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 , … , v m {\displaystyle v_{1},\dots ,v_{m}} , with v j {\displaystyle v_{j}} representing the subjective value that the agent derives from item j {\displaystyle j} . If the agent receives a set A {\displaystyle A} of items, then his total utility is given by:

u ( A ) = max j ∈ A v j {\displaystyle u(A)=\max _{j\in A}v_{j}}

since he enjoys the most valuable item from A {\displaystyle A} and ignores the rest.

Therefore, if the price of item j {\displaystyle j} is p j {\displaystyle p_{j}} , then a unit-demand buyer will typically want to buy a single item – the item j {\displaystyle j} for which the net utility v j − p j {\displaystyle v_{j}-p_{j}} is maximized.

A unit-demand valuation is formally defined by:

• For a preference relation: for every set B {\displaystyle B} there is a subset A ⊆ B {\displaystyle A\subseteq B} with cardinality | A | = 1 {\displaystyle |A|=1} , such that A ⪰ B {\displaystyle A\succeq B} .
• For a utility function: For every set A {\displaystyle A} :^[1]

u ( A ) = max x ∈ A u ( { x } ) {\displaystyle u(A)=\max _{x\in A}u(\{x\})}

A unit-demand function is an extreme case of a submodular set function.

It is characteristic of items that are pure substitute goods.

• Utility functions on indivisible goods
• Matching (graph theory)