Additive Utilities
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In
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
, additive utility is a
cardinal utility In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value u(x_i) of one ind ...
function with the sigma additivity property. 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:2^S\to\R, where 2^S is the power set of S, is additive if for any A, B\subseteq S, :u(A)+u(B)=u(A\cup B)-u(A\cap B). It follows that for any A\subseteq S, :u(A)=u(\emptyset)+\sum_\big(u(\)-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 In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask ...
functions is
weakly additive In fair division, a topic in economics, a preference relation is weakly additive if the following condition is met: : If A is preferred to B, and C is preferred to D (and the contents of A and C do not overlap) then A together with C is preferabl ...
. * A utility function is additive
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
it is both
submodular In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an ...
and
supermodular In mathematics, a function :f\colon \mathbb^k \to \mathbb is supermodular if : f(x \uparrow y) + f(x \downarrow y) \geq f(x) + f(y) for all x, y \isin \mathbb^, where x \uparrow y denotes the componentwise maximum and x \downarrow y the componentw ...
.


See also

*
Utility functions on indivisible goods Some branches of economics and game theory deal with indivisible goods, discrete items that can be traded only as a whole. For example, in combinatorial auctions there is a finite set of items, and every agent can buy a subset of the items, but an i ...
* Independent goods *
Submodular set function In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an ...
* Supermodular set function


References

Utility function types {{econ-stub