# Utilitarian rule

(Redirected from Utilitarian social choice rule)

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.

## Definition

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

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

## 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:[1]: 44

• 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 patient. 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 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 ${\displaystyle (u_{i})_{i\in I}}$  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 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

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

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),[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'.)

## The utilitarian rule and Pareto-efficiency

Every Pareto efficient choice is necessarily utilitarian. This is because every Pareto improvement necessarily increases the sum of utilities. But the utilitarian choice is not the only Pareto-efficient one. In fact, every weighted utilitarian choice (maximizing a weighted sum of utilities) is Pareto-optimal, whenever the weights of all individuals are positive.

## 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.
• A related voting rule is Nanson's method.

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

• Egalitarian rule – a different rule, that emphasizes the welfare of the worst-off individual rather than the sum of utilities.
• Proportional-fair rule – a rule that tries to balance the efficiency of the utilitarian rule and the fairness of the egalitarian rule.
• Utility maximization problem – a problem solved by an individual consumer (rather than by society).

## References

1. ^ a b Herve Moulin (2004). Fair Division and Collective Welfare. Cambridge, Massachusetts: MIT Press. ISBN 9780262134231.
2. ^ Cao, Xiren (1982-12-01). "Preference functions and bargaining solutions". 1982 21st IEEE Conference on Decision and Control. IEEE: 164–171. doi:10.1109/cdc.1982.268420. S2CID 30395654.
3. ^ Dhillon, Amrita (1998), "Extended Pareto rules and relative utilitarianism", Social Choice and Welfare, 15 (4): 521–542, doi:10.1007/s003550050121, S2CID 54899024
4. ^ Karni, Edi (1998), "Impartiality: definition and representation", Econometrica, 66 (6): 1405–1415, doi:10.2307/2999622, JSTOR 2999622
5. ^ Dhillon, Amrita; Mertens, Jean-Francois (1999), "Relative utilitarianism", Econometrica, 67 (3): 471–498, doi:10.1111/1468-0262.00033
6. ^ Segal, Uzi (2000), "Let's agree that all dictatorships are equally bad", Journal of Political Economy, 108 (3): 569–589, doi:10.1086/262129, S2CID 154610036
7. ^ Sobel, Joel (2001), "Manipulation of preferences and relative utilitarianism", Games and Economic Behavior, 37: 196–215, CiteSeerX 10.1.1.395.509, doi:10.1006/game.2000.0839
8. ^ Pivato, Marcus (2008), "Twofold optimality of the relative utilitarian bargaining solution", Social Choice and Welfare, 32 (1): 79–92, CiteSeerX 10.1.1.537.5572, doi:10.1007/s00355-008-0313-0, S2CID 15475740
9. ^ Aziz, Haris; Huang, Xin; Mattei, Nicholas; Segal-Halevi, Erel (2021-06-01). "Computing Welfare-Maximizing Fair Allocations of Indivisible Goods". arXiv:2012.03979 [cs.GT].