Results 1  10
of
20
The complexity of computing a Nash equilibrium
, 2006
"... We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of n ..."
Abstract

Cited by 287 (16 self)
 Add to MetaCart
(Show Context)
We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of normalform games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPADcomplete class of Brouwer functions. 1
COMPUTATION OF EQUILIBRIA in Finite Games
, 1996
"... We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent metho ..."
Abstract

Cited by 135 (1 self)
 Add to MetaCart
We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent methods for finding a sample equilibrium. For large problems, methods which are not globally convergent, such as sequential linear complementarity methods may be preferred on the grounds of speed. None of these methods are capable of characterizing the entire set of Nash equilibria. More computationally intensive methods, which derive from the theory of semialgebraic sets are required for finding all equilibria. These methods can also be applied to compute various equilibrium refinements.
Continuation and Path Following
, 1992
"... CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful ..."
Abstract

Cited by 84 (6 self)
 Add to MetaCart
CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (18811886), Klein (1882 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the
Computing an integer point in a class of polytopes
, 2009
"... Abstract Let P be a polytope satisfying that each row of the defining matrix has at most one positive entry. Determining whether there is an integer point in P is known to be an NPcomplete problem. By introducing an integer labeling rule on an augmented set and applying a triangulation of the Eucli ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract Let P be a polytope satisfying that each row of the defining matrix has at most one positive entry. Determining whether there is an integer point in P is known to be an NPcomplete problem. By introducing an integer labeling rule on an augmented set and applying a triangulation of the Euclidean space, we develop in this paper a variable dimension method for computing an integer point in P. The method starts from an arbitrary integer point and follows a finite simplicial path that either leads to an integer point in P or proves no such point exists.
Variational Inequality Problems with a Continuum of Solutions: Existence and Computation
, 1999
"... : In this paper three sufficient conditions are provided under each of which an upper semicontinuous pointtoset mapping defined on an arbitrary polytope has a connected set of zero points that connect two distinct faces of the polytope. Furthermore, we obtain an existence theorem of a connected s ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
: In this paper three sufficient conditions are provided under each of which an upper semicontinuous pointtoset mapping defined on an arbitrary polytope has a connected set of zero points that connect two distinct faces of the polytope. Furthermore, we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality problem over arbitrary polytopes. These results follow in a constructive way by designing a new simplicial algorithm. The algorithm operates on a triangulation of the polytope and generates a piecewise linear path of points connecting two distinct faces of the polytope. Each point on the path is an approximate zero point. As the mesh size of the triangulation goes to zero, the path converges to a connected set of zero points linking the two distinct faces. As a consequence, our results generalize Browder's fixed point theorem (1960) and an earlier result by the authors (1996) on the ndimensional unit cube. An application in economics is ...
COMPUTING AND PROVING WITH PIVOTS
, 2013
"... Abstract. A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solvin ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solving linear programs. From since, a pivoting algorithm is a method exploring subsets of a ground set and going from one subset σ to a new one σ ′ by deleting an element inside σ and adding an element outside σ: σ ′ = σ \ {v} ∪ {u}, with v ∈ σ and u / ∈ σ. This simple principle combined with other ideas appears to be quite powerful for many problems. This present paper is a survey on algorithms in operations research and discrete mathematics using pivots. We give also examples where this principle allows not only to compute but also to prove some theorems in a constructive way. A formalisation is described, mainly based on ideas by Michael J. Todd. 1.
NPHardness Results Related to PPAD
, 2009
"... Let P = {x ∈ Rn  Ax ≤ b} be a polytope satisfying that each row of A has at most one positive entry. The problem we consider is to determine whether there is an integer point in P, which is known to be an NPcomplete problem. Applying an integer labeling rule and a triangulation of an augmented int ..."
Abstract
 Add to MetaCart
Let P = {x ∈ Rn  Ax ≤ b} be a polytope satisfying that each row of A has at most one positive entry. The problem we consider is to determine whether there is an integer point in P, which is known to be an NPcomplete problem. Applying an integer labeling rule and a triangulation of an augmented integral set in R n+1, we show
Stationary Points and Equilibria
, 2009
"... Almost all existence results in mathematical economics and game theory rely on some type of fixed point theorem. However, in many cases it is much easier to apply an equivalent stationary point theorem. In this survey paper we show for various general equilibrium and game theoretical models the usef ..."
Abstract
 Add to MetaCart
(Show Context)
Almost all existence results in mathematical economics and game theory rely on some type of fixed point theorem. However, in many cases it is much easier to apply an equivalent stationary point theorem. In this survey paper we show for various general equilibrium and game theoretical models the usefulness of the stationary point approach to prove existence of equilibrium. We also consider path following methods to find a stationary point and illustrate such methods by an example. Finally we discuss some refinements of stationary points and their application to solve the problem of equilibrium selection. Keywords: game. fixed point, stationary point, general equilibrium model, noncooperative AMS subject classification: 47H10, 49J40, 90C30, 91A40, 91B50. 1 Introductory general equilibrium model The general equilibrium model introduced by Léon Walras [52] in his Eléments d’économie politique pure seeks to explain the behavior of supply, demand and prices in an economy with many markets and many individual agents. The research program laid down in his book included the investigation of the existence, uniqueness and stability of equilibria,
Combinatorial Integer Labeling Theorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations
, 2007
"... Tucker’s wellknown combinatorial lemma states that for any given symmetric triangulation of the ndimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1, ±2, · · · , ±n} with the property that antipodal vertices on the boundar ..."
Abstract
 Add to MetaCart
Tucker’s wellknown combinatorial lemma states that for any given symmetric triangulation of the ndimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1, ±2, · · · , ±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1dimensional simplex whose two vertices have opposite labels. In this paper we are concerned with an arbitrary finite set D of integral vectors in the ndimensional Euclidean
Solving Discrete Systems of Nonlinear Equations
, 2009
"... 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 ndimensional 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 sim ..."
Abstract
 Add to MetaCart
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 ndimensional 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 ndimensional 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 CournotNash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the wellknown BorsukUlam theorem and a theorem for the existence of a solution for the