In mathematics, a supermodular function is a function on a lattice that, informally, has the property of being characterized by "increasing differences." Seen from the point of set functions, this can also be viewed as a relationship of "increasing returns", where adding more elements to a subset increases its valuation. In economics, supermodular functions are often used as a formal expression of complementarity in preferences among goods. Supermodular functions are studied and have applications in game theory, economics, lattice theory, combinatorial optimization, and machine learning.

Definition

be a lattice is called supermodular

.

is strictly supermodular is (strictly) supermodular then f is called (strictly) submodular. A function that is both submodular and supermodular is called modular. This corresponds to the inequality being changed to an equality.

is supermodular if

.

If f is twice continuously differentiable, then supermodularity is equivalent to the condition

Supermodularity in economics and game theory

The concept of supermodularity is used in the social sciences to analyze how one agent's decision affects the incentives of others.

Consider a symmetric game too. Following the terminology of Bulow, Geanakoplos, and Klemperer (1985), economists call this situation strategic complementarity, because players' strategies are complements to each other. This is the basic property underlying examples of multiple equilibria in coordination games.

, called submodularity, corresponds to the situation of strategic substitutability, other players have an incentive to pick a lower.

For example, Bulow et al. consider the interactions of many imperfectly competitive firms. When an increase in output by one firm raises the marginal revenues of the other firms, production decisions are strategic complements. When an increase in output by one firm lowers the marginal revenues of the other firms, production decisions are strategic substitutes.

A supermodular utility function is often related to complementary goods. However, this view is disputed.

Supermodular set functions

Supermodularity can also be defined for set functions, which are functions defined over subsets of a larger set. Many properties of submodular set functions can be rephrased to apply to supermodular set functions.

Intuitively, a supermodular function over a set of subsets demonstrates "increasing returns". This means that if each subset is assigned a real number that corresponds to its value, the value of a subset will always be less than the value of a larger subset which contains it. Alternatively, this means that as we add elements to a set, we increase its value.

Definition

is supermodular if it satisfies the following (equivalent) conditions:

1. .

2. .

is supermodular, and modular if it is both supermodular and submodular.

Additional Facts

• is a supermodular function.

• A non-negative supermodular function is also a superadditive function.

Optimization Techniques

There are specialized techniques for optimizing submodular functions. Theory and enumeration algorithms for finding local and global maxima (minima) of submodular (supermodular) functions can be found in "Maximization of submodular functions: Theory and enumeration algorithms", B. Goldengorin.

See also

• Pseudo-Boolean function

• Topkis's theorem

• Submodular set function

• Superadditive

• Utility functions on indivisible goods

Notes and references