Additive utility

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

Additive utility

$A$	$u(A)$
$\emptyset$	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". I.e, the utility of a set of items is the sum of the utilities of each item separately. It says that for every bundle $A$:

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

An equivalent definition is: for all sets $A$ and $B$:

$u(A)+u(B)=u(A\cup B)+u(A\cap B)$

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.

See also

• 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 subadditive and superadditive.

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)