Discrete convex analysis: A tool for economics and game theory

This paper presents discrete convex analysis as a tool for use in economics and game theory. Discrete convex analysis is a new framework of discrete mathematics and optimization, developed during the last two decades. Recently, it has been recognized as a powerful tool for analyzing economic or game models with indivisibilities. The main feature of discrete convex analysis is the distinction of two convexity concepts, M-convexity and L-convexity, for functions in integer or binary variables, together with their conjugacy relationship. The crucial fact is that M-concavity in its variant is equivalent to the gross substitutes property in economics. Fundamental theorems in discrete convex analysis such as the M-L conjugacy theorems, discrete separation theorems and discrete fixed point theorems yield structural results in economics such as the existence of equilibria and the lattice structure of equilibrium price vectors. Algorithms in discrete convex analysis provide iterative auction algorithms for finding equilibria.