supermodular

Supermodular refers to a property of functions, particularly in the field of economics and optimization. A function is considered supermodular if it exhibits increasing marginal returns with respect to its arguments. More formally, a function f(x1, x2, ..., xn) is supermodular if for any two input vectors x and y such that x <= y (meaning xi <= yi for all i), the following inequality holds: f(y) - f(x) >= f(y - delta) - f(x - delta) for any delta such that x - delta >= 0 and y - delta >= 0. A more intuitive way to understand this is that the "gain" from increasing one input variable is larger when the other input variables are also larger.

This property has significant implications. For instance, in economic models, supermodularity often implies that certain decisions or investments are complementary. This means that if one factor of production or decision variable increases, it becomes more beneficial to increase other related variables as well. Supermodular functions are also important in the study of games and mechanism design, as they can lead to nice properties like the existence of pure strategy Nash equilibria. The concept is a generalization of convexity, but it focuses on the interaction between variables rather than just the curvature of the function.