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. For example, in combinatorial auctions there is a finite set of items, and every agent can buy a subset of the items, but an item cannot be divided between two or more agents.

It is usually assumed that every agent assigns subjective utility to every subset of the items. This can be represented by one of two ways:

An ordinal utility preference relation, usually marked by $\succ$ . The fact that an agent prefers a set $A$ to a set $B$ is written $A \succ B$ . If the agent only weakly prefers $A$ (i.e. either prefers $A$ or is indifferent between $A$ and $B$ ) then this is written $A \succeq B$ .
A cardinal utility function, usually marked by $u$ . The utility an agent gets from a set $A$ is written $u(A)$ . Cardinal utility functions are often normalized such that $u(\emptyset)=0$ , where $\emptyset$ is the empty set.

A cardinal utility function implies a preference relation: $u(A)>u(B)$ implies $A \succ B$ and $u(A)\geq u(B)$ implies $A \succeq B$ .

Utility functions can have several properties.^[1]

Monotonicity

Monotonicity means that an agent always (weakly) prefers to have extra items. Formally:

For a preference relation: $A\supseteq B$ implies $A \succeq B$ .
For a utility function: $A\supseteq B$ implies $u(A) \geq B$ (i.e. u is a monotone function).

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

Additive utility
$A$	$u(A)$
$\emptyset$	0
apple	5
hat	7
apple and hat	12

Additivity (also called: linearity) means that "the whole is equal to the sum of its parts". I.e, the utility of a set of items is the sum of the utilities of each item separately. This property is relevant only for cardinal utility functions. It says that for every set $A$ :

u(A)=\sum_{x\in A}u({x})

In other words, $u$ is an additive function.

An equivalent definition is: for all sets $A$ and $B$ :

u(A)+u(B) = u(A\cup B)+u(A\cap B)

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. A typical utility function for this case is given at the right.

Submodularity and Supermodularity

Submodular utility
$A$	$u(A)$
$\emptyset$	0
apple	5
bread	7
apple and bread	9

Submodularity means that "the whole is not more than the sum of its parts (but may be less)". Formally, for all sets $A$ and $B$ :

u(A)+u(B) \geq u(A\cup B)+u(A\cap B)

In other words, $u$ is a submodular set function.

An equivalent property is Diminishing marginal utility, which means that for every sets $A$ and $B$ with $A \subseteq B$ , and every $x \notin B$ :^[2]

u(A\cup \{x\})-u(A)\geq u(B\cup \{x\})-u(B)

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. A typical utility function for this case is given at the right.

Supermodular utility
$A$	$u(A)$
$\emptyset$	0
apple	5
knife	7
apple and knife	15

Supermodularity is the opposite of submodularity: it means that "the whole is not less than the sum of its parts (but may be more)". I.e, if $A$ and $B$ are sets, then:

u(A)+u(B) \leq u(A\cup B)+u(A\cap B)

In other words, $u$ is a supermodular set function.

An equivalent property is Increasing marginal utility, which means that for all sets $A$ and $B$ with $A \subseteq B$ , and every $x \notin B$ :

u(A\cup \{x\})-u(A)\leq u(B\cup \{x\})-u(B)

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 possible utility function for this case is given at the right.

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

Subadditivity and Superadditivity

Subadditive which is not submodular
$A$	$u(A)$
$\emptyset$	0
X	4
X,X	6
X,X,X	9

Subadditivity means that for all sets $A$ and $B$ :

u(A\cup B)\leq u(A)+u(B)

In other words, $u$ is a subadditive set function.

With monotone functions, every submodular function is subadditive, but the opposite is not true. For example, assume that there are 3 identical items, and the utility depends only on their quantity. The table on the right describes a utility function that is subadditive but not submodular.

Supermodular and not superadditive
$A$	$u(A)$
$\emptyset$	0
X	2
X,X	5
X,X,X	9

Superadditivity is the opposite of subadditivity and means that for all sets $A$ and $B$ :

u(A\cup B)\geq u(A)+u(B)

In other words, $u$ is a superadditive set function.

With monotone functions, every superadditive function is supermodular, but the opposite is not true. The table on the right describes a supermodular function which is not superadditive.

A utility function is additive if and only if it is both superadditive and subadditive.

Special types of submodular utilities

Because of their relation to diminishing marginal utility, submodular utility functions are very common in economics. Several sub-families of the submodular family are described below, in order of containment, from the more specific to the more general.

Unit demand

Unit demand utility
$A$	$u(A)$
$\emptyset$	0
apple	5
pear	7
apple and pear	7

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. Formally:

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 $A$ :^[3]

u(A)=\max_{x\in A}u({x})

A unit-demand function is an extreme case of a submodular function. It is characteristic of goods that are pure substitutes. 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.

