Discrete Convex Analysis

The goal of this book is to present the theory of discrete convex analysis and to describe how the concepts and theorems in discrete convex analysis can be used in economics and game theory. The book consists of three parts. Part I presents the theory of $M$-concave set functions with a particular emphasis on the connection to economic concepts. Part II is a systematic presentation of the theory of discrete convex analysis for functions defined on the integer lattice, including some latest developments. Part III deals with applications to economics and game theory. The Conclusion offers brief presentation of the discrete fixed point theorem and of applications to operations research and congestion game. Almost all fundamental theorems come with complete proofs, although for some technically complicated proofs precise references to other sources are given instead. Both researchers studying mathematical theory of convex analysis and experts in the areas where convex analysis is used can benefit from this book.