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15
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 ..."
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Cited by 227 (14 self)
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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 ..."
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Cited by 120 (1 self)
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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 ..."
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Cited by 70 (6 self)
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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
ILIN: An Implementation of the Integer Labeling Algorithm for Integer Programming
"... this paper we discuss a practical implementation of the algorithm and present a computer program (ILIN) for solving integer programming using integer labeling algorithm. We also report on the solution of a number of tested examples with up to 500 integer variables. Numerical results indicate that th ..."
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this paper we discuss a practical implementation of the algorithm and present a computer program (ILIN) for solving integer programming using integer labeling algorithm. We also report on the solution of a number of tested examples with up to 500 integer variables. Numerical results indicate that the algorithm is computationally simple, flexible, efficient and stable.
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 ..."
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: 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 ...
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 ..."
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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
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 ..."
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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 ..."
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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,
EVOLUTION, DYNAMICS, AND FIXED POINTS a by
, 1995
"... Signcompatible dynamics describe changes in the composition of a population driven by differences in fitness. A saturated equilibrium is a fixed point for signcompatible dynamics where each subgroup with positive population share has highest fitness. An evolutionary stable equilibrium is a saturat ..."
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Signcompatible dynamics describe changes in the composition of a population driven by differences in fitness. A saturated equilibrium is a fixed point for signcompatible dynamics where each subgroup with positive population share has highest fitness. An evolutionary stable equilibrium is a saturated equilibrium attracting all trajectories nearby, such that the Euclidean distance to it decreases monotonically. We address existence, multiplicity, and dynamical stability of fixed points of signcompatible dynamics. A saturated equilibrium may be approximated by using a variable dimension restart algorithm for solving the nonlinear complementarity problem.
Publication Date: 1985 Chapter Title: References, Index
"... Anderson, Robert, and Ballentine, J. Gregory. 1976. The incidence and excess burden of a profits tax under imperfect competition Public Finance 31:15976. Andrews, William D. 1974. A consumptiontype or cash flow personal income tax. Harvard Law Review 87 (April): 111388. ..."
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Anderson, Robert, and Ballentine, J. Gregory. 1976. The incidence and excess burden of a profits tax under imperfect competition Public Finance 31:15976. Andrews, William D. 1974. A consumptiontype or cash flow personal income tax. Harvard Law Review 87 (April): 111388.