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.

It is usually assumed that every agent assigns subjective utility to every subset of the items.

This can be represented in one of two ways: A cardinal utility function implies a preference relation:

[1] Monotonicity means that an agent always (weakly) prefers to have extra items.

Formally: Monotonicity is equivalent to the free disposal assumption: if an agent may always discard unwanted items, then extra items can never decrease the utility.

Additivity (also called linearity or modularity) means that "the whole is equal to the sum of its parts."

This property is relevant only for cardinal utility functions.

An equivalent definition is: for any sets of items

, 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.

An equivalent property is diminishing marginal utility, which means that for any sets

:[2] A submodular utility function is characteristic of substitute goods.

For example, an apple and a bread loaf can be considered substitutes: the utility a person receives from eating an apple is smaller if he has already ate bread (and vice versa), since he is less hungry in that case.

Supermodularity is the opposite of submodularity: it means that "the whole is not less than the sum of its parts (and may be more)".

An equivalent property is increasing marginal utility, which means that for all sets

: A supermoduler utility function is characteristic of complementary goods.

For example, an apple and a knife can be considered complementary: the utility a person receives from an apple is larger if he already has a knife (and vice versa), since it is easier to eat an apple after cutting it with a knife.

A utility function is additive if and only if it is both submodular and supermodular.

Subadditivity means that for every pair of disjoint sets

The table on the right describes a utility function that is subadditive but not submodular, since

Superadditivity means that for every pair of disjoint sets

Without any assumption on the utility from the empty set, these relations do not hold.

This utility function is submodular and supermodular and non-negative except on the empty set, but is not subadditive, since Also, if a supermodular function is not superadditive, then

This utility function is non-negative, supermodular, and submodular, but is not superadditive, since Unit demand (UD) means that the agent only wants a single good.

If the agent gets two or more goods, he uses the one of them that gives him the highest utility, and discards the rest.

For example, if there are an apple and a pear, and an agent wants to eat a single fruit, then his utility function is unit-demand, as exemplified in the table at the right.

There are many formal definitions to this property, all of which are equivalent.

See Gross substitutes (indivisible items) for more details.

A utility function describes the happiness of an individual.

Often, we need a function that describes the happiness of an entire society.