular rule for fair item allocation.
• Welfare maximization problem.
• Utilitarian stable matching