complementarities

Complementarities describe a relationship in which the benefit or output produced by one element increases with the level of another element. In other words, the marginal value of one component rises as the other component grows. The term is used in economics, game theory, optimization, and the study of complex systems, and it is generally contrasted with substitutability, where one element can replace another without losing much value.

In economics, complementary goods are pairs whose demand is jointly enhanced; a rise in the price of one tends to reduce demand for the other, with cross-price elasticity negative. Illustrative examples include printers and ink cartridges, or software and hardware accessories. In production, certain inputs exhibit joint productivity gains when used together. The presence of complementarities can justify coordinated or bundled pricing and investment decisions.

In game theory, strategic complementarities occur when players' best responses shift upward with higher actions by others, reinforcing cooperative or coordinated outcomes. Such structure can stabilize equilibria and affect diffusion of technologies or standards. In optimization, complementarities are formalized through complementarity conditions, such as the KKT conditions and variational inequalities, which link primal and dual variables through nonnegativity constraints.

Beyond economics and mathematics, the idea appears in other disciplines as well, including biology and engineering, where components are designed or evolve to work better together. Complementarity also arises in studies of institutional design and policy, where complementarities between rules, incentives, and capabilities can strengthen performance.