  This paper presents discrete convex analysis as a tool for economics and game
theory. Discrete convex analysis is a new framework of discrete mathematics and
optimization, developed during the last two decades. Recently, it is being
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, or its variant called
M-natural-concavity, 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 give iterative auction algorithms as well as
computational methods for equilibria.
