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}

:

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)

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