Additive utility

In economics, additive utility is a cardinal utility function with the sigma additivity property.^{[1]: 287–288}

Additive utility

${\displaystyle A}$	${\displaystyle u(A)}$
${\displaystyle \emptyset }$	0
apple	5
hat	7
apple and hat	12

Additivity (also called linearity or modularity) means that "the whole is equal to the sum of its parts." That is, the utility of a set of items is the sum of the utilities of each item separately. Let ${\displaystyle S}$ be a finite set of items. A cardinal utility function ${\displaystyle u:2^{S}\to \mathbb {R} }$, where ${\displaystyle 2^{S}}$ is the power set of ${\displaystyle S}$, is additive if for any ${\displaystyle A,B\subseteq S}$,

${\displaystyle u(A)+u(B)=u(A\cup B)+u(A\cap B).}$

It follows that for any ${\displaystyle A\subseteq S}$,

${\displaystyle u(A)=u(\emptyset )+\sum _{x\in A}{\big (}u(\{x\})-u(\emptyset ){\big )}.}$

An additive utility function is characteristic of independent goods. For example, an apple and a hat are considered independent: the utility a person receives from having an apple is the same whether or not he has a hat, and vice versa. A typical utility function for this case is given at the right.

Notes

• As mentioned above, additivity is a property of cardinal utility functions. An analogous property of ordinal utility functions is weakly additive.
• A utility function is additive if and only if it is both submodular and supermodular.

See also

• Utility functions on indivisible goods
• Independent goods
• Submodular set function
• Supermodular set function

References

1. Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016). Handbook of Computational Social Choice. Cambridge University Press. ISBN 9781107060432. (free online version)