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 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 utility preference relation, usually marked by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \succ} . The fact that an agent prefers a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \succ B} . If the agent only weakly prefers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} (i.e. either prefers $A$ or is indifferent between $A$ and $B$ ) then this is written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \succeq B} .
A cardinal utility function, usually denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} . The utility an agent gets from a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is written $u(A)$ . Cardinal utility functions are often normalized such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset)=0} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset} is the empty set.

A cardinal utility function implies a preference relation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A)>u(B)} implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \succ B} and $u(A)\geq u(B)$ implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\supseteq B} implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \succeq B} .
For a utility function: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\supseteq B} implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A) \geq u(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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}	$u(A)$
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset}	0
apple	5
hat	7
apple and hat	12

Additivity (also called linearity or modularity) means that "the whole is equal to the 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} of items,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A)=\sum_{x\in A}u({x})}

assuming that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset)=0} . In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is an additive function. An equivalent definition is: for any sets of items Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 (and may be less)." Formally, for all sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A)+u(B)\ge 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 any sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq B} , and every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \notin B} :^[2]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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
$A$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (and may be more)". Formally, for all sets $A$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq B} , and every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \notin B} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(B\cup \{x\})-u(B)\geq u(A\cup \{x\})-u(A)} .

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 submodular and supermodular.

Subadditivity and superadditivity

Subadditive but not submodular
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset}	0
X or Y or Z	2
X,Y or Y,Z or Z,X	3
X,Y,Z	5

Subadditivity means that for every pair of disjoint sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A,B}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A\cup B)\leq u(A)+u(B)}

In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is a subadditive set function.

Assuming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset)} 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X, Y} , 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\{X,Y\})+u(\{Y,Z\}) < u(\{X,Y\}\cup\{Y,Z\})+u(\{X,Y\}\cap\{Y,Z\}). }

Superadditive but not supermodular
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset}	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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A,B}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A\cup B)\geq u(A)+u(B)}

In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is a superadditive set function.

Assuming $u(\emptyset )$ 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X, Y} , 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\{X,Y\})+u(\{Y,Z\}) < u(\{X,Y\}\cup\{Y,Z\})+u(\{X,Y\}\cap\{Y,Z\}). }

A utility function with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset) = 0} is said to be additive if and only if it is both superadditive and subadditive.

With the typical assumption that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset) = 0} , 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset)} must be negative. For example, suppose there are two items, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X, Y} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset) = -1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\{X\}) = u(\{Y\}) = 1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\{X, Y\}) = 3} . This utility function is submodular and supermodular and non-negative except on the empty set, but is not subadditive, since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\{X,Y\}) > u(\{X\}) + u(\{Y\}).}

Also, if a supermodular function is not superadditive, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset)} must be positive. Suppose instead that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\emptyset) = u(\{X\}) = u(\{Y\}) = u(\{X, Y\}) = 1} . This utility function is non-negative, supermodular, and submodular, but is not superadditive, since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\{X,Y\}) < u(\{X\}) + u(\{Y\}).}

Unit demand

Unit demand utility
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(A)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} there is a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\subseteq B} with cardinality Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A|=1} , such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \succeq B} .
For a utility function: For every set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} :^[3]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Gross substitutes

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

Gross substitutes (GS) 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 (indivisible items) for more details.

Hence the following relations hold between the classes:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle UD \subsetneq 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 values of functions on {a}, {b} and {a,b}
Aggregate function	Property	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

References

^ Gul, F.; Stacchetti, E. (1999). "Walrasian Equilibrium with Gross Substitutes". Journal of Economic Theory. 87: 95–124. doi:10.1006/jeth.1999.2531.
^ Moulin, Hervé (1991). Axioms of cooperative decision making. Cambridge England New York: Cambridge University Press. ISBN 9780521424585.
^ Koopmans, T. C.; Beckmann, M. (1957). "Assignment Problems and the Location of Economic Activities" (PDF). Econometrica. 25 (1): 53–76. doi:10.2307/1907742. JSTOR 1907742.

[gs99-1] Gul, F.; Stacchetti, E. (1999). "Walrasian Equilibrium with Gross Substitutes". Journal of Economic Theory. 87: 95–124. doi:10.1006/jeth.1999.2531.

[2] Moulin, Hervé (1991). Axioms of cooperative decision making. Cambridge England New York: Cambridge University Press. ISBN 9780521424585.

[kb57-3] Koopmans, T. C.; Beckmann, M. (1957). "Assignment Problems and the Location of Economic Activities" (PDF). Econometrica. 25 (1): 53–76. doi:10.2307/1907742. JSTOR 1907742.

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