Journal of Mechanism and Institution Design ISSN: 2399-844X(Print), 2399-8458(Online) DOI:10.22574/jmid.2016.12.005 DISCRETE CONVEX ANALYSIS: A TOOL FOR ECONOMICS AND GAME THEORY Kazuo Murota Tokyo Metropolitan University, Japan murota@tmu.ac.jp ABSTRACT 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,-convexity and-convexity, for functions in integer or binary variables, together with their conjugacy relations­hip. The crucial fact is that-concavity in its variant is equivalent to the gross substitutes property in economics. Fundamental theorems in discrete convex analysis such as the-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. Keywords : Convex analysis, indivisibility, equilibrium, fixed point. JEL Classification Numbers : C61, C65. The author would like to thank Zaifu Yang for offering the opportunity of writing this survey paper. Special thanks go to Akiyoshi Shioura and Akihisa Tamura for carefully reading all the manuscript and making constructive comments. The author is also indebted to Satoru Fujishige, Takuya Iimura, Satoko Moriguchi, and Yu Yokoi for helpful suggestions. This work was supported by The Mitsubishi Foundation, CREST, JST, and JSPS KAKENHI Grant Number 26280004. Copyright c Kazuo Murota /(), 2016, 151–273. Licensed under the Creative Commons Attribution-NonCommercial License.0, http://creativecommons.org.