title: Unit demand
text: # 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.
  
  