supermodularityis

Supermodularity is a property of real-valued functions defined on a lattice. A function $f$ defined on a lattice $(L, \le)$ is supermodular if for any $x, y, z, w \in L$ such that $x \le z$ and $y \le w$, the following inequality holds: $f(x) + f(w) \ge f(y) + f(z)$. This condition essentially means that the "gain" from increasing the input from $x$ to $y$ is less than or equal to the "gain" from increasing the input from $z$ to $w$, provided $x \le z$ and $y \le w$. Alternatively, the condition can be expressed as $f(x \lor z) - f(x) \ge f(y \lor w) - f(y)$ when $x \le y \le z \le w$, or equivalently $f(x) - f(y) \ge f(z) - f(w)$ for $x \le y \le z \le w$.

The concept of supermodularity is fundamental in various fields, including economics, optimization, and game theory. In economics, it is used to model preferences and production technologies where complementarity exists between inputs or goods. For example, if two goods are complements, the marginal utility of one good increases as the quantity of the other good increases. This behavior is captured by supermodularity. In optimization, supermodular functions have desirable properties that simplify the search for optimal solutions, particularly in the context of lattice programming. The concavity of a function is a special case of supermodularity on the real line. Understanding supermodularity helps in analyzing strategic interactions and the incentives of agents in complex systems.