Strong no complementarities

A utility function satisfies the strong no complementarities condition (SNC) if for all sets $A$ and $B$ and for every subset $X\subseteq A$ , there is a subset $Y\subseteq B$ such that:

u(A)+u(B)\leq u(A\setminus X \cup Y)+u(A\setminus Y \cup X)

This property has the following interpretation. Suppose Alice and Bob both have utility function $u$ , and are endowed with bundles $A$ and $B$ respectively. For every subset $X$ that Alice hands Bob, there is an equivalent subset $Y$ that Bob can handle Alice, such that their total utility after the swap is preserved or increased.^[1]

Gross substitutes

An illustration of the containment relations between common classes of utility functions.

For other uses, see Gross substitutes.

The gross substitutes (GS) family^[4] is defined based on a price vector and a demand set.

A price vector $p$ is a vector containing a price for each item.
Given a utility function $u$ and a price vector $p$ , a set $A$ is called a demand if it maximizes the net utility of the agent: $u(A)-p\cdot A$ .
The demand set $D(u,p)$ is the set of all demands.

A utility function is GS if it has either one of the following properties, which are all equivalent for monotone function:^[1]

GS: When the price of some items increases, the demand for other items does not decrease. Formally, for any two price vectors $q$ and $p$ such that $q\geq p$ , and any $A \in D(u,p)$ , there is a $B \in D(u,q)$ such that $B \supseteq \{a\in A|p_a=q_a\}$ (B contains all items in A whose price remained constant).
SI (Single Improvement): A non-optimal set can be improved by adding, removing or substituting a single item. Formally, for any price vector $p$ and bundle $A \notin D(u,p)$ , there exists a bundle $B$ such that $u(B)-p\cdot B > u(A)-p\cdot A$ , $|A\setminus B|\leq 1$ and $|B\setminus A|\leq 1$ .
NC (No Complementarities): Every subset has a substitute. Formally: for any price vector $p$ and bundles $A,B \in D(u,p)$ , and for every subset $X\subseteq A$ , there is a subset $Y\subseteq B$ such that: $A\setminus X \cup Y \in D(u,p)$

Relations between families of utility functions

Every UD utility function satisfies the SNC property.

Every SNC function satisfies the NC condition, and hence also GS and SI.

Every GS utility function is submodular,^[1] but there are submodular functions which are not GS.^[5] Hence the following relations hold between the classes:

UD \subsetneq SNC\subsetneq NC = SI = GS \subsetneq Submodular \subsetneq Subadditive

See diagram on the right.

Aggregates of utility functions

A utility function describes the happiness of an individual. Often, we need a function that describes the happiness of an entire society. Such a function is called a Social welfare function, and it is usually an aggregate function of two or more utility functions. If the individual utility functions are additive, then the following is true for the aggregate functions:

Aggregate function	Property	Example^[6]
		f	g	h	aggregate(f,g,h)
Sum	Additive	1,3; 4	3,1; 4		4,4; 8
Average	Additive	1,3; 4	3,1; 4		2,2; 4
Minimum	Super-additive	1,3; 4	3,1; 4		1,1; 4
Maximum	Sub-additive	1,3; 4	3,1; 4		3,3; 4
Median	neither	1,3; 4	3,1; 4	1,1; 2	1,1; 4
Median	neither	1,3; 4	3,1; 4	3,3; 6	3,3; 4

The Average and the Sum functions are additive.
The Minimum function

References

↑ 1.0 1.1 1.2 1.3 Gul, F.; Stacchetti, E. (1999). "Walrasian Equilibrium with Gross Substitutes". Journal of Economic Theory 87: 95. doi:10.1006/jeth.1999.2531.
↑ ISBN 9780521424585
Please click here to fill in your citation. If you are still editing the main article, you may want to open the link in a new window or tab.
↑ Koopmans, T. C.; Beckmann, M. (1957). "Assignment Problems and the Location of Economic Activities". Econometrica 25: 53. doi:10.2307/1907742. JSTOR 1907742.
↑ Kelso, A. S.; Crawford, V. P. (1982). "Job Matching, Coalition Formation, and Gross Substitutes". Econometrica 50 (6): 1483. doi:10.2307/1913392. JSTOR 1913392.
↑ Ben-Zwi, Oren; Lavi, Ron; Newman, Ilan (2013). "Ascending auctions and Walrasian equilibrium". arXiv:1301.1153 [cs.GT].
↑ values of functions on {a}, {b} and {a,b}.