Introduction
Supermodularity is an important concept in economics, mathematics, and game theory. It has been studied for several decades and has gained increasing attention in recent years. Supermodular functions play an important role in various fields, including industrial organization, finance, and public economics. In this article, we will discuss the basics of supermodularity, its applications in various fields, and current research trends in the area.
What is supermodularity?
Supermodularity is a property of functions that are defined on a partially ordered set. In simple terms, a function is said to be supermodular if it satisfies the following condition:
F(A ∪ {i}) – F(A) \>= F(B ∪ {i}) – F(B)
In other words, the marginal gain of adding an element to an already chosen set A is greater than or equal to the marginal gain of adding the same element i to any other set B that contains A.
This condition can also be stated as follows: if F is a non-decreasing function, then for any two non-empty sets A and B, and any element i that is not in both A and B, we have:
F(A ∪ {i}) – F(A) \>= F(B ∪ {i}) – F(B)
If the inequality is strict, i.e., if the marginal gain of adding an element to the first set is strictly greater than the marginal gain of adding the same element to the second set, then the function is said to be strictly supermodular.
Applications of supermodularity
Supermodular functions have a wide range of applications in various fields. Here are some examples:
Industrial organization
In industrial organization, supermodular functions are used to model strategic interactions among firms. For example, consider a market with two firms that produce complementary goods. The profits of each firm depend on their output levels and the output level of the other firm. The profits of the two firms can be modeled as a supermodular function. Such models can provide insights into the behavior of firms in oligopolistic markets.
Public economics
In public economics, supermodular functions are used to model public goods that exhibit increasing returns to scale. For example, consider a park that is used by a community. The benefit that each individual derives from the park depends on the number of people who use it. The total benefit of the park can be modeled as a supermodular function. Such models can provide insights into the optimal provision of public goods.
Finance
In finance, supermodular functions are used to model portfolio optimization problems. For example, consider an investor who wants to maximize the expected return on a portfolio subject to a risk constraint. The expected return on the portfolio can be modeled as a supermodular function. Such models can provide insights into optimal portfolio allocation strategies.
Research trends
Supermodularity is an active area of research, and new developments are emerging rapidly. Here are some recent research trends in the area:
Complementarity in economic systems
Recent research has focused on the role of complementarity in economic systems. Complementarity refers to situations where the value of an activity depends on the level of other activities. This concept is closely related to supermodularity, and recent research has shown that complementarity is an important driver of economic systems.
Applications to network formation games
Network formation games are games where players choose their connections with other players. Recently, researchers have used supermodularity to analyze the behavior of players in such games. Such models can provide insights into the structure of social networks.
Generalizations of supermodularity
Researchers are also exploring generalizations of the supermodularity concept. For example, some researchers are studying supermodularity in the presence of uncertainty, while others are studying supermodularity in infinite-dimensional spaces.
Conclusion
Supermodularity is an important concept in economics, mathematics, and game theory. Supermodular functions have a wide range of applications and provide insights into various economic problems. Current research trends in the area include complementarity in economic systems, network formation games, and generalizations of the supermodularity concept. As research in the area continues to evolve, we can expect to gain deeper insights into the structure and behavior of complex economic systems.