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39
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.
Analyzing MarketBased Resource Allocation Strategies for the Computational Grid
 International Journal of High Performance Computing Applications
, 2001
"... In this paper, we investigate Gcommerce — computational economies for controlling resource allocation in Computational Grid settings. We define hypothetical resource consumers (representing users and Gridaware applications) and resource producers (representing resource owners who “sell ” their res ..."
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Cited by 101 (2 self)
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In this paper, we investigate Gcommerce — computational economies for controlling resource allocation in Computational Grid settings. We define hypothetical resource consumers (representing users and Gridaware applications) and resource producers (representing resource owners who “sell ” their resources to the Grid). We then measure the efficiency of resource allocation under two different market conditions: commodities markets and auctions. We compare both market strategies in terms of price stability, market equilibrium, consumer efficiency, and producer efficiency. Our results indicate that commodities markets are a better choice for controlling Grid resources than previously defined auction strategies. 1
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
Grid Resource Allocation and Control Using Computational Economies
 Grid Computing: Making the Global Infrastructure a Reality
, 2003
"... In this chapter, we describe the use of economic principles as the basis for Grid resource allocation policies and mechanisms. A computational economy in which users “buy ” resources from their owners is an attractive method of controlling Grid resource allocation for several reasons. Economies are ..."
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Cited by 48 (0 self)
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In this chapter, we describe the use of economic principles as the basis for Grid resource allocation policies and mechanisms. A computational economy in which users “buy ” resources from their owners is an attractive method of controlling Grid resource allocation for several reasons. Economies are intuitively easy to understand, they fit the model of flexible resource usage under local control (which is fundamental to Grid computing), and they can be analyzed through a considerable body of extant theory. We discuss many of the fundamental characteristics of computational economies, particularly as they pertain to Grid computing. We also present Gcommerce — a framework that we have used to investigate Grid resource economies — as an example of the type of results that are possible. Finally, we discuss several of the issues associated with empirical investigation of Grid economies as a motivation for future work. 1
Small gain theorems for large scale systems and construction of ISS Lyapunov functions
 SIAM JOURNAL ON CONTROL AND OPTIMIZATION
, 2009
"... We consider interconnections of n nonlinear subsystems in the inputtostate stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. ..."
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Cited by 21 (16 self)
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We consider interconnections of n nonlinear subsystems in the inputtostate stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.
A Framework for Applied Dynamic Analysis in IO
, 2006
"... This paper reviews a framework for numerically analyzing dynamic interactions in imperfectly competitive industries. The framework dates back to Ericson & Pakes (1995), but it is based on equilibrium notions that had been available for some time before, and it has been extended in many ways by diff ..."
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Cited by 21 (0 self)
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This paper reviews a framework for numerically analyzing dynamic interactions in imperfectly competitive industries. The framework dates back to Ericson & Pakes (1995), but it is based on equilibrium notions that had been available for some time before, and it has been extended in many ways by different authors since. The framework requires as input a set of primitives which describe the institutional structure in the industry to be analyzed. The framework outputs profits and policies for every incumbent and potential entrant at each possible state of the industry. These policies can be used to simulate the distribution of sample paths for all firms from any initial industry structure. The sample paths generated by the model can be quite different depending on the primitives that are fed into it, and most of the extensions were designed to enable the framework to accommodate empirically relevant cases that required modification of the initial structure. The sample paths possess similar properties to those observed in (the recently made available) panel data sets on industries. These sample paths can be used either for an analysis of the likely response to a policy or an environmental change,
S.H.: Settling the complexity of ArrowDebreu equilibria in markets with additively separable utilities
 In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science
, 2009
"... We prove that the problem of computing an ArrowDebreu market equilibrium is PPADcomplete even when all traders use additively separable, piecewiselinear and concave utility functions. In fact, our proof shows that this marketequilibrium problem does not have a fully polynomialtime approximation ..."
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Cited by 18 (3 self)
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We prove that the problem of computing an ArrowDebreu market equilibrium is PPADcomplete even when all traders use additively separable, piecewiselinear and concave utility functions. In fact, our proof shows that this marketequilibrium problem does not have a fully polynomialtime approximation scheme unless every problem in PPAD is solvable in polynomial time.
Theory of globally convergent probabilityone homotopies for nonlinear programming
 SIAM Journal on Optimization
, 2000
"... Abstract. For many years globally convergent probabilityone homotopy methods have been remarkably successful on difficult realistic engineering optimization problems, most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective. Convergence theory ..."
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Cited by 16 (4 self)
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Abstract. For many years globally convergent probabilityone homotopy methods have been remarkably successful on difficult realistic engineering optimization problems, most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective. Convergence theory has been derived for a few particular problems, and considerable fixed point theory exists, but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking. This paper derives some probabilityone homotopy convergence theorems for unconstrained and inequality constrained optimization, for linear and nonlinear inequality constraints, and with and without convexity. Some insight is provided into why the homotopies used in engineering practice are so successful, and why this success is more than dumb luck. By presenting the theory as variations on a prototype probabilityone homotopy convergence theorem, the essence of such convergence theory is elucidated.
On Algorithms for Discrete and Approximate Brouwer Fixed Points
 In STOC 2005
, 2005
"... We study the algorithmic complexity of the discrete fixed point problem and develop an asymptotic matching bound for a cube in any constantly bounded finite dimension. To obtain our upper bound, we derive a new fixed point theorem, based on a novel characterization of boundary conditions for the exi ..."
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Cited by 13 (10 self)
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We study the algorithmic complexity of the discrete fixed point problem and develop an asymptotic matching bound for a cube in any constantly bounded finite dimension. To obtain our upper bound, we derive a new fixed point theorem, based on a novel characterization of boundary conditions for the existence of fixed points. In addition, exploring a linkage with the approximation problem of the continuous fixed point problem, we obtain asymptotic matching bounds for complexity of the approximate Brouwer fixed point problem in the continuous case for Lipschitz functions that close a previous exponential gap. It settles a fifteen years old open problem of Hirsch, Papadimitriou and Vavasis by improving both the upper and lower bounds. Our new characterization for existence of a fixed point is also applicable to functions defined on nonconvex domain and makes it a potentially useful tool for design and analysis of algorithms for fixed points in general domain.