# Relative utilitarianism

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Let ${\displaystyle X}$ be a set of possible states of the world' or alternatives'; society wishes to choose some state from ${\displaystyle X}$. Let ${\displaystyle I}$ be a finite set, representing a collection of people. For each ${\displaystyle i\in I}$, let ${\displaystyle u_{i}:X\longrightarrow \mathbb {R} }$ be a utility function. A social choice rule (or voting system) 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 basic problem of social choice theory is to disambiguate the word best'.)

The classic utilitarian social choice rule selects the element ${\displaystyle x\in X}$ which maximizes the utilitarian sum

${\displaystyle U(x):=\sum _{i\in I}u_{i}(x).}$

However, for this formula to make sense, we must assume 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 Saint 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

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

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 we specify some mechanism to calibrate' the utility functions of the different individuals.

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

${\displaystyle m_{i}\ :=\ \min _{x\in X}\,u_{i}(x)\quad {\mbox{and}}\quad M_{i}\ :=\ \max _{x\in X}\,u_{i}(x)}$

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, ${\displaystyle w_{i}:X\longrightarrow \mathbb {R} }$ 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'.) When interpreted as a voting rule', it is equivalent to Range voting.

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