Additive utility

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

Additive utility  A | u(A)

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∅ | 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 S be a finite set of items. A cardinal utility function u:2S->ℝ, where 2S is the power set of S, is additive if for any A,B⊆S,

u(A)+u(B)=u(A∪B)+u(A∩B).

It follows that for any A⊆S,

u(A)=u(∅)+∑x∈A(u({x})−u(∅)).

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.[2]

## 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) (in en). Handbook of Computational Social Choice. Cambridge University Press. ISBN 9781107060432. https://books.google.com/books?id=nMHgCwAAQBAJ. (free online version)
2. ↑ Fuhrken, Gebhard; Richter, Marcel K. (1991-03-01). "Additive utility" (in en). Economic Theory 1 (1): 83–105. doi:10.1007/BF01210575. ISSN 1432-0479. https://doi.org/10.1007/BF01210575.

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