Unit demand

In economics, a unit demand agent is an agent who wants to buy a single item, which may be of one of different types. A typical example is a buyer who needs a new car. There are many different types of cars, but usually a buyer will choose only one of them, based on the quality and the price.

If there are m different item-types, then a unit-demand valuation function is typically represented by m values $$v_1,\dots,v_m$$, with $$v_j$$ representing the subjective value that the agent derives from item $$j$$. If the agent receives a set $$A$$ of items, then his total utility is given by:
 * $$u(A)=\max_{j\in A}v_j$$

since he enjoys the most valuable item from $$A$$ and ignores the rest.

Therefore, if the price of item $$j$$ is $$p_j$$, then a unit-demand buyer will typically want to buy a single item – the item $$j$$ for which the net utility $$v_j - p_j$$ is maximized.

Ordinal and cardinal definitions
A unit-demand valuation is formally defined by:
 * For a preference relation: for every set $$B$$ there is a subset $$A\subseteq B$$ with cardinality $$|A|=1$$, such that $$A \succeq B$$.
 * For a utility function: For every set $$A$$:
 * $$u(A)=\max_{x\in A}u(\{x\})$$

Connection to other classes of utility functions
A unit-demand function is an extreme case of a submodular set function.

It is characteristic of items that are pure substitute goods.