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 among two or more agents. It is usually assumed that every agent assigns subjective utility to every subset of the items. This can be represented in one of two ways:
An ordinal utilitypreference relation, usually marked by. The fact that an agent prefers a set to a set is written. If the agent only weakly prefers then this is written.
A cardinal utility function implies a preference relation: implies and implies. Utility functions can have several properties.
Monotonicity
means that an agent always prefers to have extra items. Formally:
For a preference relation: implies.
For a utility function: implies .
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
0
apple
5
hat
7
apple and hat
12
Additivity means that "the whole is equal tothe sum of its parts." That is, 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 of items, assuming that. In other words, is an additive function. An equivalent definition is: for any sets of items and, 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
0
apple
5
bread
7
apple and bread
9
Submodularity means that "the whole is not more than the sum of its parts." Formally, for all sets and, In other words, is a submodular set function. An equivalent property is diminishing marginal utility, which means that for any sets and with, and every : 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, since he is less hungry in that case. A typical utility function for this case is given at the right.
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 ". Formally, for all sets and, In other words, is a supermodular set function. An equivalent property is increasing marginal utility, which means that for all sets and with, and every : 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, 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 submodular and supermodular.
Subadditivity and superadditivity
Subadditivity means that for every pair of disjoint sets In other words, is a subadditive set function. Assuming is non-negative, every submodular function is subadditive. However, there are non-negative subadditive functions that are not submodular. For example, assume that there are 3 identical items,, and Z, and the utility depends only on their quantity. The table on the right describes a utility function that is subadditive but not submodular, since
0
X or Y or Z
1
X,Y or Y,Z or Z,X
3
X,Y,Z
4
Superadditivity means that for every pair of disjoint sets In other words, is a superadditive set function. Assuming is non-positive, every supermodular function is superadditive. However, there are non-negative superadditive functions that are not supermodular. For example, assume that there are 3 identical items,, and Z, and the utility depends only on their quantity. The table on the right describes a utility function that is non-negative and superadditive but not supermodular, since A utility function with is said to be additive if and only if it is both superadditive and subadditive. With the typical assumption that, every submodular function is subadditive and every supermodular function is superadditive. Without any assumption on the utility from the empty set, these relations do not hold. In particular, if a submodular function is not subadditive, then must be negative. For example, suppose there are two items,, with, and. This utility function is submodular and supermodular and non-negative except on the empty set, but is not subadditive, since Also, if a supermodular function is not superadditive, then must be positive. Suppose instead that. This utility function is non-negative, supermodular, and submodular, but is not superadditive, since
Unit demand 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 there is a subset with cardinality, such that.
For a utility function: For every set :
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.
Gross substitutes means that the agents regards the items as substitute goods or independent goods but not complementary goods. There are many formal definitions to this property, all of which are equivalent.
Every UD valuation is GS, but the opposite is not true.
Every GS valuation is submodular, but the opposite is not true.
See Gross substitutes for more details. Hence the following relations hold between the classes: 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:
