Solving Discrete Systems of Nonlinear Equations

Abstract

In this paper we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Z n of the n-dimensional Euclidean space IR n . It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Z n and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the 'continuity property' is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.

Computation, Multiplicity, and Comparative Statics of Cournot Equilibria in Integers

“…There are algorithms to compute the maximizers, but they are not nearly as efficient as those we will propose, especially in the integer case. We suspect the simplicial algorithm of van der Laan et al [12] is also less efficient than those below.…”

“…Integer or discrete versions are much less studied, although existence is proved in, for example, Kukushkin [4] using Tarski's fixed-point theorem and Dubey et al [2] using a potential game, following a less general treatment of Shapley [8]. Existence is also proved as a corollary of a more general theorem on discrete systems of nonlinear equations by van der Laan et al [12]; they also give a simplicial algorithm for the computation of such an equilibrium, but without a complexity analysis.…”